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# ANTORDER procedure

Assesses order of ante-dependence for repeated measures data (M.S. Ridout & R.W. Payne).

### Options

`TREATMENTSTRUCTURE` = formula Treatment formula for the model at each time; if this is not set, the default is taken from the setting (which must already have been defined) of the `TREATMENTSTRUCTURE` directive Block formula for the model at each time; if this is not set, the default is taken from any existing setting specified by the `BLOCKSTRUCTURE` directive and if neither has been set the design is assumed to be unstratified (i.e. to have a single error term) Maximum order against which to test; default is maximum possible order Limit on the number of factors in a treatment term Indicates the time of each observation when there is a single `DATA` variate

### Parameter

`DATA` = variates Data observations either in a list of variates (one for each time), or a single variate (with `TIME` set to a factor indicating the time of each observation)

### Description

A repeated measures experiment is one in which the same set of units, or subjects, is observed at a sequence of times to investigate treatment effects over a period of time. The set of variates observed at the successive times is said to have an ante-dependence structure of order r if each ith variate (i>r), given the preceding r, is independent of all further preceding variates (Gabriel 1961, 1962). Procedure `ANTORDER` calculates statistics to assist in the selection of an appropriate order of ante-dependence structure for sets of repeated measures data, using the method of Kenward (1987). Once the order of ante-dependence structure has been established, the individual variates can be analysed individually by analysis of covariance, adjusting for the r previous variates, to assess the times at which treatment effects occurred. Also, procedure `ANTTEST` can be used to perform overall tests of treatment effects.

The model for the analysis is specified by options of the procedure. `TREATMENTSTRUCTURE` specifies a model formula to define the treatment terms in the analysis; if this is unset, `ANTORDER` will use the model already defined by the `TREATMENTSTRUCTURE` directive, or will fail if that too has not been set. `BLOCKSTRUCTURE` defines the underlying structure of the design, and `ANTORDER` will use the model (if any) previously defined by the `BLOCKSTRUCTURE` directive if this is not set; these can both be omitted if there is only one error term (i.e. if the design is unstratified). Option `MAXORDER` specifies the maximum order of ante-dependence structure to be tested; by default, this is taken as the maximum possible order (the smaller of the number of times minus one or the number of residual degrees at each time; see Kenward 1987). The `FACTORIAL` option can be used to set a limit on the number of factors in the terms generated from the treatment formula.

The data are specified by the `DATA` parameter in one of two ways. The first is to supply a list of variates, each one containing the measurements made on the subjects at one of the successive occasions on which they were observed.

The second possibility is to supply a single `DATA` variate containing the data from all the times. The `TIME` option must then be set to a factor indicating the time of each observation. The block and treatment factors must be defined to match the `DATA` variate, and each subject should be represented by a unique combination of the block factors. If not, Genstat prints a warning and assumes that the subjects occur in the same order within each time.

The data may contain missing values but these should represent “dropouts”: that is, once subjects start to record missing values, their observations should continue to be missing at all subsequent times.

Options: `TREATMENTSTRUCTURE`, `BLOCKSTRUCTURE`, `MAXORDER`, `FACTORIAL`, `TIME`.

Parameter: `DATA`.

### Method

The procedure uses the method of Kenward (1987) to calculate the statistics using residual sums of squares from analysis of covariance. For further details of ante-dependence see Gabriel (1961, 1962).

### Action with `RESTRICT`

Any restriction on the `DATA` variates will be applied to all of them.

### References

Gabriel, K.R. (1961). The model of ante-dependence for data of biological growth. Bulletin Institut International Statistique (Paris), 39, 253-264, (33rd session).

Gabriel, K.R. (1962). Ante-dependence analysis of an ordered set of variables. Annals of Mathematical Statistics, 33, 201-212.

Kenward, M.G. (1987). A method for comparing profiles of repeated measurements, Applied Statistics, 36, 296-308.

Directive: `VSTRUCTURE`.

Procedures: `ANTTEST`, `ANTMVESTIMATE`.

Commands for: Repeated measurements.

### Example

```CAPTION 'ANTORDER example',\
'Data from Kenward (1987, Appl. Statist., 36, 296-308).';\
STYLE=meta,plain
POINTER [NVALUES=11] Weight
VARIATE [NVALUES=60] Weight[1...11]
FACTOR  [LEVELS=2; VALUES=30(1,2)] Treatmnt
210 215 230 244 259 266 277 292 292 290 264
230 240 258 277 277 293 300 323 327 340 343
226 233 248 277 297 313 322 340 354 365 362
233 239 253 277 292 310 318 333 336 353 338
238 241 262 282 300 314 319 331 338 348 338
225 228 237 261 271 288 300 316 319 333 330
224 225 239 257 268 290 304 313 310 318 318
237 241 255 276 293 307 312 336 336 344 328
237 224 234 239 256 266 276 300 302 293 269
233 239 259 283 294 313 320 347 348 362 352
217 222 235 256 267 285 295 317 315 308 301
228 223 246 266 277 287 300 312 308 328 333
241 247 268 290 309 323 336 348 359 372 370
221 221 240 253 273 282 292 307 306 317 318
217 220 235 259 262 276 284 305 303 315 317
214 221 237 256 271 283 287 314 316 320 298
224 231 241 256 265 283 295 314 313 328 334
200 203 221 236 248 262 276 294 291 311 310
238 232 252 268 285 298 303 320 324 320 327
230 222 243 253 268 284 290 316 314 330 330
217 224 242 265 284 302 309 324 328 338 334
209 209 221 238 256 267 281 295 301 309 289
224 227 245 267 279 294 312 328 329 297 297
230 231 244 261 272 283 294 318 320 333 338
216 218 223 243 259 270 270 290 301 314 297
231 239 254 276 294 304 317 335 333 319 307
207 216 228 255 275 285 296 314 319 330 330
227 236 251 264 276 287 297 315 309 313 294
221 232 251 274 284 295 300 323 319 333 322
233 238 254 266 282 294 295 310 320 327 326
233 224 245 258 271 287 287 287 290 293 297
231 238 260 273 290 300 311 313 317 321 326
232 237 245 265 285 298 304 319 317 334 329
239 246 268 288 308 309 327 324 327 336 341
215 216 239 264 282 299 307 321 328 332 337
236 226 242 255 263 277 290 299 300 308 310
219 229 246 265 279 292 299 299 298 300 290
231 245 270 292 302 321 322 334 323 337 337
230 228 243 255 272 276 277 289 289 300 303
232 240 247 263 275 286 294 302 308 319 326
234 237 259 289 311 324 342 347 355 368 368
237 235 258 263 282 304 318 327 336 349 353
229 234 254 276 294 315 323 341 346 352 357
220 227 248 273 290 308 322 326 330 342 343
232 241 255 276 293 309 310 330 326 329 330
210 225 242 260 272 277 273 295 292 305 306
229 241 252 265 274 285 303 308 315 328 328
204 198 217 233 251 258 272 283 279 295 298
220 221 236 260 274 295 300 301 310 318 316
233 234 250 268 280 298 308 319 318 336 333
234 234 254 274 294 306 318 334 343 349 350
200 207 217 238 252 267 284 282 282 284 288
220 213 229 252 254 273 293 289 294 292 298
225 239 254 269 289 308 313 324 327 347 344
236 245 257 271 294 307 317 327 328 328 325
231 231 237 261 274 285 291 301 307 315 320
208 211 238 254 267 287 306 312 320 337 338
232 248 261 285 292 307 312 323 318 328 329
233 241 252 273 301 316 332 336 339 348 345
221 219 231 251 270 272 287 294 292 292 299 :
ANTORDER [TREATMENTSTRUCTURE=Treatmnt; MAXORDER=10] Weight[]
```
Updated on March 11, 2019