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.
