Description Usage Arguments Details Value Similarity limits in terms of MSD T2 test for equivalence References See Also Examples
The function get_sim_lim()
estimates a similarity limit in terms of
the “Multivariate Statistical Distance” (MSD).
1 | get_sim_lim(mtad, lhs)
|
mtad |
A numeric value specifying the “maximum tolerable average
difference” (MTAD) of the profiles of two formulations at all time points
(in %). The default value is |
lhs |
A list of the estimates of Hotelling's two-sample T^2
statistic for small samples as returned by the function
|
Details about the estimation of similarity limits in terms of the “Multivariate Statistical Distance” (MSD) are explained in the corresponding section below.
A vector containing the following information is returned:
DM |
The Mahalanobis distance of the samples. |
df1 |
Degrees of freedom (number of variables or time points). |
df2 |
Degrees of freedom (number of rows - number of variables - 1). |
alpha |
The provided significance level. |
K |
Scaling factor for F to account for the distribution of the T^2 statistic. |
k |
Scaling factor for the squared Mahalanobis distance to obtain the T^2 statistic. |
T2 |
Hotelling's T^2 statistic (F-distributed). |
F |
Observed F value. |
ncp.Hoffelder |
Non-centrality parameter for calculation of the F statistic (T^2 test procedure). |
F.crit |
Critical F value (Tsong's procedure). |
F.crit.Hoffelder |
Critical F value (T^2 test procedure). |
p.F |
The p value for the Hotelling's T^2 test statistic. |
p.F.Hoffelder |
The p value for the Hotelling's T^2 statistic based on the non-central F distribution. |
MTAD |
Specified “maximum tolerable average difference” (MTAD) of the profiles of two formulations at each individual time point (in %). |
Sim.Limit |
Critical Mahalanobis distance or similarity limit (Tsong's procedure). |
For the calculation of the “Multivariate Statistical Distance” (MSD), the procedure proposed by Tsong et al. (1996) can be considered as well-accepted method that is actually recommended by the FDA. According to this method, a multivariate statistical distance, called Mahalanobis distance, is used to measure the difference between two multivariate means. This distance measure is calculated as
D_M = sqrt((x_T - x_R)^{\top} S_{pooled}^{-1} (x_T - x_R)) ,
where S_{pooled} = (S_T + S_R) / 2 is the sample variance-covariance matrix pooled across the batches, x_T and x_R are the vectors of the sample means for the test (T) and reference (R) profiles, and S_T and x_R are the variance-covariance matrices of the test and reference profiles.
In order to determine the similarity limits in terms of the MSD, i.e. the Mahalanobis distance between the two multivariate means of the dissolution profiles of the formulations to be compared, Tsong et al. (1996) proposed using the equation
D_M^{max} = sqrt(d_g^{\top} S_{pooled}^{-1} d_g) ,
where d_g is a 1 x p vector with all p elements equal to an empirically defined limit d_g, e.g., 15%, for the maximum tolerable difference at all time points, and p is the number of sampling points. By assuming that the data follow a multivariate normal distribution, the 90% confidence region (CR) bounds for the true difference between the mean vectors, μ_T - μ_R, can be computed for the resultant vector μ to satisfy the following condition:
CR = sqrt((μ - (x_T - x_R))^{\top} S_{pooled}^{-1} (μ - (x_T - x_R))) ≤q F_{p, n_T + n_R - p - 1, 0.9} ,
where K is the scaling factor that is calculated as
(n_T n_R) / (n_T + n_R) * (n_T + n_R - p - 1) / ((n_T + n_R - 2) p) ,
and F_{p, n_T + n_R - p - 1, 0.9} is the 90^{th} percentile of
the F distribution with degrees of freedom p and
n_T + n_R - p - 1. It is obvious that (n_T + n_R) must be greater
than (p + 1). The formula for CR gives a p-variate 90%
confidence region for the possible true differences.
Based on the distance measure for profile comparison that was suggested by Tsong et al. (1996), i.e. the Mahalanobis distance, Hoffelder (2016) proposed a statistical equivalence procedure for that distance, the so-called T^2 test for equivalence (T2EQ). It is used to demonstrate that the Mahalanobis distance between reference and test group dissolution profiles is smaller than the “Equivalence Margin” (EM). Decision in favour of equivalence is taken if the p value of this test statistic is smaller than the pre-specified significance level α, i.e. if p < α. The p value is calculated by aid of the formula
p = F_{p, n_T + n_R - p - 1, ncp, α} (n_T + n_R - p - 1) / ((n_T + n_R - 2) p) T^2 ,
where α is the significance level and ncp is the so-called “non-centrality parameter” that is calculated by
(n_T n_R) / (n_T + n_R) (D_M^{max})^2 .
The test statistic being used is Hotelling's T^2 that is given as
(n_T n_R) / (n_T + n_R) * (x_T - x_R)^{\top} S_{pooled}^{-1} (x_T - x_R) .
As mentioned elsewhere, d_g is a 1 x p vector with all p elements equal to an empirically defined limit d_g. Thus, the components of the vector d_g can be interpreted as upper bound for a kind of “average” allowed difference between test and reference profiles, the “global similarity limit”. Since the EMA requires that “similarity acceptance limits should be pre-defined and justified and not be greater than a 10% difference”, it is recommended to use 10%, not 15% as proposed by Tsong et al. (1996), for the maximum tolerable difference at all time points.
Tsong, Y., Hammerstrom, T., Sathe, P.M., and Shah, V.P. Statistical
assessment of mean differences between two dissolution data sets.
Drug Inf J. 1996; 30: 1105-1112.
doi: 10.1177/009286159603000427
Wellek S. (2010) Testing statistical hypotheses of equivalence and
noninferiority (2nd ed.). Chapman & Hall/CRC, Boca Raton.
doi: 10.1201/EBK1439808184
Hoffelder, T. Highly variable dissolution profiles. Comparison of
T^2-test for equivalence and f_2 based methods. Pharm Ind.
2016; 78(4): 587-592.
https://www.ecv.de/suse_item.php?suseId=Z|pi|8430
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 | # Dissolution data of one reference batch and one test batch of n = 6
# tablets each:
str(dip1)
# 'data.frame': 12 obs. of 10 variables:
# $ type : Factor w/ 2 levels "R","T": 1 1 1 1 1 1 2 2 2 2 ...
# $ tablet: Factor w/ 6 levels "1","2","3","4",..: 1 2 3 4 5 6 1 2 3 4 ...
# $ t.5 : num 42.1 44.2 45.6 48.5 50.5 ...
# $ t.10 : num 59.9 60.2 55.8 60.4 61.8 ...
# $ t.15 : num 65.6 67.2 65.6 66.5 69.1 ...
# $ t.20 : num 71.8 70.8 70.5 73.1 72.8 ...
# $ t.30 : num 77.8 76.1 76.9 78.5 79 ...
# $ t.60 : num 85.7 83.3 83.9 85 86.9 ...
# $ t.90 : num 93.1 88 86.8 88 89.7 ...
# $ t.120 : num 94.2 89.6 90.1 93.4 90.8 ...
# Estimation of the parameters for Hotelling's two-sample T2 statistic
# (for small samples)
hs <- get_hotellings(m1 = as.matrix(dip1[dip1$type == "R", c("t.15", "t.90")]),
m2 = as.matrix(dip1[dip1$type == "T", c("t.15", "t.90")]),
signif = 0.1)
# Estimation of the similarity limit in terms of the "Multivariate Statistical
# Distance" (MSD)for a "maximum tolerable average difference" (mtad) of 10
res <- get_sim_lim(mtad = 15, hs)
# Expected results in res
# DM df1 df2 alpha
# 1.044045e+01 2.000000e+00 9.000000e+00 1.000000e-01
# K k T2 F
# 1.350000e+00 3.000000e+00 3.270089e+02 1.471540e+02
# ncp.Hoffelder F.crit F.crit.Hoffelder p.F
# 2.782556e+02 3.006452e+00 8.357064e+01 1.335407e-07
# p.F.Hoffelder MTAD Sim.Limit
# 4.822832e-01 1.500000e+01 9.630777e+00
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