Scheffé Test for Testing Arbitrary Contrasts Following Anova

A parametric post hoc test for all possible pairwise and complex comparisons following a significant ANOVA, controlling the familywise error rate conservatively for any number of contrasts.

Usage

scheffeTest(x, ...)

# Default S3 method
scheffeTest(
  x,
  g = NULL,
  which = NULL,
  contrasts = NULL,
  conf.level = 0.95,
  ...
)

# S3 method for class 'formula'
scheffeTest(formula, data, subset, na.action, ...)

# S3 method for class 'aov'
scheffeTest(x, which = NULL, contrasts = NULL, conf.level = 0.95, ...)

Arguments

x: either a fitted model object, usually an aov() fit, when g is left to NULL or a response variable to be evalutated by g (which mustn't be NULL then).
...: further arguments, currently not used.
g: the grouping variable.
which: character vector listing terms in the fitted model for which the intervals should be calculated. Defaults to all the terms.
contrasts: a \(r \times c\) matrix containing the contrasts to be computed, while r is the number of factor levels and c the number of contrasts. Each column must contain a full contrast ("sum") adding up to 0. Note that the argument which must be defined, when non default contrasts are used. Default value of contrasts is NULL. In this case all pairwise contrasts will be reported.
conf.level: numeric value between zero and one giving the confidence level to use. If this is set to NA, just a matrix with the p-values will be returned.
formula: a formula of the form lhs ~ rhs, where lhs gives the response values and rhs the corresponding groups or explanatory variables.
data: an optional matrix or data frame (or similar; see model.frame()) containing the variables in the formula. By default the variables are taken from environment(formula).
subset: an optional vector specifying a subset of observations to be used in the analysis.
na.action: a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").

Value

A list of classes c("PostHocTest"), with one component for each term requested in which. Each component is a matrix with columns diff giving the difference in the observed means, lci giving the lower end point of the interval, uci giving the upper end point and pval giving the p-value after adjustment for the multiple comparisons.

There are print and plot methods for class "PostHocTest". The plot method does not accept xlab, ylab or main arguments and creates its own values for each plot.

Details

Scheffé's method applies to the set of estimates of all possible contrasts among the factor level means, not just the pairwise differences considered by Tukey's method.

References

Robert O. Kuehl, Steel R. (2000) Design of experiments. Duxbury

Steel R.G.D., Torrie J.H., Dickey, D.A. (1997) Principles and Procedures of Statistics, A Biometrical Approach. McGraw-Hill

Examples


fm1 <- aov(breaks ~ wool + tension, data = warpbreaks)
scheffeTest(x=fm1)
#> 
#>   Posthoc multiple comparisons of means: Scheffé Test 
#>     95% family-wise confidence level
#> 
#> Fit: aov(formula = breaks ~ wool + tension, data = warpbreaks)
#> 
#> $wool
#>          diff       lci       uci  pval signif
#> B-A -5.777778 -12.12841 0.5728505 0.074    .  
#> 
#> $tension
#>           diff       lci        uci  pval signif
#> M-L -10.000000 -19.76977 -0.2302286 0.044    *  
#> H-L -14.722222 -24.49199 -4.9524508 0.002    ** 
#> H-M  -4.722222 -14.49199  5.0475492 0.481       
#> 
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
scheffeTest(x=fm1, which="tension")
#> 
#>   Posthoc multiple comparisons of means: Scheffé Test 
#>     95% family-wise confidence level
#> 
#> Fit: aov(formula = breaks ~ wool + tension, data = warpbreaks)
#> 
#> $tension
#>           diff       lci        uci  pval signif
#> M-L -10.000000 -19.76977 -0.2302286 0.044    *  
#> H-L -14.722222 -24.49199 -4.9524508 0.002    ** 
#> H-M  -4.722222 -14.49199  5.0475492 0.481       
#> 
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 

# some special contrasts
y <- c(7,33,26,27,21,6,14,19,6,11,11,18,14,18,19,14,9,12,6,
       24,7,10,1,10,42,25,8,28,30,22,17,32,28,6,1,15,9,15,
       2,37,13,18,23,1,3,4,6,2)
group <- factor(c(1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,
       3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6))

r.aov <- aov(y ~ group)
scheffeTest(r.aov, contrasts=matrix( c(1,-0.5,-0.5,0,0,0,
                                       0,0,0,1,-0.5,-0.5), ncol=2) )
#> 
#>   Posthoc multiple comparisons of means: Scheffé Test 
#>     95% family-wise confidence level
#> 
#> Fit: aov(formula = y ~ group)
#> 
#> $group
#>          diff       lci      uci  pval signif
#> 1-2,3  7.2500 -6.417446 20.91745 0.637       
#> 4-5,6 14.0625  0.395054 27.72995 0.040    *  
#> 
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 

# just p-values:
scheffeTest(r.aov, conf.level=NA)
#> 
#>   Posthoc multiple comparisons of means: Scheffé Test 
#> 
#> Fit: aov(formula = y ~ group)
#> 
#> $group
#>   1     2     3     4     5    
#> 2 0.927     -     -     -     -
#> 3 0.531 0.977     -     -     -
#> 4 0.848 0.273 0.054     -     -
#> 5 0.940 1.000 0.970 0.296     -
#> 6 0.400 0.934 1.000 0.031 0.920
#> 
#>