Two population local testing procedures

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:

Usage

fdatest2(
  data1,
  data2,
  method,
  mu = 0,
  B = 1000L,
  paired = FALSE,
  dx = NULL,
  alternative = "two.sided",
  recycle = TRUE,
  partition = NULL,
  verbose = TRUE
)

Arguments

data1

First population's data. Either pointwise evaluations of the functional data set on a uniform grid, or a fd object from the package fda. If pointwise evaluations are provided, data2 is a matrix of dimensions c(n1,J), with J evaluations on columns and n1 units on rows.

data2

Second population's data. Either pointwise evaluations of the functional data set on a uniform grid, or a fd object from the package fda. If pointwise evaluations are provided, data2 is a matrix of dimensions c(n1,J), with J evaluations on columns and n2 units on rows.

method

A character string specifying the method chosen to adjust the p-value. Should be one of the following: "Global", "IWT", "TWT", "PCT", "FDR".

mu

Functional mean difference under the null hypothesis. Three possibilities are available for mu:

a constant (in this case, a constant function is used);
a J-dimensional vector containing the evaluations on the same grid which data are evaluated;
a fd object from the package fda containing one function.

Defaults to mu = 0.

B

The number of iterations of the MC algorithm to evaluate the p-values of the permutation tests. Defaults to 1000L.

paired

Flag indicating whether a paired test has to be performed. Defaults to FALSE.

dx

Used only if an fd object is provided. In this case, dx is the size of the discretization step of the gridused to evaluate functional data. If set to NULL, a grid of size 100L is used. Defaults to NULL.

alternative

A character string specifying the alternative hypothesis. Must be one of "two.sided" (default), "greater" or "less".

recycle

Flag used to decide whether the recycled version of the IWT should be used (see Pini and Vantini, 2017 for details). Defaults to TRUE.

partition

Used only if method="PCT". The partition to be used for PCT procedure. Default is NULL.

verbose

Logical: if FALSE, reduces the amount of output. Default is TRUE.

Value

An object of class fdatest2 containing at least the following components:

test: a string vector indicating the type of test performed. In this case equal to "2pop".
mu: evaluation on a grid of the functional mean difference under the null hypothesis (as entered by the user).
unadjusted_pval: evaluation on a grid of the unadjusted p-value function.
adjusted_pval: evaluation on a grid of the adjusted p-value function.
data.eval: evaluation on a grid of the functional data.
ord_labels: vector of labels indicating the group membership of data.eval.

Details

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)

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

# Importing the NASA temperatures data set
data(NASAtemp)

# Performing the TWT for two populations
TWT.result <- fdatest2(
  NASAtemp$paris, NASAtemp$milan, 
  method = "TWT", B = 10L
)
#> 
#> ── Threshold-wise tests ────────────────────────────────────────────────────────

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



# Selecting the significant components at 5% level
which(TWT.result$adjusted_pval < 0.05)
#>   [1]  20  40  49  50  53  57  61  64  65  68  69  70  71  72  73  76  77  86
#>  [19]  87  88  89  90  91  92  93  94  97 100 101 102 103 104 105 106 107 108
#>  [37] 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126
#>  [55] 127 128 129 130 131 132 133 134 135 136 138 139 140 141 142 143 144 145
#>  [73] 148 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167
#>  [91] 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186
#> [109] 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204
#> [127] 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222
#> [145] 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240
#> [163] 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258
#> [181] 259 260 261 262 263 264 266 267 268 269 270 271 272 273 274 275 276 280
#> [199] 281 282 285 286 288 289 290 291 299 304 313 314

# Performing the IWT for two populations
IWT.result <- fdatest2(
  NASAtemp$paris, NASAtemp$milan, 
  method = "IWT", B = 10L
)
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#> ── Creating the p-value matrix: end of row 235 out of 365 ──────────────────────
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#> 
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#> ── Creating the p-value matrix: end of row 348 out of 365 ──────────────────────
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#> 
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#> 
#> ── Creating the p-value matrix: end of row 351 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 352 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 353 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 354 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 355 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 356 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 357 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 358 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 359 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 360 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 361 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 362 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 363 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 364 out of 365 ──────────────────────
#> 
#> ── Creating the p-value matrix: end of row 365 out of 365 ──────────────────────
#> 
#> ── Interval-Wise Testing completed ─────────────────────────────────────────────

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



# Selecting the significant components at 5% level
which(IWT.result$adjusted_pval < 0.05)
#>   [1]  93  94  95  96 101 102 103 104 105 106 107 108 109 110 111 112 113 114
#>  [19] 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
#>  [37] 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150
#>  [55] 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168
#>  [73] 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186
#>  [91] 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204
#> [109] 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222
#> [127] 223 224 225 226 227 228 229 230 231 232 233 234 235 241 242 243 244