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Two population Global Testing procedure

Source: R/ts-global.R
Global2.Rd

The function implements the Global Testing procedure for testing mean differences between two functional populations. Functional data are tested globally and a global p-value and a constant adjusted p-value function are provided. The adjusted p-value function controls the family-wise error rate weakly.

Usage

global2(
  data1,
  data2,
  mu = 0,
  dx = NULL,
  n_perm = 1000L,
  paired = FALSE,
  alternative = c("two.sided", "less", "greater"),
  standardize = FALSE,
  verbose = FALSE,
  aggregation_strategy = c("integral", "max")
)

Global2(
  data1,
  data2,
  mu = 0,
  dx = NULL,
  B = 1000L,
  paired = FALSE,
  alternative = c("two.sided", "less", "greater"),
  statistic = c("Integral", "Max", "Integral_std", "Max_std"),
  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 \(J\).

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 step of the uniform grid on which the data are evaluated. If NULL, the step is automatically inferred from the data. Defaults to NULL.

n_perm

An integer value specifying the number of permutations to use for the local testing procedure. 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".

standardize

A boolean value specifying whether to standardize the test statistic. Defaults to FALSE.

verbose

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

aggregation_strategy

A string specifying the strategy to aggregate the point-wise test statistics for the correction procedure. Possible values are "integral" and "max". Defaults to "integral".

B

An integer value specifying the number of permutations to use for the local testing procedure. Defaults to 1000L.

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".

Value

An object of class fts 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.

  • correction_method: A string containing the correction method used to compute the adjusted p-value function.

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

  • Hall, Peter, and Nader Tajvidi. 2002. “Permutation Tests for Equality of Distributions in High-Dimensional Settings.” Biometrika 89 (2): 359–74.

  • Pini, Alessia, Aymeric Stamm, and Simone Vantini. 2018. “Hotelling’s T2 in Separable Hilbert Spaces.” Journal of Multivariate Analysis 167: 284–305.

See also

iwt2(), twt2(), pct2(), fdr2() for calling directly one of the other tests, functional_two_sample_test() for calling the interface test and plot.fts() for plotting the results.

Examples

# Performing the Global for two populations
Global_result <- global2(NASAtemp$paris, NASAtemp$milan)

# Plotting the results of the Global
plot(
  Global_result,
  xrange = c(0, 12),
  title = "Global results for testing mean differences"
)


# Selecting the significant components at 5% level
which(Global_result$adjusted_pvalues < 0.05)
#>   [1]   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18
#>  [19]  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36
#>  [37]  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54
#>  [55]  55  56  57  58  59  60  61  62  63  64  65  66  67  68  69  70  71  72
#>  [73]  73  74  75  76  77  78  79  80  81  82  83  84  85  86  87  88  89  90
#>  [91]  91  92  93  94  95  96  97  98  99 100 101 102 103 104 105 106 107 108
#> [109] 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126
#> [127] 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144
#> [145] 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162
#> [163] 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180
#> [181] 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198
#> [199] 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216
#> [217] 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234
#> [235] 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252
#> [253] 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270
#> [271] 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288
#> [289] 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306
#> [307] 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324
#> [325] 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342
#> [343] 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360
#> [361] 361 362 363 364 365