A2WAY procedure

Performs analysis of variance of a balanced or unbalanced design with up to two treatment factors (R.W. Payne).

Options

PRINT = string tokens	Controls printed output from the analysis (aovtable, information, covariates, effects, residuals, means, %cv, missingvalues); default aovt, mean
TREATMENTS = factors	Defines either one or two treatment factors
BLOCKS = factor	Can specify a blocking factor e.g. for a randomized block design
COVARIATES = variates	Specifies any covariates
FACTORIAL = scalar	Can be set to 1 to fit only the main effects of the treatments factors; default 2 also fits their interaction
FPROBABILITY = string token	Probabilities for variance ratio (yes, no); default no
PLOT = string tokens	Which residual plots to provide (fittedvalues, normal, halfnormal, histogram, absresidual); default fitt, norm, half, hist
GRAPHICS = string token	Type of graphs (lineprinter, highresolution); default high
COMBINATIONS = string token	Factor combinations for which to form predicted means (present, estimable); default esti
ADJUSTMENT = string token	Type of adjustment to be made when predicting means (marginal, equal, observed); default marg
PSE = string tokens	Types of standard errors to be printed with the means (differences, lsd, means, alldifferences, alllsd); default diff
LSDLEVEL = scalar	Significance level (%) for least significant differences; default 5
RMETHOD = string token	Type of residuals to save or display (simple, standardized); default simp
MVINCLUDE = string token	Whether to include units with missing y-values when using ANOVA (yvariate); default * i.e. not included
EXIT = scalar	Saves an exit code indicating the properties of the design

Parameters

Y = variates	Each of these contains the data values for an analysis
RESIDUALS = variates	Saves the residuals from each analysis
FITTEDVALUES = variates	Saves the fitted values from each analysis
SAVE = pointers	Save structure for each analysis (to use in A2DISPLAY or A2KEEP)

Description

A2WAY provides specialized facilities for analysis of variance with either one or two treatment factors. There can also be a blocking factor. It automatically determines the type of design and uses the appropriate method: the ANOVA directive if the design is balanced, or the regression directives (FIT, ADD and so on) if it is unbalanced. So, for example, it can analyse randomized complete block designs with one or two treatment factors, or unbalanced two-way treatment structures with or without blocking, or designs with a single treatment factor whose levels are allocated unevenly across the blocks. By default, any units with missing values in the y-variate are excluded from the analysis. Conversely, in ANOVA they need to be included to ensure balance in the more general situations that it covers, and are estimated as part of the analysis. However, you can reproduce the ANOVA analysis by setting option MVINCLUDE=yvariate.

The output is also customized. For example, if the treatments have unequal replication, a standard error is printed for each mean, rather than the summary for comparisons of means with minimum and maximum replication as given by ANOVA. Similarly, the two-way analyses show the sums of squares for the main effects both omitting and ignoring the other main effect. In addition, A2WAY provides residual plots directly, instead of requiring you to use procedure APLOT after the analysis.

The Y parameter supplies a variate containing the data values to be analysed. The RESIDUALS parameter can save the residuals from the analysis, and the FITTEDVALUES parameter can save the fitted values. The RMETHOD option controls whether simple or standardized residuals are saved or displayed; by default RMETHOD=simple.

The SAVE parameter can save a “save” structure that can be used as input to procedure A2DISPLAY to produce further output, or to procedure A2KEEP to copy output into Genstat data structures.

The treatment factor or factors are specified by the TREATMENTS option, and the BLOCKS option can be used to supply a blocking factor. Covariates can be supplied using the COVARIATES option. As in ANOVA, the FACTORIAL option sets a limit in the number of factors in each treatment term. So you can set FACTORIAL=1 to fit only the main effects when there are two treatment factors; the default FACTORIAL=2 also fits their interaction.

Printed output is controlled by the PRINT option, with settings:

aovtable	analysis-of-variance table (probabilities are given for the variance ratios if option FPROBABILITY=yes);
information	information about the design (non-orthogonality &c);
covariates	covariate regression coefficients);
effects	treatment parameters in the linear model;
means	table of means;
%cv	to print the coefficient of variation;
missingvalues	to print estimates for any missing values.

The PSE option controls the types of standard errors that are produced to accompany the tables of means, with settings:

differences	summary of standard errors for differences between pairs of means;
alldifferences	standard errors for differences between all pairs of means (unbalanced designs only);
lsd	summary of least significant differences between pairs of means;
alllsd	least significant differences between all pairs of means (unbalanced designs only);
means	standard errors of the means – for unbalanced designs, these are approximate effective standard errors formed by procedure SED2ESE with the aim of allowing good approximations to the standard errors for differences to be calculated by the usual formula of sedij = √(esei2 + esej2)

The default is differences. The LSDLEVEL option sets the significance level

(as a percentage) for the least significant differences.

For unbalanced designs, analysed using Genstat regression, the means are produced using the PREDICT directive. The first step (A) of the calculation forms the full table of predictions, classified by all the treatment and blocking factors. The second step (B) averages the full table over the factors that do not occur in the table of means. The COMBINATIONS option specifies which cells of the full table are to be formed in Step A. The default setting, estimable, fills in all the cells other than those that involve parameters that cannot be estimated. Alternatively, setting COMBINATIONS=present excludes the cells for factor combinations that do not occur in the data. The ADJUSTMENT option then defines how the averaging is done in Step B. The default setting, marginal, forms a table of marginal weights for each factor, containing the proportion of observations with each of its levels; the full table of weights is then formed from the product of the marginal tables. The setting equal weights all the combinations equally. Finally, the setting observed uses the WEIGHTS option of PREDICT to weight each factor combination according to its own individual replication in the data.

The PLOT option allows up to four of the following residual plots to be requested:

fittedvalues	for a plot of residuals against fitted values;
normal	for a Normal plot;
halfnormal	for a half-Normal plot;
histogram	for a histogram of residuals; and
absresidual	for a plot of the absolute values of the residuals against the fitted values.

By default the first four are produced. The GRAPHICS option determines the type of graphics that is used, with settings highresolution (the default) and lineprinter.

The RMETHOD option controls whether simple or standardized residuals are printed or plotted; by default RMETHOD=simple.

The EXIT option can save an exit code indicating how the analysis was done. For the exact meanings of the values see the ANOVA directive. Essentially, it has the values 0 or 1 if the analysis has been done using ANOVA (0 if design orthogonal and 1 if it is balanced). Other values indicate that it has been done using the regression directives.

Options: PRINT, TREATMENTS, BLOCKS, COVARIATES, FACTORIAL, FPROBABILITY, PLOT, GRAPHICS, COMBINATIONS, ADJUSTMENT, PSE, LSDLEVEL, RMETHOD, MVINCLUDE, EXIT.

Parameters: Y, RESIDUALS, FITTEDVALUES, SAVE.

Method

The EXIT option of the ANOVA directive is used to determine whether or not the design is unbalanced (and thus whether the Genstat regression facilities need to be used rather than the analysis of variance facilities).

Action with RESTRICT

If the Y variate is restricted, only the units not excluded by the restriction will be analysed.

See also

Procedures: A2DISPLAY, A2KEEP, A2RESULTSUMMARY
Commands for: Analysis of variance.

Example

CAPTION 'A2WAY example',\
        !t('Data from Snedecor & Cochran (1980), Statistical Methods',\
        '(7th edition), page 216 and also see page 252.');\ 
        STYLE=meta,plain
FACTOR  [LEVELS=4; VALUES=(1...4)6] Fat
VARIATE [VALUES=64,78,75,55, 72,91,93,66, 68,97,78,49,\
                77,82,71,64, 56,85,63,70, 95,77,76,68] Absorbed
A2WAY   [PRINT=aov,means; TREATMENTS=Fat; FPROBABILITY=yes] Absorbed

CAPTION !t('Experiment on foster feeding of rats from Scheffe (1959)',\
        'The Analysis of Variance; also see McConway, Jones & Taylor (1999)',\
        'Statistical Modelling using GENSTAT, Example 7.6.')
FACTOR  [NVALUES=61; LABELS=!t('A','B','I','J')] litter
READ litter; FREPRESENTATION=labels
A A A A A A A A A A A A A A A A A B B B B B B B B B B B B B B B I I I I I I
I I I I I I I I J J J J J J J J J J J J J J J :
FACTOR  [NVALUES=61; LABELS=!t('A','B','I','J')] mother
READ mother; FREPRESENTATION=labels
A A A A A B B B I I I I J J J J J A A A A B B B B B I I I I J J A A A B B B
I I I I I J J J A A A A B B B I I I J J J J J :
VARIATE [NVALUES=61] littwt
READ     littwt
61.5 68.2 64 65 59.7 55 42 60.2 52.5 61.8 49.5 52.7 42 54 61 48.2 39.6 60.3
51.7 49.3 48 50.8 64.7 61.7 64 62 56.5 59 47.2 53 51.3 40.5 37 36.3 68 56.3
69.8 67 39.7 46 61.3 55.3 55.7 50 43.8 54.5 59 57.4 54 47 59.5 52.8 56 45.2
57 61.4 44.8 51.5 53 42 54 :
A2WAY   [PRINT=aovtable,means,%cv; TREATMENTS=litter,mother;\
        PSE=differences,lsd,means,alldifferences,alllsd] littwt