Performs a parametric bootstrap of the fixed effects in a `REML`

analysis (C.J. Brien &R.W Payne).

### Options

`PRINT` = string tokens |
Controls printed output (`observedteststatistics` , `pvalues` , `vdiagnostics` , `nnotconverged` , `monitoring` , `all` , `ownstatistics` ); default `obse` , `pval` |
---|---|

`VPRINT` = string tokens |
Controls the output from the `REML` analysis of each sample (`model` , `components` , `effects` , `means` , `stratumvariances` , `monitoring` , `vcovariance` , `deviance` , `Waldtests` , `missingvalues` , `covariancemodels` ); default `*` i.e. none |

`PLOT` = string |
What to plot (`histogram` ); default `*` |

`NBOOT` = scalar |
Number of bootstrap samples to take; default 99 |

`NRETRIES` = scalar |
Maximum number of extra samples to take when some `REML` analyses fail to converge; default `NBOOT` |

`SEED` = scalar |
Seed for random number generation; default 0 continues an existing sequence or, if none, selects a seed automatically |

`METHOD` = string token |
Indicates whether to use the standard Fisher-scoring algorithm or the new AI algorithm with sparse matrix methods (`Fisher` , `AI` ); default `AI` |

`MAXCYCLE` = scalar |
Sets a limit on the number of iterations in the `REML` analyses; default 30 |

`FMETHOD` = string token |
Controls whether and how to calculate F statistics for fixed terms (`automatic` , `none` , `algebraic` , `numerical` ); default `none` |

`WMETHOD` = string token |
Controls which Wald statistics are saved (`add` , `drop` ); default `add` |

`WORKSPACE` = scalar |
Number of blocks of internal memory to be set up for use by the `REML` algorithm |

`OWNMETHOD` = string token |
Type of test required for own statistics (`twosided` , `greaterthan` , `lessthan` ); default `twos` |

`CIPROBABILITY` = scalar |
Probability level for the confidence interval for own statistics; default 0.95 |

### Parameters

`SAVE` = `REML` save structures |
Specifies the (`REML` ) save structure of the original analysis; default `*` uses the `SAVE` structure from the most recent `REML` analysis |
---|---|

`UMEANS` = variates |
Specifies the expected values for the units under the null hypothesis of no effects from the `FIXEDTERMS` |

`UVCOVARIANCE` = symmetric matrices |
Specifies the variances and covariances of the units under the null hypothesis of no effects from the `FIXEDTERMS` |

`FIXEDTERMS` = formula |
Specifies the fixed terms to test; default `*` tests all the fixed terms in the original analysis |

`FSTATISTICS` = pointers |
Saves a pointer with a variate for each of the `FIXEDTERMS` , containing the F statistics from the bootstrap samples |

`PVALUES` = pointers |
Saves a pointer with a scalar for each of the `FIXEDTERMS` , containing the test probability obtained from the position of its F statistic within those from the bootstrap samples |

`NNOTCONVERGED` = scalars |
Saves the number of bootstrap samples whose `REML` analysis failed to converge |

`OWNDATA` = pointers |
Data required to calculate own statistics |

`OWNOBSERVEDVALUES` = variates |
Saves observed values of the own statistics |

`OWNPROBABILITIES` = variates |
Saves bootstrap probabilities for the own statistics |

`OWNESTIMATES` = variates |
Saves boostrap estimates for the own statistics |

`OWNSES` = variates |
Saves boostrap standard errors for the own statistics |

`OWNLOWERCIS` = variates |
Saves boostrap lower values of the confidence intervals for the own statistics |

`OWNUPPERCIS` = variates |
Saves boostrap upper values of the confidence intervals for the own statistics |

`OWNSTATISTICS` = pointers |
Saves the own statistics obtained from the bootstrap samples, in a pointer with a variate for each statistic |

### Description

`VBOOTSTRAP`

performs a parametric bootstrap for fixed effects in a `REML`

analysis. The model to be fitted must be defined using the `VCOMPONENTS`

and `VSTRUCTURE`

directives, in the usual way. The `SAVE`

parameter supplies the save structure from the original analysis; if this is not set, the most recent `REML`

analysis is used.

The bootstrap samples are generated from a multivariate Normal distribution with dimension equal to the number of units in the analysis. The `UMEANS`

parameter supplies the expected values for the distribution, Usually, this contains the fitted values under the null model for the terms being tested. If `UMEANS`

is not set, a variate containing the grand mean of the response is used. The `UVCOVARIANCE`

parameter supplies the variances and covariances of the units. If this is not set, the unit-by-unit variance-covariance matrix from the original analysis is used (see the `UVCOVARIANCE`

option of `VKEEP`

). Note: you can use the `VUVCOVARIANCE`

procedure to form the variance-covariance matrix, if you know the variance components for a `REML`

model that contains no covariance models.

By default all the fixed terms in the original analysis are tested simultaneously. However, you can set the `FIXEDTERMS`

parameter to test a smaller model, and you should then also set `UMEANS`

to specify the expected values under the null model.

The `NBOOT`

option specifies the number of bootstrap samples to take (default 99). The `NRETRIES`

option specifies the maximum number of extra samples to take when some `REML`

analyses fail to converge; the default is to use the same number as specified by `NBOOT`

. The `SEED`

option supplies the seed for the random number generator used to make the permutations; default 0 continues from the previous generation or (if none) initializes the seed automatically. The `NNOTCONVERGED`

parameter can save the number of samples whose analyses did not converge, in a scalar.

The bootstrap p-values are calculated by taking the proportion of F statistics in the bootstrap samples that are larger than the observed F statistic of each fixed term. The `WMETHOD`

option controls whether these statistics are obtained from the table where terms are added sequentially (the default), or from the table where suitable terms are dropped from the full fixed model. Note that, if you use the table where terms are dropped, the only terms that can be tested are those that are not marginal to any other term in the fixed model: for example, the main effect `A`

cannot be tested if the model contains an interaction, such as `A.B`

.

The bootstrap F statistics can be saved, in a pointer with a variate for each of the `FIXEDTERMS`

, using the `FSTATISTICS`

parameter. The p-values can be saved, in a pointer with a scalar for each of the `FIXEDTERMS`

, using the `PVALUES`

parameter. You can obtain a plot of a histogram showing the position of the observed F statistic, compared to those from the bootstrap samples, by setting option `PLOT=histogram`

.

You can define your own statistics to be assessed by the bootstrap. They are calculated by a procedure `VBOOTownstatistics`

, which is called by `VBOOTSTRAP`

following the `REML`

analysis of each bootstrap sample. Its use is shown in the `VBOOTSTRAP`

example, which can be modified to calculate your own statistics instead. The information required by `VBOOTownstatistics`

to do the calculations is supplied, in a pointer, by the `OWNDATA`

parameter. The `OWNMETHOD`

option specifies the type of test to be made. The default, twosided tests whether the statistics differ from zero. The greaterthan setting tests whether they are greater than zero, and the lessthan setting tests whether they are less than zero. Bootstrap estimates, standard errors and confidence intervals are also calculated, The `CIPROBABILITY`

option specifies the probability for the confidence intervals (default 0.95). The `OWNOBSERVEDVALUES`

parameter can save a variate containing the values of the own statistics from the original data set. The `OWNPROBABILITIES`

can save a variate containing the probabilities from the tests. The `OWNESTIMATES`

can save a variate containing the bootstrap estimates of the statistics (calculated as the mean of the values obtained from the bootstrap samples) The `OWNSES`

can save a variate containing standard errors of bootstrap estimates. The `OWNLOWERCIS`

and `OWNUPPERCIS`

parameters can save variates containing the lower and upper values, respectively, of the confidence intervals. Finally, the `OWNSTATISTICS`

can save the values of the own statistics obtained from the bootstrap samples, in a pointer with a variate for each statistic.

Printed output is controlled by the `PRINT`

option, with settings:

`observedteststatistics` |
to print the values of the observed Wald or F statistics for the fixed terms in the original `REML` analysis, |
---|---|

`pvalues` |
to print the bootstrap p-values of the observed Wald or F statistics for the fixed terms, |

`vdiagnostics` |
to print the diagnostics from the `REML` analyses performed on the bootstrap samples, |

`nnotconverged` |
to print the number of samples whose analyses did not converge, |

`monitoring` |
to print the progress of the bootstrapping, |

`ownstatistics` |
to print the estimates, standard errors and confidence intervals for the own statistics, and |

`all` |
to print all the information other than the own statistics. |

By default, the observed statistics and the p-values are printed.

The `VPRINT`

option controls the output from the `REML`

analyses of the bootstrap samples, with the same settings as the `PRINT`

option of `REML`

. By default, nothing is printed.

The `MAXCYCLE`

option sets a limit on the number of iterations in the `REML`

analyses (default 30). The `METHOD`

option controls whether `REML`

uses the standard Fisher-scoring algorithm, or the new AI algorithm with sparse matrix methods (the default). The `FMETHOD`

option controls whether and how to calculate F statistics for fixed terms; the default is not to calculate the statistics. (This is relevant if tests for fixed effects are being printed in the `REML`

analyses of the bootstrap samples.) The `WORKSPACE`

option specifies the number of blocks of internal memory to be set up for use by the `REML`

algorithm; the default is to use the same value as in the original `REML`

analysis.

Options: `PRINT`

, `VPRINT`

, `PLOT`

, `NBOOT`

, `NRETRIES`

, `SEED`

, `METHOD`

, `MAXCYCLE`

, `FMETHOD`

, `WMETHOD`

, `WORKSPACE`

, `OWNMETHOD`

, `CIPROBABILITY`

.

Parameters: `SAVE`

, `UMEANS`

, `UVCOVARIANCE`

, `FIXEDTERMS`

, `FSTATISTICS`

, `PVALUES`

, `NNOTCONVERGED`

, `OWNDATA`

, `OWNOBSERVEDVALUES`

, `OWNPROBABILITIES`

, `OWNESTIMATES`

, `OWNSES`

, `OWNLOWERCIS`

, `OWNUPPERCIS`

, `OWNSTATISTICS`

.

### See also

Directives: `REML`

, `VCOMPONENTS`

.

Procedures: `BOOTSTRAP`

, `VCRITICAL`

, `VFLC`

, `VPERMTEST`

, `VRPERMTEST`

. `VUVCOVARIANCE`

.

Commands for: REML analysis of linear mixed models.

### Example

CAPTION 'VBOOTSTRAP example',!t('Split plot design, see the',\ 'Guide to Genstat, Part 2, Section 4.2.1.'); STYLE=meta,plain SPLOAD [PRINT=*] '%gendir%/data/Oats.gsh' " Fit a model with no interaction, and get the fitted values." VCOMPONENTS [FIXED=variety+nitrogen]\ RANDOM=blocks/wplots/subplots REML yield; FITTED=fit " Fit full model to get variances & covariances of the units." VCOMPONENTS [FIXED=variety*nitrogen]\ RANDOM=blocks/wplots/subplots REML [PRINT=model,comp,Wald] yield; SAVE=fullfixed VKEEP [UVCOVARIANCE=V] " Parameteric bootstrap to test the interaction." VBOOTSTRAP [PLOT=histogram; NBOOT=999; SEED=265600] SAVE=fullfixed;\ UMEANS=fit; UVCOVARIANCE=V; FIXEDTERMS=!f(variety.nitrogen)