1. 1 Rob Woodruff Battelle Memorial Institute, Health & Analytics Email: woodruffr@battelle.org Cynthia Ferre Centers for Disease Control and Prevention Conditional Stereotype Logistic Regression A new estimation command

2. 2 Overview What is it? - Stereotype Logistic Regression - Conditional on what? What‘s it good for? Syntax and Examples

3. 3 Constrained Multinomial Logistic Regression Multinomial Model -Categorical Outcome Variable -Vector of Explanatory Variables -Related through the m logits:

4. 4 Constrained Multinomial (continued) -The stereotype model imposes the constraints: Note: The phi’s are scalar quantities

5. 5 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

6. 6 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?

7. 7 Conditional vs. Unconditional Likelihood

8. 8

9. 9 CSTEREO cstereo command Basic syntax:. cstereo depvar indepvars [if] [in], group(varname) [options]

10. 10 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

11. 11 Example Continued (nolog option):

12. 12 Example Continued:

13. 13 Interpretation of cstereo output: Estimated beta coefficient of ohd25_total = -0.0074 with 95% confidence interval (-0.0358, 0.0210) 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

14. 14 Interpretation continued: 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)

15. 15 Constraints: Are the 36 week and 32-35 weeks case categories distinguishable?

16. 16 Constraint Output

17. 17 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

18. 18 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

19. 19 Limitations Convergence Issues Currently only a one dimensional stereotype model Cannot currently force an ordering on the stereotype parameters Additional dependence structure

20. 20 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 case- control 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 Maximum- Likelihood 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