rmfanova {rmfanova}

Repeated measures functional analysis of variance

Description

The function rmfanova() calculates the tests based on three test statistics \mathcal{C}_n, \mathcal{D}_n, and \mathcal{E}_n for the problem of comparing \ell-samples of repeated measures for functional data. The tests are based on five resampling methods, i.e., two permutation and three bootstrap ones. The overall and local hypotheses are considered.

Usage

rmfanova(
  x,
  method = "bonferroni",
  n_perm = 1000,
  n_boot = 1000,
  parallel = FALSE,
  n_cores = NULL,
  multi_gen = FALSE
)

Arguments

x

a list of length \ell with elements being n\times p matrices of data corresponding to n functional observations measured in p design time points under given experimental conditions.

method

the correction method to be used for pairwise comparisons. Options are "bonferroni" (default) and those given in the vector p.adjust.methods (as for the p.adjust() function).

n_perm

a number of permutation replicates. The default is 1000.

n_boot

a number of bootstrap replicates. The default is 1000.

parallel

a logical indicating of whether to use parallel computing. The default is FALSE.

n_cores

if parallel = TRUE, a number of processes used in parallel computation. Its default value (NULL) means that it will be equal to a number of cores of a computer used.

multi_gen

a logical indicating of whether to use separate multiple generations of Gaussian processes for the parametric bootstrap tests. The default is FALSE, which means that the processes will be generated once in a big matrix. This method is much faster, but for larger n and p the generated data can be too large for RAM. In such a case, we suggest using separate generation (multi_gen = TRUE), which is slower, but possible to calculate.

Details

The function rmfanova() concerns the tests for the functional repeated measures analysis problem. The details are presented in Kurylo and Smaga (2023), where in particular, some recommendations for using tests are given. Here we present only some summary of the problem and its solutions implemented in the package.

We have n subjects subjected to \ell\geq 2 (possibly) different conditions. The results of the experiments are functional observations. Let the subjects be represented by a functional sample consisting of independent stochastic processes Y_1,\dots,Y_n defined on the interval [0,\ell], which satisfy the following model proposed by Martinez-Camblor and Corral (2011):

Y_j(t)=\mu(t)+e_j(t),\ j=1,\dots,n,\ t\in[0,\ell],

where \mu is a fixed mean function, and e_j is a random process with zero mean function. In this notation, t\in[0,1] corresponds to the first experimental condition, t\in[1,2] to the second, and so on. Thus, in this model, we ignore the possible time periods between repetitions of the experiment, but this does not mean that they do not exist. We are interested in testing the equality of \ell mean functions corresponding to experimental conditions; namely, the global null hypothesis is as follows:

\mathcal{H}_0:\mu(t)=\mu(t+1)=\dots=\mu(t+(\ell-1))\ \ \forall t\in[0,1].

For the global null hypothesis \mathcal{H}_0, the tests given by Martinez-Camblor and Corral (2011) used the pointwise sum of squares due to the hypothesis:

\mathrm{SSA}_{point}(t)=n\sum_{i=1}^\ell(\bar{Y}_{i\cdot}(t)-\bar{Y}(t))^2,\ t\in[0,1],

where

\bar{Y}_{i\cdot}(t)=n^{-1}\sum_{j=1}^nY_j(t+(i-1)),\ \bar{Y}(t)=N^{-1}\sum_{i=1}^\ell\sum_{j=1}^nY_j(t+(i-1)),

i=1,\dots,\ell. In the package, it is calculated and drawn by the pointwise_ssa_test_statistic() function. The other option is the following pointwise F-type test statistic proposed in Kurylo and Smaga (2023):

F_{point}(t)=\frac{\mathrm{SSA}_{point}(t)/(\ell-1)}{\mathrm{SSR}_{point}(t)/((\ell-1)(n-1))},\ t\in[0,1],

where

\mathrm{SSR}_{point}(t)=\sum_{i=1}^\ell\sum_{j=1}^n(Y_j(t+(i-1))-\bar{Y}_{i\cdot}(t)-\bar{Y}_{\cdot j}(t)+\bar{Y}(t))^2

is the pointwise sum of squares due to residuals, and

\bar{Y}_{\cdot j}(t)=\ell^{-1}\sum_{i=1}^\ell Y_j(t+(i-1)),\ j=1,\dots,n.

F_{point} is calculated and drawn by the pointwise_f_test_statistic() function.

To obtain global test statistics for \mathcal{H}_0, Martinez-Camblor and Corral (2011) proposed the following test statistic:

\mathcal{C}_n(\ell)=\int_0^1\mathrm{SSA}_{point}(t)dt.

On the other hand, Kurylo and Smaga (2023) proposed the following two test statistics:

\mathcal{D}_n(\ell)=\int_0^1F_{point}(t)dt,\quad\mathcal{E}_n(\ell)=\sup\limits_{t\in[0,1]}F_{point}(t).

To construct the tests, five resampling strategies are proposed by Kurylo and Smaga (2023). For details, we refer to this paper. Here we just note the two permutation tests and three bootstrap tests are denoted by P1, P2, B1, B2, and B3 in the output of the summary.rmfanova() function.

When \ell>2, by rejecting the global null hypothesis \mathcal{H}_0, we determine the presence of significant differences in the mean functions corresponding to the experimental conditions. However, we do not know which conditions are significantly different and which are not. To solve this problem, one needs to perform a post hoc analysis. More precisely, we would like to test the family of hypotheses:

\left\{\begin{array}{l} \mathcal{H}_0^{rs}:\mu(t+(r-1))=\mu(t+(s-1))\ \forall t\in[0,1],\\ \mathcal{H}_1^{rs}:\mu(t+(r-1))\neq\mu(t+(s-1))\ \text{for some}\ t\in[0,1],\\ \end{array}\right.

for r,s=1,\dots,\ell, r\neq s. These hypotheses are also named pairwise comparisons. To test this family of local hypotheses, we propose the following procedure:

1. Test each of the hypotheses \mathcal{H}_0^{rs} using the data for the r-th and s-th objects, i.e., Y_1(t),\dots,Y_n(t) for t\in[r-1,r] and t\in[s-1,s] respectively, and the chosen test from those presented above. Let p_{rs} denote the p-values obtained.

2. Make a final decision using the Bonferroni method, i.e., reject \mathcal{H}_0^{rs} if p_{rs}^{Bonf}\leq \alpha, where p_{rs}^{Bonf}=m\cdot p_{rs} are the corrected p-values, \alpha is the significance level and m is the number of null hypotheses considered.

In the paper Kurylo and Smaga (2023), the Bonferroni method was used only. However, in the package, there is a possibility to use other correction methods, which are available in the vector p.adjust.methods.

The results of testing the global and local hypotheses are given separately in the output of the summary.rmfanova() function for the convenience of the user.

Value

A list of class rmfanova containing the following 7 components:

n

a number n of functional observations.

p

a number p of design time points.

l

a number \ell of repeated samples.

method

an argument method.

test_stat

values of the test statistics \mathcal{C}_n, \mathcal{D}_n, and \mathcal{E}_n.

p_values

p-values for the global null hypothesis.

p_values_pc

p-values of the pairwise comparisons.

References

Martinez-Camblor P., Corral N. (2011) Repeated Measures Analysis for Functional Data. Computational Statistics & Data Analysis 55, 3244–3256.

Kurylo K., Smaga L. (2023) Functional repeated measures analysis of variance and its application. Preprint https://arxiv.org/abs/2306.03883

Ramsay J.O., Silverman B.W. (2005) Functional Data Analysis, 2nd Edition. New York: Springer.

Zhang J.T. (2013) Analysis of Variance for Functional Data. London: Chapman & Hall.

Examples

# Some of the examples may run some time.
# preparation of the DTI data set, for details see Kurylo and Smaga (2023)
library(refund)
data(DTI)
# MS patients
DTI_ms <- DTI[DTI$case == 1, ]
miss_data <- c()
for (i in 1:340) if (any(is.na(DTI_ms$cca[i, ]))) miss_data <- c(miss_data, i)
DTI_ms <- DTI_ms[-miss_data, ]
DTI_ms_2 <- DTI_ms[DTI_ms$Nscans == 4, ]
xx <- vector("list", 4)
for (i in 1:4) {
  xx[[i]] <- DTI_ms_2$cca[DTI_ms_2$visit == i, ]
}
xx[[1]] <- xx[[1]][-14, ]
xx[[3]] <- xx[[3]][-14, ]
yy <- xx
for (i in seq_len(4)) yy[[i]] <- yy[[i]][1:17, ]
# data trajectories for four visits
oldpar <- par(mfrow = c(1, 4), mar = c(4, 4, 4, 0.1))
matplot(t(yy[[1]]), type = "l", col = 1, lty = 1, xlab = "t", ylab = "FA",
        main = "Visit 1", xaxt = "n", ylim = c(0.29, 0.73))
axis(1, c(1, 15, 30, 45, 60, 75, 93), labels = c(1, 15, 30, 45, 60, 75, 93))
matplot(t(yy[[2]]), type = "l", col = 1, lty = 1, xlab = "t", ylab = "FA",
        main = "Visit 2", xaxt = "n", ylim = c(0.29, 0.73))
axis(1, c(1, 15, 30, 45, 60, 75, 93), labels = c(1, 15, 30, 45, 60, 75, 93))
matplot(t(yy[[3]]), type = "l", col = 1, lty = 1, xlab = "t", ylab = "FA",
        main = "Visit 3", xaxt = "n", ylim = c(0.29, 0.73))
axis(1, c(1, 15, 30, 45, 60, 75, 93), labels = c(1, 15, 30, 45, 60, 75, 93))
matplot(t(yy[[4]]), type = "l", col = 1, lty = 1, xlab = "t", ylab = "FA",
        main = "Visit 4", xaxt = "n", ylim = c(0.29, 0.73))
axis(1, c(1, 15, 30, 45, 60, 75, 93), labels = c(1, 15, 30, 45, 60, 75, 93))
par(oldpar)
# sample mean functions
oldpar <- par(mfrow = c(1, 1), mar = c(4, 4, 2, 0.1))
pointwise_sample_mean_fun(yy, values = FALSE,
                          col = 1:4, xlab = "t", ylab = "FA", xaxt = "n")
axis(1, c(1, 15, 30, 45, 60, 75, 93), labels = c(1, 15, 30, 45, 60, 75, 93))
legend(x = 36, y = 0.64, legend = 1:4, lty = 1, col = 1:4, title = "Visit")
par(oldpar)
# pointwise SSA and F-type test statistics
oldpar <- par(mfrow = c(1, 2), mar = c(4, 2, 2, 0.1))
pointwise_ssa_test_statistic(yy, xlab = "t", xaxt = "n")
axis(1, c(1, 15, 30, 45, 60, 75, 93), labels = c(1, 15, 30, 45, 60, 75, 93))
pointwise_f_test_statistic(yy, xlab = "t", xaxt = "n")
axis(1, c(1, 15, 30, 45, 60, 75, 93), labels = c(1, 15, 30, 45, 60, 75, 93))
par(oldpar)

# testing without parallel computing and multiple generation of Gaussian processes
res <- rmfanova(yy)
summary(res, digits = 3)
# testing without parallel computing and with multiple generation of Gaussian processes
res <- rmfanova(yy, multi_gen = TRUE)
summary(res, digits = 3)
# testing with parallel computing and without multiple generation of Gaussian processes
res <- rmfanova(yy, parallel = TRUE, n_cores = 2)
summary(res, digits = 3)
# testing with parallel computing and with multiple generation of Gaussian processes
res <- rmfanova(yy, parallel = TRUE, multi_gen = TRUE, n_cores = 2)
summary(res, digits = 3)