Conditional logistic regression: Difference between revisions

Revision as of 17:02, 21 April 2019

Revision as of 09:44, 29 April 2019

Line 2:

==Motivation==

Observational studies use [[stratification (clinical trials)|stratification]] or [[Matching (statistics)|matching]] as a way to control for [[confounding]]. Several tests existed before conditional logistic regression for matched data as shown in [[Conditional logistic regression#Related tests|related tests]]. However, they did not allow for the analysis of continuous predictors with arbitrary stratastratum size. All of those procedures also lack the flexibility of conditional logistic regression and in particular the possibility to control for covariates.

Logistic regression can take into account stratification by having a different constant term for each stratastratum. 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 stratastratum 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 stratastratum. 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>

==Conditional likelihood==

The conditional likelihood approach deals with the above pathological behavior by conditioning on the number of cases in each stratastratum and therefore eliminating the need to estimate the strata parameters. In the case where the strata are pairs, where the first observation is a case and the second is a control, this can be seen as follows

:<math>

\begin{align}

Line 21:

</math>

With similar computations, the conditional likelihood of a stratastratum of size <math>m</math>, with the <math>k</math> first observations being the cases, is

:<math>

\mathbb{P}(Y_{ij}=1\text{ for }j\leq k,Y_{ij}=0\text{ for } k<j\leq m|X_{i1},...,X_{im},\sum_{j=1}^m Y_{ij}=k)=\frac{\exp(\sum_{j=1}^k \boldsymbol{\beta}^\top X_{ij})}{\sum_{J\in \mathcal{C} _{k}^{m}} \exp(\sum_{j\in J}\boldsymbol{\beta}^\top X_{ij})},

Line 27:

where <math>\mathcal{C} _{k}^{m}</math> is the set of all subsets of size <math>k</math> of the set <math>\{1,...,m\}</math>.

The full conditional log likelihood is then simply the sum of the log likelihoods for each stratastratum. The estimator is then defined as the <math>\beta</math> that maximizes the conditional log likelihood.

==Implementation==