VFLC procedure

Performs an F-test of random effects in a linear mixed model based on linear combinations of the responses, i.e. an FLC test (V.M. Cave).

PRINT = string tokens	Controls printed output (summary, monitoring); default summ
PLOT = string tokens	What graphs to plot for the bootstrap and fast double bootstrap FLC tests (kerneldensity, histogram); default * i.e. none
TEST = string tokens	Type(s) of test to perform; (flc, bootstrap, fastdoublebootstrap); default flc
NBOOT = scalar	Number of bootstrap samples to take; default 99
SEED = scalar	Seed for random number generation; default 0 continues an existing sequence or, if none, selects a seed automatically
WINDOW = scalar	Window to use for the graphs; default 3
SAVE = REML save structure	Specifies the save structure of the original analysis; default is to use the save structure from the most recent REML analysis

Parameters

TERMS = formula	Random terms to test
STATISTIC = scalar	Saves the FLC test statistic
BOOTSTATISTICS = variate	Saves the FLC test statistics from the original data set (i.e. the observed FLC test statistic), and then the bootstrap samples
FASTDOUBLE = pointer	Pointer to scalars and variates to save the first-level bootstrap probability value and FLC test statistics, and the second-level fast double bootstrap FLC test statistics and resulting critical value
PROBABILITIES = pointer	Pointer to scalar(s) to save the probability value(s) from the test(s)
TITLE = text	Title for the graphs

Description

The VFLC procedure performs an FLC test to assess whether random terms can be dropped from a linear mixed model, that has been fitted by REML. The FLC test is an F-test based on linear combinations of the responses. VFLC offers the standard FLC test as well its bootstrapped and fast double bootstrapped counterparts.

The original linear mixed model must be fitted using the REML, VCOMPONENTS and VSTRUCTURE directives, in the usual way. The random effects may be correlated, but the model must not contain any spline terms. The SAVE option supplies the save structure from the original analysis; if this is not set, the most recent REML analysis is used. The random term(s) to drop from the original model are defined by a model formula supplied by the TERMS parameter.

The types of FLC test to be performed are specified by the TEST option, with settings flc, bootstrap and fastdoublebootstrap. The default is to use the standard FLC test. For the bootstrap and fast double bootstrap FLC tests, the NBOOT option specifies the number of bootstrap samples to take (default 99), and the SEED option supplies the seed for the random number generator used to generate the bootstrap samples. The default SEED of zero continues the sequence of random numbers from a previous generation or, if this is the first use of the generator in this run of Genstat, it initializes the seed automatically. If you use the same (non-zero) seed more than once, you will get the same random numbers, and hence the same bootstrap samples.

Printed output is controlled by the PRINT option, with the following settings.

summary prints a summary of the test results. For the standard FLC test, this is a table giving the test statistic (i.e. an F-value), its degrees of freedom and corresponding probability value. For the bootstrap and fast double bootstrap FLC tests, this is a table giving the number of bootstrap samples, the seed, the test statistic (i.e. the observed F-value) and the corresponding probability value.

monitoring prints monitoring information, showing the progress of the bootstrapping.
The default is to print the summary.

The PLOT option controls the graphical output from the bootstrap and fast double bootstrap FLC tests, with settings:

histogram to plot a histogram of the bootstrap FLC test statistics, and

kerneldensity to produce a kernel density plot of the bootstrap FLC test statistics.

By default, nothing is plotted. If TEST=bootstrap, the observed FLC test statistic is included in the set of bootstrap FLC test statistics that are plotted. In addition, a reference line is added to indicate where it sits compared to those from the bootstrap samples. Conversely, if TEST=fastdoublebootstrap, the observed FLC test statistic is not included in the set of bootstrap FLC test statistics plotted, and the reference line indicates where the estimated fast double bootstrap critical value, QB, falls. The WINDOW option defines the window to use for the plots; default 3. The TITLE parameter can supply a title for the plots.

Results can be saved using the STATISTIC, BOOTSTATISTICS, FASTDOUBLE and PROBABILITIES parameters. The STATISTIC parameter saves the FLC test statistic in a scalar. The BOOTSTATISTICS parameter saves the bootstrap FLC statistics in a variate, whose first value is the test statistic from the original data set (i.e. the observed FLC test statistic). The FASTDOUBLE parameter saves the results from the fast double bootstrap FLC test in a pointer. The first element of the pointer, labelled ‘B_FLC pr.‘, is a scalar storing the first-level bootstrap probability value. The second element, labelled ‘B_FLC F‘, is a variate storing the first-level bootstrap FLC test statistics. The third element, labelled ‘FDB_FLC F‘, is a variate storing the second-level fast double bootstrap FLC test statistics. The fourth element, labelled ‘QB‘, is a scalar the storing the critical value from the fast double bootstrap FLC test.

Options: PRINT, PLOT, TEST, NBOOT, SEED, WINDOW, SAVE.
Parameters: TERMS, STATISTIC, BOOTSTATISTICS, FASTDOUBLE, PROBABILITIES, TITLE.

Method

VFLC uses the methods described in Hui et al. (2019) and O’Shaughnessy et al. (2018).

Action with RESTRICT

The REML analysis may be restricted in the usual way.

References

Hui, F.K.C., Müller, S., & Welsh, A.H. (2019). Testing random effects in linear mixed models: another look at the F-test. Australia & New Zealand Journal of Statistics, 61, 61-84.

O’Shaughnessy, P.Y., Hui, F.K.C., Müller, S., & Welsh, A.H. (2018). Bootstrapping F-test for random effects in linear mixed models. arXiv:1812.03428.

See also

Directives: REML, VCOMPONENTS, VSTRUCTURE
Procedures: VBOOTSTRAP, VRPERMTEST.
Commands for: REML analysis of linear mixed models.

Example

CAPTION     'VFLC example',\
            !T('Random coefficient regression: An experiment to study',\
               'the effect of drugs on the growth rates of rats.'),\
            !T('Guide to REML in Genstat, Section 4.3.'); \
            STYLE=meta,plain,plain
SPLOAD      [PRINT=*] '%gendir%/data/Boxrat.gsh'
"Full model:  Linear plus quadratic regression coefficients"
CALC        timesq = time*time
VCOMPONENTS [FIXED=drug*(time+timesq)] RANDOM=rat/(time+timesq)
VSTRUCTURE  [TERMS=rat/(time+timesq); CORRELATE=unrest; FORM=whole]
REML        [PRINT=model,components,waldTests] weight
"Assessing whether the quadratic effect of time is heterogeneous between rats"
VFLC        [PLOT=histogram; TEST=flc,bootstrap,fastdouble; SEED=2231225] \
            TERMS=rat.timesq