Report

Conditional Stereotype Logistic Regression A new estimation command Rob Woodruff Battelle Memorial Institute, Health & Analytics Email: woodruffr@battelle.org Cynthia Ferre Centers for Disease Control and Prevention 1 Overview • What is it? - Stereotype Logistic Regression - Conditional on what? • What‘s it good for? • Syntax and Examples 2 Constrained Multinomial Logistic Regression • Multinomial Model -Categorical Outcome Variable -Vector of Explanatory Variables -Related through the m logits: 3 Constrained Multinomial (continued) -The stereotype model imposes the constraints: Note: The phi’s are scalar quantities 4 It’s all about the phi’s • Full multinomial has m(p+1) parameters • Stereotype model has m-1 + m + p = 2m-1+p • The phi parameters give a way to quantify ordinality of the outcome variable. If Then we have evidence of ordinal effect. • Also allow tests of distinguishability of outcome categories 5 So what’s the condition? • The multinomial and stereotype logistic regression models are implemented in Stata by mlogit and slogit • Assume independence of observations, not true for matched case-control data • For matched case control study, only independence of matched groups (strata, panels, clusters, etc) • For 1:M matching, condition on stratum total for outcome variable and focus instead on conditional likelihood Do I have to? Why condition on this particular event? 6 Conditional vs. Unconditional Likelihood 7 Conditional vs. Unconditional Likelihood 8 CSTEREO cstereo command Basic syntax: . cstereo depvar indepvars [if] [in], group(varname) [options] 9 Example with Real Data: Preterm Birth and Vitamin D • 1:2 (some 1:1) Pooled, Matched Case-Control Study of 2,583 Mothers in 870 matched groups • A case defined as gestational age at delivery of <37 weeks outcome4=3 (<32 weeks), outcome4=2, (32-35 weeks), outcome4=1 (36 weeks) and outcome4=0 (control: 37+ weeks) • Primary exposure variable of interest: Vitamin D levels, ohd25_total: blood serum concentration of (25)OHD in ng/ml • Sample of other covariates measured: edu = 0/1 indicator of post-high school education vitamin = 0/1 indicator of vitamin use during pregnancy 10 Example Continued (nolog option): . cstereo outcome4 ohd25_total edu vitamin, group(matchgroup) nolog note: 77 groups (139 obs) dropped because of all positive or all negative outcomes. Log-Likelihood from Conditional Multinomial Model: -835.83679 Chi2 value on 4 degrees of freedom: 10.604861 P-value: .03138281 11 Example Continued: Number of obs Wald chi2(3) Prob > chi2 Log likelihood = -841.13922 Std. Err. z P>|z| = = = 2322 1.85 0.6048 [95% Conf. Interval] outcome4 Coef. xb ohd25_total edu vitamin -.0073684 -.4010391 .1301369 .0144916 .431587 .1954516 -0.51 -0.93 0.67 0.611 0.353 0.506 -.0357714 -1.246934 -.2529413 .0210346 .4448559 .5132151 _cons .8764578 1.268331 0.69 0.490 -1.609424 3.36234 _cons .9398113 1.206139 0.78 0.436 -1.424178 3.3038 phi1 phi2 12 Interpretation of cstereo output: • Estimated beta coefficient of ohd25_total = -0.0074 95% confidence interval (-0.0358, 0.0210) with • Odds ratio of being in <32 weeks gestational age compared to control is exp(-0.0074) = 0.993 (0.965, 1.021) • Now for odds ratios for the 32-35 weeks and 36 week case categories, we need the products of the parameters: • For standard errors, use Delta Method via nlcom 13 Interpretation continued: . nlcom [xb]ohd25_total*[phi2]_cons _nl_1: [xb]ohd25_total*[phi2]_cons outcome4 Coef. _nl_1 -.0069249 Std. Err. .0072757 z -0.95 P>|z| [95% Conf. Interval] 0.341 -.021185 .0073351 Exponentiating gives the odds ratio of being in the 32-35 weeks case category compare to controls of 0.994 with a 95% C.I. of (0.983, 1.004) 14 Constraints: • Are the 36 week and 32-35 weeks case categories distinguishable? . constraint 1 [phi1]_cons=[phi2]_cons . cstereo outcome4 ohd25_total edu vitamin, group(matchgroup) nolog constraints(1) note: 77 groups (139 obs) dropped because of all positive or all negative outcomes. 15 Constraint Output Number of obs Wald chi2(3) Prob > chi2 Log likelihood = -841.1454 = = = 2322 1.73 0.6293 ( 1) [phi1]_cons - [phi2]_cons = 0 Std. Err. z P>|z| [95% Conf. Interval] outcome4 Coef. xb ohd25_total edu vitamin -.0068382 -.3909924 .1294806 .013172 .4289348 .1888154 -0.52 -0.91 0.69 0.604 0.362 0.493 -.0326548 -1.231689 -.2405908 .0189784 .4497043 .4995519 _cons .9417836 1.24291 0.76 0.449 -1.494276 3.377843 _cons .9417836 1.24291 0.76 0.449 -1.494276 3.377843 phi1 phi2 16 Constraint Output • The log-likelihood from the constrained model is -841.145 compared to -841.139 for the unconstrained stereotype model • Difference of 0.006 gives a chi2 value of 0.012 on 1 degree of freedom • P-value = 0.91 • Unconstrained stereotype model does not fit significantly better than the constrained and the two case categories are indistinguishable 17 Relationship to Other Models for Ordered/Categorical Outcomes • Constrained Multinomial • Not as parsimonious as the proportional odds model (ologit) but not valid in outcome dependent sampling • Adjacent category model is (basically) a constrained stereotype model. Also valid under outcome dependent sampling 18 Limitations • Convergence Issues • Currently only a one dimensional stereotype model • Cannot currently force an ordering on the stereotype parameters • Additional dependence structure 19 References: • Ferre C, et al; Maternal 25-Hydroxyvitamin D Status and the Risk of Preterm Delivery: A Multi-Center Nested Case Control Study; preprint • Mukherjee B, Liu I, Sinha S; Analysis of matched casecontrol data with multiple ordered disease states; Statistics in Medicine 2007 • Ahn J et. al.; Missing Exposure Date in Stereotype Regression Model; Biometrics 2011 • Andersen EB; Asymptotic Properties of Conditional MaximumLikelihood Estimators; Journal of the Royal Statistical Society 1970 • Liang KY, Stewart WF; Polychotomous Logistic Regression Methods for Matched Case-Control Studies with Multiple Case or Control Groups; American Journal of Epidemiology 1987 • Scott AJ, Wild CJ; Fitting Regression Models to Case-Contro Data by Maximum Likelihood; Biometrika 1997 • Anderson JA; Regression and Ordered Categorical Variable; Journal of the Royal Statistical Society 1984\ • Greenland S; Alternative Models for Ordinal Logistic Regression; Statistics in Medicine 1994 20