Analysis of Non-commensurate Outcomes

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

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The city of PORTO

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The city of PORTO

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The city of PORTO

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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…

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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

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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

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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

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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

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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

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Multiple outcomes

EQ-5D indexEQ-5D index

D-VASD-VAS

age

previous health state

Apache II

age


Apache II

n=485

n=366

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Multiple outcomes

EQ-5D indexEQ-5D index

D-VASD-VAS

age


Apache II

age


Apache II

n=485

n=366

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Instrument EQ-5D

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Instrument EQ-5D

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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

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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

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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

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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

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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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)1(

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NNu

uXy

uXyPprobit

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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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)1(

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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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)1(

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NNu

uXy

uXyPprobit

21 u

b

IMPORTANT NOTE: The models are for yb |u and yc |u . I omit the conditional from the equations for simplification.

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Latent model

A nice feature of this model is that it can be easily implemented in commercial stats software With SAS, use PROC NLMIXED

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)1(

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NNu

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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

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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;

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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


<0.01 -0.49(0.11)

<0.01

Apache II -0.018(0.011)

0.09 -0.027(0.010)

<0.01

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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


<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


<0.01 -0.49(0.11)

<0.01

Apache II -0.018(0.011)

0.09 -0.027(0.010)

<0.01

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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

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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

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Extention to more than two outcomes

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“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

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Thank you for your attention!

Slides available at:

http://users.med.up.pt/tpinto/ahealth.ppt

Documents

Analysis of Non-commensurate Outcomes