ANCOVA Modelling

[gkmzinsou]

Hello,

Is it possible to have to an ANCOVA with a change score as a dependent variable with adjustment for baseline?

Y1 = pre-test

Y2 = post-test

Grp 1 = Treatment 1 (ref)

Grp 2 = Treatment 2

Grp 3 = Treatment 3

Change = Y2 - Y1 

ANCOVA model :  Change = Bo + B1*Y1 + B2*Grp2 +B2*Grp3 

(Knowing that I have 3 treatment groups)

Thanks!

[Ksharp]

No. ANOVA hypothesis is iid (i.e. independent individual ). But your 'change' is correlation or paired .
I think you should use PROC MIXED or PROC GLIMMIX .

or try MANOVA , put both y1 and y2 as dependent variables in model .

[SteveDenham]

From Frank Harrell's online version of Regression Modeling Strategies, I pulled this from the annotated bibliography:

Kung-Yee Liang and Scott L. Zeger.“Longitudinal Data Analysis of Continuous and Discrete Responses for Pre-Post
Designs”. In: Sankhy¯a 62 (2000). makes an error in assuming the baseline variable will have the same univariate
distribution as the response except for a shift; Baseline may have for example a truncated distribution based on a
trial’s inclusion criteria. If correlation between baseline and response is zero, ANCOVA will be twice as efficient as
simple analysis of change scores;if correlation is one they may be equally efficient, pp. 134–148 (cit. on p. 7-5).

That last sentence lets you know that fitting a change score, with the baseline as a covariate, is never as good as fitting the actual data.  Once you get your marginal means for treatment groups, you can calculate change from baseline.

Another way to think about this is to just do some simple algebraic rearrangement.  You have this model:

Change = Bo + B1*Y1 + B2*Grp2 +B2*Grp3.  Plugging in the definition of Change, you get:

Y2 - Y1 = Bo + B1*Y1 + B2*Grp2 +B2*Grp3.  Now add Y1 to both sides, and get:

Y2 = Bo + B1*Y1 + B2*Grp2 +B2*Grp3 + Y1.  Rearranging terms, this gives:

Y2 = Bo + (B1 + 1)*Y1 + B2*Grp2 +B2*Grp3.  Redefine B1 + 1 as B1' and you get:

Y2 = Bo + B1'*Y1 + B2*Grp2 +B2*Grp3.  But this holds if and only if there is no correlation between Y1 and Y2.  If there is correlation (and there usually is in most pre/post designs), then the estimates for B2 and B3 will be biased as a function of the amount of correlation in the response variables after removing the true effects of B2 and B3..

SteveDenham

[jiltao]

This is interesting. Thanks for sharing! What I am not sure about. is what exactly is the issue if the dependent variable and one of the covariates is correlated? 

[SteveDenham]

Of course we would like to see that the dependent variable is correlated with the independent covariate. Otherwise, it really doesn't add anything to the analysis. The problem I see in this case is that some of the correlation is artificially induced mathematically.  I believe the problem gets worse when there is an interaction between the covariate and any categorical variables in the model - there is a greater chance of calling regression to the mean a covariate by class level interaction.

Shoot - it's just that I was taught someplace along the line to not fit both a change score (equivalent to a regression coefficient of -1) and the pretreatment baseline (with regression coefficient = beta_hat).

SteveDenham

[Ksharp]

That is lead to Analysis of Covariance ? Change Y1 to the right side ? Great idea !

[jiltao]

You can use PROC GLM for this --

proc glm data=yourdata;

   class group (ref="1") ;

   model change = baseline group;   ** for an unequal slope model, add the interaction baseline*group;

run;

quit;