two_mean_vector_test: Tests for Equality of Two Normal Mean Vectors

In fhernanb/stests: Package with useful statistical tests

Tests for Equality of Two Normal Mean Vectors

Description

The function implements the test for H_0: \mu_1 = \mu_2 versus H_1: \mu_1 not = \mu_2 when two random samples are obtained from two p-variate normal populations Np(\mu_1, \Sigma_1) and Np(\mu_2, \Sigma_2) respectively. By default, the function performs the Hotelling test for two normal mean vectors assumming equality in the covariance matrices. Other tests for the multivariate Behrens-Fisher problem are also implemented, see the argument method below.

Usage

two_mean_vector_test(
  xbar1,
  s1,
  n1,
  xbar2,
  s2,
  n2,
  delta0 = NULL,
  method = "T2",
  alpha = 0.05
)

Arguments

xbar1	a vector with the sample mean from population 1.
s1	a matrix with sample variances and covariances from population 1.
n1	sample size 1.
xbar2	a vector with the sample mean from population 2.
s2	a matrix with sample variances and covariances from population 2.
n2	sample size 2.
delta0	a number indicating the true value of the difference in means.
method	a character string specifying the method, "T2" (default), "james" (James' first order test), please see the details section for other methods.
alpha	the significance level only for method "james", by default its value is 0.05.

Details

the "method" argument must be one of "T2" (default), "james" (James' first order test), "yao" (Yao's test), "johansen" (Johansen's test), "nvm" (Nel and Van der Merwe test), "mnvm" (modified Nel and Van der Merwe test), "gamage" (Gamage's test), "yy" (Yanagihara and Yuan test), "byy" (Bartlett Correction test), "mbyy" (modified Bartlett Correction test), "ks1" (Second Order Procedure), "ks2" (Bias Correction Procedure). For James test the critic value is reported, we reject H_0 if T2 > critic_value.

Value

A list with class "htest" containing the following components:

statistic	the value of the statistic.
parameter	the degrees of freedom for the test.
p.value	the p-value for the test.
estimate	the estimated mean vectors.
method	a character string indicating the type of test performed.

Author(s)

Freddy Hernandez, Jean Paul Piedrahita, Valentina Garcia.

Examples

# Example 5.4.2 from Rencher & Christensen (2012) page 137,
# using Hotelling's test
n1 <- 32
xbar1 <- c(15.97, 15.91, 27.19, 22.75)
s1 <- matrix(c(5.192, 4.545, 6.522, 5.25,
               4.545, 13.18, 6.76, 6.266,
               6.522, 6.76, 28.67, 14.47,
               5.25, 6.266, 14.47, 16.65), ncol = 4)

n2 <- 32
xbar2 <- c(12.34, 13.91, 16.66, 21.94)
s2 <- matrix(c(9.136, 7.549, 4.864, 4.151,
               7.549, 18.6, 10.22, 5.446,
               4.864, 10.22, 30.04, 13.49,
               4.151, 5.446, 13.49, 28), ncol = 4)

res1 <- two_mean_vector_test(xbar1 = xbar1, s1 = s1, n1 = n1,
                             xbar2 = xbar2, s2 = s2, n2 = n2,
                             method = "T2")
res1
plot(res1, from=21, to=25, shade.col='tomato')

# Example 3.7 from Seber (1984) page 116.
# using the James first order test (1954).
n1 <- 16
xbar1 <- c(9.82, 15.06)
s1 <- matrix(c(120, -16.3,
               -16.3, 17.8), ncol = 2)

n2 <- 11
xbar2 <- c(13.05, 22.57)
s2 <- matrix(c(81.8, 32.1,
               32.1, 53.8), ncol = 2)

res2 <- two_mean_vector_test(xbar1 = xbar1, s1 = s1, n1 = n1,
                             xbar2 = xbar2, s2 = s2, n2 = n2,
                             method = 'james', alpha=0.05)
res2
plot(res2, from=5, to=10, shade.col="lightgreen")

# Example from page 141 from Yao (1965),
# using Yao's test

n1 <- 16
xbar1 <- c(9.82, 15.06)
s1 <- matrix(c(120, -16.3,
               -16.3, 17.8), ncol = 2)

n2 <- 11
xbar2 <- c(13.05, 22.57)
s2 <- matrix(c(81.8, 32.1,
               32.1, 53.8), ncol = 2)

res3 <- two_mean_vector_test(xbar1 = xbar1, s1 = s1, n1 = n1,
                             xbar2 = xbar2, s2 = s2, n2 = n2,
                             method = 'yao')
res3
plot(res3, from=2, to=6, shade.col="pink")


# Example for Johansen's test using the data from
# Example from page 141 from Yao (1965)

res4 <- two_mean_vector_test(xbar1 = xbar1, s1 = s1, n1 = n1,
                             xbar2 = xbar2, s2 = s2, n2 = n2,
                             method = 'johansen')
res4
plot(res4, from=2, to=6, shade.col="aquamarine1")

# Example 4.1 from Nel and Van de Merwe (1986) page 3729
# Test H0: mu1 = mu2 versus H1: mu1 != mu2
n1 <- 45
xbar1 <- c(204.4, 556.6)
s1 <- matrix(c(13825.3, 23823.4,
               23823.4, 73107.4), ncol=2)

n2 <- 55
xbar2 <- c(130.0, 355.0)
s2 <- matrix(c(8632.0, 19616.7,
               19616.7, 55964.5), ncol=2)

res5 <- two_mean_vector_test(xbar1 = xbar1, s1 = s1, n1 = n1,
                             xbar2 = xbar2, s2 = s2, n2 = n2,
                             method = 'nvm')
res5
plot(res5, from=6, to=10, shade.col='cyan2')

# using mnvm method for same data
res6 <- two_mean_vector_test(xbar1 = xbar1, s1 = s1, n1 = n1,
                             xbar2 = xbar2, s2 = s2, n2 = n2,
                             method = 'mnvm')
res6
plot(res6, from=6, to=10, shade.col='lightgoldenrodyellow')

# using gamage method for same data
res7 <- two_mean_vector_test(xbar1 = xbar1, s1 = s1, n1 = n1,
                             xbar2 = xbar2, s2 = s2, n2 = n2,
                             method = 'gamage')
res7
plot(res7, from=0, to=20, shade.col='mediumpurple4')
text(x=10, y=0.30, "The curve corresponds to an empirical density",
     col='orange')

# using Yanagihara and Yuan method for same data
res8 <- two_mean_vector_test(xbar1 = xbar1, s1 = s1, n1 = n1,
                             xbar2 = xbar2, s2 = s2, n2 = n2,
                             method = 'yy')
res8
plot(res8, from=6, to=10, shade.col='gold2')

# using Bartlett Correction method for same data
res9 <- two_mean_vector_test(xbar1 = xbar1, s1 = s1, n1 = n1,
                             xbar2 = xbar2, s2 = s2, n2 = n2,
                             method = 'byy')
res9
plot(res9, from=12, to=20, shade.col='hotpink2')

# using modified Bartlett Correction method for same data
res10 <- two_mean_vector_test(xbar1 = xbar1, s1 = s1, n1 = n1,
                              xbar2 = xbar2, s2 = s2, n2 = n2,
                              method = 'mbyy')
res10
plot(res10, from=12, to=20, shade.col='green3')

# using Second Order Procedure (Kawasaki and Seo (2015)) method for same data
res11 <- two_mean_vector_test(xbar1 = xbar1, s1 = s1, n1 = n1,
                              xbar2 = xbar2, s2 = s2, n2 = n2,
                              method = 'ks1')
res11
plot(res11, from=6, to=10, shade.col='hotpink1')

# using Bias Correction Procedure (Kawasaki and Seo (2015)) method for same data
res12 <- two_mean_vector_test(xbar1 = xbar1, s1 = s1, n1 = n1,
                              xbar2 = xbar2, s2 = s2, n2 = n2,
                              method = 'ks2')
res12
plot(res12, from=6, to=10, shade.col='seagreen2')

fhernanb/stests documentation built on March 29, 2025, 10:36 a.m.