get_sim_lim {disprofas} | R Documentation |
Similarity limit
Description
The function get_sim_lim()
estimates a similarity limit in terms of
the “Multivariate Statistical Distance” (MSD).
Usage
get_sim_lim(mtad, lhs)
Arguments
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 |
Details
Details about the estimation of similarity limits in terms of the “Multivariate Statistical Distance” (MSD) are explained in the corresponding section below.
Value
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 |
k |
Scaling factor for the squared Mahalanobis distance to obtain
the |
T2 |
Hotelling's |
F |
Observed |
ncp.Hoffelder |
Non-centrality parameter for calculation of the |
F.crit |
Critical |
F.crit.Hoffelder |
Critical |
p.F |
The |
p.F.Hoffelder |
The |
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). |
Similarity limits in terms of MSD
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{ \left( \bm{x_T} - \bm{x_R} \right)^{\top}
\bm{S}_{pooled}^{-1} \left( \bm{x_T} - \bm{x_R} \right)} ,
where \bm{S}_{pooled} = \frac{\left( \bm{S}_T + \bm{S}_R \right)}{2}
is the sample variance-covariance matrix
pooled across the batches, \bm{x}_T
and \bm{x}_R
are
the vectors of the sample means for the test (T
) and reference
(R
) profiles, and \bm{S}_T
and \bm{S}_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{ \bm{d}_g^{\top} \bm{S}_{pooled}^{-1} \bm{d}_g} ,
where \bm{d}_g
is a 1 \times p
vector with all
p
elements equal to an empirically defined limit \bm{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,
\bm{\mu}_T - \bm{\mu}_R
, can be computed for the
resultant vector \bm{\mu}
to satisfy the following condition:
\bm{CR} = K \left( \bm{\mu} - \left( \bm{x_T} - \bm{x_R} \right)
\right)^{\top} \bm{S}_{pooled}^{-1} \left( \bm{\mu} - \left( \bm{x_T} -
\bm{x_R} \right) \right) \leq F_{p, n_T + n_R - p - 1, 0.9} ,
where K
is the scaling factor that is calculated as
K = \frac{n_T n_R}{n_T + n_R} \cdot \frac{n_T + n_R - p - 1}{
\left( n_T + n_R - 2 \right) \cdot 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.
T2 test for equivalence
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 \alpha
, i.e. if
p < \alpha
. The p
value is calculated by aid of the formula
p = F_{p, n_T + n_R - p - 1, ncp, \alpha}
\frac{n_T + n_R - p - 1}{(n_T + n_R - 2) p} T^2 ,
where \alpha
is the significance level and ncp
is the so-called
“non-centrality parameter” that is calculated by
\frac{n_T n_R}{n_T + n_R} \left( D_M^{max} \right)^2 .
The test statistic being used is Hotelling's T^2
that is given as
T^2 = \frac{n_T n_R}{n_T + n_R} \left( \bm{x_T} - \bm{x_R}
\right)^{\top} \bm{S}_{pooled}^{-1} \left( \bm{x_T} - \bm{x_R} \right) .
As mentioned elsewhere, \bm{d}_g
is a 1 \times p
vector with all p
elements equal to an empirically defined limit
d_g
. Thus, the components of the vector \bm{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.
References
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
See Also
Examples
# 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