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Logistic regression can take into account stratification by having a different constant term for each strata. Let us denote <math>Y_{i\ell}\in\{0,1\}</math> the label (e.g. case status) of the <math>\ell</math>th observation of the <math>i</math>th strata and <math>X_{i\ell}\in\mathbb{R}^p</math> the values of the corresponding predictors. Then, the likelihood of one observation is
:<math> \mathbb{P}(Y_{i\ell}=1|X_{i\ell})=\frac{\exp(\alpha_i + \boldsymbol\beta^\top X_{i\ell})}{1+\exp(\alpha_i + \boldsymbol\beta^\top X_{i\ell})}</math>
where <math>\alpha_i</math> is the constant term for the <math>i</math>th strata. While this works satisfactorily for a limited number of strata, pathological behavior occurs when the strata are small. When the strata are pairs, the number of variables grows with the number of observations <math>N</math> (it equals <math>\frac{N}{2}+p</math>). The asymptotic results on which [[maximum likelihood estimation]] is based on are therefore not valid and the estimation is biased. In fact, it can be shown that the unconditional analysis of matched pair data results in an estimate of the odds ratio which is the square of the correct, conditional one.<ref>{{cite book |last1=Breslow |first1=N.E. |last2=Day|first2=N.E.|date=1980 |title=Statistical Methods in Cancer Research. Volume 1-The Analysis of Case-Control Studies |url=http://www.iarc.fr/en/publications/pdfs-online/stat/sp32/ |___location=Lyon, France |publisher= IARC |pages=249–251 }}</ref>
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