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<title>Local case-control sampling</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Local case-control sampling</span></span>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Machine_learning" title="Machine learning">machine learning</a>, <b>local case-control sampling</b> <sup id="cite_ref-LCC_1-0" class="reference"><a href="#cite_note-LCC-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> is an <a href="Algorithm" title="Algorithm">algorithm</a> used to reduce the complexity of training a <a href="Logistic_regression" title="Logistic regression">logistic regression</a> classifier. The algorithm reduces the training complexity by selecting a small subsample of the original dataset for training. It assumes the availability of a (unreliable) pilot estimation of the parameters. It then performs a single pass over the entire dataset using the pilot estimation to identify the most "surprising" samples. In practice, the pilot may come from prior knowledge or training using a subsample of the dataset. The algorithm is most effective when the underlying dataset is imbalanced. It exploits the structures of conditional imbalanced datasets more efficiently than alternative methods, such as <a href="Logistic_regression#Case-control_sampling" title="Logistic regression">case control sampling</a> and weighted case control sampling.

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<div class="mw-heading mw-heading2"><h2 id="Imbalanced_datasets">Imbalanced datasets</h2></div>

<p>In <a href="Statistical_classification" title="Statistical classification">classification</a>, a dataset is a set of <i>N</i> data points <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 (x_{i},y_{i})_{i=1}^{N}}">

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<annotation encoding="application/x-tex">{\displaystyle (x_{i},y_{i})_{i=1}^{N}}</annotation>

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</math></span><img src="./6be021862dbd361a6698e0a3568f074e5722d2b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.812ex; height:3.176ex;" alt="{\displaystyle (x_{i},y_{i})_{i=1}^{N}}" loading="lazy"></span>, where <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 x_{i}\in \mathbb {R} ^{d}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{i}\in \mathbb {R} ^{d}}</annotation>

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</math></span><img src="./c075fe55b0ba068fc53eb1e867304432c038e22f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.74ex; height:3.009ex;" alt="{\displaystyle x_{i}\in \mathbb {R} ^{d}}" loading="lazy"></span> is a feature vector, <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 y_{i}\in \{0,1\}}">

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<annotation encoding="application/x-tex">{\displaystyle y_{i}\in \{0,1\}}</annotation>

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</math></span><img src="./e5c2ebeb5127a5306baf27c9c622a86aa35625f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.463ex; height:2.843ex;" alt="{\displaystyle y_{i}\in \{0,1\}}" loading="lazy"></span> is a label. Intuitively, a dataset is imbalanced when certain important statistical patterns are rare. The lack of observations of certain patterns does not always imply their irrelevance. For example, in medical studies of rare diseases, the small number of infected patients (cases) conveys the most valuable information for diagnosis and treatments.

</p><p>Formally, an imbalanced dataset exhibits one or more of the following properties:

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<ul><li><i>Marginal Imbalance</i>. A dataset is marginally imbalanced if one class is rare compared to the other class. In other words, <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 \mathbb {P} (Y=1)\approx 0}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (Y=1)\approx 0}</annotation>

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</math></span><img src="./758cf22414d80d532428360788c12b534b9b3983.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.525ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} (Y=1)\approx 0}" loading="lazy"></span>.</li>

<li><i>Conditional Imbalance</i>. A dataset is conditionally imbalanced when it is easy to predict the correct labels in most cases. For example, if <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 X\in \{0,1\}}">

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<annotation encoding="application/x-tex">{\displaystyle X\in \{0,1\}}</annotation>

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</math></span><img src="./a9ca6b78592b27f74b2ee3e0e0912b263be602e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.504ex; height:2.843ex;" alt="{\displaystyle X\in \{0,1\}}" loading="lazy"></span>, the dataset is conditionally imbalanced if <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 \mathbb {P} (Y=1\mid X=0)\approx 0}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (Y=1\mid X=0)\approx 0}</annotation>

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</math></span><img src="./190919ce128f82a7836ed0ef15bd4e2fa897c1ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.703ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} (Y=1\mid X=0)\approx 0}" loading="lazy"></span> and <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 \mathbb {P} (Y=1\mid X=1)\approx 1}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (Y=1\mid X=1)\approx 1}</annotation>

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</math></span><img src="./966daa04ef6e5588f572355e12efd5d8e140c916.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.703ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} (Y=1\mid X=1)\approx 1}" loading="lazy"></span>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Algorithm_outline">Algorithm outline</h2></div>

<p>In logistic regression, given the model <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 =(\alpha ,\beta )}">

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<annotation encoding="application/x-tex">{\displaystyle \theta =(\alpha ,\beta )}</annotation>

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</math></span><img src="./ed8bfb170a41d3ed5dfd57adce9753aa194652b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.852ex; height:2.843ex;" alt="{\displaystyle \theta =(\alpha ,\beta )}" loading="lazy"></span>, the prediction is made according to <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 \mathbb {P} (Y=1\mid X;\theta )={\tilde {p}}_{\theta }(x)={\frac {\exp(\alpha +\beta ^{T}x)}{1+\exp(\alpha +\beta ^{T}x)}}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (Y=1\mid X;\theta )={\tilde {p}}_{\theta }(x)={\frac {\exp(\alpha +\beta ^{T}x)}{1+\exp(\alpha +\beta ^{T}x)}}}</annotation>

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</math></span><img src="./0fd1aa2f4b0cfff51fefd9c472159383c01d5860.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:45.589ex; height:6.676ex;" alt="{\displaystyle \mathbb {P} (Y=1\mid X;\theta )={\tilde {p}}_{\theta }(x)={\frac {\exp(\alpha +\beta ^{T}x)}{1+\exp(\alpha +\beta ^{T}x)}}}" loading="lazy"></span>. The local-case control sampling algorithm assumes the availability of a pilot model <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 {\tilde {\theta }}=({\tilde {\alpha }},{\tilde {\beta }})}">

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {\theta }}=({\tilde {\alpha }},{\tilde {\beta }})}</annotation>

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</math></span><img src="./ec80675882c56e401fbd8c2da245f777f9c6f20c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.237ex; height:3.176ex;" alt="{\displaystyle {\tilde {\theta }}=({\tilde {\alpha }},{\tilde {\beta }})}" loading="lazy"></span>. Given the pilot model, the algorithm performs a single pass over the entire dataset to select the subset of samples to include in training the logistic regression model. For a sample <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 (x,y)}">

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<annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation>

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</math></span><img src="./41cf50e4a314ca8e2c30964baa8d26e5be7a9386.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}" loading="lazy"></span>, define the acceptance probability as <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 a(x,y)=|y-{\tilde {p}}_{\tilde {\theta }}(x)|}">

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<annotation encoding="application/x-tex">{\displaystyle a(x,y)=|y-{\tilde {p}}_{\tilde {\theta }}(x)|}</annotation>

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</math></span><img src="./ee02c4fdbd88c1f6453dccd6cf580cd418efa3de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.692ex; height:3.009ex;" alt="{\displaystyle a(x,y)=|y-{\tilde {p}}_{\tilde {\theta }}(x)|}" loading="lazy"></span>. The algorithm proceeds as follows:

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<ol><li>Generate independent <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 z_{i}\sim {\text{Bernoulli}}(a(x_{i},y_{i}))}">

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<mtext>Bernoulli</mtext>

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<annotation encoding="application/x-tex">{\displaystyle z_{i}\sim {\text{Bernoulli}}(a(x_{i},y_{i}))}</annotation>

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</math></span><img src="./781fee56ae1ce3800529b30cdd3460dc95716d33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.208ex; height:2.843ex;" alt="{\displaystyle z_{i}\sim {\text{Bernoulli}}(a(x_{i},y_{i}))}" loading="lazy"></span> for <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 i\in \{1,\ldots ,N\}}">

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<annotation encoding="application/x-tex">{\displaystyle i\in \{1,\ldots ,N\}}</annotation>

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</math></span><img src="./95f9e7f38c3a1671e1b43d0dc4215d11d8f64968.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.372ex; height:2.843ex;" alt="{\displaystyle i\in \{1,\ldots ,N\}}" loading="lazy"></span>.</li>

<li>Fit a logistic regression model to the subsample <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 S=\{(x_{i},y_{i}):z_{i}=1\}}">

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<annotation encoding="application/x-tex">{\displaystyle S=\{(x_{i},y_{i}):z_{i}=1\}}</annotation>

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</math></span><img src="./2881dfebd187edb6e7a7fc45a4063743d40b3467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.913ex; height:2.843ex;" alt="{\displaystyle S=\{(x_{i},y_{i}):z_{i}=1\}}" loading="lazy"></span>, obtaining the unadjusted estimates <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 {\hat {\theta }}_{S}=({\hat {\alpha }}_{S},{\hat {\beta }}_{S})}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\theta }}_{S}=({\hat {\alpha }}_{S},{\hat {\beta }}_{S})}</annotation>

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</math></span><img src="./7b48a8f6b2dd1794a4f36f267e0b93dea88bec6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.114ex; height:3.343ex;" alt="{\displaystyle {\hat {\theta }}_{S}=({\hat {\alpha }}_{S},{\hat {\beta }}_{S})}" loading="lazy"></span>.</li>

<li>The output model is <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 {\hat {\theta }}=({\hat {\alpha }},{\hat {\beta }})}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\theta }}=({\hat {\alpha }},{\hat {\beta }})}</annotation>

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</math></span><img src="./4dcc3a0c886d29cde3f01ce8dceb118662304bf6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.237ex; height:3.343ex;" alt="{\displaystyle {\hat {\theta }}=({\hat {\alpha }},{\hat {\beta }})}" loading="lazy"></span>, where <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 {\hat {\alpha }}\leftarrow {\hat {\alpha }}_{S}+{\tilde {\alpha }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\alpha }}\leftarrow {\hat {\alpha }}_{S}+{\tilde {\alpha }}}</annotation>

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</math></span><img src="./d07bb0dac27a09c30cc609d3067ea238f5556027.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.21ex; height:2.509ex;" alt="{\displaystyle {\hat {\alpha }}\leftarrow {\hat {\alpha }}_{S}+{\tilde {\alpha }}}" loading="lazy"></span> and <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 {\hat {\beta }}\leftarrow {\hat {\beta }}_{S}+{\tilde {\beta }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\beta }}\leftarrow {\hat {\beta }}_{S}+{\tilde {\beta }}}</annotation>

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</math></span><img src="./3fd576d3ab0c3fb151fbfc30f724e639bb7dbe08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.101ex; height:3.343ex;" alt="{\displaystyle {\hat {\beta }}\leftarrow {\hat {\beta }}_{S}+{\tilde {\beta }}}" loading="lazy"></span>.</li></ol>

<p>The algorithm can be understood as selecting samples that surprises the pilot model. Intuitively these samples are closer to the <a href="Decision_boundary" title="Decision boundary">decision boundary</a> of the classifier and is thus more informative.

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<div class="mw-heading mw-heading3"><h3 id="Obtaining_the_pilot_model">Obtaining the pilot model</h3></div>

<p>In practice, for cases where a pilot model is naturally available, the algorithm can be applied directly to reduce the complexity of training. In cases where a natural pilot is nonexistent, an estimate using a subsample selected through another sampling technique can be used instead. In the original paper describing the algorithm, the authors propose to use weighted case-control sampling with half the assigned sampling budget. For example, if the objective is to use a subsample with size <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 N=1000}">

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<annotation encoding="application/x-tex">{\displaystyle N=1000}</annotation>

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</math></span><img src="./61bed31d4342cb97d4d8853c5c7ddd4d5b93adaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.812ex; height:2.176ex;" alt="{\displaystyle N=1000}" loading="lazy"></span>, first estimate a model <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 {\tilde {\theta }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {\theta }}}</annotation>

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</math></span><img src="./49469d232a862cff02082ead6773b64c1df1fed2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.356ex; height:2.676ex;" alt="{\displaystyle {\tilde {\theta }}}" loading="lazy"></span> using <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 N_{h}=500}">

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</math></span><img src="./52274d827b606bb90518ef7929c7f4659599c2bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.631ex; height:2.509ex;" alt="{\displaystyle N_{h}=500}" loading="lazy"></span> samples from weighted case control sampling, then collect another <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 N_{h}=500}">

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</math></span><img src="./52274d827b606bb90518ef7929c7f4659599c2bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.631ex; height:2.509ex;" alt="{\displaystyle N_{h}=500}" loading="lazy"></span> samples using local case-control sampling.

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<div class="mw-heading mw-heading3"><h3 id="Larger_or_smaller_sample_size">Larger or smaller sample size</h3></div>

<p>It is possible to control the sample size by multiplying the acceptance probability with a constant <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 c}">

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<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

</semantics>

</math></span><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span>. For a larger sample size, pick <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 c>1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c>1}</annotation>

</semantics>

</math></span><img src="./bd1d6b5c226641ad3def8e65631f0eb463d9c67a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c>1}" loading="lazy"></span> and adjust the acceptance probability to <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 \min(ca(x_{i},y_{i}),1)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \min(ca(x_{i},y_{i}),1)}</annotation>

</semantics>

</math></span><img src="./3c5a913e60d34ce7a8f77453417c287a2c0fb89a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.029ex; height:2.843ex;" alt="{\displaystyle \min(ca(x_{i},y_{i}),1)}" loading="lazy"></span>. For a smaller sample size, the same strategy applies. In cases where the number of samples desired is precise, a convenient alternative method is to uniformly downsample from a larger subsample selected by local case-control sampling.

</p>

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

<p>The algorithm has the following properties. When the pilot is <a href="Consistency_(statistics)" title="Consistency (statistics)">consistent</a>, the estimates using the samples from local case-control sampling is consistent even under <a href="Statistical_model_specification" title="Statistical model specification">model misspecification</a>. If the model is correct then the algorithm has exactly twice the asymptotic variance of logistic regression on the full data set. For a larger sample size with <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 c>1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c>1}</annotation>

</semantics>

</math></span><img src="./bd1d6b5c226641ad3def8e65631f0eb463d9c67a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c>1}" loading="lazy"></span>, the factor 2 is improved to <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 1+{\frac {1}{c}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1+{\frac {1}{c}}}</annotation>

</semantics>

</math></span><img src="./30fd4d09f648846e152bb1837a48cc6f4db858dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.001ex; height:5.176ex;" alt="{\displaystyle 1+{\frac {1}{c}}}" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFFithianHastie2014" class="citation journal cs1">Fithian, William; Hastie, Trevor (2014). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4258397">"Local case-control sampling: Efficient subsampling in imbalanced data sets"</a>. <i>The Annals of Statistics</i>. <b>42</b> (5): <span class="nowrap">1693–</span>1724. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1306.3706">1306.3706</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2F14-aos1220">10.1214/14-aos1220</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4258397">4258397</a></span>. <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/25492979">25492979</a>.</cite></span>

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