# Differences between logistic regression parameter estimates and Cox-proportional hazard parameters

We have been working on a survival analysis. We are examining tree seedling survival over a decade with annual to biannual census intervals. We have been using the package coxme in R for a mixed effects Cox model. Covariates include time-dependent variables and time-independent covariates and a random intercept term for species to account for inherent life history differences among species. All continuous covariates are standardized to one standard deviation unit of change. The model returns sensible parameter estimates from what we know about the system. The strongest predictor covariate of survival has a hazard ratio around 4. This covariate is continuous and time-dependent, but happens to only take one or two values for all individuals per census period though it varies widely between census periods. We wanted to examine how mortality rate related to this strongest covariate. When we regress census period mortality against this variable we find no linear relationship.

We tried logistic regression to look at how this variable behaves in terms of odd ratios. We used the package lme4 and included random intercept terms for species and census in the glmer function. We also used a offset of log(time) and the binomial family log-log link. This model suggests that there is no significant effect of this variable on the odd ratio for survival.

My question concerns the inherent differences between hazard ratio and odds ratio. I have read that the hazard ratio is an instantaneous event rate. I think the odds ratio is more of an overall exposure variable. But I haven't been able to discern how that might result in such different parameter estimate between the models. All of the other variables included in the models are consistent in relative effect size and significance. Which leads me to think that the behavior has something to do with the lack of variability within a census period.

Can someone help me understand the differences between hazard rates and odds in these two models and offer any suggestions why they may return such different results?

• how is time entering your logistic regression? As an offset-term, for instance? And how is a time-dependent variable entering? – swmo Mar 5 '15 at 23:42
• The logistic model is: logistic.model <- glmer(mortality ~ covariates... + (1|species)+ (1|census), offset=log((stop-start)/365.25), family = binomial(link = "cloglog"), data = df, verbose = T) The time dependent variable is enters as a covariate and the data frame is set up in long format with repeated records for each individual for each census interval. – user2246528 Mar 6 '15 at 15:22
• I'm still not sure I understand the model, but for one thing, you're assuming independence even within individuals - that's not a good assumption, if I'm understanding what you're doing. Furthermore, you're assuming a certain relationship between odds and time. The Cox model is much more flexible in that regard. I suggest you upload data (or some example data), along with your code. – swmo Mar 6 '15 at 15:46
• A little confused. You mention that you estimate a logistic regression using the log-log link. This can't be true, by definition a logistic regression uses a logit link. Since you're not using the logit link, you're not dealing with odds ratios. Furthermore, in the glmer() statement above, you use a complimentary log-log link. I don't think this is the same as a log-log model. – AOGSTA Jun 23 '16 at 22:02