RcmdrPlugin.aRnova is a R Commander add-on for repeated-measures and mixed-design ('split-plot') analysis of variance (ANOVA). In essence it adds a new menu entry for repeated measures that allows to deal with up to three within-subject factors and optionally with one or several between-subject factors.
Besides, it also provides supplementary options and outputs to the existing
One Way ANOVA and
Multi Way ANOVA entries, such as the choice of type of sum of squares (II or III), the calculation of effect sizes, or multiway ANOVA post-hoc analysis.
This tutorial (not finished yet...) aims to illustrate typical use of this plugin by replicating ANOVAs described in particular in John Fox' R and S-plus Companion to Applied Regession (SAGE Publications Inc. 2002). While all steps are illustrated in the following, the user is expected to have minimal proficiency in using Rcmdr.
The aRnova plugin can be downloaded and installed using:
To load the package the fastest way is to use the following R command line which causes the package to load Rcmdr simultaneously with the aRnova plugin:
Note that if aRnova has just been installed, restart a new R session before launching this command (else the following error may occur:
Error : .onAttach failed in attachNamespace() for [....]).
Alternatively, after loading Rcmdr without the aRnova plugin using:
The aRnova package can then be loaded using the
Load Packages... menu entry in Rcmdr. The user is then asked to restart Rcmdr so that the Plug-in is available.
It can be verified that aRnova has been effectively loaded by checking that the supplementary entry
Repeated measures ANOVA... has been added in the
Means Rcmr menu.
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).
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
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
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
conformity. This will be checked in the following using a two-way ANOVA of
fcategory as factors.
The multiway ANOVA dialog box is accessed by the
Multi-way ANOVA... menu entry. It presents two tabs (
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
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:
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
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 (
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.
To conduct an ANOVA using a repeated measures design, activate the
dialog box entitleded "Definition of Within-subject factors" in the menu
Statistics -> Means -> Repeated Measures ANOVA...
In this dialog box, you have to supply a name and a number of levels for each of the within-subject (repeated -measures) variable. A valid within-factor entry must consist in a syntactically valid name (see make.names) and 2 levels or more. On clicking the OK button, the first valid entries are kept and used
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