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14: ANOVA Designs, Multiple Factors

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    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

    1. Reviewing the one-way ANOVA, and providing a few examples of the one-way design.
    2. Reviewing and setting 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 (e.g. 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: \[Y_{ik} = \mu + A_{i} + \epsilon_{ik} \nonumber\]

    where \(\mu\) is the grand mean, \(Y\) is the response variable and \(A\) is the independent variable, or factor, with \(k = 1, 2, \ldots K\) levels, groups, or treatments. The total number of experimental units (e.g., subjects) is given by \(i = 1, 2, 3, ldots n\). 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

    Biology 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.

    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 (e.g., on DASH diet, weight loss is accelerated when subjects exercise regularly).

    \[Y_{ijk} = \mu + A_{i} + B_{j} + AB_{ij} + \epsilon_{ijk} \nonumber\]

    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.

    • 14.1: Crossed, balanced, fully replicated designs
      Introduction to models with multiple levels of interaction and two-factor ANOVA with replicates.
    • 14.2: Sources of variation
      Sources of variation in the two-way ANOVA, namely factors, the interaction between factors, and residual or within-cell error.
    • 14.3: Fixed effects, random effects
      Introduction to two-factor ANOVA models that have one or two random factors. Calculating the \(F\)-test statistic for different ANOVA models.
    • 14.4: Randomized block design
      Introduction to randomized block design, as a special form of two-way ANOVA with both a blocking factor that groups experimental units and a treatment factor.
    • 14.5: Nested designs
      Nested experimental designs, how they differ from crossed designs, and their advantages and limitations as compared to the standard two-way ANOVA model.
    • 14.6: Some other ANOVA designs
      Discussion of additional considerations to be made when planning different types of experiments, and other ANOVA experimental designs that might be applied. Includes repeated-measures ANOVA, three-way ANOVA, Latin square design, and split-plot design.
    • 14.7: Rcmdr Multiway ANOVA
      Worked example of using two methods of conducting the multiway ANOVA in R Commander: the Multiway ANOVA and the Linear Model.
    • 14.8: More on the linear model in rcmdr
      How the Linear Model procedure in R and R Commander can be used to analyze previously discussed multifactorial ANOVA problems.
    • 14.9: Chapter 14 References

    This page titled 14: ANOVA Designs, Multiple Factors is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Michael R Dohm via source content that was edited to the style and standards of the LibreTexts platform.