## Thursday, March 06, 2014

### Intra-Subject Coefficient of Variation (CV%) for Sample Size Estimation for Crossover Design

To calculate the sample size for a crossover design for bioequivalence study, a key assumption is the intra-subject variation. The intra-subject variation is usually expressed with coefficient of variation (COV).

In sample size calculation software ‘PASS’, “Equivalence Test for Two Means in 2x2 Crossover Design - Specify Using Ratio” procedure requires COV (Coefficient of Variation) as the following:

COV (COEFFICENT OF VARIATION):“...For log-normal data, the following relationship exists: COV(Y) = SQR(Exp(SX*SX)-1) where SX is the square root of the within mean square error in the ANOVA table of the log-transformed values.”
In Sample size calculation software Nquery, “T-tests (TOST) of equivalence in ratio of means for crossover design (natural log scale)” procedure specifies that
“the square root of the mean square error is needed, the square root of the mean squared error can be obtained from the crossover ANOVA computed using the natural log scale. sqrt(MSE) is equal to sd/sqrt(2), where sd is the standard deviation of the period difference computed using the natural log scale. To compare results from this table to those in the Diletti, E., et al (1991) paper note that CV = sqrt(exp(s^2-1)”

Suppose we analyze the data using Proc Mixed as following:

proc mixed data=pk ;
class sequence period treatment subject;
model logauc = sequence period treatment;
random subject(sequence);
lsmeans treatment/pdiff cl alpha=0.1;
run;

Covariance parameter estimates in the SAS outputs will provide the information for calculating intra-subject coeffiicient.

`                                                                The Mixed Procedure`
`[...]`

`           `
 Covariance Parameter Estimates Cov Parm Estimate SUBJECT(SEQUENCE) 0.06800 Residual 0.1856

`[...]`

Here, Subject(Sequence) is the inter-subject variability s2inter and residual the within-subject (or intra-subject) variability s2within.

If we need to design a new study with crossover design, we will convert the intra-subject variability to CV for sample size calculation. CVintra can be calculated with the formula CV=100*sqrt(exp(S2
within)-1) or CV=100*sqrt(exp(Residual)-1). From the table above, s2within=0.1856, CV can be calculated as 45.16%

If we use Proc GLM (for a balanced study with no missing data) as specified below, the within-subject (or intra-subject) variability s2within is the Mean Square Error (0.1856) – identical to the residual in Proc Mixed. Intra-subject CV is then calculated using CV=100*sqrt(exp(MSE)-1).

PROC GLM data=pk;
CLASS trtsEQ period trtgrp patno;
MODEL lauc = trtSEQ patno(trtSEQ) period TRtgrp; * TRtgrp*PERIOD;
LSMEANS TRTgrp / PDIFF CL alpha=0.1;
RUN;

 Source DF Sum of Squares Mean Square F Value Pr > F Model 23 6.93036415 0.30132018 1.62 0.1387 Error 20 3.71219721 0.18560986 Corrected Total 43 10.64256136