get_sim_lim function

Similarity limit

The function get_sim_lim() estimates a similarity limit in terms of the Multivariate Statistical Distance (MSD).

get_sim_lim(mtad, lhs)

Arguments

• mtad: A numeric value that specifies the maximum tolerable average difference (MTAD) of the profiles of two formulations at all time points (in %). The default value is 10. It determines the size of the similarity limit $d_g$ (see the details section for more information).

• lhs: A list of the estimates of Hotelling's two-sample $T^2$

  statistic for small samples as returned by the function get_T2_two().

Returns

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).

Details

Details about the estimation of similarity limits in terms of the Multivariate Statistical Distance (MSD) are explained in the corresponding section below.

#s 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 $\bm{S}_{pooled}$ is the sample variance-covariance matrix pooled across the comparative groups, $x_T$ 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. The pooled variance-covariance matrix $S_{pooled}$ is calculated by

\bm{S}_{pooled} = \frac{(n_R - 1) \bm{S}_R + (n_T - 1) \bm{S}_T}{%n_R + n_T - 2} .S_{pooled} = ((n_R - 1) S_R + (n_T - 1) S_T) /(n_R + n_T - 2) .

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 $d_g$ 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 ($\textit{CR}$) bounds for the true difference between the mean vectors, $\mu_T - \mu_R$, can be computed for the resultant vector $\mu$ to satisfy the following condition:

$$\bm{\textit{CR}} = K \left( \bm{\mu} - \left( \bm{x}_T -\bm{x}_R \right) \right)^{\top} \bm{S}_{pooled}^{-1} \left( \bm{\mu} -\left( \bm{x}_T - \bm{x}_R \right) \right) \leqF_{p, n_T + n_R - p - 1, 0.9} ,$$

where $K$ is the scaling factor that is calculated as

$$K = \frac{n_T n_R}{n_T + n_R} \; \frac{n_T + n_R - p - 1}{\left( n_T + n_R - 2 \right) p} ,$$

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 $n_T + n_R - p - 1$, where $n_T$ and $n_R$ are the number of observations of the reference and the test group, respectively, and $p$

is the number of sampling or time points, as mentioned already. It is obvious that $(n_T + n_R)$ must be greater than $(p + 1)$. The formula for $\textit{CR}$ 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 two-sample $T^2$ test 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 in paragraph Similarity limits in terms of MSD , $d_g$ 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 andnot 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.

Examples

# Estimation of the parameters for Hotelling's two-sample T2 statistic
# (for small samples)
hs <- get_T2_two(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

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.

tools:::Rd_expr_doi("10.1177/009286159603000427")

Wellek S. (2010) Testing statistical hypotheses of equivalence and noninferiority (2nd ed.). Chapman & Hall/CRC, Boca Raton.

tools:::Rd_expr_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

mimcr, get_T2_two.