1. Armando Teixeira-Pinto AcademyHealth, Orlando ‘07 Analysis of Non-commensurate Outcomes

2. A. Teixeira-Pinto AcademyHealth, Orlando 2007 Agenda Introduction Example: HRQOL after intensive care Common approach to multiple outcomes The latent variable model HRQOL results Discussion and summary

3. A. Teixeira-Pinto AcademyHealth, Orlando 2007 The city of PORTO

4. A. Teixeira-Pinto AcademyHealth, Orlando 2007 The city of PORTO

5. A. Teixeira-Pinto AcademyHealth, Orlando 2007 The city of PORTO

6. A. Teixeira-Pinto AcademyHealth, Orlando 2007 Introduction Multiple outcomes are often collected in health studies Longitudinal data, repeated measurements, multiple informants, multi-dimension outcome (health related quality of life), multiple surrogates for an outcome of interest For outcomes measured in the same scale there are several multivariate methods implemented in commercial software Generalized linear mixed model, GEE, GLM, MANOVA… Often the outcomes are non-commensurate (mixed type) as for example a binary and a continuous outcome

7. A. Teixeira-Pinto AcademyHealth, Orlando 2007 Introduction These outcomes are often correlated: Common approach: Analyze each outcome separately (univariate framework) ignoring the correlation A multivariate approach will: Use the additional information contained in the correlation between outcomes Permit better control over Type I error rates Answer intrinsically multivariate questions Be helpful in some situations of missing data

8. A. Teixeira-Pinto AcademyHealth, Orlando 2007 Motivation example Quality of life after Intensive Care Objective: evaluate health related quality of life (HRQOL) of patients 6 months after ICU discharge. Study the association with: Age Previous health state Non-chronic disease Chronic disease with no disability Chronic disease with disability Apache II score Severity score at ICU admission

9. A. Teixeira-Pinto AcademyHealth, Orlando 2007 Instrument EQ-5D Measuring HRQOL EQ-5D is a standardized instrument for use as a measure of health outcome. Applicable to a wide range of health conditions and treatments, it provides a simple descriptive profile and a single index value for health status based on 5 health related dimensions. Includes a question about patient’s perception of his/hers HRQOL

10. A. Teixeira-Pinto AcademyHealth, Orlando 2007

11. A. Teixeira-Pinto AcademyHealth, Orlando 2007 Instrument EQ-5D We’ll consider two outcomes EQ-5D index Summarizes the 5 dimensions of the EQ5D Continuous outcome D-VAS (visual analogue scale) VAS Dichotomized 50 Binary outcome And the three covariates: Age ; Previous health state; Apache II

12. A. Teixeira-Pinto AcademyHealth, Orlando 2007 Common approach Data for the HRQOL after ICU stay: 4 years of data collection One intensive care unit from a tertiary hospital in Portugal 485 patients participated in the study The EQ-5D index was available for all the patients Only 366 patients answered the question associated with the D-VAS Common approach: Linear model for the EQ-5D index Logistic or probit regression for D-VAS

13. A. Teixeira-Pinto AcademyHealth, Orlando 2007 Multiple outcomes EQ-5D index D-VAS age previous health state Apache II age previous health state Apache II n=485 n=366

14. A. Teixeira-Pinto AcademyHealth, Orlando 2007 Multiple outcomes EQ-5D index D-VAS age previous health state Apache II age previous health state Apache II n=485 n=366

15. A. Teixeira-Pinto AcademyHealth, Orlando 2007 Instrument EQ-5D

16. A. Teixeira-Pinto AcademyHealth, Orlando 2007 Instrument EQ-5D

17. A. Teixeira-Pinto AcademyHealth, Orlando 2007 Why should we use a multivariate method? Missing values of D-VAS are associated with lower HRQOL For a separate model for D-VAS we have missing not a random (MNAR) and the regression estimates might be biased Because the two outcomes are correlated, in a joint model, we can ‘borrow’ information from the EQ-5d index and reduce the bias for the estimates associated with D-VAS

18. A. Teixeira-Pinto AcademyHealth, Orlando 2007 Multiple outcomes If the outcomes are of the same type, we could assume a multivariate distribution for the outcomes For example, two continuous outcomes

19. A. Teixeira-Pinto AcademyHealth, Orlando 2007 Binary and continuous outcomes For mixed type of outcomes there is no obvious multivariate distribution Strategy: Avoid direct specification of the joint distribution Latent variable model for y b, y c Introduce a latent variable, u, and assume that conditional on u the outcomes are independent f(y b, y c )=  f(y b, y c,u) du = =  f(y b, y c |u) f(u) du =  f(y b |u) f(y c | u) f(u) du

20. A. Teixeira-Pinto AcademyHealth, Orlando 2007 Binary and continuous outcomes Latent variable model  f(y b |u) f(y c | u) f(u) du We can specify separate equations for the outcomes conditional on u. The latent variable is modeling the correlation between the outcomes

21. A. Teixeira-Pinto AcademyHealth, Orlando 2007 Latent model Mathematically speaking:  b and c are scale factors “adjusting” the latent variable to the different scales of the outcomes

22. A. Teixeira-Pinto AcademyHealth, Orlando 2007 Latent model However this models has parameters that are non- identifiable and we have to fix some of them It can be shown that the correct way to fix some of the parameters is:

23. A. Teixeira-Pinto AcademyHealth, Orlando 2007 Latent model The interpretation of  b ’s referring to the effect of the covariates on the outcome y b is conditional on u, i.e., y b |u The ‘marginal’ effect can be obtained: IMPORTANT NOTE: The models are for y b |u and y c |u. I omit the conditional from the equations for simplification.

24. A. Teixeira-Pinto AcademyHealth, Orlando 2007 Latent model A nice feature of this model is that it can be easily implemented in commercial stats software With SAS, use PROC NLMIXED The same is true for  c ’s, but because of the linear link the interpretation is the same for y c |u and y c

25. A. Teixeira-Pinto AcademyHealth, Orlando 2007 SAS code to fit the Latent Model #SAS code to maximize the likelihood resulting from the latent variable model for the HRQOL example; proc nlmixed data=Icu.Euroqolreduced technique=newrap; #initial values; parms a1=-0.9 b1=.02 c1=-1 d1=0 a2=104 b2=-.2 c2=-9 d2=-4 sigmau=1 sigma2=15 ; bounds sigma2>0, sigmau>0; #likelihood; part1=a1 + b1*age + c1*apache +d1*pstate+ u; part2=eq5d - (a2 + b2*age +c2*apache + d2*pstate) - u*sigma2; if missing(dvas) then loglik=-log(sigma2)-.5*1/(sigma2**2)*(part2)**2; else loglik =dvas*log(PROBNORM (part1))+(1-dvas)*log(PROBNORM (-part1))-log(sigma2) - 5*1/(sigma2**2)*(part2)**2; #model (actually you can put any variable other than eq5d with complete observations; model eq5d ~ general(loglik) ; random u ~ normal(0,sigmau**2) subject=idnumb; #computes the ‘marginalized’ parameters for the probit model; estimate ‘intercept' a1/sqrt(1+sigmau**2); estimate 'age_marg' b1/sqrt(1+sigmau**2); estimate 'apache_marg' c1/sqrt(1+sigmau**2); estimate ‘pstate_marg’ d1/sqrt(1+sigmau**2); run;

26. A. Teixeira-Pinto AcademyHealth, Orlando 2007 Results of the HRQOL study UnivariateLatent model CoefficientP-valueCoefficientP-value EQ-5D Index (n=485) Age -0.24 (0.06) <0.01-0.24 (0.06) <0.01 Previous state -8.12 (1.53) <0.01-8.12 (1.53) <0.01 Apache II ~0 (0.15) ~1~0 (0.16) ~1 D-VAS (n=366) Age -0.01 (0.005) 0.01-0.01 (0.005) 0.03 Previous state -0.46 (0.11) <0.01-0.49 (0.11) <0.01 Apache II -0.018 (0.011) 0.09-0.027 (0.010) <0.01

27. A. Teixeira-Pinto AcademyHealth, Orlando 2007 Results of the HRQOL study UnivariateLatent model CoefficientP-valueCoefficientP-value EQ-5D Index (n=485) Age -0.24 (0.06) <0.01-0.24 (0.06) <0.01 Previous state -8.12 (1.53) <0.01-8.12 (1.53) <0.01 Apache II ~0 (0.15) ~1~0 (0.16) ~1 D-VAS (n=366) Age -0.01 (0.005) 0.01-0.01 (0.005) 0.03 Previous state -0.46 (0.11) <0.01-0.49 (0.11) <0.01 Apache II -0.018 (0.011) 0.09-0.027 (0.010) <0.01

28. A. Teixeira-Pinto AcademyHealth, Orlando 2007 Results of the HRQOL study The analysis suggests that the severity of the episode leading to the ICU admission is associated with the patients perception of his/hers HRQOL but not with the EQ-5D index This effect would not be noticed with univariate analysis Taking into account the correlation between the two outcomes (crude  = 0.42) helped to reduce the bias of the effects estimates

29. A. Teixeira-Pinto AcademyHealth, Orlando 2007 Other approaches Other strategies presented in the literature: Factorization method: f(y b, y c ) = f(y b )f(y c | y b ) or f(y b, y c ) = f(y c )f(y b | y c ) Extension of weighted GEEs to non- commensurate outcomes Other strategies for the missing data can also be used, e.g., multiple imputation

30. A. Teixeira-Pinto AcademyHealth, Orlando 2007 Extention to more than two outcomes For k outcomes:

31. A. Teixeira-Pinto AcademyHealth, Orlando 2007 “Take home” message Complete cases + Same covariates for all the outcomes Univariate approach  Multivariate approach Complete cases + Different covariates for the the outcomes Univariate approach less efficient (larger std. errors) Multivariate approach more efficient (smaller std. errors) Missing data on the outcomes Univariate approach may lead to biased estimates Multivariate approach may reduce the bias

32. A. Teixeira-Pinto AcademyHealth, Orlando 2007 Thank you for your attention! Slides available at: http://users.med.up.pt/tpinto/ahealth.ppt