VCRITICAL procedure

Uses a parametric bootstrap to estimate critical values for a fixed term in a REML analysis (R.W. Payne & C.J. Brien).

Options

PRINT = string tokens	Prints the critical values (critical, fcritical, tcritical, wcritical); default crit, fcri, tcri, wcri
VPRINT = string tokens	Controls the output from the REML analyses (model, components, effects, means, stratumvariances, monitoring, vcovariance, deviance, Waldtests, missingvalues, covariancemodels); default * i.e. none
TERM = formula	Fixed term to be tested
UMEANS = variate	Specifies the expected values for the units under the null hypothesis of no effects from the TERM; default is to use the constant from the SAVE structure
UVCOVARIANCE = symmetric matrix	Specifies the variances and covariances of the units under the null hypothesis of no effects from the TERM; default is to take this from the SAVE structure
WCRITICAL = variate	Saves the critical values of the Wald statistic
FCRITICAL = variate	Saves the critical values of the F statistic
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
PROBABILITIES = scalar or variate	Significance levels for which critical values are required; default 0.05
METHOD = string token	Indicates whether to use the Fisher-scoring algorithm or the 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 how to calculate estimated denominator degrees of freedom when these are to be saved (automatic, none, algebraic, numerical); default auto
WMETHOD = string token	Controls which Wald statistics are saved (add, drop); default add
TMETHOD = string token	Type of test to be made for the contrasts (twosided, greaterthan, lessthan, equivalence, noninferiority); default twos
WALD = variate	Saves the Wald statistics from the samples
FSTATISTIC = variate	Saves the F statistics from the samples
NDF = scalar	Saves the numerator degrees of freedom for the Wald and F statistics
DDF = variate	Saves the estimated denominator degrees of freedom for the F statistics
NNOTCONVERGED = scalar	Saves the number of bootstrap samples whose REML analysis failed to converge
WORKSPACE = scalar	Number of blocks of internal memory to be set up for use by the REML algorithm
SAVE = vsave	REML save structure to provide the information about the analysis

Parameters

XCONTRASTS = variates or tables	X-variate defining a contrast to be detected
CONTRASTTYPE = string tokens	Type of contrast (regression, comparison) default rege
ESTIMATE = variates	Saves the estimated values of the contrasts from the samples
SE = variates	Saves the standard errors for the estimates of the contrasts from the samples
CRITICAL = variates	Saves the critical values for the contrasts
TCRITICAL = variates	Saves the critical values for the t-statistics of the contrasts

Description

The conventional way to assess fixed terms in a REML analysis is to use either the Wald or the F tests, in the table of tests for fixed effects that is produced by setting option PRINT=wald in either REML or VDISPLAY. The Wald tests have the disadvantage of being biased, i.e. they tend to generate significant results too frequently. The F tests are more reliable. However, their denominator degrees of freedom need to be estimated, using the method of Kenward & Roger (1997), and this may not be feasible for some data sets. These denominator degrees of freedom can also be used in t-tests to assess contrasts amongst the effects of a term; see procedure VTCOMPARISONS. However, those tests must be used with caution, as the degrees of freedom are relevant for assessing the fixed term as a whole, and may differ over the various contrasts.

VCRITICAL provides an alternative method of assessment, that may be useful if the decision from the conventional tests is not clear-cut, or if contrasts are to be assessed. It uses a parametric bootstrap, in the same way as the VBOOTSTRAP procedure. However, it differs from VBOOTSTRAP, in that it generates critical values, rather than assessing the significance of terms in a specific data set. These critical values can be used test hypotheses with a specific data set, and the critical values for the F, Wald and t-statistics may be useful with similar data sets. The critical values for the t-statistics also allow you to determine the size of the contrast that may be detectable in these investigations.

The model to be fitted must be defined using the VCOMPONENTS and VSTRUCTURE directives, in the usual way. The bootstrap samples are generated from a multivariate Normal distribution with dimension equal to the number of units in the analysis. The UMEANS option supplies the expected values for the distribution. This should contain the fitted values under the null model for the term being tested. The UVCOVARIANCE option supplies the variances and covariances of the units. If either UMEANS or UVCOVARIANCE is not specified, defaults are taken from the REML analysis supplied by the SAVE option, or from the most recent REML if SAVE is not set. For UMEANS the default is a variate containing the constant estimated in that analysis. For UVCOVARIANCE it is the unit-by-unit variance-covariance matrix from the analysis (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.

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 form the samples; default 0 continues from the previous generation or (if none) initializes the seed automatically. The NNOTCONVERGED option can save the number of samples whose analyses did not converge, in a scalar.

The fixed term to be assessed is specified by the TERM option. If the term is a main effect (i.e. if TERM contains just one factor) you can use the XCONTRASTS parameter to specify variates or tables containing the coefficients defining the contrasts amongst the effects of the term. The CONTRASTTYPE option indicates whether each of these is a regression contrast (as specified in analysis of variance by the REG function) or a comparison (as specified by the COMPARISON function).

The TMETHOD option specifies the type of test that is to be used to assess the contrasts, with the following settings.

twosided	assumes a two-sided test to assess whether the contrast differs from zero (default).
lessthan	assumes a one-sided test to assess whether the contrast is less than zero.
greaterthan	assumes a one-sided test to assess whether the contrast is greater than zero.
noninferiority	assumes a test to check that the contrast is not significantly less then zero. (See Method for more details.)
equivalence	assumes a one-sided test to check that the contrast does not differ significantly from zero; see Method for more details.

The PROBABILITIES option specifies the significance levels for which you want to obtain critical values; the default is 0.05, i.e. 5%.

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

critical	prints critical values for the contrasts,
fcritical	prints critical values for the F statistics,
tcritical	prints critical values for the t-statistics of the contrasts,
wcritical	prints critical values for the Wald statistics,
nnotconverged	prints the number of bootstrap samples whose analysis failed to converge, and
monitoring	prints monitoring information, showing the progress of the bootstrap sampling.

By default, all the critical values printed.

The critical values for the contrasts and their t-statistics can be saved, in variates, by the CRITICAL and TCRITICAL parameters, respectively. The critical values for the F and Wald statistics can be saved, again in variates by the FCRITICAL and WCRITICAL options.

You can also save the values estimated for the various statistics, in the analyses of the bootstrap samples, in variates (with a unit for each sample). Those for the contrasts and their standard errors can be saved the ESTIMATES and SE parameters, respectively. The F and Wald statistics can be saved by the FSTATISTIC and WALD options. The degrees of freedom for the Wald statistics and numerator degrees for the F statistics can be saved, in a scalar, using the NDF option. The estimated denominator degrees of freedom for the F tests can be saved, in a variate, using the DDF option.

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 Fisher-scoring algorithm, or the AI algorithm with sparse matrix methods (the default). The WMETHOD option controls whether the Wald and F 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 TERM must not be 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 FMETHOD option controls how to estimate the denominator degrees of freedom for the F tests. (This is relevant if tests for fixed effects are being printed in the REML analyses of the bootstrap samples, or if the DDF option is set.) 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 SAVE structure, if SAVE has been set. Otherwise, it uses the value from the most recent REML analysis, or the standard REML default if there has been no analysis.

Options: PRINT, VPRINT, TERM, UMEANS, UVCOVARIANCE, WCRITICAL, FCRITICAL, NBOOT, NRETRIES, SEED, PROBABILITIES, METHOD, MAXCYCLE, FMETHOD, WMETHOD, TMETHOD, WALD, FSTATISTIC, NDF, DDF, NNOTCONVERGED, WORKSPACE, SAVE.

Parameters: XCONTRASTS, CONTRASTTYPE, ESTIMATE, SE, CRITICAL, TCRITICAL.

Method

The critical values are calculated by taking appropriate quantiles of the statistics obtained from the bootstrap samples. For the Wald and F statistics, and the “greater-than” tests of the contrasts or their t-statistics, this is the quantile for one minus the probability. For the “less-than” tests of the contrasts or their t-statistics, it is the quantile for the probability. For the two-sided tests, the quantiles are taken over the absolute values of the contrasts and their t-statistics, and are for one minus the probability.

With an equivalence test, you define a threshold h below which two treatments can be assumed to be equivalent. The contrast c would be the difference between the treatments, and the null hypothesis that the treatments are not equivalent is that either

c ≤ –t

or

c ≥ t

with the alternative hypothesis that they are equivalent, i.e.

–t < c < t

This defines an intersection-union test, in which each component of the null hypothesis must be rejected separately. This implies performing two one-sided t-tests (this is known as a TOST procedure). If the significance level for the full test is to be α, each t-test must have significance level α (see Berger & Hsu 1996). The critical values are thus given by quantiles that are taken over the absolute values of the contrasts and their t-statistics, and are for one minus twice the probability. The hypothesis that the treatments are equivalent would be rejected if the absolute value of the estimated contrast was less than the critical value.

With a non-inferiority test, you again define the threshold t for the effect of the new treatment to be inferior to the standard treatment, and a contrast representing the effect of the new test minus the effect of the standard treatment. The null hypothesis is

–c ≥ t

which represents a one-sided “less-than” t-test.

Reference

Berger, M.L. & Hsu, J.C. (1996). Bioequivalence trials, intersection-union tests and equivalence confidence sets. Statistical Science, 11, 283-319.

See also

Directive: REML, VCOMPONENTS, VSTRUCTURE.

Procedure: VBOOTSTRAP, VPOWER, VUVCOVARIANCE.

Commands for: REML analysis of linear mixed models.

Example

CAPTION     'VCRITICAL 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 variety and nitrogen, to 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]
" Parametric bootstrap to get critical values for variety.nitrogen"
VCRITICAL   [PRINT=critical,fcritical,tcritical,wcritical; NBOOT=199;\
            SEED=265600; UMEANS=fit; UVCOVARIANCE=V; TERM=variety]