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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 Bikin, Pharm.D.
Good Samaritan Regional Medical Center
1111 E. McDowell, Phoenix, AZ 85006
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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
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
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
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
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
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