EP 521, Spring 2007 Vol II, Part 4 Copyright © 2006 Trustees of the University of Pennsylvania 1§6. Conditional Logistic Regression (CLR) for Matched or Stratified Data §6.1 Overview Have considered “unconditional” regression, e.g., logistic Results in estimate for intercept (α) which corresponds to the baseline risk (risk in group with no risk factors present) and estimates of “slope” (β’s) which represent departure from baseline risk when one or more factors is present. If there 5 centers in a study, unconditional regression will need 4 dummy variables to account for differences across those centers. For example, suppose we have a simple two-arm randomized study. 0 1 1 2 2 3 3 4 4ˆln( )ˆ1phhhhtxp=α+α⋅+α⋅+α⋅+α⋅+β⋅− Then we assume that there are 5 possible “intercepts” or baseline risks in the control groups across the 5 hospitals (0,…,4) and then a common treatment effect. But this method requires that we estimate 5 parameters for the hospital. For 30 hospitals, this means 30 parameters. EP 521, Spring 2006, Vol II, Part 4 2 “Conditional” regression – what do we mean? (a) Have levels of strata (age category) or centers (hospitals) or pairs (matched) (b) Wish to estimate association of a within-strata exposure (drug exposure) and outcome (c) Have different intercept for each stratum, but these levels are “nuisance” parameters – we do not care about them, and we cannot estimate them using this method. They are “conditioned out” of the analysis (d) We want to condition these out in order to address the problem of heterogeneity in the baseline risk (the risk of outcome in the reference group of patients) Example: In a group of 30 ICUs we are analyzing the association between a new antibiotic and outcome, and we have randomly selected equal numbers of treated and controls within each clinic. Each ICU serves a somewhat different population, and the different populations each has a somewhat different risk of outcome among the controls. But we do not care about estimating or modeling the baseline risk. We are concerned only about estimating the effect of treatment. These ICUs form “strata”EP 521, Spring 2007 Vol II, Part 4 Copyright © 2006 Trustees of the University of Pennsylvania 3For this ICU the baseline (reference) risk =0.5. OR = 0.33 * * * For this ICU the baseline (reference) risk =0.2. But we are not going to estimate the baseline risks. We are interested only in the overall OR of the association of antibiotic and infection. If we were to use unconditional logistic regression (logit or logistic), then we would have to include 29 indicator (dummy) variables for the 30 ICUs to allow for differences in baseline (reference group) risks across the ICUs. Conditional logistic regression works in nearly the same way as regular logistic regression, except we need to specify which individuals belong to which matched set (e.g., which pair) or stratum. The theory is similar: we can derive a likelihood and maximize it, etc. From a practical perspective, the only difference is the need to specify the matched set or the stratum to which each person belongs. ICU=1 Antibiotic + - + 25 50 Infection - 75 50 ICU=30 Antibiotic + - + 20 Infection - 80 EP 521, Spring 2006, Vol II, Part 4 4 Comparisons within the matched set Recall from stratified analysis what subtables (matched set) are informative: E+ E- E+ E D+ 1 0 0 1 D- 0 1 0 1 Informative Uninformative For the conditional analyses to be able to estimate effect of exposure (E), must have variation of E within the matched set. So, for CLR we are making comparisons of E+ vs E- within the matched sets In unconditional regression, the comparisons are across (among) sets.EP 521, Spring 2007 Vol II, Part 4 Copyright © 2006 Trustees of the University of Pennsylvania 5§6.2 Example: (Kelsey et al—Table 7.11) Layout for a pair-matched case-control study of prolapsed lumbar disc, with place of residence and whether or not a person drives considered as risk factors. Control Suburban residence City residence Does drive Does not drive Does drive Does not drive Does drive 63 4 32 22 Suburban residence Does not drive 1 2 1 2 Does drive 29 1 20 14 Case City residence Does not drive 7 0 10 9 McNemar's Test (collasping the table) ORMLE = 41/19 = 2.16 Case Does not drive Does drive Does not drive 13 41 54 Control Does drive 19 144 163 32 185 217 EP 521, Spring 2006, Vol II, Part 4 6 STATA commands and output for this example: Check the data . tab driving driving| Freq. Percent Cum. ------------|----------------------------------- 0 | 86 19.82 19.82 1 | 348 80.18 100.00 ------------|----------------------------------- Total | 434 100.00 . tab suburbs suburbs| Freq. Percent Cum. ------------|----------------------------------- 0 | 200 46.08 46.08 1 | 234 53.92 100.00 ------------|----------------------------------- Total | 434 100.00 . tab casecon casecon| Freq. Percent Cum. ------------|----------------------------------- 0 | 217 50.00 50.00 1 | 217 50.00 100.00 ------------|----------------------------------- Total | 434 100.00EP 521, Spring 2007 Vol II, Part 4 Copyright © 2006 Trustees of the University of Pennsylvania 7 Running the regression models Generating the coefficients . clogit casecon driving, strata(pairid) Iteration 0: Log Likelihood =-150.41294 Iteration 1: Log Likelihood =-146.29234 Iteration 2: Log Likelihood =-146.28399 Iteration 3: Log Likelihood =-146.28399 Conditional logistic regression Number of obs = 434 chi2(1) = 8.26 Prob > chi2 = 0.0041 Log Likelihood = -146.28399 Pseudo R2 = 0.0275 ------------------------------------------------------------------------------ casecon | Coef. Std. Err. z P>|z| [95% Conf. Interval] ---------+-------------------------------------------------------------------- driving | .7691331 .2775281 2.771 0.006 .2251881 1.313078 ------------------------------------------------------------------------------ EP 521, Spring 2006, Vol II, Part 4 8 Estimating