Bioequivalence and Crossover Analysis with sasLM

Kyun-Seop Bae

The standard 2x2 crossover analysis

Average bioequivalence with a 2x2 crossover design is conventionally analyzed with SAS PROC GLM:

PROC GLM DATA=be;
  CLASS SEQ SUBJ PRD TRT;
  MODEL LNCMAX = SEQ SUBJ(SEQ) PRD TRT;
  RANDOM SUBJ(SEQ) / TEST;
  LSMEANS TRT / CL ALPHA=0.1;
  ESTIMATE 'T - R' TRT -1 1 / CL ALPHA=0.1;
RUN;

The package ships BEdata, real data from a 2x2 bioequivalence study with three hospitalization groups. The same analysis with sasLM:

BEdata = af(BEdata, c("ADM", "SEQ", "PRD", "TRT", "SUBJ")) # columns as factors
formula1 = log(CMAX) ~ SEQ/SUBJ + PRD + TRT
GLM(formula1, BEdata)
$ANOVA
Response : log(CMAX)
                Df  Sum Sq Mean Sq F value    Pr(>F)    
MODEL           48 23.1924 0.48317  5.6278 4.395e-08 ***
RESIDUALS       42  3.6059 0.08585                      
CORRECTED TOTAL 90 26.7983                              
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$Fitness
  Root MSE log(CMAX) Mean Coef Var  R-square  Adj R-sq
 0.2930098       6.071036 4.826355 0.8654428 0.7116631

$`Type I`
         Df  Sum Sq Mean Sq F value    Pr(>F)    
SEQ       1  0.6454 0.64544  7.5178  0.008938 ** 
SEQ:SUBJ 45 22.4395 0.49866  5.8081 3.359e-08 ***
PRD       1  0.0969 0.09686  1.1281  0.294242    
TRT       1  0.0106 0.01057  0.1231  0.727410    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$`Type II`
         Df  Sum Sq Mean Sq F value    Pr(>F)    
SEQ       1  0.6440 0.64395  7.5005  0.009011 ** 
SEQ:SUBJ 45 22.5232 0.50052  5.8298 3.173e-08 ***
PRD       1  0.0996 0.09958  1.1599  0.287632    
TRT       1  0.0106 0.01057  0.1231  0.727410    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$`Type III`
         Df  Sum Sq Mean Sq F value    Pr(>F)    
SEQ       1  0.3368 0.33679  3.9228   0.05421 .  
SEQ:SUBJ 45 22.5232 0.50052  5.8298 3.173e-08 ***
PRD       1  0.0996 0.09958  1.1599   0.28763    
TRT       1  0.0106 0.01057  0.1231   0.72741    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

SEQ/SUBJ denotes subjects nested within sequence, equivalent to SEQ SUBJ(SEQ) of SAS. The Type I, II, and III tables above match SAS PROC GLM for this unbalanced data set (91 subjects).

Testing SEQ against the correct error term

The sequence effect must be tested against the subject-within-sequence mean square, not the residual. This is what the RANDOM / TEST statement of SAS does:

RanTest(formula1, BEdata, Random="SUBJ")
$SEQ
Error: 0.9669*SEQ:SUBJ + 0.0331*MSE
          Df  Sum Sq Mean Sq F value Pr(>F)
SEQ    1.000  0.3368 0.33679  0.6919 0.4099
Error 45.529 22.1626 0.48679               

$`The Rest Terms`
          Df  Sum Sq Mean Sq F value    Pr(>F)    
PRD        1  0.0996 0.09958  1.1599    0.2876    
TRT        1  0.0106 0.01057  0.1231    0.7274    
SEQ:SUBJ  45 22.5232 0.50052  5.8298 3.173e-08 ***
RESIDUALS 42  3.6059 0.08585                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$EMS
         SEQ PRD TRT SEQ:SUBJ eps
SEQ        Q           1.8693   1
PRD            Q                1
TRT                Q            1
SEQ:SUBJ               1.9333   1

The 90% confidence interval of the geometric mean ratio

The two-one-sided-tests (TOST) procedure at the 5% level is operationally the 90% confidence interval of T/R in the log scale:

ci0 = CIest(formula1, BEdata, "TRT", c(-1, 1), conf.level=0.90)
ci0
     Estimate  Lower CL Upper CL Std. Error t value Df Pr(>|t|)
[1,] 0.021944 -0.083236  0.12712   0.062535  0.3509 42   0.7274
exp(ci0[, c("Estimate", "Lower CL", "Upper CL")]) # GMR and its 90% CI
 Estimate  Lower CL  Upper CL 
1.0221866 0.9201339 1.1355579

The exponentiated estimate and confidence limits are the geometric mean ratio (GMR) and its 90% confidence interval. Bioequivalence is concluded when the interval is contained in [0.80, 1.25].

Least squares means of treatments

LSM(formula1, BEdata, "TRT", conf.level=0.90)
  Group   LSmean  LowerCL  UpperCL         SE Df
T     A 6.060638 5.988736 6.132541 0.04274955 42
R     A 6.038694 5.961929 6.115459 0.04564037 42

Notes