Analysis of
Non-commensurate Outcomes
Armando Teixeira-Pinto
AcademyHealth, Orlando ‘07
Agenda
Introduction
 Example: HRQOL after intensive care
 Common approach to multiple outcomes
 The latent variable model
 HRQOL results
 Discussion and summary

A. Teixeira-Pinto
AcademyHealth, Orlando 2007
The city of PORTO
A. Teixeira-Pinto
AcademyHealth, Orlando 2007
The city of PORTO
A. Teixeira-Pinto
AcademyHealth, Orlando 2007
The city of PORTO
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

Typically these outcomes are correlated.
 For outcomes measured in the same scale
there are several multivariate methods
implemented in commercial software

Generalized linear mixed model, GEE, GLM,
MANOVA…
A. Teixeira-Pinto
AcademyHealth, Orlando 2007
Introduction

Often the outcomes are non-commensurate
(mixed type) as for example a binary and a
continuous outcome
 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
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
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
A. Teixeira-Pinto
AcademyHealth, Orlando 2007
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 and >50
 Binary outcome


And the three covariates:

Age ; Previous health state; Apache
II
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
A. Teixeira-Pinto
AcademyHealth, Orlando 2007
Multiple outcomes
age
EQ-5D index
previous health state
n=485
Apache II
age
D-VAS
previous health state
n=366
Apache II
A. Teixeira-Pinto
AcademyHealth, Orlando 2007
Multiple outcomes
age
EQ-5D index
previous health state
n=485
Apache II
age
D-VAS
previous health state
n=366
Apache II
A. Teixeira-Pinto
AcademyHealth, Orlando 2007
Instrument EQ-5D
A. Teixeira-Pinto
AcademyHealth, Orlando 2007
Instrument EQ-5D
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
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
  1    12
MVN   , 
   2   
 1 2

 1  2  

2
 2  
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 yb, yc

Introduce a latent variable, u, and assume that
conditional on u the outcomes are independent
f(yb, yc)=  f(yb, yc ,u) du =
=  f(yb, yc |u) f(u) du
=  f(yb |u) f(yc| u) f(u) du
A. Teixeira-Pinto
AcademyHealth, Orlando 2007
Binary and continuous outcomes

Latent variable model
 f(yb |u) f(yc| 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
A. Teixeira-Pinto
AcademyHealth, Orlando 2007
Latent model

Mathematically speaking:
probit P( yb  1)    X b  bu
T
b
yc   cT X c  cu   c
u ~ N (0,  u2 ),

 c ~ N (0,  c2 )
b and c are scale factors “adjusting” the latent variable
to the different scales of the outcomes
A. Teixeira-Pinto
AcademyHealth, Orlando 2007
Latent model


However this models has parameters that are nonidentifiable and we have to fix some of them
It can be shown that the correct way to fix some of the
parameters is:
probit P( yb  1)    bT X b  u
yc   cT X c   c u   c
ui ~ N (0,  ),
2
u
 c ~ N (0,  )
2
c
A. Teixeira-Pinto
AcademyHealth, Orlando 2007
Latent model

IMPORTANT NOTE: The models are for yb |u and yc |u . I
omit the conditional from the equations for simplification.
probit P( yb  1)    bT X b  u
yc   cT X c   c u   c
ui ~ N (0,  u2 ),


 c ~ N (0,  c2 )
The interpretation of b ’s referring to the effect of the
covariates on the outcome yb is conditional on u,
i.e., yb |u
The ‘marginal’ effect can be obtained:
b
1   u2
A. Teixeira-Pinto
AcademyHealth, Orlando 2007
Latent model

The same is true for c ’s, but because of the linear link
the interpretation is the same for yc |u and yc
probit P( yb  1)    X b  u
T
b
yc   cT X c   c u   c
ui ~ N (0,  u2 ),

 c ~ N (0,  c2 )
A nice feature of this model is that it can be easily
implemented in commercial stats software

With SAS, use PROC NLMIXED
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;
A. Teixeira-Pinto
AcademyHealth, Orlando 2007
Results of the HRQOL study
Univariate
Latent model
Coefficient P-value
Coefficient P-value
EQ-5D Index (n=485)
Age
-0.24
<0.01
(0.06)
Previous state
-8.12
~0
<0.01
(0.06)
<0.01
(1.53)
Apache II
-0.24
-8.12
<0.01
(1.53)
~1
(0.15)
~0
~1
(0.16)
D-VAS (n=366)
Age
-0.01
0.01
(0.005)
Previous state
-0.46
-0.018
(0.011)
0.03
(0.005)
<0.01
(0.11)
Apache II
-0.01
-0.49
<0.01
(0.11)
0.09
-0.027
<0.01
(0.010)
A. Teixeira-Pinto
AcademyHealth, Orlando 2007
Results of the HRQOL study
Univariate
Latent model
Coefficient P-value
Coefficient P-value
EQ-5D Index (n=485)
Age
-0.24
<0.01
(0.06)
Previous state
-8.12
~0
<0.01
(0.06)
<0.01
(1.53)
Apache II
-0.24
-8.12
<0.01
(1.53)
~1
(0.15)
~0
~1
(0.16)
D-VAS (n=366)
Age
-0.01
0.01
(0.005)
Previous state
-0.46
-0.018
(0.011)
0.03
(0.005)
<0.01
(0.11)
Apache II
-0.01
-0.49
<0.01
(0.11)
0.09
-0.027
<0.01
(0.010)
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
A. Teixeira-Pinto
AcademyHealth, Orlando 2007
Other approaches
Other strategies presented in the literature:
 Factorization method:
f(yb, yc) = f(yb)f(yc| yb) or
 f(yb, yc) = f(yc)f(yb| yc)


Extension of weighted GEEs to noncommensurate outcomes

Other strategies for the missing data can
also be used, e.g., multiple imputation
A. Teixeira-Pinto
AcademyHealth, Orlando 2007
Extention to more than two outcomes
For k outcomes:
g1 E ( y1 )   1T X 1  1u
g 2 E ( y2 )    2T X 2  2u
g 3 E ( y3 )    3T X 3  3u

g k E ( yk )    kT X k  k u
u ~ N (0,  u2 )
A. Teixeira-Pinto
AcademyHealth, Orlando 2007
“Take home” message
Complete cases
Univariate approach
+

Same covariates for all the
outcomes
Multivariate approach
Complete cases
+
Different covariates for the
the outcomes
Missing data on the
outcomes
Univariate approach less
efficient (larger std. errors)
Multivariate approach more
efficient (smaller std. errors)
Univariate approach may
lead to biased estimates
Multivariate approach may
reduce the bias
A. Teixeira-Pinto
AcademyHealth, Orlando 2007
Thank you for your attention!
Slides available at:
http://users.med.up.pt/tpinto/ahealth.ppt
A. Teixeira-Pinto
AcademyHealth, Orlando 2007