The multiway ANOVA is illustrated here using the data form Moore and
Krupat (1971) on conformity. The dataset can be found in the
car package (Moore.rda) and in the present package
for convenience. The treatment below reproduces the treatmant of section
4.3 of John Fox R and S-plus Companion to Applied Regession
(SAGE Publications, 2002).
Figure 3 below illustrates how to (1) load and (2) view the Moore
dataset which first entries are shown in (3). In this experiment, the
conformity (DV, 1st columbns) of each subject interacting
with a partner of either ‘low’ or ‘high’ status
(partner.status, 1st IV, 2nd column) and’low’, ‘medium’ or
‘high’ authoritarism (fcategory, 2nd IV, 3rd column).
4.1. levels reordering
The replication of this analysis starts by re-ordering the levels of
fcategory in the order low < medium < high (the
default ordering is the alphabetical order). This reordering is
essentially for graphical purpose. This can be done in Rcmdr by using
the menu entry Data >
Manage variables in the active dataset >
Reorder factor levels... as illustrated in Figure 2.
After re-ordering, the means of all combinations of factors
(cells) can be plot using the menu entry Graphs
> Plot of means... which yields the dialog box shown in
Figure 3.1. The graph in Figure 3.2 is obtained after selecting the two
factors and conformity as the response variable.
This graph shows a clear effect of partner.status on the
conformity (whereby high status yields higher conformity),
providing the authoritarism (fcategory) remains
low or medium. Overall, this suggest a main effect of
partner.status and an interaction between
partner.status and fcategory on
conformity. This will be checked in the following using a
two-way ANOVA of conformity using
partner.status and fcategory as factors.
4.2. two-way analysis of variance
The multiway ANOVA dialog box is accessed by the
Statistics > Means >
Multi-way ANOVA... menu entry. It presents two tabs
(Data and Options) shown in Figure 4.
With the settings shown in Figure X, a two-way ANOVA model is fitted
to the data employing sum-to-zero contrasts, and a type II test which is
the default in the car::ANOVA() function used in aRnova
(see J Fox textbook for details on the choice of contrasts and type of
sum of squares). The R commands and results are displayed in the output
window as shown in Figure Y and reproduced below:
> options(contrasts = c('contr.sum', 'contr.poly'))
> MooreModel.1 <- lm(conformity ~ fcategory*partner.status, data=Moore)
> .myAnova <- Anova(MooreModel.1, type = 2)
> .myAnova
Anova Table (Type II tests)
Response: conformity
Sum Sq Df F value Pr(>F)
fcategory 11.61 2 0.2770 0.759564
partner.status 212.21 1 10.1207 0.002874 **
fcategory:partner.status 175.49 2 4.1846 0.022572 *
Residuals 817.76 39
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
These results indicate a significant effect of
partner.status on the conformity, as well as an interaction
between fcategory and partner.status affecting
conformity, as was anticipated in this study. The use of Type III sum of
squares (widely used and the default in many other statistics packages)
leads to similar results.
Depending on the options chosen, supplementary output is given:
Efffect size
When selecting Effect size in the Options
tab, partial eta squared (\(\eta^2\))
are computed and displayed as follows. In the present case, these values
show that both the main effect and interaction effect have a large
effect size according to Cohen’s guidelines:
> \# EFFECT SIZE
> .effectSize <- data.frame(format(round(.myAnova$`Sum Sq` / sum(.myAnova$`Sum Sq`), 4), nsmall = 4), row.names = rownames(.myAnova))
> colnames(.effectSize) <- c("part. eta sq.")
> .effectSize
part. eta sq.
fcategory 0.0095
partner.status 0.1744
fcategory:partner.status 0.1442
Residuals 0.6719
summary statistics of groups
When this option is selected in the Options tab, the
mean, standard deviations and number of observations for each
combination of factor levels (cells):
> \# NUMERIC SUMMARY OF GROUPS
> with(Moore, (tapply(conformity, list(fcategory, partner.status), mean, na.rm=TRUE))) # means
high low
low 17.40000 8.900
medium 14.27273 7.250
high 11.85714 12.625
> with(Moore, (tapply(conformity, list(fcategory, partner.status), sd, na.rm=TRUE))) # std. deviations
high low
low 4.505552 2.643651
medium 3.951985 3.947573
high 3.933979 7.347254
> with(Moore, (tapply(conformity, list(fcategory, partner.status), function(x) sum(!is.na(x))))) # counts
high low
low 5 10
medium 11 4
high 7 8
post-hoc analysis When this option is
selected in the Options tab, post-hoc analysis on
significant or close to significant effects is carried out. To this aim
an ‘extended’ dataset (with the extension .extd) is
generated. This dataset contains additional factor combinations that
allows particularizing each combination of factor levels (cell)
and is useful for plotting means, data mining, and post-hoc analysis.
Hence, in the present case, a supplementary column
fcategory.partner.status is generated. This allows, for
instance, to generate the boxplots corresponding to each cell
(Graphs > Boxplot...), as examplified in
Figure.
The post-hoc analysis consist in pairwise comparison between cells
using Tukey’s HSD test when interactions are significant (p < .05) or
close to significant (p < 0.1) . In the present case, the interaction
between the two factors is sginficant (p = 0.02, see above), which
triggers the pairwise comparison:
> summary(glht(aov(conformity ~ fcategory.partner.status, data = Moore.extd), linfct = mcp(fcategory.partner.status = "Tukey")))
Simultaneous Tests for General Linear Hypotheses
Multiple Comparisons of Means: Tukey Contrasts
Fit: aov(formula = conformity ~ fcategory.partner.status, data = Moore.extd)
Linear Hypotheses:
Estimate Std. Error t value Pr(>|t|)
medium.high - low.high == 0 -3.1273 2.4698 -1.266 0.7975
high.high - low.high == 0 -5.5429 2.6813 -2.067 0.3202
low.low - low.high == 0 -8.5000 2.5081 -3.389 0.0183 *
medium.low - low.high == 0 -10.1500 3.0718 -3.304 0.0227 *
high.low - low.high == 0 -4.7750 2.6105 -1.829 0.4543
high.high - medium.high == 0 -2.4156 2.2140 -1.091 0.8795
low.low - medium.high == 0 -5.3727 2.0008 -2.685 0.0989 .
medium.low - medium.high == 0 -7.0227 2.6736 -2.627 0.1124
high.low - medium.high == 0 -1.6477 2.1277 -0.774 0.9695
low.low - high.high == 0 -2.9571 2.2566 -1.310 0.7737
medium.low - high.high == 0 -4.6071 2.8701 -1.605 0.5950
high.low - high.high == 0 0.7679 2.3699 0.324 0.9995
medium.low - low.low == 0 -1.6500 2.7090 -0.609 0.9895
high.low - low.low == 0 3.7250 2.1721 1.715 0.5252
high.low - medium.low == 0 5.3750 2.8041 1.917 0.4024
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Adjusted p values reported -- single-step method)
This post-hoc analysis show that the significant interaction comes
from differences between low authoritarism/low status and low
authoritarism/high status, as well as between medium authoritarism/low
status and low authoritarism/high status. As shown in the previous
plots, these two pairs of cells present indeed the largest differences
of conformity.
