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# CAT ANALYSIS EPIDEM BIOST 536

UW

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This 11 page Class Notes was uploaded by Ramona Leannon on Wednesday September 9, 2015. The Class Notes belongs to BIOST 536 at University of Washington taught by Staff in Fall. Since its upload, it has received 14 views. For similar materials see /class/192294/biost-536-university-of-washington in Biostatistics at University of Washington.

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Date Created: 09/09/15

Lecture 13 More on Matched Models Outline u Interactions with the matching variables a Additive models a Diagnostics u Efficiency Underlying model logitPji log Pji 1 Pji 05 1 Xjil 2 inz k ink for the j th matched set i th person in that set Unconditional model Need to estimate 061062 306 Conditional model Remove 061 062 pm from the model As J gets large the need to fit a conditional model becomes more acute Results for 3 s potentially biased BIOST 536 Lecture 13 Leisure world data Neither ignoring the matching nor fitting individual ori for each matched set is a good idea Leisure world 1 to 1 data GALL V ESTROGEN W Conditional matched 200 911 Unconditional breaking the matches 234 895 Unconditional fitting estimates for the matched pairs 401 8293 Leisure world 1 to 4 data GALL V ESTROGEN W Conditional matched 358 829 Unconditional breaking the matches 318 755 Unconditional fitting estimates for the matched pairs 603 1485 BIOST 536 Lecture 13 Effect modi cation by the matching or stratifying variables What do you lose in the conditional approach the ability to look at main effects of the variables used for matching Can we still fit interaction terms involving the matching variables Yes In this case we do not want to use Strata but rather Age since it is one of the matching variables stratum outcome age gall gall age est est age Conjugated estrogen dose 1 1 74 0 0 1 74 4 1 0 75 0 0 0 0 0 1 0 74 0 0 0 0 0 1 0 74 0 0 0 0 0 1 0 75 0 0 1 75 1 2 1 67 0 0 1 67 6 2 0 67 0 0 1 67 6 2 0 67 0 0 0 0 0 2 0 67 0 0 1 67 3 2 0 68 0 0 1 68 3 3 1 76 0 0 1 76 1 3 0 76 0 0 1 76 2 3 0 76 0 0 1 76 0 3 0 76 0 0 1 76 3 3 0 77 0 0 1 77 0 Note that the first three sets have no variation in gall so would not contribute to estimation of that odds ratio third set has no variation in estrogen use either BIOST 536 Lecture 13 3 Interactions with the matching variables Significant interactions indicate effect modification by the matching variable For example estrogen may have effects only on one part of the age spectrum Could create age group variable but will only try age as a continuous variable here Est Age and Gall Age groupset clogit case gall estrogen Conditional fixedeeffects logistic regression Number of obs 315 LR chi22 4505 Prob gt chi2 00000 Log likelihood 778871308 Pseudo R2 02221 case 1 Coef Std Err z Pgt1z1 95 Conf Interval gall 1 1274654 4108678 310 0002 4693683 2079941 estrogen 1 2114785 4397942 481 0000 1252804 2976766 est store A Now add the interaction terms to the model gen estage estrogentage clogit case gall estrogen estage groupset Conditional fixedeeffects logistic regression Number of obs 315 LR chi23 4506 Prob gt chi2 00000 Log likelihood 778864691 Pseudo R2 02222 case 1 Coef Std Err z Pgt1z1 95 Conf Interval gall 1 1276811 4111677 311 0002 470937 2082685 estrogen 1 1594645 4535164 035 0725 77294114 104834 estage 1 007207 0626158 012 0908 71155177 1299318 BIOST 536 Lecture 13 Interactions with the matching variables lrtest A Likelihooderatio test LR Chi21 001 Assumption A nested in Prob gt Chi2 09084 gen gallagegallage Clogit case gall estrogen gallage groupset Conditional fixedeeffects logistic regression Number of obs 315 LR chi23 4525 Prob gt chi2 00000 Log likelihood 778768358 Pseudo R2 02232 case Coef Std Err z Pgtlzl 95 Conf Interval gall 1 78909208 4825236 7018 0854 71034821 8566369 estrogen 1 2130138 4447269 479 0000 1258489 3001787 gallage 1 0305221 0678171 045 0653 71023969 1634412 lrtest A Likelihooderatio test LR Chi21 021 Assumption A nested in Prob gt Chi2 06500 No evidence of effect modification by age Underlying model for the interaction with age logit Pji a39 1estrogenji 2 gallji 3 ageji estmgenji for the j th matched set i th person in that set Condition out the 061 062 m 06 from the model Matching variable age appears in the interaction term but the main effect is removed by matching BIOST 536 Lecture 13 5 Additive models Additive models Can also consider a linear relative risk model General conditional likelihood for matched set j controls i1 0 In Where r is the risk function Multiplicative exponential risk 7 e A X1 2 X2 m k Xk Additive linear risk r161 X1 162 X2 6k Xk Have to impose some constraints so that the risk is positive Can be more biologically plausible than a standard multiplicative model Compare additive and multiplicative models for leisure 1 to 4 data This is what we found before with no interaction in the multiplicative model BIOST 536 Lecture 13 Additive models Estimated OR No estrogen Estrogen No gall bladder 10 829 Gall bladder 358 2965 And with the interaction term Estimated OR No estrogen Estrogen No gall bladder 10 1488 Gall bladder 1807 3452 a As is often the case the interaction term allows OR A1 amp B1 lt ORA1 x ORB1 a Suggests an additive model without interaction might be as good u Stata cannot fit the additive model without writing a special routine some specialized packages Egret Epicure can do these model fits BIOST 536 Lecture 13 Additive models Fitting the linear additive model in Egret gives a two parameter model whose fit is almost as good as the three parameter multiplicative model Estimated OR No estrogen Estrogen No gall bladder 10 1495 Gall bladder 1923 1013951823 3318 PE ULT CLRa77 TERM ODDS RATIO 95 CONFIDENCE BOUNDS gallbladdr 1923 79103 4757 estuse 1495 72949 3284 Note the Wald confidence bounds are nonsensical The two parameter additive model fits the data most simply however despite the poor statistical properties associated with additive models a Additive models can be attractive scientifically but in practice are difficult to fit and have poor statistical properties BIOST 536 Lecture 13 8 Diagnostics Diagnostics Based on the score contribution to the conditional log likelihood Cain and Lange 1982 developed for Cox regression Adapted by Barlow and Prentice 1988 to conditional logistic regression and Cox regression with timedependent covariates One covariate X A Vdr YipiXiYweighted Vdrlt gt Score ji Actual outcome Yi ele i Zexjk l k Conditional expected probability P Observed covariate X i Xweighted Z ij xjk Expected covarlate k Multivariate version slightly more complicated BIOST 536 Lecture 13 Diagnostics Robust empirical Variance Based on the delta beta diagnostic from the multivariate model Robust variance A3 T A3 Stata modifies this slightly since it adjusts for the df minor change Stata does not give the delta beta s but it does give the robust variance estimates Clogit case gall estrogen groupset robust Conditional fixedeeffects logistic regression Number of obs 315 Wald Chi22 3609 Prob gt Chi2 00000 Log pseudolikelihood 778871308 Pseudo R2 02221 Std Err adjusted for Clustering on set Robust case Coef Std Err z Pgtizi 95 Conf Interval gall 1274654 4465026 285 0004 3995255 2149783 estrogen 2114785 397369 532 0000 1335956 2893614 Clogit case gall estrogen groupset Conditional fixedeeffects logistic regression Number of obs 315 LR Chi22 4505 Prob gt Chi2 00000 Log likelihood 778871308 Pseudo R2 02221 case Coef Std Err z Pgtizi 95 Conf Interval gall 1274654 4108678 310 0002 4693683 2079941 estrogen 2114785 4397942 481 0000 1252804 2976766 BIOST 536 Lecture 13 Ef ciency What if you match when the matching variable is unimportant gt Loss of efficiency compared to unconditional logistic regression except when the true odds ratio is one Degree of loss depends on the true odds ratio and how common the binary exposure is Figure 71 in Breslow amp Day Volume 1 shows the expected efficiency For 3 1 the loss can be as great as 10 For 3 2 the loss can be as much as 40 when the prevalence of exposure in controls is 30 0 Matching on known risk factors is important 0 Matching on well measured characteristics is also important BIOST 536 Lecture 13

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