14.7 – Rcmdr Multiway ANOVA

Introduction

We have been talking about the two-way randomized, balanced, replicated design.  Here, we take you step by step through use of R to conduct the multiway ANOVA.

R code: Multiway ANOVA

Rcmdr: Statistics → Means → Multiway ANOVA… we will review this as Option 1

or

Rcmdr: Statistics → Fit Models → Linear model… we will review this as Option 2

In either case, as a reminder, your data set must be stacked worksheet, like the data in this worksheet

Table 1. Data set, example.14.7†

Diet Population Response
A 1 4
A 1 6
A 1 5
A 2 5
A 2 8
A 2 9
B 1 12
B 1 15
B 1 11
B 2 5
B 2 7
B 2 8

Option 1

Your first option is to use the ANOVA menus via “Means.” This is a perfectly good way to handle a standard two-way, fully-crossed, fixed effects model.  However, other designs will not run with this command and R will return a report of errors for ANOVA models that do not conform to the replicated, balanced, crossed design.

Rcmdr: Statistics → Means → Multiway Analysis of variance …

Factors: highlight “Diet” AND “Population”

Response variable: pick one (in this window, all we see is “Response”)

Screenshot Rcmdr multi-way ANOVA

Figure 1. Screenshot Rcmdr multi-way ANOVA.

†Note 1: Don’t forget to convert numeric Population to factor. Assuming Population is part of a data.frame called example.14.7, then example.14.7$Population <- factor(example.14.7$Population).

Interpret the output

AnovaModel.2 <- (lm(Response ~ Diet*Population, data=example.14.7))

Anova(AnovaModel.2)
 
Anova Table (Type II tests)
Response: Response
                  Sum Sq    Df    F value      Pr(>F) 
Diet              36.750     1    12.2500    0.008079 **
Population        10.083     1     3.3611    0.104104 
Diet:Population   52.083     1    17.3611    0.003136 **
Residuals         24.000     8
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
tapply(example.14.7 Response, list(Diet=example.14.7 Diet, 
+ Population=example.14.7 Population), mean, na.rm=TRUE) # means
Population
Diet 1 2
A 5.00000 7.333333
B 12.66667 6.666667

tapply(example.14.7 Response, list(Diet=example.14.7 Diet, 
+ Population=example.14.7 Population), sd, na.rm=TRUE) # std. deviations
Population
Diet 1 2
A 1.000000 2.081666
B 2.081666 1.527525

tapply(example.14.7 Response, list(Diet=example.14.71 Diet, 
+ Population=example.14.7 Population), function(x) sum(!is.na(x))) # counts
Population
Diet 1 2
A 3 3
B 3 3

End R output.

Note 2: What is the “Type II tests”? When you fit a model with categorical predictors using lm() or aov(), you test hypothesis for each effect (main effects and interactions). These tests differ depending on how the model partitions the variance among predictors. See Note 2 in Chapter 12.7 for description of Type I, Type II, and Type III sums of squares.

Summary of multi-way ANOVA command

The multi-way ANOVA command returns our ANOVA table plus the adjusted means, along with standard deviations and number of observations (counts).  The adjusted means would then be good to put into a chart to present group comparisons following adjustments from the effects of levels within groups.

Rcmdr: Models → Graphs → Predictor effect plots …

Here’s the chart (hint: \pm SEM = \frac{SD}{\sqrt{count}})

predictor chart

Figure 2. Predictor effect plots, Diet and Population on Response variable.

Option 2

A more general approach is to use the General linear model. This approach can handle the standard 2-way fixed effects ANOVA (above), but any other model as well. The model is Response ~ Diet*Population.

Rcmdr: Statistics → Fit Models → Linear model… 

Screenshot Rcmdr linear model

Figure 3. Screenshot Rcmdr linear model menu.

Interpret the output

LinearModel.1 <- lm(Response ~ Diet * Population, data=example.14.7)

summary(LinearModel.1)

Call:
lm(formula = Response ~ Diet * Population, data = example.14.7)

Residuals:
Min 1Q Median 3Q Max 
-2.3333 -1.1667 0.1667 1.0833 2.3333 

Coefficients:
                        Estimate    Std. Error ..t value .  Pr(>|t|) 
(Intercept) .              5.000         1.000 .   5.000    .0.00105 ** 
Diet[T.B] .                7.667       ..1.414 .   5.421    .0.00063 ***
Population[T.2]           .2.333       ..1.414 .   1.650    .0.13757 
Diet[T.B]:Population[T.2] -8.333       ..2.000    -4.167    .0.00314 ** 
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

Residual standard error: 1.732 on 8 degrees of freedom
Multiple R-squared: 0.8047, Adjusted R-squared: 0.7315 
F-statistic: 10.99 on 3 and 8 DF, p-value: 0.003285

End R output

Lots to sort through, so let’s begin with what is in common between the two approaches, the Multi-way ANOVA command versus the linear model command.

Compare R output from ANOVA and linear model

As a direct output, the linear model option does not provide an ANOVA summary table. Instead of our ANOVA table the linear model returns estimates of coefficients along with t-test results for each coefficient of the model from the lm() command output

Recall that we can get ANOVA tables through the following R commands via Rcmdr.

Rcmdr: Models → Hypothesis tests → ANOVA Table.

Let’s do so for this linear model (accept the default for type of tests = “Type II”).

And the output from lm() stored in the object LinearModel.1 is

Anova(LinearModel.1, type="II")
Anova Table (Type II tests)

Response: Response
                Sum Sq Df F value   Pr(>F)   
Diet            36.750  1 12.2500 0.008079 **
Population      10.083  1  3.3611 0.104104   
Diet:Population 52.083  1 17.3611 0.003136 **
Residuals       24.000  8                    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

End R output

Now, we’re in business, and, using the lm() function, we have the estimates for each model coefficient plus our ANOVA table.

Same answer! Of course. Which to choose, Option 1 or Option 2? Use the lm() options; more flexible, covers more designs than the multiway ANOVA which is strictly for the crossed fully replicated design.

Note 3: Rcmdr uses Anova(), not anova(). anova() is a generic function, part of the base statistics package, and it returns an ANOVA table. Anova() is not a wrapper to anova(), it is a different function, part of the car package. It uses different default methods and different types of sums of squares: Type-II or Type-III analysis-of-variance tables for model objects produced by lm(), glm(), etc. See Note 2 in Chapter 12.7 for description of Type I, Type II, and Type III sums of squares.

Questions

  1. Write out the two-way model described for the data in Table 1.
  2. Write the null hypotheses and provide a summary of the statistical significance of the model.

Quiz Chapter 14.7

Rcmdr Multiway ANOVA

Chapter 14 contents

14 – ANOVA designs, multiple factors

Introduction

In our previous discussions about t-tests and ANOVA we focused on procedures with one dependent (response) variable and a single independent (predictor) factor variable that may cause variation in the response variable. In this chapter we extend our discussions about the general linear model by reviewing

  1. One-way ANOVA and provided a few examples of the one-way design.
  2. To review and set the stage for adding a second independent variable to the model.

Additional one-way ANOVA examples

1. In a plants, we may have a response variable like height and one factor variable (location: sun vs. shade) thought to influence plant height (eg, Aphalo et al 1999).

2. Pulmonary macrophage phagocytosis behavior (response variable) after exposure of toads to clean air or ozone (factor with 2 levels) (Dohm et al. 2005).

3. Monitor weight change on subjects after 6 weeks eating different diet (DASH, control) (Elmer et al. 2006).

All three of the examples are based on the same statistical model which may be written as:

    \begin{align*} Y_{ik}=\mu + A_{i} + \epsilon _{ik} \end{align*}

where μ is the grand mean, Y is the response variable and A is the independent variable, or factor, with k = 1, 2, …. K levels, groups, or treatments. The total number of experimental units (eg, subjects) is given by i = 1, 2, 3, … n. Note that in the first and third examples, because there were only two groups (example 1: k = location, shade; example 3: k = DASH, control) note that this problem could have been evaluated as an independent sample t-test. For the second example, there were three groups so k = clean air, first ozone level, second ozone level).

Two-way ANOVA with replication

As useful as the one-way ANOVA design can be, it is a “one-factor-at-a-time” approach (Radzilani et al 2025), experiments are typically more complicated than a single t-test or one-way ANOVA design can handle; rarely would we conduct an experiment that reflects only one source of variation. Looking only at variability for one factor at a time while while keeping all other factors constant ignores interactions between factors, resulting in potentially serious model bias (eg, discussed in Vatcheva et al 2015).

For example, while diet has a profound effect on weight, clearly, activity levels are also important. At a minimum, when considering a weight loss program, we would want to control or monitor activity of the subjects. This is a two-factor model, and the main effects, the two factors, were diet (factor A) and activity, (factor B). Both are expected to affect weight loss, and, perhaps, they may do so in complicated ways — an interaction (eg, on DASH diet, weight loss is accelerated when subjects exercise regularly).

    \begin{align*} Y_{ijk}=\mu + A_{i} + B_{j} + AB_{ij} + \epsilon _{ijk} \end{align*}

The subject of this chapter is the introduction to two-way ANOVA designs. In fact, to many, ANOVA design is practically synonymous to a statistician when they think about experimental design (Lindman 1992; Quinn and Keough 2002). As noted by Quinn and Keough (2002) in the preface to their book, “… many biological hypotheses, even deceptively simple ones, are matched by complex statistical models” (p. xv). Once you start adding factor variables there becomes a number of ways in which the groups and experimental units can be distributed, and thus impact the inferences one can make from the ANOVA results. The first statistical model we introduced was the one-way ANOVA. Next, we begin the two-way ANOVA with the crossed, balanced, fully replicated design. Along the way we introduce model symbols to help us communicate the design structure and implications of the statistical models.

Quizzes in this chapter

A total of 71 questions among the several subchapters, a mix of true or false and multiple choice question format.

Chapter 14 contents