Local testing procedures for the functional two-sample test

The function implements local testing procedures for testing mean differences between two functional populations. Functional data are tested locally and unadjusted and adjusted p-value functions are provided. The unadjusted p-value function controls the point-wise error rate. The adjusted p-value function can be computed according to the following methods:

global testing (controlling the FWER weakly)
interval-wise testing (controlling the interval-wise error rate)
threshold-wise testing (controlling the FWER asymptotically)
partition closed testing (controlling the FWER on a partition)
functional Benjamini Hochberg (controlling the FDR)

Usage

functional_two_sample_test(
  data1,
  data2,
  correction,
  mu = 0,
  dx = NULL,
  B = 1000L,
  paired = FALSE,
  alternative = c("two.sided", "less", "greater"),
  statistic = c("Integral", "Max", "Integral_std", "Max_std"),
  recycle = TRUE,
  partition = NULL,
  verbose = FALSE
)

Arguments

data1

Either a numeric matrix or an object of class fda::fd specifying the data in the first sample. If the data is provided within a matrix, it should be of shape \(n_1 \times J\) and it should contain in each row one of the \(n_1\) functions in the sample and in columns the evaluation of each function on a same uniform grid of size \(J\).

data2

Either a numeric matrix or an object of class fda::fd specifying the data in the second sample. If the data is provided within a matrix, it should be of shape \(n_2 \times J\) and it should contain in each row one of the \(n_2\) functions in the sample and in columns the evaluation of each function on a same uniform grid of size \(J\).

correction

A string specifying the correction method to perform the local functional testing procedure and adjust the p-value function. Choices are "Global, "IWT", "TWT", "PCT" or "FDR".

mu

Either a numeric value or a numeric vector or an object of class fda::fd specifying the functional mean difference under the null hypothesis. If mu is a constant, then a constant function is used. If mu is a numeric vector, it must correspond to evaluation of the mean difference function on the same grid that has been used to evaluate the data samples. Defaults to 0.

dx

A numeric value specifying the discretization step of the grid used to evaluate functional data when it is provided as objects of class fda::fd. Defaults to NULL, in which case a default value of 0.01 is used which corresponds to a grid of size 100L. Unused if functional data is provided in the form of matrices.

B

An integer value specifying the number of iterations of the MC algorithm to evaluate the p-value of the permutation tests. Defaults to 1000L.

paired

A boolean value specifying whether a paired test should be performed. Defaults to FALSE.

alternative

A string specifying the type of alternative hypothesis. Choices are "two.sided", "less" or "greater". Defaults to "two.sided".

statistic

A string specifying the test statistic to use. Possible values are:

"Integral": Integral of the squared sample mean difference.
"Max": Maximum of the squared sample mean difference.
"Integral_std": Integral of the squared t-test statistic.
"Max_std": Maximum of the squared t-test statistic.

Defaults to "Integral".

recycle

A boolean value specifying whether the recycled version of the interval-wise testing procedure should be used. See Pini and Vantini (2017) for details. Defaults to TRUE.

partition

An integer vector of length \(J\) specifying the membership of each point of the domain to an element of the partition. Only used and must be set if the correction argument is set to "PCT".

verbose

A boolean value specifying whether to print the progress of the computation. Defaults to FALSE.

Value

An object of class ftwosample containing the following components:

data: A numeric matrix of shape \(n \times J\) containing the evaluation of the \(n = n_1 + n_2\) functions on a common uniform grid of size \(p\).
group_labels: An integer vector of size \(n = n_1 + n_2\) containing the group membership of each function.
mu: A numeric vector of shape \(J\) containing the evaluation of the functional mean difference under the null hypothesis on the same uniform grid used to evaluate the functional samples.
unadjusted_pvalues: A numeric vector of size \(J\) containing the evaluation of the unadjusted p-value function on the same uniform grid used to evaluate the functional samples.
adjusted_pvalues: A numeric vector of size \(J\) containing the evaluation of the adjusted p-value functione on the same uniform grid used to evaluate the functional samples.

Optionally, the list may contain the following components:

global_pvalue: A numeric value containing the global p-value. Only present if the correction argument is set to "Global".
pvalue_matrix: A numeric matrix of shape \(p \times p\) containing the p-values of the interval-wise tests. Element \(i, j\) contains the p-value of the test performed on the interval indexed by \(j, j+1 , \dots, j+(p-i)\). Only present if the correction argument is set to "IWT".

References

Abramowicz, K., Pini, A., Schelin, L., Stamm, A., & Vantini, S. (2022). “Domain selection and familywise error rate for functional data: A unified framework. Biometrics 79(2), 1119-1132.

Pini, A., & Vantini, S. (2017). Interval-wise testing for functional data. Journal of Nonparametric Statistics, 29(2), 407-424

A. Pini and S. Vantini (2017). The Interval Testing Procedure: Inference for Functional Data Controlling the Family Wise Error Rate on Intervals. Biometrics 73(3): 835–845.

Lundtorp Olsen, N., Pini, A., & Vantini, S. (2021). False discovery rate for functional data TEST 30, 784–809.

Examples

# Performing the TWT for two populations
TWT_result <- functional_two_sample_test(
  NASAtemp$paris, NASAtemp$milan,
  correction = "TWT", B = 10L
)

# Plotting the results of the TWT
plot(
  TWT_result,
  xrange = c(0, 12),
  title = 'TWT results for testing mean differences'
)


# Selecting the significant components at 5% level
which(TWT_result$adjusted_pval < 0.05)
#>   [1]  29  49  50  53  61  65  66  68  69  70  71  72  73  81  82  83  86  87
#>  [19]  88  89  91  92  93  94  95  96  97  98  99 100 101 102 103 104 105 106
#>  [37] 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124
#>  [55] 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142
#>  [73] 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160
#>  [91] 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178
#> [109] 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196
#> [127] 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214
#> [145] 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232
#> [163] 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250
#> [181] 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 269
#> [199] 270 272 273 274 275 276 281 285 286 287 288 289 290 291 295 298 299 304
#> [217] 321 328 364

# Performing the IWT for two populations
IWT_result <- functional_two_sample_test(
  NASAtemp$paris, NASAtemp$milan,
  correction = "IWT", B = 10L
)

# Plotting the results of the IWT
plot(
  IWT_result,
  xrange = c(0, 12),
  title = 'IWT results for testing mean differences'
)


# Selecting the significant components at 5% level
which(IWT_result$adjusted_pval < 0.05)
#>   [1] 101 102 103 104 105 106 111 112 113 114 115 116 117 118 119 120 121 122
#>  [19] 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140
#>  [37] 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158
#>  [55] 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176
#>  [73] 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194
#>  [91] 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212
#> [109] 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230
#> [127] 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248
#> [145] 249 250 251 252 253 254 255 256 257 258 259 260 261 265 266 267 268 269
#> [163] 270 271 272 273 274 275 276 277 278 281

Usage

Arguments

Value

References

See also

Examples