Two-way or multi-way data often come from experiments with afactorial design. A factorial design has at least two factor variablesfor its independent variables, and multiple observation for every combinationof these factors.

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The weight gain example below show factorial data. In thisexample, there are three observations for each combination of Diet and Country.

With this kind of data, we are usually interested in testingthe effect of each factor variable (main effects) and then the effect oftheir combination (interaction effect).

For two-way data, an interaction plot shows the mean ormedian value for the response variable for each combination of the independentvariables. This type of plot, especially if it includes error bars to indicatethe variability of data within each group, gives us some understanding of theeffect of the main factors and their interaction.

When main effects or interaction effects are statisticallysignificant, post-hoc testing can be conducted to determine which groups differsignificantly from other groups. With a factorial experiment, there are a fewguidelines for determining when to do post-hoc testing. The followingguidelines are presented for two-way data for simplicity.

• When neither the main effects nor the interaction effectis statistically significant, no post-hoc mean-separation testing should beconducted.

• When one or more of the main effects are statisticallysignificant and the interaction effect is not, post-hoc mean-separationtesting should be conducted on significant main effects only.
(This is shown in the first weight gain example below)

• When the interaction effect is statistically significant,post-hoc mean-separation testing should be conducted on the interaction effectonly. This is the case even when the main effects are also statisticallysignificant.
(This is shown in the second weight gain example below)

The final guideline above is not always followed. There aretimes when people will present the mean-separation tests for significant maineffects even when the interaction effect is significant. In general, though,if there is a significant interaction, the mean-separation tests forinteraction will better explain the results of the analysis, and themean-separation tests for the main effects will be of less interest.

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Packages used in this chapter

The packages used in this chapter include:

• psych

• car

• multcompView

• lsmeans

• FSA

• ggplot2

• phia

The following commands will install these packages if theyare not already installed:

if(!require(car)){install.packages('car')}
if(!require(psych)){install.packages('psych')}
if(!require(multcompView)){install.packages('multcompView')}
if(!require(lsmeans)){install.packages('lsmeans')}
if(!require(FSA)){install.packages('FSA')}
if(!require(ggplot2)){install.packages('ggplot2')}
if(!require(phia)){install.packages('phia')}

Two-way ANOVA example with interaction effect

Imagine for this example an experiment in which people wereput on one of three diets to encourage weight gain. The amount of weightgained will be the dependent variable, and will be considered an interval/ratiovariable. For independent variables, there are three different diets and thecountry in which subjects live.

The model will be fit with the lm function, whosesyntax is similar to that of the clm function.

This kind of analysis makes certain assumptions about thedistribution of the data, but for simplicity, this example will ignore the needto determine that the data meet these assumptions.

Input =('
Diet Country Weight_change
A USA 0.120
A USA 0.125
A USA 0.112
A UK 0.052
A UK 0.055
A UK 0.044
B USA 0.096
B USA 0.100
B USA 0.089
B UK 0.025
B UK 0.029
B UK 0.019
C USA 0.149
C USA 0.150
C USA 0.142
C UK 0.077
C UK 0.080
C UK 0.066
')
Data = read.table(textConnection(Input),header=TRUE)

### Order levels of the factor; otherwise R will alphabetize them

Data$Country = factor(Data$Country,
levels=unique(Data$Country))
### Check the data frame
library(psych)
headTail(Data)
str(Data)

summary(Data)
### Remove unnecessary objects
rm(Input)

Simple interaction plot

The interaction.plot function creates a simpleinteraction plot for two-way data. The options shown indicate which variableswill used for the x-axis, trace variable, and response variable. The fun=meanoption indicates that the mean for each group will be plotted. For the meaningof other options, see ?interaction.plot.

This style of interaction plot does not show the variabilityof each group mean, so it is difficult to use this style of plot to determineif there are significant differences among groups.

The plot shows that mean weight gain for each diet was lowerfor the UK compared with USA. And that this difference was relatively constantfor each diet, as is evidenced by the lines on the plot being parallel. Thissuggests that there is no large or significant interaction effect. That is,the difference among diets is consistent across countries. And vice-versa, thedifference in countries is consistent across diets.

A couple of other styles of interaction plot are shown atthe end of this chapter.

interaction.plot(x.factor = Data$Country,
trace.factor = Data$Diet,
response = Data$Weight_change,
fun = mean,
type='b',
col=c('black','red','green'), ### Colors for levels of trace var.
pch=c(19, 17, 15), ###Symbols for levels of trace var.
fixed=TRUE, ###Order by factor order in data
leg.bty = 'o')

Specify the linear model and conduct an analysis ofvariance

A linear model is specified with the lm function. Weight_changeis the dependent variable. Country and Diet are the independentvariables, and including Country:Diet in the formula adds theinteraction term for Country and Diet to the model.

The ANOVA table indicates that the main effects aresignificant, but that the interaction effect is not.

model = lm(Weight_change ~ Country + Diet + Country:Diet,
data = Data)
library(car)
Anova(model,
type = 'II')

https://tradeclever.amebaownd.com/posts/19721463.
Anova Table (Type II tests)
Sum Sq Df F value Pr(>F)

Factorial And Graphic Display Pdf