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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Design of experiments</span></span>
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<p>The <b>design of experiments</b> (<b>DOE</b>),<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> also known as <b>experiment design</b> or <b>experimental design</b>, is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. The term is generally associated with <a href="Experiments" class="mw-redirect" title="Experiments">experiments</a> in which the design introduces conditions that directly affect the variation, but may also refer to the design of <a href="Quasi-experiment" title="Quasi-experiment">quasi-experiments</a>, in which <a href="Naturalistic_observation" title="Naturalistic observation">natural</a> conditions that influence the variation are selected for observation.
</p><p>In its simplest form, an experiment aims at predicting the outcome by introducing a change of the preconditions, which is represented by one or more <a href="Dependent_and_independent_variables" title="Dependent and independent variables">independent variables</a>, also referred to as "input variables" or "predictor variables." The change in one or more independent variables is generally hypothesized to result in a change in one or more <a href="Dependent_and_independent_variables" title="Dependent and independent variables">dependent variables</a>, also referred to as "output variables" or "response variables." The experimental design may also identify <a href="Control_variable" title="Control variable">control variables</a> that must be held constant to prevent external factors from affecting the results. Experimental design involves not only the selection of suitable independent, dependent, and control variables, but planning the delivery of the experiment under statistically optimal conditions given the constraints of available resources. There are multiple approaches for determining the set of design points (unique combinations of the settings of the independent variables) to be used in the experiment.
</p><p>Main concerns in experimental design include the establishment of <a href="Validity_(statistics)" title="Validity (statistics)">validity</a>, <a href="Reliability_(statistics)" title="Reliability (statistics)">reliability</a>, and <a href="Reproducibility" title="Reproducibility">replicability</a>. For example, these concerns can be partially addressed by carefully choosing the independent variable, reducing the risk of measurement error, and ensuring that the documentation of the method is sufficiently detailed. Related concerns include achieving appropriate levels of <a href="Statistical_power" class="mw-redirect" title="Statistical power">statistical power</a> and <a href="Sensitivity_and_specificity" title="Sensitivity and specificity">sensitivity</a>.
</p><p>Correctly designed experiments advance knowledge in the natural and social sciences and engineering, with design of experiments methodology recognised as a key tool in the successful implementation of a <a href="Quality_by_design" title="Quality by design">Quality by Design</a> (QbD) framework.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Other applications include marketing and policy making. The study of the design of experiments is an important topic in <a href="Metascience" title="Metascience">metascience</a>.
</p>
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<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Statistical_experiments,_following_Charles_S._Peirce">Statistical experiments, following Charles S. Peirce</h3></div>
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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Frequentist_statistics" class="mw-redirect" title="Frequentist statistics">Frequentist statistics</a></div>
<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Randomization" title="Randomization">Randomization</a></div>
<p>A theory of <a href="Statistical_inference" title="Statistical inference">statistical inference</a> was developed by <a href="Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles S. Peirce</a> in "<a href="Charles_Sanders_Peirce_bibliography#illus" title="Charles Sanders Peirce bibliography">Illustrations of the Logic of Science</a>" (1877–1878)<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> and "<a href="Charles_Sanders_Peirce_bibliography#SIL" title="Charles Sanders Peirce bibliography">A Theory of Probable Inference</a>" (1883),<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> two publications that emphasized the importance of randomization-based inference in statistics.<sup id="cite_ref-Stigler78_5-0" class="reference"><a href="#cite_note-Stigler78-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading4"><h4 id="Randomized_experiments">Randomized experiments</h4></div>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Random_assignment" title="Random assignment">Random assignment</a></div>
<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Repeated_measures_design" title="Repeated measures design">Repeated measures design</a></div>
<p>Charles S. Peirce randomly assigned volunteers to a <a href="Blinding_(medicine)" class="mw-redirect" title="Blinding (medicine)">blinded</a>, <a href="Repeated_measures_design" title="Repeated measures design">repeated-measures design</a> to evaluate their ability to discriminate weights.<sup id="cite_ref-smalldiff_6-0" class="reference"><a href="#cite_note-smalldiff-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-telepathy_7-0" class="reference"><a href="#cite_note-telepathy-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-stigler_8-0" class="reference"><a href="#cite_note-stigler-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-dehue_9-0" class="reference"><a href="#cite_note-dehue-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>
Peirce's experiment inspired other researchers in psychology and education, which developed a research tradition of randomized experiments in laboratories and specialized textbooks in the 1800s.<sup id="cite_ref-smalldiff_6-1" class="reference"><a href="#cite_note-smalldiff-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-telepathy_7-1" class="reference"><a href="#cite_note-telepathy-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-stigler_8-1" class="reference"><a href="#cite_note-stigler-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-dehue_9-1" class="reference"><a href="#cite_note-dehue-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading4"><h4 id="Optimal_designs_for_regression_models">Optimal designs for regression models</h4></div>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Response_surface_methodology" title="Response surface methodology">Response surface methodology</a></div>
<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></div>
<p><a href="Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles S. Peirce</a> also contributed the first English-language publication on an <a href="Optimal_design" class="mw-redirect" title="Optimal design">optimal design</a> for <a href="Regression_analysis" title="Regression analysis">regression</a> <a href="Statistical_model" title="Statistical model">models</a> in 1876.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> A pioneering <a href="Optimal_design" class="mw-redirect" title="Optimal design">optimal design</a> for <a href="Polynomial_regression" title="Polynomial regression">polynomial regression</a> was suggested by <a href="Joseph_Diaz_Gergonne" class="mw-redirect" title="Joseph Diaz Gergonne">Gergonne</a> in 1815. In 1918, <a href="Kirstine_Smith" title="Kirstine Smith">Kirstine Smith</a> published optimal designs for polynomials of degree six (and less).<sup id="cite_ref-GL2009_11-0" class="reference"><a href="#cite_note-GL2009-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-polynomials_12-0" class="reference"><a href="#cite_note-polynomials-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading3"><h3 id="Sequences_of_experiments">Sequences of experiments</h3></div>
<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Sequential_analysis" title="Sequential analysis">Sequential analysis</a></div>
<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Multi-armed_bandit_problem" class="mw-redirect" title="Multi-armed bandit problem">Multi-armed bandit problem</a>, <a href="Gittins_index" title="Gittins index">Gittins index</a>, and <a href="Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></div>
<p>The use of a sequence of experiments, where the design of each may depend on the results of previous experiments, including the possible decision to stop experimenting, is within the scope of <a href="Sequential_analysis" title="Sequential analysis">sequential analysis</a>, a field that was pioneered<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> by <a href="Abraham_Wald" title="Abraham Wald">Abraham Wald</a> in the context of sequential tests of statistical hypotheses.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> <a href="Herman_Chernoff" title="Herman Chernoff">Herman Chernoff</a> wrote an overview of optimal sequential designs,<sup id="cite_ref-ref3_15-0" class="reference"><a href="#cite_note-ref3-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> while <a href="Adaptive_design_(medicine)" title="Adaptive design (medicine)">adaptive designs</a> have been surveyed by S. Zacks.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> One specific type of sequential design is the "two-armed bandit", generalized to the <a href="Multi-armed_bandit" title="Multi-armed bandit">multi-armed bandit</a>, on which early work was done by <a href="Herbert_Robbins" title="Herbert Robbins">Herbert Robbins</a> in 1952.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Fisher's_principles">Fisher's principles</h2></div>
<p>A methodology for designing experiments was proposed by <a href="Ronald_Fisher" title="Ronald Fisher">Ronald Fisher</a>, in his innovative books: <i>The Arrangement of Field Experiments</i> (1926) and <i><a href="The_Design_of_Experiments" title="The Design of Experiments">The Design of Experiments</a></i> (1935). Much of his pioneering work dealt with agricultural applications of statistical methods. As a mundane example, he described how to test the <a href="Lady_tasting_tea" title="Lady tasting tea">lady tasting tea</a> <a href="Hypothesis" title="Hypothesis">hypothesis</a>, that a certain lady could distinguish by flavour alone whether the milk or the tea was first placed in the cup. These methods have been broadly adapted in biological, psychological, and agricultural research.<sup id="cite_ref-Miller00_18-0" class="reference"><a href="#cite_note-Miller00-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup>
</p>
<dl><dt>Comparison</dt>
<dd>In some fields of study it is not possible to have independent measurements to a traceable <a href="Standard_(metrology)" title="Standard (metrology)">metrology standard</a>. Comparisons between treatments are much more valuable and are usually preferable, and often compared against a <a href="Scientific_control" title="Scientific control">scientific control</a> or traditional treatment that acts as baseline.</dd></dl>
<dl><dt><a href="Randomization" title="Randomization">Randomization</a></dt>
<dd>Random assignment is the process of assigning individuals at random to groups or to different groups in an experiment, so that each individual of the population has the same chance of becoming a participant in the study. The random assignment of individuals to groups (or conditions within a group) distinguishes a rigorous, "true" experiment from an observational study or "quasi-experiment".<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> There is an extensive body of mathematical theory that explores the consequences of making the allocation of units to treatments by means of some random mechanism (such as tables of random numbers, or the use of randomization devices such as playing cards or dice). Assigning units to treatments at random tends to mitigate <a href="Confounding" title="Confounding">confounding</a>, which makes effects due to factors other than the treatment to appear to result from the treatment.</dd></dl>
<dl><dd>The risks associated with random allocation (such as having a serious imbalance in a key characteristic between a treatment group and a control group) are calculable and hence can be managed down to an acceptable level by using enough experimental units. However, if the population is divided into several subpopulations that somehow differ, and the research requires each subpopulation to be equal in size, stratified sampling can be used. In that way, the units in each subpopulation are randomized, but not the whole sample. The results of an experiment can be generalized reliably from the experimental units to a larger <a href="Statistical_population" title="Statistical population">statistical population</a> of units only if the experimental units are a <a href="Sampling_(statistics)" title="Sampling (statistics)">random sample</a> from the larger population; the probable error of such an extrapolation depends on the sample size, among other things.</dd></dl>
<dl><dt><a href="Replication_(statistics)" title="Replication (statistics)">Statistical replication</a></dt>
<dd>Measurements are usually subject to variation and <a href="Measurement_uncertainty" title="Measurement uncertainty">measurement uncertainty</a>; thus they are repeated and full experiments are replicated to help identify the sources of variation, to better estimate the true effects of treatments, to further strengthen the experiment's reliability and validity, and to add to the existing knowledge of the topic.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> However, certain conditions must be met before the replication of the experiment is commenced: the original research question has been published in a <a href="Peer-review" class="mw-redirect" title="Peer-review">peer-reviewed</a> journal or widely cited, the researcher is independent of the original experiment, the researcher must first try to replicate the original findings using the original data, and the write-up should state that the study conducted is a replication study that tried to follow the original study as strictly as possible.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup></dd></dl>
<dl><dt><a href="Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></dt>
<dd>Blocking is the non-random arrangement of experimental units into groups (blocks) consisting of units that are similar to one another. Blocking reduces known but irrelevant sources of variation between units and thus allows greater precision in the estimation of the source of variation under study.</dd></dl>
<dl><dt></dt>
<dt></dt>
<dt></dt>
<dt><a href="Orthogonality#Statistics.2C_econometrics.2C_and_economics" title="Orthogonality">Orthogonality</a></dt></dl>
<dl><dd>Orthogonality concerns the forms of comparison (contrasts) that can be legitimately and efficiently carried out. Contrasts can be represented by vectors and sets of orthogonal contrasts are uncorrelated and independently distributed if the data are normal. Because of this independence, each orthogonal treatment provides different information to the others. If there are <i>T</i> treatments and <i>T</i> − 1 orthogonal contrasts, all the information that can be captured from the experiment is obtainable from the set of contrasts.</dd></dl>
<dl><dt>Multifactorial experiments</dt>
<dd>Use of multifactorial experiments instead of the <a href="One-factor-at-a-time_method" title="One-factor-at-a-time method">one-factor-at-a-time method</a>. These are efficient at evaluating the effects and possible <a href="Interaction_(statistics)" title="Interaction (statistics)">interactions</a> of several factors (independent variables). Analysis of <a href="Experiment" title="Experiment">experiment</a> design is built on the foundation of the <a href="Analysis_of_variance" title="Analysis of variance">analysis of variance</a>, a collection of models that partition the observed variance into components, according to what factors the experiment must estimate or test.</dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Example">Example</h2></div>
<p>This example of design experiments is attributed to <a href="Harold_Hotelling" title="Harold Hotelling">Harold Hotelling</a>, building on examples from <a href="Frank_Yates" title="Frank Yates">Frank Yates</a>.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ref3_15-1" class="reference"><a href="#cite_note-ref3-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> The experiments designed in this example involve <a href="Combinatorial_design" title="Combinatorial design">combinatorial designs</a>.<sup id="cite_ref-yout_Howt_24-0" class="reference"><a href="#cite_note-yout_Howt-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup>
</p><p>Weights of eight objects are measured using a <a href="Pan_balance" class="mw-redirect" title="Pan balance">pan balance</a> and set of standard weights. Each weighing measures the weight difference between objects in the left pan and any objects in the right pan by adding calibrated weights to the lighter pan until the balance is in equilibrium. Each measurement has a <a href="Errors_and_residuals_in_statistics" class="mw-redirect" title="Errors and residuals in statistics">random error</a>. The average error is zero; the <a href="Standard_deviation" title="Standard deviation">standard deviations</a> of the <a href="Probability_distribution" title="Probability distribution">probability distribution</a> of the errors is the same number σ on different weighings; errors on different weighings are <a href="Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a>. Denote the true weights by
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{1},\dots ,\theta _{8}.\,}">
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<p>We consider two different experiments:
</p>
<ol><li>Weigh each object in one pan, with the other pan empty. Let <i>X</i><sub><i>i</i></sub> be the measured weight of the object, for <i>i</i> = 1, ..., 8.</li>
<li>Do the eight weighings according to the following schedule—a <a href="Weighing_matrix" title="Weighing matrix">weighing matrix</a>:</li></ol>
<dl><dd><dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{lcc}&{\text{left pan}}&{\text{right pan}}\\\hline {\text{1st weighing:}}&1\ 2\ 3\ 4\ 5\ 6\ 7\ 8&{\text{(empty)}}\\{\text{2nd:}}&1\ 2\ 3\ 8\ &4\ 5\ 6\ 7\\{\text{3rd:}}&1\ 4\ 5\ 8\ &2\ 3\ 6\ 7\\{\text{4th:}}&1\ 6\ 7\ 8\ &2\ 3\ 4\ 5\\{\text{5th:}}&2\ 4\ 6\ 8\ &1\ 3\ 5\ 7\\{\text{6th:}}&2\ 5\ 7\ 8\ &1\ 3\ 4\ 6\\{\text{7th:}}&3\ 4\ 7\ 8\ &1\ 2\ 5\ 6\\{\text{8th:}}&3\ 5\ 6\ 8\ &1\ 2\ 4\ 7\end{array}}}">
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<mtr>
<mtd></mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>left pan</mtext>
</mrow>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>right pan</mtext>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>1st weighing:</mtext>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
<mtext> </mtext>
<mn>2</mn>
<mtext> </mtext>
<mn>3</mn>
<mtext> </mtext>
<mn>4</mn>
<mtext> </mtext>
<mn>5</mn>
<mtext> </mtext>
<mn>6</mn>
<mtext> </mtext>
<mn>7</mn>
<mtext> </mtext>
<mn>8</mn>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>(empty)</mtext>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>2nd:</mtext>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
<mtext> </mtext>
<mn>2</mn>
<mtext> </mtext>
<mn>3</mn>
<mtext> </mtext>
<mn>8</mn>
<mtext> </mtext>
</mtd>
<mtd>
<mn>4</mn>
<mtext> </mtext>
<mn>5</mn>
<mtext> </mtext>
<mn>6</mn>
<mtext> </mtext>
<mn>7</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>3rd:</mtext>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
<mtext> </mtext>
<mn>4</mn>
<mtext> </mtext>
<mn>5</mn>
<mtext> </mtext>
<mn>8</mn>
<mtext> </mtext>
</mtd>
<mtd>
<mn>2</mn>
<mtext> </mtext>
<mn>3</mn>
<mtext> </mtext>
<mn>6</mn>
<mtext> </mtext>
<mn>7</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>4th:</mtext>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
<mtext> </mtext>
<mn>6</mn>
<mtext> </mtext>
<mn>7</mn>
<mtext> </mtext>
<mn>8</mn>
<mtext> </mtext>
</mtd>
<mtd>
<mn>2</mn>
<mtext> </mtext>
<mn>3</mn>
<mtext> </mtext>
<mn>4</mn>
<mtext> </mtext>
<mn>5</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>5th:</mtext>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
<mtext> </mtext>
<mn>4</mn>
<mtext> </mtext>
<mn>6</mn>
<mtext> </mtext>
<mn>8</mn>
<mtext> </mtext>
</mtd>
<mtd>
<mn>1</mn>
<mtext> </mtext>
<mn>3</mn>
<mtext> </mtext>
<mn>5</mn>
<mtext> </mtext>
<mn>7</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>6th:</mtext>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
<mtext> </mtext>
<mn>5</mn>
<mtext> </mtext>
<mn>7</mn>
<mtext> </mtext>
<mn>8</mn>
<mtext> </mtext>
</mtd>
<mtd>
<mn>1</mn>
<mtext> </mtext>
<mn>3</mn>
<mtext> </mtext>
<mn>4</mn>
<mtext> </mtext>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>7th:</mtext>
</mrow>
</mtd>
<mtd>
<mn>3</mn>
<mtext> </mtext>
<mn>4</mn>
<mtext> </mtext>
<mn>7</mn>
<mtext> </mtext>
<mn>8</mn>
<mtext> </mtext>
</mtd>
<mtd>
<mn>1</mn>
<mtext> </mtext>
<mn>2</mn>
<mtext> </mtext>
<mn>5</mn>
<mtext> </mtext>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>8th:</mtext>
</mrow>
</mtd>
<mtd>
<mn>3</mn>
<mtext> </mtext>
<mn>5</mn>
<mtext> </mtext>
<mn>6</mn>
<mtext> </mtext>
<mn>8</mn>
<mtext> </mtext>
</mtd>
<mtd>
<mn>1</mn>
<mtext> </mtext>
<mn>2</mn>
<mtext> </mtext>
<mn>4</mn>
<mtext> </mtext>
<mn>7</mn>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{lcc}&{\text{left pan}}&{\text{right pan}}\\\hline {\text{1st weighing:}}&1\ 2\ 3\ 4\ 5\ 6\ 7\ 8&{\text{(empty)}}\\{\text{2nd:}}&1\ 2\ 3\ 8\ &4\ 5\ 6\ 7\\{\text{3rd:}}&1\ 4\ 5\ 8\ &2\ 3\ 6\ 7\\{\text{4th:}}&1\ 6\ 7\ 8\ &2\ 3\ 4\ 5\\{\text{5th:}}&2\ 4\ 6\ 8\ &1\ 3\ 5\ 7\\{\text{6th:}}&2\ 5\ 7\ 8\ &1\ 3\ 4\ 6\\{\text{7th:}}&3\ 4\ 7\ 8\ &1\ 2\ 5\ 6\\{\text{8th:}}&3\ 5\ 6\ 8\ &1\ 2\ 4\ 7\end{array}}}</annotation>
</semantics>
</math></span><img src="./32ef87a62d262ad9ce0231e139dd525344692202.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.338ex; width:42.99ex; height:29.843ex;" alt="{\displaystyle {\begin{array}{lcc}&{\text{left pan}}&{\text{right pan}}\\\hline {\text{1st weighing:}}&1\ 2\ 3\ 4\ 5\ 6\ 7\ 8&{\text{(empty)}}\\{\text{2nd:}}&1\ 2\ 3\ 8\ &4\ 5\ 6\ 7\\{\text{3rd:}}&1\ 4\ 5\ 8\ &2\ 3\ 6\ 7\\{\text{4th:}}&1\ 6\ 7\ 8\ &2\ 3\ 4\ 5\\{\text{5th:}}&2\ 4\ 6\ 8\ &1\ 3\ 5\ 7\\{\text{6th:}}&2\ 5\ 7\ 8\ &1\ 3\ 4\ 6\\{\text{7th:}}&3\ 4\ 7\ 8\ &1\ 2\ 5\ 6\\{\text{8th:}}&3\ 5\ 6\ 8\ &1\ 2\ 4\ 7\end{array}}}" loading="lazy"></span></dd></dl></dd></dl>
<dl><dd>Let <i>Y</i><sub><i>i</i></sub> be the measured difference for <i>i</i> = 1, ..., 8. Then the estimated value of the weight <i>θ</i><sub>1</sub> is</dd></dl>
<dl><dd><dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\theta }}_{1}={\frac {Y_{1}+Y_{2}+Y_{3}+Y_{4}-Y_{5}-Y_{6}-Y_{7}-Y_{8}}{8}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>θ<!-- θ --></mi>
<mo>^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>6</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>7</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>8</mn>
</mrow>
</msub>
</mrow>
<mn>8</mn>
</mfrac>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\widehat {\theta }}_{1}={\frac {Y_{1}+Y_{2}+Y_{3}+Y_{4}-Y_{5}-Y_{6}-Y_{7}-Y_{8}}{8}}.}</annotation>
</semantics>
</math></span><img src="./0ef2bb2f0c5fe9b2f90551d13baa46ba126bbdea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:46.113ex; height:5.343ex;" alt="{\displaystyle {\widehat {\theta }}_{1}={\frac {Y_{1}+Y_{2}+Y_{3}+Y_{4}-Y_{5}-Y_{6}-Y_{7}-Y_{8}}{8}}.}" loading="lazy"></span></dd></dl></dd></dl>
<dl><dd>Similar estimates can be found for the weights of the other items:</dd></dl>
<dl><dd><dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\widehat {\theta }}_{2}&={\frac {Y_{1}+Y_{2}-Y_{3}-Y_{4}+Y_{5}+Y_{6}-Y_{7}-Y_{8}}{8}}.\\[5pt]{\widehat {\theta }}_{3}&={\frac {Y_{1}+Y_{2}-Y_{3}-Y_{4}-Y_{5}-Y_{6}+Y_{7}+Y_{8}}{8}}.\\[5pt]{\widehat {\theta }}_{4}&={\frac {Y_{1}-Y_{2}+Y_{3}-Y_{4}+Y_{5}-Y_{6}+Y_{7}-Y_{8}}{8}}.\\[5pt]{\widehat {\theta }}_{5}&={\frac {Y_{1}-Y_{2}+Y_{3}-Y_{4}-Y_{5}+Y_{6}-Y_{7}+Y_{8}}{8}}.\\[5pt]{\widehat {\theta }}_{6}&={\frac {Y_{1}-Y_{2}-Y_{3}+Y_{4}+Y_{5}-Y_{6}-Y_{7}+Y_{8}}{8}}.\\[5pt]{\widehat {\theta }}_{7}&={\frac {Y_{1}-Y_{2}-Y_{3}+Y_{4}-Y_{5}+Y_{6}+Y_{7}-Y_{8}}{8}}.\\[5pt]{\widehat {\theta }}_{8}&={\frac {Y_{1}+Y_{2}+Y_{3}+Y_{4}+Y_{5}+Y_{6}+Y_{7}+Y_{8}}{8}}.\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="0.8em 0.8em 0.8em 0.8em 0.8em 0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>θ<!-- θ --></mi>
<mo>^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>6</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>7</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>8</mn>
</mrow>
</msub>
</mrow>
<mn>8</mn>
</mfrac>
</mrow>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>θ<!-- θ --></mi>
<mo>^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>6</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>7</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>8</mn>
</mrow>
</msub>
</mrow>
<mn>8</mn>
</mfrac>
</mrow>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>θ<!-- θ --></mi>
<mo>^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msub>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>6</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>7</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>8</mn>
</mrow>
</msub>
</mrow>
<mn>8</mn>
</mfrac>
</mrow>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>θ<!-- θ --></mi>
<mo>^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msub>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>6</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>7</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>8</mn>
</mrow>
</msub>
</mrow>
<mn>8</mn>
</mfrac>
</mrow>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>θ<!-- θ --></mi>
<mo>^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>6</mn>
</mrow>
</msub>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>6</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>7</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>8</mn>
</mrow>
</msub>
</mrow>
<mn>8</mn>
</mfrac>
</mrow>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>θ<!-- θ --></mi>
<mo>^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>7</mn>
</mrow>
</msub>
</mtd>
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\widehat {\theta }}_{2}&={\frac {Y_{1}+Y_{2}-Y_{3}-Y_{4}+Y_{5}+Y_{6}-Y_{7}-Y_{8}}{8}}.\\[5pt]{\widehat {\theta }}_{3}&={\frac {Y_{1}+Y_{2}-Y_{3}-Y_{4}-Y_{5}-Y_{6}+Y_{7}+Y_{8}}{8}}.\\[5pt]{\widehat {\theta }}_{4}&={\frac {Y_{1}-Y_{2}+Y_{3}-Y_{4}+Y_{5}-Y_{6}+Y_{7}-Y_{8}}{8}}.\\[5pt]{\widehat {\theta }}_{5}&={\frac {Y_{1}-Y_{2}+Y_{3}-Y_{4}-Y_{5}+Y_{6}-Y_{7}+Y_{8}}{8}}.\\[5pt]{\widehat {\theta }}_{6}&={\frac {Y_{1}-Y_{2}-Y_{3}+Y_{4}+Y_{5}-Y_{6}-Y_{7}+Y_{8}}{8}}.\\[5pt]{\widehat {\theta }}_{7}&={\frac {Y_{1}-Y_{2}-Y_{3}+Y_{4}-Y_{5}+Y_{6}+Y_{7}-Y_{8}}{8}}.\\[5pt]{\widehat {\theta }}_{8}&={\frac {Y_{1}+Y_{2}+Y_{3}+Y_{4}+Y_{5}+Y_{6}+Y_{7}+Y_{8}}{8}}.\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./8b6970abd1a6e69f062cddd667a1ea60088e94c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -21.894ex; margin-bottom: -0.277ex; width:46.864ex; height:45.509ex;" alt="{\displaystyle {\begin{aligned}{\widehat {\theta }}_{2}&={\frac {Y_{1}+Y_{2}-Y_{3}-Y_{4}+Y_{5}+Y_{6}-Y_{7}-Y_{8}}{8}}.\\[5pt]{\widehat {\theta }}_{3}&={\frac {Y_{1}+Y_{2}-Y_{3}-Y_{4}-Y_{5}-Y_{6}+Y_{7}+Y_{8}}{8}}.\\[5pt]{\widehat {\theta }}_{4}&={\frac {Y_{1}-Y_{2}+Y_{3}-Y_{4}+Y_{5}-Y_{6}+Y_{7}-Y_{8}}{8}}.\\[5pt]{\widehat {\theta }}_{5}&={\frac {Y_{1}-Y_{2}+Y_{3}-Y_{4}-Y_{5}+Y_{6}-Y_{7}+Y_{8}}{8}}.\\[5pt]{\widehat {\theta }}_{6}&={\frac {Y_{1}-Y_{2}-Y_{3}+Y_{4}+Y_{5}-Y_{6}-Y_{7}+Y_{8}}{8}}.\\[5pt]{\widehat {\theta }}_{7}&={\frac {Y_{1}-Y_{2}-Y_{3}+Y_{4}-Y_{5}+Y_{6}+Y_{7}-Y_{8}}{8}}.\\[5pt]{\widehat {\theta }}_{8}&={\frac {Y_{1}+Y_{2}+Y_{3}+Y_{4}+Y_{5}+Y_{6}+Y_{7}+Y_{8}}{8}}.\end{aligned}}}" loading="lazy"></span></dd></dl></dd></dl>
<p>The question of design of experiments is: which experiment is better?
</p><p>The variance of the estimate <i>X</i><sub>1</sub> of <i>θ</i><sub>1</sub> is <i>σ</i><sup>2</sup> if we use the first experiment. But if we use the second experiment, the variance of the estimate given above is <i>σ</i><sup>2</sup>/8. Thus the second experiment gives us 8 times as much precision for the estimate of a single item, and estimates all items simultaneously, with the same precision. What the second experiment achieves with eight would require 64 weighings if the items are weighed separately. However, note that the estimates for the items obtained in the second experiment have errors that correlate with each other.
</p><p>Many problems of the design of experiments involve <a href="Combinatorial_design" title="Combinatorial design">combinatorial designs</a>, as in this example and others.<sup id="cite_ref-yout_Howt_24-1" class="reference"><a href="#cite_note-yout_Howt-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Avoiding_false_positives">Avoiding false positives</h2></div>
<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Metascience" title="Metascience">Metascience</a></div>
<p><a href="False_positive" class="mw-redirect" title="False positive">False positive</a> conclusions, often resulting from the <a href="Publish_or_perish" title="Publish or perish">pressure to publish</a> or the author's own <a href="Confirmation_bias" title="Confirmation bias">confirmation bias</a>, are an inherent hazard in many fields.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup>
</p><p>Use of <a href="Double-blind" class="mw-redirect" title="Double-blind">double-blind designs</a> can prevent <a href="Bias" title="Bias">biases</a> potentially leading to <a href="False_positives" class="mw-redirect" title="False positives">false positives</a> in the <a href="Data_collection" title="Data collection">data collection</a> phase. When a double-blind design is used, participants are randomly assigned to experimental groups but the researcher is unaware of what participants belong to which group. Therefore, the researcher can not affect the participants' response to the intervention.<sup id="cite_ref-:0_26-0" class="reference"><a href="#cite_note-:0-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup>
</p><p>Experimental designs with undisclosed <a href="Degrees_of_freedom" title="Degrees of freedom">degrees of freedom</a> are a problem,<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> in that they can lead to conscious or unconscious "<a href="P-hacking" class="mw-redirect" title="P-hacking">p-hacking</a>": trying multiple things until you get the desired result. It typically involves the manipulation – perhaps unconsciously – of the process of <a href="Statistical_analysis" class="mw-redirect" title="Statistical analysis">statistical analysis</a> and the <a href="Degrees_of_freedom" title="Degrees of freedom">degrees of freedom</a> until they return a figure below the <a href="P-value" title="P-value">p<.05 level</a> of <a href="Statistical_significance" title="Statistical significance">statistical significance</a>.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup>
</p><p>P-hacking can be prevented by <a href="Preregistration_(science)" title="Preregistration (science)">preregistering</a> researches, in which researchers have to send their data analysis plan to the journal they wish to publish their paper in before they even start their data collection, so no data manipulation is possible.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup>
</p><p>Another way to prevent this is taking a double-blind design to the data-analysis phase, making the study triple-blind, where the data are sent to a data-analyst unrelated to the research who scrambles up the data so there is no way to know which participants belong to before they are potentially taken away as outliers.<sup id="cite_ref-:0_26-1" class="reference"><a href="#cite_note-:0-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup>
</p><p>Clear and complete <a href="Documentation" title="Documentation">documentation</a> of the experimental <a href="Methodology" title="Methodology">methodology</a> is also important in order to support <a href="Replication_of_results" class="mw-redirect" title="Replication of results">replication of results</a>.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Discussion_topics_when_setting_up_an_experimental_design">Discussion topics when setting up an experimental design</h2></div>
<p>An experimental design or randomized <a href="Clinical_trial" title="Clinical trial">clinical trial</a> requires careful consideration of several factors before actually doing the experiment.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> An experimental design is the laying out of a detailed experimental plan in advance of doing the experiment. Some of the following topics have already been discussed in the principles of experimental design section:
</p>
<ol><li>How many factors does the design have, and are the levels of these factors fixed or random?</li>
<li>Are control conditions needed, and what should they be?</li>
<li>Manipulation checks: did the manipulation really work?</li>
<li>What are the background variables?</li>
<li>What is the sample size? How many units must be collected for the experiment to be generalisable and have enough <a href="Statistical_power" class="mw-redirect" title="Statistical power">power</a>?</li>
<li>What is the relevance of interactions between factors?</li>
<li>What is the influence of delayed effects of substantive factors on outcomes?</li>
<li>How do response shifts affect self-report measures?</li>
<li>How feasible is repeated administration of the same measurement instruments to the same units at different occasions, with a post-test and follow-up tests?</li>
<li>What about using a proxy pretest?</li>
<li>Are there <a href="Confounding" title="Confounding">confounding variables</a>?</li>
<li>Should the client/patient, researcher or even the analyst of the data be blind to conditions?</li>
<li>What is the feasibility of subsequent application of different conditions to the same units?</li>
<li>How many of each control and noise factors should be taken into account?</li></ol>
<p>The independent variable of a study often has many levels or different groups. In a true experiment, researchers can have an experimental group, which is where their intervention testing the hypothesis is implemented, and a control group, which has all the same element as the experimental group, without the interventional element. Thus, when everything else except for one intervention is held constant, researchers can certify with some certainty that this one element is what caused the observed change. In some instances, having a control group is not ethical. This is sometimes solved using two different experimental groups. In some cases, independent variables cannot be manipulated, for example when testing the difference between two groups who have a different disease, or testing the difference between genders (obviously variables that would be hard or unethical to assign participants to). In these cases, a quasi-experimental design may be used.
</p>
<div class="mw-heading mw-heading2"><h2 id="Causal_attributions">Causal attributions</h2></div>
<p>In the pure experimental design, the independent (predictor) variable is manipulated by the researcher – that is – every participant of the research is chosen randomly from the population, and each participant chosen is assigned randomly to conditions of the independent variable. Only when this is done is it possible to certify with high probability that the reason for the differences in the outcome variables are caused by the different conditions. Therefore, researchers should choose the experimental design over other design types whenever possible. However, the nature of the independent variable does not always allow for manipulation. In those cases, researchers must be aware of not certifying about causal attribution when their design doesn't allow for it. For example, in observational designs, participants are not assigned randomly to conditions, and so if there are differences found in outcome variables between conditions, it is likely that there is something other than the differences between the conditions that causes the differences in outcomes, that is – a third variable. The same goes for studies with correlational design.
</p>
<div class="mw-heading mw-heading2"><h2 id="Statistical_control">Statistical control</h2></div>
<p>It is best that a process be in reasonable statistical control prior to conducting designed experiments. When this is not possible, proper blocking, replication, and randomization allow for the careful conduct of designed experiments.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup>
To control for nuisance variables, researchers institute <b>control checks</b> as additional measures. Investigators should ensure that uncontrolled influences (e.g., source credibility perception) do not skew the findings of the study. A <a href="Manipulation_checks" class="mw-redirect" title="Manipulation checks">manipulation check</a> is one example of a control check. Manipulation checks allow investigators to isolate the chief variables to strengthen support that these variables are operating as planned.
</p><p>One of the most important requirements of experimental research designs is the necessity of eliminating the effects of <a href="Spurious_relationship" title="Spurious relationship">spurious</a>, intervening, and <a href="Antecedent_variable" title="Antecedent variable">antecedent variables</a>. In the most basic model, cause (X) leads to effect (Y). But there could be a third variable (Z) that influences (Y), and X might not be the true cause at all. Z is said to be a spurious variable and must be controlled for. The same is true for <a href="Intervening_variable" class="mw-redirect" title="Intervening variable">intervening variables</a> (a variable in between the supposed cause (X) and the effect (Y)), and anteceding variables (a variable prior to the supposed cause (X) that is the true cause). When a third variable is involved and has not been controlled for, the relation is said to be a zero order relationship. In most practical applications of experimental research designs there are several causes (X1, X2, X3). In most designs, only one of these causes is manipulated at a time.
</p>
<div class="mw-heading mw-heading2"><h2 id="Experimental_designs_after_Fisher">Experimental designs after Fisher</h2></div>
<p>Some efficient designs for estimating several main effects were found independently and in near succession by <a href="Raj_Chandra_Bose" title="Raj Chandra Bose">Raj Chandra Bose</a> and <a href="Kishen" title="Kishen">K. Kishen</a> in 1940 at the <a href="Indian_Statistical_Institute" title="Indian Statistical Institute">Indian Statistical Institute</a>, but remained little known until the <a href="Plackett%E2%80%93Burman_design" title="Plackett–Burman design">Plackett–Burman designs</a> were published in <i><a href="Biometrika" title="Biometrika">Biometrika</a></i> in 1946. About the same time, <a href="C._R._Rao" title="C. R. Rao">C. R. Rao</a> introduced the concepts of <a href="Orthogonal_array" title="Orthogonal array">orthogonal arrays</a> as experimental designs. This concept played a central role in the development of <a href="Taguchi_methods" title="Taguchi methods">Taguchi methods</a> by <a href="Genichi_Taguchi" title="Genichi Taguchi">Genichi Taguchi</a>, which took place during his visit to Indian Statistical Institute in early 1950s. His methods were successfully applied and adopted by Japanese and Indian industries and subsequently were also embraced by US industry albeit with some reservations.
</p><p>In 1950, <a href="Gertrude_Mary_Cox" title="Gertrude Mary Cox">Gertrude Mary Cox</a> and <a href="William_Gemmell_Cochran" title="William Gemmell Cochran">William Gemmell Cochran</a> published the book <i>Experimental Designs,</i> which became the major reference work on the design of experiments for statisticians for years afterwards.
</p><p>Developments of the theory of <a href="Linear_model" title="Linear model">linear models</a> have encompassed and surpassed the cases that concerned early writers. Today, the theory rests on advanced topics in <a href="Linear_algebra" title="Linear algebra">linear algebra</a>, <a href="Algebraic_statistics" title="Algebraic statistics">algebra</a> and <a href="Combinatorial_design" title="Combinatorial design">combinatorics</a>.
</p><p>As with other branches of statistics, experimental design is pursued using both <a href="Frequentist_statistics" class="mw-redirect" title="Frequentist statistics">frequentist</a> and <a href="Bayesian_experimental_design" title="Bayesian experimental design">Bayesian</a> approaches: In evaluating statistical procedures like experimental designs, <a href="Frequentist_statistics" class="mw-redirect" title="Frequentist statistics">frequentist statistics</a> studies the <a href="Sampling_distribution" title="Sampling distribution">sampling distribution</a> while <a href="Bayesian_statistics" title="Bayesian statistics">Bayesian statistics</a> updates a <a href="Bayesian_probability" title="Bayesian probability">probability distribution</a> on the parameter space.
</p><p>Some important contributors to the field of experimental designs are <a href="Charles_Sanders_Peirce" title="Charles Sanders Peirce">C. S. Peirce</a>, <a href="R._A._Fisher" class="mw-redirect" title="R. A. Fisher">R. A. Fisher</a>, <a href="Frank_Yates" title="Frank Yates">F. Yates</a>, <a href="R._C._Bose" class="mw-redirect" title="R. C. Bose">R. C. Bose</a>, A. C. Atkinson, <a href="Rosemary_A._Bailey" title="Rosemary A. Bailey">R. A. Bailey</a>, <a href="David_R._Cox" class="mw-redirect" title="David R. Cox">D. R. Cox</a>, <a href="G._E._P._Box" class="mw-redirect" title="G. E. P. Box">G. E. P. Box</a>, <a href="William_G._Cochran" class="mw-redirect" title="William G. Cochran">W. G. Cochran</a>, W. T. Federer, V. V. Fedorov, A. S. Hedayat, <a href="Jack_Kiefer_(mathematician)" class="mw-redirect" title="Jack Kiefer (mathematician)">J. Kiefer</a>, <a href="Oscar_Kempthorne" title="Oscar Kempthorne">O. Kempthorne</a>, <a href="John_Nelder" title="John Nelder">J. A. Nelder</a>, <a href="Andrej_P%C3%A1zman" class="mw-redirect" title="Andrej Pázman">Andrej Pázman</a>, Friedrich Pukelsheim, <a href="D._Raghavarao" title="D. Raghavarao">D. Raghavarao</a>, <a href="C._R._Rao" title="C. R. Rao">C. R. Rao</a>, <a href="Shrikhande_S._S." class="mw-redirect" title="Shrikhande S. S.">Shrikhande S. S.</a>, <a href="Jagdish_N._Srivastava" class="mw-redirect" title="Jagdish N. Srivastava">J. N. Srivastava</a>, William J. Studden, <a href="Genichi_Taguchi" title="Genichi Taguchi">G. Taguchi</a> and <a href="Henry_Wynn" title="Henry Wynn">H. P. Wynn</a>.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup>
</p><p>The textbooks of D. Montgomery, R. Myers, and G. Box/W. Hunter/J.S. Hunter have reached generations of students and practitioners.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> Furthermore, there is ongoing discussion of experimental design in the context of model building for models either static or dynamic models, also known as <a href="System_identification" title="System identification">system identification</a>. <sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Human_participant_constraints">Human participant constraints</h2></div>
<p>Laws and ethical considerations preclude some carefully designed
experiments with human subjects. Legal constraints are dependent on
<a href="Human_subject_research" title="Human subject research">jurisdiction</a>. Constraints may involve
<a href="Institutional_review_board" title="Institutional review board">institutional review boards</a>, <a href="Informed_consent" title="Informed consent">informed consent</a>
and <a href="Confidentiality" title="Confidentiality">confidentiality</a> affecting both clinical (medical) trials and
behavioral and social science experiments.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup>
In the field of toxicology, for example, experimentation is performed
on laboratory <i>animals</i> with the goal of defining safe exposure limits
for <i>humans</i>.<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> Balancing
the constraints are views from the medical field.<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> Regarding the randomization of patients,
"... if no one knows which therapy is better, there is no ethical
imperative to use one therapy or another." (p 380) Regarding
experimental design, "...it is clearly not ethical to place subjects
at risk to collect data in a poorly designed study when this situation
can be easily avoided...". (p 393)
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Adversarial_collaboration" title="Adversarial collaboration">Adversarial collaboration</a> – Method of research</li>
<li><a href="Bayesian_experimental_design" title="Bayesian experimental design">Bayesian experimental design</a> – Experimental design framework</li>
<li><a href="Block_design" title="Block design">Block design</a> – Structure in combinatorial mathematics</li>
<li><a href="Box%E2%80%93Behnken_design" title="Box–Behnken design">Box–Behnken design</a> – Experimental designs for response surface methodology</li>
<li><a href="Central_composite_design" title="Central composite design">Central composite design</a> – Experimental design in statistical mathematics</li>
<li><a href="Clinical_study_design" title="Clinical study design">Clinical study design</a> – Plan for research in clinical medicine</li>
<li><a href="Computer_experiment" title="Computer experiment">Computer experiment</a> – Experiment used to study computer simulation</li>
<li><a href="Controlling_for_a_variable" title="Controlling for a variable">Controlling for a variable</a> – Binning data according to measured values of the variable</li>
<li><a href="Experimetrics" title="Experimetrics">Experimetrics</a> (<a href="Econometrics" title="Econometrics">econometrics</a>-related experiments)</li>
<li><a href="Factor_analysis" title="Factor analysis">Factor analysis</a> – Statistical method</li>
<li><a href="Fractional_factorial_design" title="Fractional factorial design">Fractional factorial design</a> – Statistical experimental design approach</li>
<li><a href="Glossary_of_experimental_design" title="Glossary of experimental design">Glossary of experimental design</a></li>
<li><a href="Grey_box_model" title="Grey box model">Grey box model</a> – Mathematical data production model with limited structure</li>
<li><a href="Industrial_engineering" title="Industrial engineering">Industrial engineering</a> – Branch of engineering which deals with the optimization of complex processes or systems</li>
<li><a href="Instrument_effect" title="Instrument effect">Instrument effect</a></li>
<li><a href="Law_of_large_numbers" title="Law of large numbers">Law of large numbers</a> – Averages of repeated trials converge to the expected value</li>
<li><a href="Manipulation_checks" class="mw-redirect" title="Manipulation checks">Manipulation checks</a></li>
<li><a href="Multifactor_design_of_experiments_software" title="Multifactor design of experiments software">Multifactor design of experiments software</a></li>
<li><a href="One-factor-at-a-time_method" title="One-factor-at-a-time method">One-factor-at-a-time method</a> – Method of designing experiments</li>
<li><a href="Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a> – Experimental design that is optimal with respect to some statistical criterion<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li>
<li><a href="Plackett%E2%80%93Burman_design" title="Plackett–Burman design">Plackett–Burman design</a> – Type of experimental design</li>
<li><a href="Probabilistic_design" title="Probabilistic design">Probabilistic design</a> – Discipline within engineering design</li>
<li><a href="Protocol_(natural_sciences)" class="mw-redirect" title="Protocol (natural sciences)">Protocol (natural sciences)</a> – Procedural method for the design and implementation of an experiment<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li>
<li><a href="Quasi-experimental_design" class="mw-redirect" title="Quasi-experimental design">Quasi-experimental design</a> – Empirical interventional study<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li>
<li><a href="Randomized_block_design" class="mw-redirect" title="Randomized block design">Randomized block design</a> – Design of experiments to collect similar contexts together<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li>
<li><a href="Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a> – Form of scientific experiment</li>
<li><a href="Research_design" title="Research design">Research design</a> – Overall strategy utilized to carry out research</li>
<li><a href="Robust_parameter_design_(RPD)" class="mw-redirect" title="Robust parameter design (RPD)">Robust parameter design</a></li>
<li><a href="Sample_size_determination" title="Sample size determination">Sample size determination</a> – Statistical considerations on how many observations to make</li>
<li><a href="Plackett%E2%80%93Burman_design#Supersaturated_designs" title="Plackett–Burman design">Supersaturated designs</a> – Type of experimental design</li>
<li><a href="Royal_Commission_on_Animal_Magnetism" title="Royal Commission on Animal Magnetism">Royal Commission on Animal Magnetism</a> – 1784 French scientific bodies' investigations involving systematic controlled trials</li>
<li><a href="Survey_sampling" title="Survey sampling">Survey sampling</a> – Statistical selection process</li>
<li><a href="System_identification" title="System identification">System identification</a> – Statistical methods to build mathematical models of dynamical systems from measured data</li>
<li><a href="Taguchi_methods" title="Taguchi methods">Taguchi methods</a> – Statistical methods to improve the quality of manufactured goods</li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://asq.org/quality-resources/design-of-experiments?srsltid=AfmBOoqGNe13QlU1WGcx1ABznp_0sVoAdwVX3jHd_Hq_a9iaqVTQ9p1u">"What Is Design of Experiments (DOE)?"</a>. <i>asq.org</i>. American Society for Quality<span class="reference-accessdate">. Retrieved <span class="nowrap">20 February</span> 2025</span>.</cite></span>
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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://prismtc.co.uk/resources/blogs-and-articles/the-sequential-nature-of-classical-design-of-experiments">"The Sequential Nature of Classical Design of Experiments | Prism"</a>. <i>prismtc.co.uk</i><span class="reference-accessdate">. Retrieved <span class="nowrap">10 March</span> 2023</span>.</cite></span>
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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Peirce, Charles Sanders (1887). "Illustrations of the Logic of Science". Open Court (10 June 2014). <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0812698495</bdi>.</span>
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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Peirce, Charles Sanders (1883). "A Theory of Probable Inference". In C. S. Peirce (Ed.), Studies in logic by members of the Johns Hopkins University (p. 126–181). Little, Brown and Co (1883)</span>
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<li id="cite_note-Stigler78-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Stigler78_5-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFStigler1978" class="citation journal cs1"><a href="Stephen_Stigler" title="Stephen Stigler">Stigler, Stephen M.</a> (1978). <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.aos/1176344123">"Mathematical statistics in the early States"</a>. <i>Annals of Statistics</i>. <b>6</b> (2): 239–65 [248]. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176344123">10.1214/aos/1176344123</a></span>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2958876">2958876</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0483118">0483118</a>. <q>Indeed, Pierce's work contains one of the earliest explicit endorsements of mathematical randomization as a basis for inference of which I am aware (Peirce, 1957, pages 216–219</q></cite></span>
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<li id="cite_note-smalldiff-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-smalldiff_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-smalldiff_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFPeirceJastrow1885" class="citation journal cs1"><a href="Charles_Sanders_Peirce" title="Charles Sanders Peirce">Peirce, Charles Sanders</a>; <a href="Joseph_Jastrow" title="Joseph Jastrow">Jastrow, Joseph</a> (1885). <a rel="nofollow" class="external text" href="http://psychclassics.yorku.ca/Peirce/small-diffs.htm">"On Small Differences in Sensation"</a>. <i>Memoirs of the National Academy of Sciences</i>. <b>3</b>: <span class="nowrap">73–</span>83.</cite></span>
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<li id="cite_note-telepathy-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-telepathy_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-telepathy_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">of
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<li id="cite_note-stigler-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-stigler_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-stigler_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">
<cite id="CITEREFStephen_M._Stigler1992" class="citation journal cs1"><a href="Stephen_M._Stigler" class="mw-redirect" title="Stephen M. Stigler">Stephen M. Stigler</a> (November 1992). "A Historical View of Statistical Concepts in Psychology and Educational Research". <i>American Journal of Education</i>. <b>101</b> (1): <span class="nowrap">60–</span>70. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F444032">10.1086/444032</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1085417">1085417</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:143685203">143685203</a>.</cite></span>
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<li id="cite_note-dehue-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-dehue_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-dehue_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">
<cite id="CITEREFTrudy_Dehue1997" class="citation journal cs1">Trudy Dehue (December 1997). <a rel="nofollow" class="external text" href="https://www.rug.nl/research/portal/en/publications/deception-efficiency-and-random-groups(459e54f0-1e56-4390-876a-46a33e80621d).html">"Deception, Efficiency, and Random Groups: Psychology and the Gradual Origination of the Random Group Design"</a>. <i><a href="Isis_(journal)" title="Isis (journal)">Isis</a></i>. <b>88</b> (4): <span class="nowrap">653–</span>673. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F383850">10.1086/383850</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/9519574">9519574</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:23526321">23526321</a>.</cite></span>
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<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><cite id="CITEREFPeirce,_C._S.1876" class="citation journal cs1"><a href="Charles_Sanders_Peirce" title="Charles Sanders Peirce">Peirce, C. S.</a> (1876). "Note on the Theory of the Economy of Research". <i>Coast Survey Report</i>: <span class="nowrap">197–</span>201.</cite>, actually published 1879, NOAA <a rel="nofollow" class="external text" href="http://docs.lib.noaa.gov/rescue/cgs/001_pdf/CSC-0025.PDF#page=222">PDF Eprint</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170302071239/https://docs.lib.noaa.gov/rescue/cgs/001_pdf/CSC-0025.PDF#page=222">Archived</a> 2 March 2017 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.<br> Reprinted in <i><a href="Charles_Sanders_Peirce_bibliography#CP" title="Charles Sanders Peirce bibliography">Collected Papers</a></i> <b>7</b>, paragraphs 139–157, also in <i><a href="Charles_Sanders_Peirce_bibliography#W" title="Charles Sanders Peirce bibliography">Writings</a></i> <b>4</b>, pp. 72–78, and in <cite id="CITEREFPeirce,_C._S.1967" class="citation journal cs1"><a href="Charles_Sanders_Peirce" title="Charles Sanders Peirce">Peirce, C. S.</a> (July–August 1967). "Note on the Theory of the Economy of Research". <i>Operations Research</i>. <b>15</b> (4): <span class="nowrap">643–</span>648. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1287%2Fopre.15.4.643">10.1287/opre.15.4.643</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/168276">168276</a>.</cite></span>
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<li id="cite_note-GL2009-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-GL2009_11-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFGuttorpLindgren2009" class="citation journal cs1">Guttorp, P.; Lindgren, G. (2009). "Karl Pearson and the Scandinavian school of statistics". <i>International Statistical Review</i>. <b>77</b>: 64. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.368.8328">10.1.1.368.8328</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1751-5823.2009.00069.x">10.1111/j.1751-5823.2009.00069.x</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121294724">121294724</a>.</cite></span>
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<li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Wald, A. (1945) "Sequential Tests of Statistical Hypotheses", <a href="Annals_of_Mathematical_Statistics" title="Annals of Mathematical Statistics">Annals of Mathematical Statistics</a>, 16 (2), 117–186.</span>
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<li id="cite_note-ref3-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-ref3_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ref3_15-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="Herman_Chernoff" title="Herman Chernoff">Herman Chernoff</a>, <i>Sequential Analysis and Optimal Design</i>, <a href="Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">SIAM</a> Monograph, 1972.</span>
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<li id="cite_note-Miller00-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-Miller00_18-0">^</a></b></span> <span class="reference-text"><a href="Geoffrey_Miller_(psychologist)" title="Geoffrey Miller (psychologist)">Miller, Geoffrey</a> (2000). <i>The Mating Mind: how sexual choice shaped the evolution of human nature</i>, London: Heineman, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-434-00741-2</bdi> (also Doubleday, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-385-49516-1</bdi>) "To biologists, he was an architect of the 'modern synthesis' that used mathematical models to integrate Mendelian genetics with Darwin's selection theories. To psychologists, Fisher was the inventor of various statistical tests that are still supposed to be used whenever possible in psychology journals. To farmers, Fisher was the founder of experimental agricultural research, saving millions from starvation through rational crop breeding programs." p.54.</span>
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<cite id="CITEREFBoxHunterHunter1978" class="citation book cs1">Box, George E.P.; Hunter, William G.; Hunter, J. Stuart (1978). <a rel="nofollow" class="external text" href="https://archive.org/details/statisticsforexp00geor"><i>Statistics for Experimenters : An Introduction to Design, Data Analysis, and Model Building</i></a>. New York: Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-09315-2</bdi>.</cite></span>
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</ol></div></div>
<div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3></div>
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<ul><li><a href="Charles_Sanders_Peirce" title="Charles Sanders Peirce">Peirce, C. S.</a> (1877–1878), "Illustrations of the Logic of Science" (series), <i>Popular Science Monthly</i>, vols. 12–13. Relevant individual papers:
<ul><li>(1878 March), "The Doctrine of Chances", <i>Popular Science Monthly</i>, v. 12, March issue, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZKMVAAAAYAAJ">604</a>–615. <i>Internet Archive</i> <a rel="nofollow" class="external text" href="https://archive.org/stream/popscimonthly12yoummiss#page/612/mode/1up">Eprint</a>.</li>
<li>(1878 April), "The Probability of Induction", <i>Popular Science Monthly</i>, v. 12, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZKMVAAAAYAAJ">705</a>–718. <i>Internet Archive</i> <a rel="nofollow" class="external text" href="https://archive.org/stream/popscimonthly12yoummiss#page/715/mode/1up">Eprint</a>.</li>
<li>(1878 June), "The Order of Nature", <i>Popular Science Monthly</i>, v. 13, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=u8sWAQAAIAAJ">203</a>–217.<i>Internet Archive</i> <a rel="nofollow" class="external text" href="https://archive.org/stream/popularsciencemo13newy#page/203/mode/1up">Eprint</a>.</li>
<li>(1878 August), "Deduction, Induction, and Hypothesis", <i>Popular Science Monthly</i>, v. 13, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=u8sWAQAAIAAJ">470</a>–482. <i>Internet Archive</i> <a rel="nofollow" class="external text" href="https://archive.org/stream/popularsciencemo13newy#page/470/mode/1up">Eprint</a>.</li>
<li>(1883), "A Theory of Probable Inference", <i>Studies in Logic</i>, pp. <a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_V7oIAAAAQAAJ/page/n134">126–181</a>, Little, Brown, and Company. (Reprinted 1983, John Benjamins Publishing Company, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>90-272-3271-7</bdi>)</li></ul></li></ul>
</div>
<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
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<div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Design_of_experiments" class="extiw external" title="commons:Category:Design of experiments">Design of experiments</a></span>.</div></div>
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<ul><li>A <a rel="nofollow" class="external text" href="http://www.itl.nist.gov/div898/handbook/pri/section1/pri1.htm">chapter</a> from a <a rel="nofollow" class="external text" href="http://www.itl.nist.gov/div898/handbook/">"NIST/SEMATECH Handbook on Engineering Statistics"</a> at <a href="National_Institute_of_Standards_and_Technology" title="National Institute of Standards and Technology">NIST</a></li>
<li><a rel="nofollow" class="external text" href="http://www.itl.nist.gov/div898/handbook/pri/section3/pri3362.htm">Box–Behnken designs</a> from a <a rel="nofollow" class="external text" href="http://www.itl.nist.gov/div898/handbook/">"NIST/SEMATECH Handbook on Engineering Statistics"</a> at <a href="National_Institute_of_Standards_and_Technology" title="National Institute of Standards and Technology">NIST</a></li></ul>
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</style><div id="Design_of_experiments708" style="font-size:114%;margin:0 4em"></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Scientific_method" title="Scientific method">Scientific<br>method</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Experiment" title="Experiment">Scientific experiment</a></li>
<li><a href="Scientific_control" title="Scientific control">Control</a></li>
<li><a href="Internal_validity" title="Internal validity">Internal</a> and <a href="External_validity" title="External validity">external</a> <a href="Validity_(statistics)" title="Validity (statistics)">validity</a></li>
<li><a href="Experimental_unit" class="mw-redirect" title="Experimental unit">Experimental unit</a></li>
<li><a href="Blind_experiment" class="mw-redirect" title="Blind experiment">Blinding</a></li></ul>
<ul><li><b><a href="Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></b>: <a href="Bayesian_experimental_design" title="Bayesian experimental design">Bayesian</a></li>
<li><a href="Random_assignment" title="Random assignment">Random assignment</a></li>
<li><a href="Randomization" title="Randomization">Randomization</a></li>
<li><a href="Restricted_randomization" title="Restricted randomization">Restricted randomization</a></li>
<li><a href="Replication_(statistics)" title="Replication (statistics)">Replication versus subsampling</a></li>
<li><a href="Sample_size" class="mw-redirect" title="Sample size">Sample size</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Glossary_of_experimental_design" title="Glossary of experimental design">Treatment</a><br> and <a href="Blocking_(statistics)" title="Blocking (statistics)">blocking</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><b><a href="Glossary_of_experimental_design" title="Glossary of experimental design">Treatment</a></b></li>
<li><a href="Effect_size" title="Effect size">Effect size</a></li>
<li><a href="Contrast_(statistics)" title="Contrast (statistics)">Contrast</a></li>
<li><a href="Interaction_(statistics)" title="Interaction (statistics)">Interaction</a></li>
<li><a href="Confounding" title="Confounding">Confounding</a></li>
<li><a href="Orthogonality#Statistics,_econometrics,_and_economics" title="Orthogonality">Orthogonality</a></li>
<li><b><a href="Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></b></li>
<li><a href="Covariate" class="mw-redirect" title="Covariate">Covariate</a></li>
<li><a href="Nuisance_variable" title="Nuisance variable">Nuisance variable</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Statistical_model" title="Statistical model">Models</a> <br> and <a href="Statistical_inference" title="Statistical inference">inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Linear_regression" title="Linear regression">Linear regression</a></li>
<li><a href="Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li>
<li><a href="Bayesian_linear_regression" title="Bayesian linear regression">Bayesian</a></li></ul>
<ul><li><b><a href="Random_effect" class="mw-redirect" title="Random effect">Random effect</a></b></li>
<li><a href="Mixed_model" title="Mixed model">Mixed model</a></li>
<li><a href="Hierarchical_linear_modeling" class="mw-redirect" title="Hierarchical linear modeling">Hierarchical model:</a> <a href="Hierarchical_Bayes_model" class="mw-redirect" title="Hierarchical Bayes model">Bayesian</a></li></ul>
<ul><li><b><a href="Analysis_of_variance" title="Analysis of variance">Analysis of variance (Anova)</a></b></li>
<li><a href="Cochran's_theorem" title="Cochran's theorem">Cochran's theorem</a></li>
<li><a href="Multivariate_analysis_of_variance" title="Multivariate analysis of variance"><b>Manova</b> (<i>multivariate</i>)</a></li>
<li><a href="Analysis_of_covariance" title="Analysis of covariance"><b>Ancova</b> (<i>covariance</i>)</a></li></ul>
<ul><li><a href="Comparing_means" class="mw-redirect" title="Comparing means">Compare means</a></li>
<li><a href="Multiple_comparison" class="mw-redirect" title="Multiple comparison">Multiple comparison</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"> <br> <br><a href="Completely_randomized_design" title="Completely randomized design">Completely<br>randomized</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><b><a href="Factorial_experiment" title="Factorial experiment">Factorial</a></b></li>
<li><a href="Fractional_factorial_design" title="Fractional factorial design">Fractional factorial</a></li>
<li><a href="Plackett-Burman_design" class="mw-redirect" title="Plackett-Burman design">Plackett–Burman</a></li>
<li><a href="Taguchi_methods" title="Taguchi methods">Taguchi</a></li></ul>
<ul><li><b><a href="Response_surface_methodology" title="Response surface methodology">Response surface methodology</a></b></li>
<li><a href="Polynomial_and_rational_function_modeling" title="Polynomial and rational function modeling">Polynomial and rational modeling</a></li>
<li><a href="Box%E2%80%93Behnken_design" title="Box–Behnken design">Box–Behnken</a></li>
<li><a href="Central_composite_design" title="Central composite design">Central composite</a></li></ul>
<ul><li><b><a href="Randomized_block_design" class="mw-redirect" title="Randomized block design">Block</a></b></li>
<li><a href="Generalized_randomized_block_design" title="Generalized randomized block design">Generalized randomized block design</a> (GRBD)</li>
<li><a href="Latin_square" title="Latin square">Latin square</a></li>
<li><a href="Graeco-Latin_square" class="mw-redirect" title="Graeco-Latin square">Graeco-Latin square</a></li>
<li><a href="Orthogonal_array" title="Orthogonal array">Orthogonal array</a></li>
<li><a href="Latin_hypercube_sampling" title="Latin hypercube sampling">Latin hypercube</a> <br> <b><a href="Repeated_measures_design" title="Repeated measures design">Repeated measures design</a></b></li>
<li><a href="Crossover_study" title="Crossover study">Crossover study</a></li></ul>
<ul><li><b><a href="Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></b></li>
<li><a href="Sequential_analysis" title="Sequential analysis">Sequential analysis</a></li>
<li><a href="Sequential_probability_ratio_test" title="Sequential probability ratio test">Sequential probability ratio test</a></li></ul>
</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>
<ul><li><a href="Glossary_of_experimental_design" title="Glossary of experimental design">Glossary</a></li>
<li>Category</li>
<li><span class="nowrap"><span class="skin-invert-image noviewer" typeof="mw:File"></span> </span><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></li>
<li><a href="Outline_of_statistics" title="Outline of statistics">Statistical outline</a></li>
<li><a href="List_of_statistics_articles" title="List of statistics articles">Statistical topics</a></li></ul>
</div></td></tr></tbody></table></div>
<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Statistics654" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistics654" style="font-size:114%;margin:0 4em"><a href="Statistics" title="Statistics">Statistics</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>
<ul><li><a href="Outline_of_statistics" title="Outline of statistics">Outline</a></li>
<li><a href="List_of_statistics_articles" title="List of statistics articles">Index</a></li></ul>
</div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Descriptive_statistics654" style="font-size:114%;margin:0 4em"><a href="Descriptive_statistics" title="Descriptive statistics">Descriptive statistics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Continuous_probability_distribution" class="mw-redirect" title="Continuous probability distribution">Continuous data</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Central_tendency" title="Central tendency">Center</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Mean" title="Mean">Mean</a>
<ul><li><a href="Arithmetic_mean" title="Arithmetic mean">Arithmetic</a></li>
<li><a href="Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">Arithmetic-Geometric</a></li>
<li><a href="Contraharmonic_mean" title="Contraharmonic mean">Contraharmonic</a></li>
<li><a href="Cubic_mean" title="Cubic mean">Cubic</a></li>
<li><a href="Generalized_mean" title="Generalized mean">Generalized/power</a></li>
<li><a href="Geometric_mean" title="Geometric mean">Geometric</a></li>
<li><a href="Harmonic_mean" title="Harmonic mean">Harmonic</a></li>
<li><a href="Heronian_mean" title="Heronian mean">Heronian</a></li>
<li><a href="Heinz_mean" title="Heinz mean">Heinz</a></li>
<li><a href="Lehmer_mean" title="Lehmer mean">Lehmer</a></li></ul></li>
<li><a href="Median" title="Median">Median</a></li>
<li><a href="Mode_(statistics)" title="Mode (statistics)">Mode</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Statistical_dispersion" title="Statistical dispersion">Dispersion</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Average_absolute_deviation" title="Average absolute deviation">Average absolute deviation</a></li>
<li><a href="Coefficient_of_variation" title="Coefficient of variation">Coefficient of variation</a></li>
<li><a href="Interquartile_range" title="Interquartile range">Interquartile range</a></li>
<li><a href="Percentile" title="Percentile">Percentile</a></li>
<li><a href="Range_(statistics)" title="Range (statistics)">Range</a></li>
<li><a href="Standard_deviation" title="Standard deviation">Standard deviation</a></li>
<li><a href="Variance#Sample_variance" title="Variance">Variance</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Shape_of_the_distribution" class="mw-redirect" title="Shape of the distribution">Shape</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li>
<li><a href="Moment_(mathematics)" title="Moment (mathematics)">Moments</a>
<ul><li><a href="Kurtosis" title="Kurtosis">Kurtosis</a></li>
<li><a href="L-moment" title="L-moment">L-moments</a></li>
<li><a href="Skewness" title="Skewness">Skewness</a></li></ul></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Count_data" title="Count data">Count data</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Index_of_dispersion" title="Index of dispersion">Index of dispersion</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Summary tables</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Contingency_table" title="Contingency table">Contingency table</a></li>
<li><a href="Frequency_distribution" class="mw-redirect" title="Frequency distribution">Frequency distribution</a></li>
<li><a href="Grouped_data" title="Grouped data">Grouped data</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Dependence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Partial_correlation" title="Partial correlation">Partial correlation</a></li>
<li><a href="Pearson_correlation_coefficient" title="Pearson correlation coefficient">Pearson product-moment correlation</a></li>
<li><a href="Rank_correlation" title="Rank correlation">Rank correlation</a>
<ul><li><a href="Kendall_rank_correlation_coefficient" title="Kendall rank correlation coefficient">Kendall's τ</a></li>
<li><a href="Spearman's_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's ρ</a></li></ul></li>
<li><a href="Scatter_plot" title="Scatter plot">Scatter plot</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Statistical_graphics" title="Statistical graphics">Graphics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Bar_chart" title="Bar chart">Bar chart</a></li>
<li><a href="Biplot" title="Biplot">Biplot</a></li>
<li><a href="Box_plot" title="Box plot">Box plot</a></li>
<li><a href="Control_chart" title="Control chart">Control chart</a></li>
<li><a href="Correlogram" title="Correlogram">Correlogram</a></li>
<li><a href="Fan_chart_(statistics)" title="Fan chart (statistics)">Fan chart</a></li>
<li><a href="Forest_plot" title="Forest plot">Forest plot</a></li>
<li><a href="Histogram" title="Histogram">Histogram</a></li>
<li><a href="Pie_chart" title="Pie chart">Pie chart</a></li>
<li><a href="Q%E2%80%93Q_plot" title="Q–Q plot">Q–Q plot</a></li>
<li><a href="Radar_chart" title="Radar chart">Radar chart</a></li>
<li><a href="Run_chart" title="Run chart">Run chart</a></li>
<li><a href="Scatter_plot" title="Scatter plot">Scatter plot</a></li>
<li><a href="Stem-and-leaf_display" title="Stem-and-leaf display">Stem-and-leaf display</a></li>
<li><a href="Violin_plot" title="Violin plot">Violin plot</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible uncollapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Data_collection654" style="font-size:114%;margin:0 4em"><a href="Data_collection" title="Data collection">Data collection</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Effect_size" title="Effect size">Effect size</a></li>
<li><a href="Missing_data" title="Missing data">Missing data</a></li>
<li><a href="Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></li>
<li><a href="Statistical_population" title="Statistical population">Population</a></li>
<li><a href="Replication_(statistics)" title="Replication (statistics)">Replication</a></li>
<li><a href="Sample_size_determination" title="Sample size determination">Sample size determination</a></li>
<li><a href="Statistic" title="Statistic">Statistic</a></li>
<li><a href="Statistical_power" class="mw-redirect" title="Statistical power">Statistical power</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Survey_methodology" title="Survey methodology">Survey methodology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Sampling_(statistics)" title="Sampling (statistics)">Sampling</a>
<ul><li><a href="Cluster_sampling" title="Cluster sampling">Cluster</a></li>
<li><a href="Stratified_sampling" title="Stratified sampling">Stratified</a></li></ul></li>
<li><a href="Opinion_poll" title="Opinion poll">Opinion poll</a></li>
<li><a href="Questionnaire" title="Questionnaire">Questionnaire</a></li>
<li><a href="Standard_error" title="Standard error">Standard error</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Experiment" title="Experiment">Controlled experiments</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></li>
<li><a href="Factorial_experiment" title="Factorial experiment">Factorial experiment</a></li>
<li><a href="Interaction_(statistics)" title="Interaction (statistics)">Interaction</a></li>
<li><a href="Random_assignment" title="Random assignment">Random assignment</a></li>
<li><a href="Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></li>
<li><a href="Randomized_experiment" title="Randomized experiment">Randomized experiment</a></li>
<li><a href="Scientific_control" title="Scientific control">Scientific control</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Adaptive designs</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Adaptive_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a></li>
<li><a href="Stochastic_approximation" title="Stochastic approximation">Stochastic approximation</a></li>
<li><a href="Up-and-Down_Designs" class="mw-redirect" title="Up-and-Down Designs">Up-and-down designs</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Observational_study" title="Observational study">Observational studies</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Cohort_study" title="Cohort study">Cohort study</a></li>
<li><a href="Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a></li>
<li><a href="Natural_experiment" title="Natural experiment">Natural experiment</a></li>
<li><a href="Quasi-experiment" title="Quasi-experiment">Quasi-experiment</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistical_inference654" style="font-size:114%;margin:0 4em"><a href="Statistical_inference" title="Statistical inference">Statistical inference</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Statistical_theory" title="Statistical theory">Statistical theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Population_(statistics)" class="mw-redirect" title="Population (statistics)">Population</a></li>
<li><a href="Statistic" title="Statistic">Statistic</a></li>
<li><a href="Probability_distribution" title="Probability distribution">Probability distribution</a></li>
<li><a href="Sampling_distribution" title="Sampling distribution">Sampling distribution</a>
<ul><li><a href="Order_statistic" title="Order statistic">Order statistic</a></li></ul></li>
<li><a href="Empirical_distribution_function" title="Empirical distribution function">Empirical distribution</a>
<ul><li><a href="Density_estimation" title="Density estimation">Density estimation</a></li></ul></li>
<li><a href="Statistical_model" title="Statistical model">Statistical model</a>
<ul><li><a href="Model_specification" class="mw-redirect" title="Model specification">Model specification</a></li>
<li><a href="Lp_space" title="Lp space">L<sup><i>p</i></sup> space</a></li></ul></li>
<li><a href="Statistical_parameter" title="Statistical parameter">Parameter</a>
<ul><li><a href="Location_parameter" title="Location parameter">location</a></li>
<li><a href="Scale_parameter" title="Scale parameter">scale</a></li>
<li><a href="Shape_parameter" title="Shape parameter">shape</a></li></ul></li>
<li><a href="Parametric_statistics" title="Parametric statistics">Parametric family</a>
<ul><li><a href="Likelihood_function" title="Likelihood function">Likelihood</a> <a href="Monotone_likelihood_ratio" title="Monotone likelihood ratio"><span style="font-size: 85%;">(monotone)</span></a></li>
<li><a href="Location%E2%80%93scale_family" title="Location–scale family">Location–scale family</a></li>
<li><a href="Exponential_family" title="Exponential family">Exponential family</a></li></ul></li>
<li><a href="Completeness_(statistics)" title="Completeness (statistics)">Completeness</a></li>
<li><a href="Sufficient_statistic" title="Sufficient statistic">Sufficiency</a></li>
<li><a href="Plug-in_principle" class="mw-redirect" title="Plug-in principle">Statistical functional</a>
<ul><li><a href="Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li>
<li><a href="U-statistic" title="U-statistic">U</a></li>
<li><a href="V-statistic" title="V-statistic">V</a></li></ul></li>
<li><a href="Optimal_decision" title="Optimal decision">Optimal decision</a>
<ul><li><a href="Loss_function" title="Loss function">loss function</a></li></ul></li>
<li><a href="Efficiency_(statistics)" title="Efficiency (statistics)">Efficiency</a></li>
<li><a href="Statistical_distance" title="Statistical distance">Statistical distance</a>
<ul><li><a href="Divergence_(statistics)" title="Divergence (statistics)">divergence</a></li></ul></li>
<li><a href="Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">Asymptotics</a></li>
<li><a href="Robust_statistics" title="Robust statistics">Robustness</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Frequentist_inference" title="Frequentist inference">Frequentist inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Point_estimation" title="Point estimation">Point estimation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Estimating_equations" title="Estimating equations">Estimating equations</a>
<ul><li><a href="Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">Maximum likelihood</a></li>
<li><a href="Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></li>
<li><a href="M-estimator" title="M-estimator">M-estimator</a></li>
<li><a href="Minimum_distance_estimation" class="mw-redirect" title="Minimum distance estimation">Minimum distance</a></li></ul></li>
<li><a href="Bias_of_an_estimator" title="Bias of an estimator">Unbiased estimators</a>
<ul><li><a href="Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">Mean-unbiased minimum-variance</a>
<ul><li><a href="Rao%E2%80%93Blackwell_theorem" title="Rao–Blackwell theorem">Rao–Blackwellization</a></li>
<li><a href="Lehmann%E2%80%93Scheff%C3%A9_theorem" title="Lehmann–Scheffé theorem">Lehmann–Scheffé theorem</a></li></ul></li>
<li><a href="Median-unbiased_estimator" class="mw-redirect" title="Median-unbiased estimator">Median unbiased</a></li></ul></li>
<li><a href="Plug-in_principle" class="mw-redirect" title="Plug-in principle">Plug-in</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Interval_estimation" title="Interval estimation">Interval estimation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Confidence_interval" title="Confidence interval">Confidence interval</a></li>
<li><a href="Pivotal_quantity" title="Pivotal quantity">Pivot</a></li>
<li><a href="Likelihood_interval" class="mw-redirect" title="Likelihood interval">Likelihood interval</a></li>
<li><a href="Prediction_interval" title="Prediction interval">Prediction interval</a></li>
<li><a href="Tolerance_interval" title="Tolerance interval">Tolerance interval</a></li>
<li><a href="Resampling_(statistics)" title="Resampling (statistics)">Resampling</a>
<ul><li><a href="Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li>
<li><a href="Jackknife_resampling" title="Jackknife resampling">Jackknife</a></li></ul></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">Testing hypotheses</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="One-_and_two-tailed_tests" title="One- and two-tailed tests">1- & 2-tails</a></li>
<li><a href="Power_(statistics)" title="Power (statistics)">Power</a>
<ul><li><a href="Uniformly_most_powerful_test" title="Uniformly most powerful test">Uniformly most powerful test</a></li></ul></li>
<li><a href="Permutation_test" title="Permutation test">Permutation test</a>
<ul><li><a href="Randomization_test" class="mw-redirect" title="Randomization test">Randomization test</a></li></ul></li>
<li><a href="Multiple_comparisons" class="mw-redirect" title="Multiple comparisons">Multiple comparisons</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Parametric_statistics" title="Parametric statistics">Parametric tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio</a></li>
<li><a href="Score_test" title="Score test">Score/Lagrange multiplier</a></li>
<li><a href="Wald_test" title="Wald test">Wald</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="List_of_statistical_tests" title="List of statistical tests">Specific tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Z-test" title="Z-test"><i>Z</i>-test <span style="font-size: 85%;">(normal)</span></a></li>
<li><a href="Student's_t-test" title="Student's t-test">Student's <i>t</i>-test</a></li>
<li><a href="F-test" title="F-test"><i>F</i>-test</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Chi-squared_test" title="Chi-squared test">Chi-squared</a></li>
<li><a href="G-test" title="G-test"><i>G</i>-test</a></li>
<li><a href="Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov</a></li>
<li><a href="Anderson%E2%80%93Darling_test" title="Anderson–Darling test">Anderson–Darling</a></li>
<li><a href="Lilliefors_test" title="Lilliefors test">Lilliefors</a></li>
<li><a href="Jarque%E2%80%93Bera_test" title="Jarque–Bera test">Jarque–Bera</a></li>
<li><a href="Shapiro%E2%80%93Wilk_test" title="Shapiro–Wilk test">Normality <span style="font-size: 85%;">(Shapiro–Wilk)</span></a></li>
<li><a href="Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio test</a></li>
<li><a href="Model_selection" title="Model selection">Model selection</a>
<ul><li><a href="Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross validation</a></li>
<li><a href="Akaike_information_criterion" title="Akaike information criterion">AIC</a></li>
<li><a href="Bayesian_information_criterion" title="Bayesian information criterion">BIC</a></li></ul></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Rank_statistics" class="mw-redirect" title="Rank statistics">Rank statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Sign_test" title="Sign test">Sign</a>
<ul><li><a href="Sample_median" class="mw-redirect" title="Sample median">Sample median</a></li></ul></li>
<li><a href="Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Signed rank <span style="font-size: 85%;">(Wilcoxon)</span></a>
<ul><li><a href="Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann estimator</a></li></ul></li>
<li><a href="Mann%E2%80%93Whitney_U_test" title="Mann–Whitney U test">Rank sum <span style="font-size: 85%;">(Mann–Whitney)</span></a></li>
<li><a href="Nonparametric_statistics" title="Nonparametric statistics">Nonparametric</a> <a href="Analysis_of_variance" title="Analysis of variance">anova</a>
<ul><li><a href="Kruskal%E2%80%93Wallis_test" title="Kruskal–Wallis test">1-way <span style="font-size: 85%;">(Kruskal–Wallis)</span></a></li>
<li><a href="Friedman_test" title="Friedman test">2-way <span style="font-size: 85%;">(Friedman)</span></a></li>
<li><a href="Jonckheere's_trend_test" title="Jonckheere's trend test">Ordered alternative <span style="font-size: 85%;">(Jonckheere–Terpstra)</span></a></li></ul></li>
<li><a href="Van_der_Waerden_test" title="Van der Waerden test">Van der Waerden test</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Bayesian_inference" title="Bayesian inference">Bayesian inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Bayesian_probability" title="Bayesian probability">Bayesian probability</a>
<ul><li><a href="Prior_probability" title="Prior probability">prior</a></li>
<li><a href="Posterior_probability" title="Posterior probability">posterior</a></li></ul></li>
<li><a href="Credible_interval" title="Credible interval">Credible interval</a></li>
<li><a href="Bayes_factor" title="Bayes factor">Bayes factor</a></li>
<li><a href="Bayes_estimator" title="Bayes estimator">Bayesian estimator</a>
<ul><li><a href="Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">Maximum posterior estimator</a></li></ul></li></ul>
</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="CorrelationRegression_analysis654" style="font-size:114%;margin:0 4em"><div class="hlist"><ul><li><a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></li><li><a href="Regression_analysis" title="Regression analysis">Regression analysis</a></li></ul></div></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment</a></li>
<li><a href="Partial_correlation" title="Partial correlation">Partial correlation</a></li>
<li><a href="Confounding" title="Confounding">Confounding variable</a></li>
<li><a href="Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Regression_analysis" title="Regression analysis">Regression analysis</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li>
<li><a href="Regression_validation" title="Regression validation">Regression validation</a></li>
<li><a href="Mixed_model" title="Mixed model">Mixed effects models</a></li>
<li><a href="Simultaneous_equations_model" title="Simultaneous equations model">Simultaneous equations models</a></li>
<li><a href="Multivariate_adaptive_regression_splines" class="mw-redirect" title="Multivariate adaptive regression splines">Multivariate adaptive regression splines (MARS)</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Linear_regression" title="Linear regression">Linear regression</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Simple_linear_regression" title="Simple linear regression">Simple linear regression</a></li>
<li><a href="Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li>
<li><a href="General_linear_model" title="General linear model">General linear model</a></li>
<li><a href="Bayesian_linear_regression" title="Bayesian linear regression">Bayesian regression</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Non-standard predictors</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li>
<li><a href="Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li>
<li><a href="Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li>
<li><a href="Isotonic_regression" title="Isotonic regression">Isotonic</a></li>
<li><a href="Robust_regression" title="Robust regression">Robust</a></li>
<li><a href="Homoscedasticity_and_heteroscedasticity" title="Homoscedasticity and heteroscedasticity">Homoscedasticity and Heteroscedasticity</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Exponential_family" title="Exponential family">Exponential families</a></li>
<li><a href="Logistic_regression" title="Logistic regression">Logistic <span style="font-size: 85%;">(Bernoulli)</span></a> / <a href="Binomial_regression" title="Binomial regression">Binomial</a> / <a href="Poisson_regression" title="Poisson regression">Poisson regressions</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Partition_of_sums_of_squares" title="Partition of sums of squares">Partition of variance</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Analysis_of_variance" title="Analysis of variance">Analysis of variance (ANOVA, anova)</a></li>
<li><a href="Analysis_of_covariance" title="Analysis of covariance">Analysis of covariance</a></li>
<li><a href="Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Multivariate ANOVA</a></li>
<li><a href="Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">Degrees of freedom</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Categorical_/_multivariate_/_time-series_/_survival_analysis654" style="font-size:114%;margin:0 4em"><a href="Categorical_variable" title="Categorical variable">Categorical</a> / <a href="Multivariate_statistics" title="Multivariate statistics">multivariate</a> / <a href="Time_series" title="Time series">time-series</a> / <a href="Survival_analysis" title="Survival analysis">survival analysis</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Categorical_variable" title="Categorical variable">Categorical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Cohen's_kappa" title="Cohen's kappa">Cohen's kappa</a></li>
<li><a href="Contingency_table" title="Contingency table">Contingency table</a></li>
<li><a href="Graphical_model" title="Graphical model">Graphical model</a></li>
<li><a href="Poisson_regression" title="Poisson regression">Log-linear model</a></li>
<li><a href="McNemar's_test" title="McNemar's test">McNemar's test</a></li>
<li><a href="Cochran%E2%80%93Mantel%E2%80%93Haenszel_statistics" title="Cochran–Mantel–Haenszel statistics">Cochran–Mantel–Haenszel statistics</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Multivariate_statistics" title="Multivariate statistics">Multivariate</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="General_linear_model" title="General linear model">Regression</a></li>
<li><a href="Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Manova</a></li>
<li><a href="Principal_component_analysis" title="Principal component analysis">Principal components</a></li>
<li><a href="Canonical_correlation" title="Canonical correlation">Canonical correlation</a></li>
<li><a href="Linear_discriminant_analysis" title="Linear discriminant analysis">Discriminant analysis</a></li>
<li><a href="Cluster_analysis" title="Cluster analysis">Cluster analysis</a></li>
<li><a href="Statistical_classification" title="Statistical classification">Classification</a></li>
<li><a href="Structural_equation_modeling" title="Structural equation modeling">Structural equation model</a>
<ul><li><a href="Factor_analysis" title="Factor analysis">Factor analysis</a></li></ul></li>
<li><a href="Multivariate_distribution" class="mw-redirect" title="Multivariate distribution">Multivariate distributions</a>
<ul><li><a href="Elliptical_distribution" title="Elliptical distribution">Elliptical distributions</a>
<ul><li><a href="Multivariate_normal_distribution" title="Multivariate normal distribution">Normal</a></li></ul></li></ul></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Time_series" title="Time series">Time-series</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Decomposition_of_time_series" title="Decomposition of time series">Decomposition</a></li>
<li><a href="Trend_estimation" class="mw-redirect" title="Trend estimation">Trend</a></li>
<li><a href="Stationary_process" title="Stationary process">Stationarity</a></li>
<li><a href="Seasonal_adjustment" title="Seasonal adjustment">Seasonal adjustment</a></li>
<li><a href="Exponential_smoothing" title="Exponential smoothing">Exponential smoothing</a></li>
<li><a href="Cointegration" title="Cointegration">Cointegration</a></li>
<li><a href="Structural_break" title="Structural break">Structural break</a></li>
<li><a href="Granger_causality" title="Granger causality">Granger causality</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Specific tests</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Dickey%E2%80%93Fuller_test" title="Dickey–Fuller test">Dickey–Fuller</a></li>
<li><a href="Johansen_test" title="Johansen test">Johansen</a></li>
<li><a href="Ljung%E2%80%93Box_test" title="Ljung–Box test">Q-statistic <span style="font-size: 85%;">(Ljung–Box)</span></a></li>
<li><a href="Durbin%E2%80%93Watson_statistic" title="Durbin–Watson statistic">Durbin–Watson</a></li>
<li><a href="Breusch%E2%80%93Godfrey_test" title="Breusch–Godfrey test">Breusch–Godfrey</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Time_domain" title="Time domain">Time domain</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Autocorrelation" title="Autocorrelation">Autocorrelation (ACF)</a>
<ul><li><a href="Partial_autocorrelation_function" title="Partial autocorrelation function">partial (PACF)</a></li></ul></li>
<li><a href="Cross-correlation" title="Cross-correlation">Cross-correlation (XCF)</a></li>
<li><a href="Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">ARMA model</a></li>
<li><a href="Box%E2%80%93Jenkins_method" title="Box–Jenkins method">ARIMA model <span style="font-size: 85%;">(Box–Jenkins)</span></a></li>
<li><a href="Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH)</a></li>
<li><a href="Vector_autoregression" title="Vector autoregression">Vector autoregression (VAR)</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Frequency_domain" title="Frequency domain">Frequency domain</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li>
<li><a href="Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li>
<li><a href="Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li>
<li><a href="Wavelet" title="Wavelet">Wavelet</a></li>
<li><a href="Whittle_likelihood" title="Whittle likelihood">Whittle likelihood</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Survival_analysis" title="Survival analysis">Survival</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Survival_function" title="Survival function">Survival function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Kaplan%E2%80%93Meier_estimator" title="Kaplan–Meier estimator">Kaplan–Meier estimator (product limit)</a></li>
<li><a href="Proportional_hazards_model" title="Proportional hazards model">Proportional hazards models</a></li>
<li><a href="Accelerated_failure_time_model" title="Accelerated failure time model">Accelerated failure time (AFT) model</a></li>
<li><a href="First-hitting-time_model" title="First-hitting-time model">First hitting time</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Failure_rate" title="Failure rate">Hazard function</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Nelson%E2%80%93Aalen_estimator" title="Nelson–Aalen estimator">Nelson–Aalen estimator</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Test</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Log-rank_test" class="mw-redirect" title="Log-rank test">Log-rank test</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Applications654" style="font-size:114%;margin:0 4em"><a href="List_of_fields_of_application_of_statistics" title="List of fields of application of statistics">Applications</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Biostatistics" title="Biostatistics">Biostatistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Bioinformatics" title="Bioinformatics">Bioinformatics</a></li>
<li><a href="Clinical_trial" title="Clinical trial">Clinical trials</a> / <a href="Clinical_study_design" title="Clinical study design">studies</a></li>
<li><a href="Epidemiology" title="Epidemiology">Epidemiology</a></li>
<li><a href="Medical_statistics" title="Medical statistics">Medical statistics</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Engineering_statistics" title="Engineering statistics">Engineering statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Chemometrics" title="Chemometrics">Chemometrics</a></li>
<li><a href="Methods_engineering" title="Methods engineering">Methods engineering</a></li>
<li><a href="Probabilistic_design" title="Probabilistic design">Probabilistic design</a></li>
<li><a href="Statistical_process_control" title="Statistical process control">Process</a> / <a href="Quality_control" title="Quality control">quality control</a></li>
<li><a href="Reliability_engineering" title="Reliability engineering">Reliability</a></li>
<li><a href="System_identification" title="System identification">System identification</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Social_statistics" title="Social statistics">Social statistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Actuarial_science" title="Actuarial science">Actuarial science</a></li>
<li><a href="Census" title="Census">Census</a></li>
<li><a href="Crime_statistics" title="Crime statistics">Crime statistics</a></li>
<li><a href="Demographic_statistics" title="Demographic statistics">Demography</a></li>
<li><a href="Econometrics" title="Econometrics">Econometrics</a></li>
<li><a href="Jurimetrics" title="Jurimetrics">Jurimetrics</a></li>
<li><a href="National_accounts" title="National accounts">National accounts</a></li>
<li><a href="Official_statistics" title="Official statistics">Official statistics</a></li>
<li><a href="Population_statistics" class="mw-redirect" title="Population statistics">Population statistics</a></li>
<li><a href="Psychometrics" title="Psychometrics">Psychometrics</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Spatial_analysis" title="Spatial analysis">Spatial statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Cartography" title="Cartography">Cartography</a></li>
<li><a href="Environmental_statistics" title="Environmental statistics">Environmental statistics</a></li>
<li><a href="Geographic_information_system" title="Geographic information system">Geographic information system</a></li>
<li><a href="Geostatistics" title="Geostatistics">Geostatistics</a></li>
<li><a href="Kriging" title="Kriging">Kriging</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>
<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span><b>Category</b></li>
<li><b><span class="nowrap"><span class="skin-invert-image noviewer" typeof="mw:File"></span> </span><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></b></li>
<li><span class="noviewer" typeof="mw:File"><span title="Commons page"></span></span><b><a href="https://commons.wikimedia.org/wiki/Category:Statistics" class="extiw external" title="commons:Category:Statistics">Commons</a></b></li>
<li><span class="noviewer" typeof="mw:File"><span title="WikiProject"></span></span> <b>WikiProject</b></li></ul>
</div></td></tr></tbody></table></div>
<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Clinical_research_and_experimental_design505" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Clinical_research_and_experimental_design505" style="font-size:114%;margin:0 4em"><a href="Clinical_research" title="Clinical research">Clinical research</a> and </div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Overview</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Clinical_trial" title="Clinical trial">Clinical trial</a>
<ul><li><a href="Protocol_(science)" title="Protocol (science)">Trial protocols</a></li>
<li><a href="Adaptive_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a></li></ul></li>
<li><a href="Academic_clinical_trials" class="mw-redirect" title="Academic clinical trials">Academic clinical trials</a></li>
<li><a href="Clinical_study_design" title="Clinical study design">Clinical study design</a></li>
<li><a href="Evidence-based_medicine" title="Evidence-based medicine">Evidence-based medicine</a></li>
<li><a href="Real_world_evidence" title="Real world evidence">Real world evidence</a></li>
<li><a href="Patient_and_public_involvement" title="Patient and public involvement">Patient and public involvement</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Scientific_control" title="Scientific control">Controlled study</a><br>(<a href="Hierarchy_of_evidence#USA" title="Hierarchy of evidence">EBM I to II-1</a>)</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a>
<ul><li><a href="Experiment" title="Experiment">Scientific experiment</a></li>
<li><a href="Blind_experiment" class="mw-redirect" title="Blind experiment">Blind experiment</a></li>
<li><a href="Open-label_trial" title="Open-label trial">Open-label trial</a></li></ul></li>
<li><a href="Adaptive_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a>
<ul><li><a href="Platform_trial" title="Platform trial">Platform trial</a></li></ul></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Observational_study" title="Observational study">Observational study</a><br>(<a href="Hierarchy_of_evidence#USA" title="Hierarchy of evidence">EBM II-2 to II-3</a>)</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a> vs. <a href="Longitudinal_study" title="Longitudinal study">Longitudinal study</a>, <a href="Ecological_study" title="Ecological study">Ecological study</a></li>
<li><a href="Cohort_study" title="Cohort study">Cohort study</a>
<ul><li><a href="Retrospective_cohort_study" title="Retrospective cohort study">Retrospective</a></li>
<li><a href="Prospective_cohort_study" title="Prospective cohort study">Prospective</a></li></ul></li>
<li><a href="Case%E2%80%93control_study" title="Case–control study">Case–control study</a> (<a href="Nested_case%E2%80%93control_study" title="Nested case–control study">Nested case–control study</a>)</li>
<li><a href="Case_series" title="Case series">Case series</a></li>
<li><a href="Case_study" title="Case study">Case study</a></li>
<li><a href="Case_report" title="Case report">Case report</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Outcome_measure" title="Outcome measure">Measures</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Epidemiological_methods" class="mw-redirect" title="Epidemiological methods">Occurrence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><a href="Incidence_(epidemiology)" title="Incidence (epidemiology)">Incidence</a>, <a href="Cumulative_incidence" class="mw-redirect" title="Cumulative incidence">Cumulative incidence</a>, <a href="Prevalence" title="Prevalence">Prevalence</a>, <a href="Point_prevalence" class="mw-redirect" title="Point prevalence">Point prevalence</a>, <a href="Period_prevalence" class="mw-redirect" title="Period prevalence">Period prevalence</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Epidemiological_methods" class="mw-redirect" title="Epidemiological methods">Association</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><a href="Risk_difference" title="Risk difference">Risk difference</a>, <a href="Number_needed_to_treat" title="Number needed to treat">Number needed to treat</a>, <a href="Number_needed_to_harm" title="Number needed to harm">Number needed to harm</a>, <a href="Risk_ratio" class="mw-redirect" title="Risk ratio">Risk ratio</a>, <a href="Relative_risk_reduction" title="Relative risk reduction">Relative risk reduction</a>, <a href="Odds_ratio" title="Odds ratio">Odds ratio</a>, <a href="Hazard_ratio" title="Hazard ratio">Hazard ratio</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Population_impact_measure" title="Population impact measure">Population impact</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><a href="Attributable_fraction_among_the_exposed" title="Attributable fraction among the exposed">Attributable fraction among the exposed</a>, <a href="Attributable_fraction_for_the_population" title="Attributable fraction for the population">Attributable fraction for the population</a>, <a href="Preventable_fraction_among_the_unexposed" title="Preventable fraction among the unexposed">Preventable fraction among the unexposed</a>, <a href="Preventable_fraction_for_the_population" title="Preventable fraction for the population">Preventable fraction for the population</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Epidemiological_methods" class="mw-redirect" title="Epidemiological methods">Other</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><a href="Clinical_endpoint" title="Clinical endpoint">Clinical endpoint</a>, <a href="Virulence" title="Virulence">Virulence</a>, <a href="Infectivity" title="Infectivity">Infectivity</a>, <a href="Mortality_rate" title="Mortality rate">Mortality rate</a>, <a href="Morbidity" class="mw-redirect" title="Morbidity">Morbidity</a>, <a href="Case_fatality_rate" title="Case fatality rate">Case fatality rate</a>, <a href="Specificity_and_sensitivity" class="mw-redirect" title="Specificity and sensitivity">Specificity and sensitivity</a>, <a href="Likelihood_ratios_in_diagnostic_testing" title="Likelihood ratios in diagnostic testing">Likelihood-ratios</a>, <a href="Pre-_and_post-test_probability" title="Pre- and post-test probability">Pre- and post-test probability</a></div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Trial/test types</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="In_vitro" title="In vitro">In vitro</a></li>
<li><a href="In_vivo" title="In vivo">In vivo</a></li>
<li><a href="Animal_testing" title="Animal testing">Animal testing</a></li>
<li><a href="Animal_testing_on_non-human_primates" title="Animal testing on non-human primates">Animal testing on non-human primates</a></li>
<li><a href="First-in-man_study" class="mw-redirect" title="First-in-man study">First-in-man study</a></li>
<li><a href="Multicenter_trial" title="Multicenter trial">Multicenter trial</a></li>
<li><a href="Seeding_trial" title="Seeding trial">Seeding trial</a></li>
<li><a href="Vaccine_trial" title="Vaccine trial">Vaccine trial</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Analysis_of_clinical_trials" title="Analysis of clinical trials">Analysis of clinical trials</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Risk%E2%80%93benefit_ratio" title="Risk–benefit ratio">Risk–benefit ratio</a></li>
<li><a href="Systematic_review" title="Systematic review">Systematic review</a></li>
<li><a href="Reproducibility" title="Reproducibility">Replication</a></li>
<li><a href="Meta-analysis" title="Meta-analysis">Meta-analysis</a></li>
<li><a href="Intention-to-treat_analysis" title="Intention-to-treat analysis">Intention-to-treat analysis</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Interpretation of results</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Selection_bias" title="Selection bias">Selection bias</a></li>
<li><a href="Survivorship_bias" title="Survivorship bias">Survivorship bias</a></li>
<li><a href="Correlation_does_not_imply_causation" title="Correlation does not imply causation">Correlation does not imply causation</a></li>
<li><a href="Null_result" title="Null result">Null result</a></li>
<li><a href="Sex_as_a_biological_variable" title="Sex as a biological variable">Sex as a biological variable</a></li></ul>
</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>
<ul><li><b>Category</b></li>
<li><b><a href="Glossary_of_clinical_research" title="Glossary of clinical research">Glossary</a></b></li>
<li><b><a href="List_of_clinical_research_topics" class="mw-redirect" title="List of clinical research topics">List of topics</a></b></li></ul>
</div></td></tr></tbody></table></div>
<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Six_Sigma_tools92" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Six_Sigma_tools92" style="font-size:114%;margin:0 4em"><a href="Six_Sigma" title="Six Sigma">Six Sigma</a> tools</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Define phase</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Project_charter" title="Project charter">Project charter</a></li>
<li><a href="Voice_of_the_customer" title="Voice of the customer">Voice of the customer</a></li>
<li><a href="Value-stream_mapping" title="Value-stream mapping">Value-stream mapping</a></li>
<li><a href="SIPOC" title="SIPOC">SIPOC</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Measure phase</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Business_process_mapping" title="Business process mapping">Business process mapping</a></li>
<li><a href="Process_capability" title="Process capability">Process capability</a></li>
<li><a href="Pareto_chart" title="Pareto chart">Pareto chart</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Analyse phase</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Root_cause_analysis" title="Root cause analysis">Root cause analysis</a></li>
<li><a href="Failure_mode_and_effects_analysis" title="Failure mode and effects analysis">Failure mode and effects analysis</a></li>
<li><a href="Multi-vari_chart" title="Multi-vari chart">Multi-vari chart</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Improve phase</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul>
<li><a href="Kaizen" title="Kaizen">Kaizen</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Control phase</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li>Control plan</li>
<li><a href="Statistical_process_control" title="Statistical process control">Statistical process control</a></li>
<li><a href="5S_(methodology)" title="5S (methodology)">5S</a></li>
<li><a href="Poka-yoke" title="Poka-yoke">Poka-yoke</a></li></ul>
</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><a href="DMAIC" title="DMAIC">DMAIC</a></div></td></tr></tbody></table></div>
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