| SE.EQ.dissolution.profiles {SE.EQ} | R Documentation |
The SE-test for equivalence for dissolution profile similarity analyses
Description
The function SE.EQ.dissolution.profiles() implements a variant of the
SE-test for equivalence with a concrete equivalence margin for analyses of
dissolution profiles. It is a multivariate two-sample equivalence procedure.
Distance measure of the test is the sum of standardized differences between the
expected values or in other words: the sum of effect sizes of all components
of the two multivariate samples.
Usage
SE.EQ.dissolution.profiles(X, Y, alpha = 0.05, print.results = TRUE)
Arguments
X |
numeric data matrix of the first sample (REF). The rows of |
Y |
numeric data matrix of the second sample (TEST). The rows of |
alpha |
numeric (0< |
print.results |
logical; if TRUE (default) summary statistics and test results are printed in the output. If FALSE no output is created |
Details
The function SE.EQ.dissolution.profiles() implements a variant
of the SE-test for equivalence for similarity analyses of dissolution
profiles as mentioned in Suarez-Sharp et al.(2020)
<DOI:10.1208/s12248-020-00458-9>). The equivalence margin
is analogically defined as for the T2EQ
approach according to Hoffelder (2019) <DOI:10.1002/bimj.201700257>)
by means of a systematic shift in location
of 10 [% of label claim] of both dissolution profile populations.
SE.EQ.dissolution.profiles() checks whether the weighted mean of the
differences between the expected values of both dissolution profile populations
is statistically significantly smaller than 10 [% of label claim]. The
weights are built up by the inverse variances.
The current regulatory standard approach for comparing dissolution profiles is the similarity factor f2 (see FDA, 1997, EMA, 2010, among others) with which the type I error cannot be controlled. According to EMA (2010) "similarity acceptance limits should be pre-defined and justified and not be greater than a 10% difference". The functions
-
SE.EQ.dissolution.profiles
and f2 have in common that they all check wether a kind of average difference between the expected values is smaller than 10 [% of label claim] (see Suarez-Sharp et al., 2020). Thus, all three methods
-
SE.EQ.dissolution.profiles
are compliant with current regulatory requirements. In contrast to the standard approach f2 they all allow (at least approximate) type I error control.
Value
a data frame; three columns containing the results of the test
p.value |
numeric; the p-value of the SE test for equivalence |
testresult.num |
numeric; 0 (null hypothesis of nonequivalence not rejected) or 1 (null hypothesis of nonequivalence rejected, decision in favor of equivalence) |
testresult.text |
character; test result of the test in text mode |
Author(s)
Thomas Hoffelder <thomas.hoffelder at boehringer-ingelheim.com>
References
EMA (2010). Guidance on the Investigation of Bioequivalence. European Medicines Agency, CHMP, London. Doc. Ref.: CPMP/EWP/QWP/1401/98 Rev. 1/ Corr **. URL: https://www.ema.europa.eu/en/documents/scientific-guideline/guideline-investigation-bioequivalence-rev1_en.pdf
FDA (1997). Guidance for Industry: Dissolution Testing of Immediate Release Solid Oral Dosage Forms. Food and Drug Administration FDA, CDER, Rockville. URL: https://www.fda.gov/media/70936/download
Hoffelder, T., Goessl, R., Wellek, S. (2015). Multivariate Equivalence Tests for Use in Pharmaceutical Development. Journal of Biopharmaceutical Statistics, 25:3, 417-437. URL: http://dx.doi.org/10.1080/10543406.2014.920344
Hoffelder, T. (2019) Equivalence analyses of dissolution profiles with the Mahalanobis distance. Biometrical Journal, 61:5, 1120-1137. URL: https://doi.org/10.1002/bimj.201700257
Suarez-Sharp, S., Abend, A., Hoffelder, T., Leblond, D., Delvadia, P., Kovacs, E., Diaz, D.A. (2020). In Vitro Dissolution Profiles Similarity Assessment in Support of Drug Product Quality: What, How, When - Workshop Summary Report. The AAPS Journal, 22:74. URL: http://dx.doi.org/10.1208/s12248-020-00458-9
Examples
# Apart from simulation errors, a recalculation of the SE results
# of some parts (normal distribution only) of the simulation study in
# Hoffelder et al. (2015) can be done with the following code. Please note that
# the simulation takes approximately 20 minutes for 50.000 simulation
# runs (number_of_simu_runs <- 50000). To shorten calculation time for
# test users, number_of_simu_runs is set to 100 here and can/should be adapted.
# In the result of the simulation the variable empirical.size.se presents the
# simulated size obtained by function \code{SE.EQ()} whereas variable
# empirical.size.se.disso shows the
# simulated size obtained by function \code{SE.EQ.dissolution.profiles()}.
# A detailed analysis of the operating characteristics of the SE variant
# implemented in \code{SE.EQ.dissolution.profiles()} is the content of
# a future paper.
library(MASS)
number_of_simu_runs <- 100
set.seed(2020)
mu1 <- c(41,76,97)
mu2 <- mu1 - c(10,10,10)
SIGMA_1 <- matrix(data = c(537.4 , 323.8 , 91.8 ,
323.8 , 207.5 , 61.7 ,
91.8 , 61.7 , 26.1) ,ncol = 3)
SIGMA_2 <- matrix(data = c(324.1 , 233.6 , 24.5 ,
233.6 , 263.5 , 61.4 ,
24.5 , 61.4 , 32.5) ,ncol = 3)
SIGMA <- matrix(data = c(430.7 , 278.7 , 58.1 ,
278.7 , 235.5 , 61.6 ,
58.1 , 61.6 , 29.3) ,ncol = 3)
SIMULATION_SIZE_SE <- function(disttype , Hom , Var , mu_1 , mu_2
, n_per_group , n_simus ) {
n_success_SE <- 0
n_success_SE_disso <- 0
if ( Hom == "Yes" ) {
COVMAT_1 <- SIGMA
COVMAT_2 <- SIGMA
}
else {
COVMAT_1 <- SIGMA_1
COVMAT_2 <- SIGMA_2
}
if ( Var == "Low" ) {
COVMAT_1 <- COVMAT_1 / 4
COVMAT_2 <- COVMAT_2 / 4
}
d <- ncol(COVMAT_1)
Mean_diff <- mu_1 - mu_2 # Difference of both exp. values
vars_X <- diag(COVMAT_1) # variances of first sample
vars_Y <- diag(COVMAT_2) # variances of second sample
dist_SE <- sum( (Mean_diff * Mean_diff) / (0.5 * (vars_X + vars_Y) ) )
# true SE distance and equivalence margin for SE.EQ
if ( n_per_group == 10 ) {
cat("Expected value sample 1:",mu_1,"\n",
"Expected value sample 2:",mu_2,"\n",
"Covariance matrix sample 1:",COVMAT_1,"\n",
"Covariance matrix sample 2:",COVMAT_2,"\n",
"EM_SE:",dist_SE,"\n")
}
for (i in 1:n_simus) {
if ( disttype == "Normal" ) {
REF <- mvrnorm(n = n_per_group, mu=mu_1, Sigma=COVMAT_1)
TEST<- mvrnorm(n = n_per_group, mu=mu_2, Sigma=COVMAT_2)
}
n_success_SE_disso <- n_success_SE_disso +
SE.EQ.dissolution.profiles( X = REF ,
Y = TEST ,
print.results = FALSE
)$testresult.num
n_success_SE <- n_success_SE +
SE.EQ( X=REF ,
Y=TEST ,
eq_margin = dist_SE ,
print.results = FALSE
)$testresult.num
}
empirical_succ_prob_SE <- n_success_SE / n_simus
empirical_succ_prob_SE_disso <- n_success_SE_disso / n_simus
simuresults <- data.frame(dist = disttype , Hom = Hom , Var = Var
, dimension = d , em_se = dist_SE
, sample.size = n_per_group
, empirical.size.se = empirical_succ_prob_SE
, empirical.size.se.disso = empirical_succ_prob_SE_disso)
}
SIMULATION_LOOP_SAMPLE_SIZE <- function(disttype , Hom , Var
, mu_1 , mu_2 , n_simus ) {
run_10 <- SIMULATION_SIZE_SE(disttype = disttype , Hom = Hom , Var = Var
, mu_1 = mu_1 , mu_2 = mu_2
, n_per_group = 10 , n_simus = n_simus)
run_30 <- SIMULATION_SIZE_SE(disttype = disttype , Hom = Hom , Var = Var
, mu_1 = mu_1 , mu_2 = mu_2
, n_per_group = 30 , n_simus = n_simus)
run_50 <- SIMULATION_SIZE_SE(disttype = disttype , Hom = Hom , Var = Var
, mu_1 = mu_1 , mu_2 = mu_2
, n_per_group = 50 , n_simus = n_simus)
run_100 <- SIMULATION_SIZE_SE(disttype = disttype , Hom = Hom , Var = Var
, mu_1 = mu_1 , mu_2 = mu_2
, n_per_group = 100 , n_simus = n_simus)
RESULT_MATRIX <- rbind(run_10 , run_30 , run_50 , run_100)
RESULT_MATRIX
}
simu_1 <- SIMULATION_LOOP_SAMPLE_SIZE(disttype = "Normal", Hom = "Yes"
, Var = "High" , mu_1 = mu1 , mu_2 = mu2
, n_simus = number_of_simu_runs)
simu_2 <- SIMULATION_LOOP_SAMPLE_SIZE(disttype = "Normal", Hom = "Yes"
, Var = "Low" , mu_1 = mu1 , mu_2 = mu2
, n_simus = number_of_simu_runs)
simu_3 <- SIMULATION_LOOP_SAMPLE_SIZE(disttype = "Normal", Hom = "No"
, Var = "High" , mu_1 = mu1 , mu_2 = mu2
, n_simus = number_of_simu_runs)
simu_4 <- SIMULATION_LOOP_SAMPLE_SIZE(disttype = "Normal", Hom = "No"
, Var = "Low" , mu_1 = mu1 , mu_2 = mu2
, n_simus = number_of_simu_runs)
FINAL_RESULT <- rbind(simu_1 , simu_2 , simu_3 , simu_4)
cat("****** Simu results n_simu_runs: ",number_of_simu_runs," ***** \n")
FINAL_RESULT