ph_test_statistic {multiFANOVA}

Pointwise Hotelling's T^2-test statistic

Description

The function ph_test_statistic() calculates the pointwise Hotelling's T^2-test statistic.

Usage

ph_test_statistic(x, gr_label, h)

Arguments

x

matrix of observations n\times j (n = n_1 + ... + n_k, j is a number of design time points).

gr_label

a vector with group labels; the integer labels (from 1 to a number of groups) should be used.

h

contrast matrix. For Dunnett’s and Tukey’s contrasts, it can be created by the contr_mat() function from the package GFDmcv (see examples).

Details

For details, see the documentation of the multiFANOVA() function or the paper Munko et al. (2023).

Value

A vector of values of the pointwise Hotelling's T^2-test statistic.

References

Dunnett C. (1955) A multiple comparison procedure for comparing several treatments with a control. Journal of the American Statistical Association 50, 1096-1121.

Munko M., Ditzhaus M., Pauly M., Smaga L., Zhang J.T. (2023) General multiple tests for functional data. Preprint https://arxiv.org/abs/2306.15259

Tukey J.W. (1953) The problem of multiple comparisons. Princeton University.

Examples

# Some of the examples may run some time.

# Canadian weather data set
# There are three samples of mean temperatures for
# fifteen weather stations in Eastern Canada,
# another fifteen in Western Canada, and
# the remaining five in Northern Canada.
library(fda)
data_set <- t(CanadianWeather$dailyAv[,, "Temperature.C"])
k <- 3
gr_label <- rep(c(1, 2, 3), c(15, 15, 5))
# trajectories of mean temperatures
matplot(t(data_set), type = "l", col = gr_label, lty = 1,
        xlab = "Day", ylab = "Temperature (C)",
        main = "Canadian weather data set")
legend("bottom", legend = c("Eastern Canada", "Western Canada", "Northern Canada"),
       col = 1:3, lty = 1)

# Tukey's contrast matrix
h_tukey <- GFDmcv::contr_mat(k, type = "Tukey")
# testing without parallel computing
res <- multiFANOVA(data_set, gr_label, h_tukey)
summary(res, digits = 3)
# plots for pointwise Hotelling's T^2-test statistics
oldpar <- par(mfrow = c(2, 2), mar = c(2, 2, 2, 0.1))
plot(ph_test_statistic(data_set, gr_label, h_tukey), type = "l",
     ylim = c(0, max(ph_test_statistic(data_set, gr_label, h_tukey))),
     main = "Global hypothesis")
plot(ph_test_statistic(data_set, gr_label, matrix(h_tukey[1, ], 1)), type = "l",
     ylim = c(0, max(ph_test_statistic(data_set, gr_label, h_tukey))),
     main = "Contrast 1")
plot(ph_test_statistic(data_set, gr_label, matrix(h_tukey[2, ], 1)), type = "l",
     ylim = c(0, max(ph_test_statistic(data_set, gr_label, h_tukey))),
     main = "Contrast 2")
plot(ph_test_statistic(data_set, gr_label, matrix(h_tukey[3, ], 1)), type = "l",
     ylim = c(0, max(ph_test_statistic(data_set, gr_label, h_tukey))),
     main = "Contrast 3")
par(oldpar)