# PharmPK Discussion - Covariates...

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• On 22 Sep 1999 at 22:32:46, "Dale Bikin, Pharm.D." (daleb.aaa.samaritan.edu) sent the message
`I have a number of patients that receive between 2-4 doses of drugand generally have only 2 blood levels obtained for each patient. Iwant to generate a population model and evaluate covariate impact onthe model parameters (clearance and volume). I would include changedcovariate values for each dose, but do I include the covariate valuefor each dose if it remains unchanged?DaleDale Bikin, Pharm.D.Good Samaritan Regional Medical Center1111 E. McDowell, Phoenix, AZ  85006daleb.aaa.samaritan.edu`
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• On 26 Sep 1999 at 23:07:40, ml11439.aaa.goodnet.com (Michael J. Leibold) sent the message
`Hello Dale,     Analysis of covariance is a combination of regression analysis withan analysis of variance. Covariance is used when the response variable y,in addition to be affected by treatment, is also linearly related toanother 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 ofone-way ANOVA and linear regression. In the following example, by virtueof a linear relationhip with Kel, CrCl serves as a covariate in severaltreatment 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 treatmentgroups when corrected for the variance caused by the linear relationshipbetween CrCl and Kel, analysis of covariance will find if there isany significant effect of the various treatments on Kel. That is, CrClwould be the covariate in this example for which the ANOVA is adjustedso the treatment effects can be studied.   Also, in this "simple" model an examination of the linear relationshipbetween Kel and creatinine clearance among various treatment groupscould be obtained. The null hypothesis (Ho) would be that the variancesobserved in Kel between groups are not significantly different. If the nullhypothesis is not rejected, than the regression equation for Kel andcreatinine clearance explains the variances in the Kel observed amongthe various treatment groups. If the null hypthesis is rejected, thana separate regression equation for each treatment group should beconsidered. The above statistical model results in a series of ANOVAcalculations in which the treatment and error sum of squares arecompared with an F statistic, to test for a significant differencein variance among the treatment groups.     It appears that this model could entertain multiple linear regressionas well, but the calculations would be more complicated.The aboveANOVA-linear regression analysis could complement the standard ANOVAstatistical analysisof linear regression where regression and residual sum of squares arecompared. However, the above is analysis of covariance since the regressionrelationship (Kel= B x CrCl + a) makes CrCl a covariate of the treatmentgroups.     Sheiner also discusses multivariate responses and population modelsin his paper in Journ of Pharmacokinetics and Biopharm 1984;12(1):93-117.In the multivariate case, the covariances among multiple measured responsesin patients are examined. That is, the covariance between Cp and Curine couldbe examined:     cov(x,y)= E(x-E(x))(y-E(y))     However, I think that the covariance among parameters could alsobe 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 mustbe an abundance of statistical software as well. So, if this doesn't reallyanswer your question, it will at least provide some additional information.       Mike Leibold, PharmD, RPh       ML11439.at.goodnet.com`
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