Different Distributions

1. Different Distributions David Purdie

2. Topics Application of GEE to: • Binary outcomes: • logistic regression • Events over time (rate): • Poisson regression • Survival data: • Cox regression

3. General form for distributions from the exponential family Outcome for subject i at time j = Yij E(Yij)=ij Generalized linear model g(ij)=Xi where Xi=(xi1,…,xij) is the matrix of covariates for subject i

4. Binary outcomes: logistic regression Outcome: Pr(Yij = 1) = ij (probability of an event) Pr(Yij = 0) = 1- ij. Logit link function: Logistic model: where ij = E(Yij|Xi)

5. Events over time: Poisson regression • Outcome: Yi= number of events in time period ti • E(Yi): iti Var(Yiti)= iti (were i is the event rate) • Log link function: log (i) • Poisson model:

6. Parameter: tij (time to event yij) Based on a hazard function: ht Outcome: Tij= time till event yij Log link function: log (ht) Cox model: Survival data: Cox regression where t is the baseline hazard rate.

7. Alternating logistic regression If the responses are binary, it may make more sense to use a matrix of odds ratios rather than correlations. Replace corr(Yij, Yik) with: The ALR algorithm models ijk = log{OR(Yij,Yik)} as: ijk=zijk  where  are regression parameters and z is fixed and needs to be specified

8. Mixed Models for Non-Normal Data • E(y|u)=, var(y|u)=V(), g()=X+Zu • Random coefficients u have dist f(u) • y|u has the usual glm distribution • Binary outcome: • binomial for y|u and beta for u • Count outcome: • Poisson for y|u and gamma for u

9. Example - binary • Study of bladder cancer • All patients had superficial bladder tumours on entry which were removed • Two randomly allocated treatments (group): • Placebo (n=47), Thiotepa (n=38) • Many multiple recurrences of tumours • Month is month since treatment (1 to 53) • Baseline covariates of number of initial tumours (number) & size of largest tumour (size) • Lots of missing data: 3585 out of 4505 potential observations (80%) are missing • Model missing data (yes/no) using a binomial GEE to assess if data is missing at random (logit link function)  Name in data set

11. Visits per subject

12. Plot of missing proportion over time

13. Format for the data in SAS

14. Logistic GEE in SAS procgenmod data=tumour_miss descending; class group subject month; model missing=group month size number / dist=binomial type3; repeated subject=subject / type=ind corrw within=month; estimate 'effect of thiotepa' group -11/ exp; run;

15. ORs for group (Thiotepa vs plac)

16. Example - Poisson • Response: number of new tumours (count) • Month is month since treatment (1 to 53) • Baseline covariates of number of initial tumours (number) & size of largest tumour (size) • Timesince is the number of months since the last visit • Missing data are dependent upon treatment group and time • Model new tumour counts using a Poisson GEE to assess treatment effect (log link function)  Name in data set

17. Count of tumours by treatment group

18. New tumour counts over time by treatment group

20. Plot of observed means over time

21. Poisson GEE in SAS procgenmod data=tumour_count; class group subject month; model count=group size number timesince / dist=poisson scale=deviance; repeated subject=subject / type=exch withinsubject=month corrw; estimate 'effect of thiotepa' group -11/ exp; run;

22. RRs for group (Thiotepa vs plac) *WARNING: The number of response pairs for estimating correlation is less than or equal to the number of regression parameters. A simpler correlation model might be more appropriate.

23. Using an offset data tumour_count; set tumour_count; off=log(timesince+1); run; proc genmod data=tumour_count; class group subject month; model count=group size number / dist=poisson scale=deviance offset=off type3; repeated subject=subject / type=unstr withinsubject=month; estimate 'effect of thiotepa' group -1 1/ exp; run;

24. RRs for group (Thiotepa vs plac) *WARNING: The number of response pairs for estimating correlation is less than or equal to the number of regression parameters. A simpler correlation model might be more appropriate.

25. Interpretation and Presentation • Descriptive: plots of means or tables of means (percentages, etc.) • Tables of parameter estimates and confidence intervals (odds ratios or relative risks) • P-values for effects or interactions (possibly just in the text) • Emphasize results from descriptive analysis and effect estimates.

26. Statistical Methods • What is the distribution of the outcome? • How were the data summarized? • Due to the repeated nature of the data, a generalized estimated equations (GEE) approach was used to estimate parameters and test for differences between groups. • What was the form of the correlation structure? • What hypotheses were being tested? • How were missing data handled? • How were variances calculated? • What statistical package was used?

27. Example: Statistical Methods Mean numbers of new tumours were used to summarise the data. Poisson regression was used to model tumour counts using the time between successive observations as an offset. Due to the repeated nature of the data, a generalized estimated equations (GEE) approach was used to estimate parameters and test for differences between groups. The main hypothesis being tested was whether Thiotepa affected the numbers of new tumours. The correlation between successive observations was examined and an appropriate correlation structure was specified. Drop outs and non-attendance was examined to assess for differences between the treatment groups. Robust variance estimate techniques were used to calculate standard errors and confidence intervals. All analysis were performed using SAS version 8.2.