- On 22 Sep 1999 at 22:32:46, "Dale Bikin, Pharm.D." (daleb.aaa.samaritan.edu) sent the message

Back to the Top

I have a number of patients that receive between 2-4 doses of drug

and generally have only 2 blood levels obtained for each patient. I

want to generate a population model and evaluate covariate impact on

the model parameters (clearance and volume). I would include changed

covariate values for each dose, but do I include the covariate value

for each dose if it remains unchanged?

Dale

Dale Bikin, Pharm.D.

Good Samaritan Regional Medical Center

1111 E. McDowell, Phoenix, AZ 85006

daleb.aaa.samaritan.edu - On 26 Sep 1999 at 23:07:40, ml11439.aaa.goodnet.com (Michael J. Leibold) sent the message

Back to the Top

Hello Dale,

Analysis of covariance is a combination of regression analysis with

an analysis of variance. Covariance is used when the response variable y,

in addition to be affected by treatment, is also linearly related to

another variable x. The textbook "Statistics for Research" by Dowdy,

Shirley and Wearden, Stanley and published by Wiley Interscience 1983,

gives a brief discussion of the analysis of covariance as combination of

one-way ANOVA and linear regression. In the following example, by virtue

of a linear relationhip with Kel, CrCl serves as a covariate in several

treatment groups studing the effect of the treatments on Kel.

Using Kel as an example, each observation would be modeled as:

Yij= u + alpha(i) + Beta(Xij - Xave) + epislon(ij)

Yij= observation ij (eg Kel)

Xij= covariate ij (eg creatinine clearance)

u= overall mean for all studies involving the specified treatments

alpha(i)=the deviation due to the ith treatment after allowance

for the relationship of y to x.(eg Kel= B x CrCl)

Beta= The true common slope of the a regression lines(eg Ke= B x CrCl)

Xave= The average of the covariate observations in the study(eg CrCl)

epsilon(ij)= A random effect of the jth element in the ith treatment

group.

Note: The above equation could be modified to include the regression

relationship Kel= B x CrCl + a, where there is a Y intercept in

the regression relationship.

By examination of the variances in the Kel in the various treatment

groups when corrected for the variance caused by the linear relationship

between CrCl and Kel, analysis of covariance will find if there is

any significant effect of the various treatments on Kel. That is, CrCl

would be the covariate in this example for which the ANOVA is adjusted

so the treatment effects can be studied.

Also, in this "simple" model an examination of the linear relationship

between Kel and creatinine clearance among various treatment groups

could be obtained. The null hypothesis (Ho) would be that the variances

observed in Kel between groups are not significantly different. If the null

hypothesis is not rejected, than the regression equation for Kel and

creatinine clearance explains the variances in the Kel observed among

the various treatment groups. If the null hypthesis is rejected, than

a separate regression equation for each treatment group should be

considered. The above statistical model results in a series of ANOVA

calculations in which the treatment and error sum of squares are

compared with an F statistic, to test for a significant difference

in variance among the treatment groups.

It appears that this model could entertain multiple linear regression

as well, but the calculations would be more complicated.The above

ANOVA-linear regression analysis could complement the standard ANOVA

statistical analysis

of linear regression where regression and residual sum of squares are

compared. However, the above is analysis of covariance since the regression

relationship (Kel= B x CrCl + a) makes CrCl a covariate of the treatment

groups.

Sheiner also discusses multivariate responses and population models

in his paper in Journ of Pharmacokinetics and Biopharm 1984;12(1):93-117.

In the multivariate case, the covariances among multiple measured responses

in patients are examined. That is, the covariance between Cp and Curine could

be examined:

cov(x,y)= E(x-E(x))(y-E(y))

However, I think that the covariance among parameters could also

be assessed, such as between Cl and Vd.

I hope that this was of some help, but I think that Sheiner's litera-

ture and various statistics texts would be the sources you need. There must

be an abundance of statistical software as well. So, if this doesn't really

answer your question, it will at least provide some additional information.

Mike Leibold, PharmD, RPh

ML11439.at.goodnet.com

Want to post a follow-up message on this topic? If this link does not work with your browser send a follow-up message to PharmPK@boomer.org with "Covariates..." as the subject

PharmPK Discussion List Archive Index page

Copyright 1995-2010 David W. A. Bourne (david@boomer.org)