| get_T2_two {disprofas} | R Documentation |
Hotelling's statistics (for two independent (small) samples)
Description
The function get_T2_two() estimates the parameters for Hotelling's
two-sample T^2 statistic for small samples.
Usage
get_T2_two(m1, m2, signif, na_rm = FALSE)
Arguments
m1 |
A matrix with the data of the reference group, e.g. a matrix representing dissolution profiles, i.e. with rows for the different dosage units and columns for the different time points, or a matrix for the different model parameters (columns) of different dosage units (rows). |
m2 |
A matrix with the same dimensions as matrix |
signif |
A positive numeric value between |
na_rm |
A logical value that indicates whether observations containing
|
Details
The two-sample Hotelling's T^2 test statistic is given by
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) ,
where \bm{x}_T and \bm{x}_R are the vectors of the
sample means of the test (T) and reference (R) group, e.g.
vectors of the average dissolution per time point or of the average model
parameters, n_T and n_R are the numbers of observations of the
reference and the test group, respectively (i.e. the number of rows in
matrices m1 and m2 handed over to the get_T2_two()
function), and \bm{S}_{pooled} is the pooled
variance-covariance matrix which is calculated by
\bm{S}_{pooled} = \frac{(n_R - 1) \bm{S}_R + (n_T - 1) \bm{S}_T}{%
n_R + n_T - 2} ,
where \bm{S}_R and \bm{S}_T are the estimated
variance-covariance matrices which are calculated from the matrices of the
two groups being compared, i.e. m1 and m2. The matrix
\bm{S}_{pooled}^{-1} is the inverted
variance-covariance matrix. As the number of columns of matrices m1
and m2 increases, and especially as the correlation between the
columns increases, the risk increases that the pooled variance-covariance
matrix \bm{S}_{pooled} is ill-conditioned or even singular
and thus cannot be inverted. The term
D_M = \sqrt{ \left( \bm{x}_T - \bm{x}_R \right)^{\top}
\bm{S}_{pooled}^{-1} \left( \bm{x}_T - \bm{x}_R \right) }
is the Mahalanobis distance which is used to measure the difference between
two multivariate means. For large samples, T^2 is approximately
chi-square distributed with p degrees of freedom, where p is
the number of variables, i.e. the number of dissolution profile time points
or the number of model parameters. In terms of the Mahalanobis distance,
Hotelling's T^2 statistic can be expressed has
\frac{n_T n_R}{n_T + n_R} \; D_M^2 = k \; D_M^2 .
To transform the Hotelling's T^2 statistic into an F-statistic,
a conversion factor is necessary, i.e.
K = k \; \frac{n_T + n_R - p - 1}{\left( n_T + n_R - 2 \right) p} .
With this transformation, the following test statistic can be applied:
K \; D_M^2 \leq F_{p, n_T + n_R - p - 1, \alpha} .
Under the null hypothesis, H_0: \bm{\mu}_T = \bm{\mu}_R, this F-statistic is F-distributed with
p and n_T + n_R - p - 1 degrees of freedom. H_0 is
rejected at significance level \alpha if the F-value exceeds
the critical value from the F-table evaluated at \alpha, i.e.
F > F_{p, n_T + n_R - p - 1, \alpha}. The null hypothesis is satisfied
if, and only if, the population means are identical for all variables. The
alternative is that at least one pair of these means is different.
The following assumptions concerning the data are made:
The data from population
iis a sample from a population with mean vector\mu_i. In other words, it is assumed that there are no sub-populations.The data from both populations have common variance-covariance matrix
\Sigma.The elements from both populations are independently sampled, i.e. the data values are independent.
Both populations are multivariate normally distributed.
Confidence intervals:
Confidence intervals for the mean differences at each time point or
confidence intervals for the mean differences between the parameter
estimates of the reference and the test group are calculated by aid of the
formula
\left( \bm{x}_T - \bm{x}_R \right) \pm \sqrt{\frac{1}{K} \;
F_{p, n_T + n_R - p - 1, \alpha} \; \bm{s}_{pooled}} ,
where \bm{s}_{pooled} is the vector of the diagonal
elements of the pooled variance-covariance matrix
\bm{S}_{pooled}. With (1 - \alpha)100\% confidence,
this interval covers the respective linear combination of the differences
between the means of the two sample groups. If not the linear combination
of the variables is of interest but rather the individual variables, then
the Bonferroni corrected confidence intervals should be used instead which
are given by the expression
\left( \bm{x}_T - \bm{x}_R \right) \pm
t_{n_T + n_R - 2, \frac{\alpha}{2 p}} \;
\sqrt{\frac{1}{k} \; \bm{s}_{pooled}} .
Value
A list with the following elements is returned:
Parameters |
Parameters determined for the estimation of Hotelling's
|
S.pool |
Pooled variance-covariance matrix. |
covs |
A list with the elements |
means |
A list with the elements |
CI |
A list with the elements |
The Parameters element contains the following information:
dm |
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 |
Provided significance level. |
K |
Scaling factor for |
k |
Scaling factor for the squared Mahalanobis distance to obtain
the |
T2 |
Hotelling's |
F |
Observed |
F.crit |
Critical |
t.crit |
Critical |
p.F |
|
References
Hotelling, H. The generalisation of Student's ratio. Ann Math Stat. 1931; 2(3): 360-378.
Hotelling, H. (1947) Multivariate quality control illustrated by air testing of sample bombsights. In: Eisenhart, C., Hastay, M.W., and Wallis, W.A., Eds., Techniques of Statistical Analysis, McGraw Hill, New York, 111-184.
See Also
get_T2_one, get_sim_lim,
mimcr.
Examples
# Estimation of the parameters for Hotelling's two-sample T2 statistic
# (for small samples)
res1 <- 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)
res1$S.pool
res1$Parameters
# Results in res1$S.pool
# t.15 t.90
# t.15 3.395808 1.029870
# t.90 1.029870 4.434833
# Results in res1$Parameters
# dm df1 df2 signif K
# 1.044045e+01 2.000000e+00 9.000000e+00 1.000000e-01 1.350000e+00
# k T2 F F.crit t.crit
# 3.000000e+00 3.270089e+02 1.471540e+02 3.006452e+00 2.228139e+00
# p.F
# 1.335407e-07
# The results above correspond to the values that are shown in Tsong (1996)
# (see reference of dip1 data set) under paragraph "DATA1 data (Comparing
# the 15- and 90-minute sample time points only).
# For the second assessment shown in Tsong (1996) (see reference of dip1 data
# set) under paragraph "DATA2 data (Comparing all eight time points), the
# following results are obtained.
res2 <- get_T2_two(m1 = as.matrix(dip1[dip1$type == "R", 3:10]),
m2 = as.matrix(dip1[dip1$type == "T", 3:10]),
signif = 0.1)
res2$Parameters
# Results in res2$Parameters
# dm df1 df2 signif K
# 2.648562e+01 8.000000e+00 3.000000e+00 1.000000e-01 1.125000e-01
# k T2 F F.crit t.crit
# 3.000000e+00 2.104464e+03 7.891739e+01 5.251671e+00 3.038243e+00
# p.F
# 2.116258e-03
# In Tsong (1997) (see reference of dip7), the model-dependent approach is
# illustrated with an example data set of alpha and beta parameters obtained
# by fitting the Weibull curve function to a data set of dissolution profiles
# of three reference batches and one new batch (12 profiles per batch).
res3 <-
get_T2_two(m1 = as.matrix(dip7[dip7$type == "ref", c("alpha", "beta")]),
m2 = as.matrix(dip7[dip7$type == "test", c("alpha", "beta")]),
signif = 0.05)
res3$Parameters
# Results in res3$Parameters
# dm df1 df2 signif K
# 3.247275e+00 2.000000e+00 4.500000e+01 5.000000e-02 4.402174e+00
# k T2 F F.crit t.crit
# 9.000000e+00 9.490313e+01 4.642001e+01 3.204317e+00 2.317152e+00
# p.F
# 1.151701e-11
# In Sathe (1996) (see reference of dip8), the model-dependent approach is
# illustrated with an example data set of alpha and beta parameters obtained
# by fitting the Weibull curve function to a data set of dissolution profiles
# of one reference batch and one new batch with minor modifications and another
# new batch with major modifications (12 profiles per batch). Note that the
# assessment is performed on the (natural) logarithm scale.
res4.minor <- get_T2_two(m1 = log(as.matrix(dip8[dip8$type == "ref",
c("alpha", "beta")])),
m2 = log(as.matrix(dip8[dip8$type == "minor",
c("alpha", "beta")])),
signif = 0.1)
res4.major <- get_T2_two(m1 = log(as.matrix(dip8[dip8$type == "ref",
c("alpha", "beta")])),
m2 = log(as.matrix(dip8[dip8$type == "major",
c("alpha", "beta")])),
signif = 0.1)
res4.minor$Parameters
res4.minor$CI$Hotelling
res4.major$Parameters
res4.major$CI$Hotelling
# Expected results in res4.minor$Parameters
# dm df1 df2 signif K
# 1.462603730 2.000000000 21.000000000 0.100000000 2.863636364
# k T2 F F.crit t.crit
# 6.000000000 12.835258028 6.125918604 2.574569390 2.073873068
# p.F
# 0.008021181
# Results in res4.minor$CI$Hotelling
# LCL UCL
# alpha -0.2553037 -0.02814098
# beta -0.1190028 0.01175691
# Expected results in res4.major$Parameters
# dm df1 df2 signif K
# 4.508190e+00 2.000000e+00 2.100000e+01 5.000000e-02 2.863636e+00
# k T2 F F.crit t.crit
# 6.000000e+00 1.219427e+02 5.819992e+01 2.574569e+00 2.073873e+00
# p.F
# 2.719240e-09
# Expected results in res4.major$CI$Hotelling
# LCL UCL
# alpha -0.4864736 -0.2360966
# beta 0.1954760 0.3035340