analysis of non-commensurate outcomes
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Armando Teixeira-PintoAcademyHealth, Orlando ‘07
Analysis of Non-commensurate Outcomes
A. Teixeira-Pinto
AcademyHealth, Orlando 2007
Agenda
IntroductionExample: HRQOL after intensive careCommon approach to multiple outcomesThe latent variable modelHRQOL resultsDiscussion 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 HRQOLEQ-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 outcomesEQ-5D index
Summarizes the 5 dimensions of the EQ5D
Continuous outcomeD-VAS (visual analogue scale)
VAS Dichotomized <=50 and >50Binary 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
EQ-5D index
D-VAS
age
previous health state
Apache II
age
previous health state
Apache II
n=485
n=366
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
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
2221
2121
2
1 ,
MVN
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 independentf(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:
b and c are scale factors “adjusting” the latent variable to the different scales of the outcomes
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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:
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A. Teixeira-Pinto
AcademyHealth, Orlando 2007
Latent model
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:
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b
IMPORTANT NOTE: The models are for yb |u and yc |u . I omit the conditional from the equations for simplification.
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
),0(~ ),,0(~
)1(
22ccui
cccTcc
bTbb
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uXy
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The same is true for c ’s, but because of the linear link the interpretation is the same for yc |u and yc
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
(0.06)<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
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
(0.06)<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
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 non-commensurate 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
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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
A. Teixeira-Pinto
AcademyHealth, Orlando 2007
Thank you for your attention!
Slides available at:
http://users.med.up.pt/tpinto/ahealth.ppt
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