Two-way anova
When to use it
You use a two-way anova (also known as a factorial anova, with two factors) when you have one measurement variable and two nominal variables. The nominal variables (often called "factors" or "main effects") are found in all possible combinations. For example, let's say you are testing the null hypothesis that stressed and unstressed rats have the same glycogen content in their gastrocnemius muscle, and you are worried that there might be sex-related differences in glycogen content as well. The two factors are stress level (stressed vs. unstressed) and sex (male vs. female). Unlike a nested anova, each grouping extends across the other grouping. In a nested anova, you might have "cage 1" and "cage 2" nested entirely within the stressed group, while "cage 3" and "cage 4" were nested within the unstressed group. In a two-way anova, the stressed group contains both male and female rats, and the unstressed group also contains both male and female rats. The factors used to group the observations may both be model I, may both be model II, or may be one of each ("mixed model").
A two-way anova may be done with replication (more than one observation for each combination of the nominal variables) or without replication (only one observation for each combination of the nominal variables).
Assumptions
Two-way anova, like all anovas, assumes that the observations within each cell are normally distributed and have equal variances.
Two-way anova with replication
Null hypotheses: The results of a two-way anova with replication include tests of three null hypotheses: that the means of observations grouped by one factor are the same; that the means of observations grouped by the other factor are the same; and that there is no interaction between the two factors. The interaction test tells you whether the effects of one factor depend on the other factor. In the rat example, imagine that stressed and unstressed female rats have about the same glycogen level, while stressed male rats had much lower glycogen levels than unstressed male rats. The different effects of stress on female and male rats would result in a significant interaction term in the anova. When the interaction term is significant, the usual advice is that you should not test the effects of the individual factors. In this example, it would be misleading to examine the individual factors and conclude "Stressed rats have lower glycogen than unstressed," when that is only true for male rats, or "Male rats have lower glycogen than female rats," when that is only true when they are stressed.
What you can do, if the interaction term is significant, is look at each factor separately, using a one-way anova. In the rat example, you might be able to say that for female rats, the mean glycogen levels for stressed and unstressed rats are not significantly different, while for male rats, stresed rats have a significantly lower mean glycogen level than unstressed rats. Or, if you're more interested in the sex difference, you might say that male rats have a significantly lower mean glycogen level than female rats under stress conditions, while the mean glycogen levels do not differ significantly under unstressed conditions.
How the test works: When the sample sizes in each subgroup are equal (a "balanced design"), the mean square is calculated for each of the two factors (the "main effects"), for the interaction, and for the variation within each combination of factors. Each F-statistic is found by dividing a mean square by the within-subgroup mean square.
When the sample sizes for the subgroups are not equal (an "unbalanced design"), the analysis is much more complicated, and there are several different techniques for testing the main and interaction effects. The details of this are beyond the scope of this handbook. If you're doing a two-way anova, your statistical life will be a lot easier if you make it a balanced design.
Two-way anova without replication
Null hypotheses: When there is only a single observation for each combination of the nominal variables, there are only two null hypotheses: that the means of observations grouped by one factor are the same, and that the means of observations grouped by the other factor are the same. It is impossible to test the null hypothesis of no interaction. Testing the two null hypotheses about the main effects requires assuming that there is no interaction.
How the test works: The mean square is calculated for each of the two main effects, and a total mean square is also calculated by considering all of the observations as a single group. The remainder mean square (also called the discrepance or error mean square) is found by subtracting the two main effect mean squares from the total mean square. The F-statistic for a main effect is the main effect mean square divided by the remainder mean square.
Repeated measures: One experimental design that is analyzed by a two-way anova is repeated measures, where an observation has been made on the same individual more than once. This usually involves measurements taken at different time points. For example, you might measure running speed before, one week into, and three weeks into a program of exercise. Because individuals would start with different running speeds, it is better to analyze using a two-way anova, with "individual" as one of the factors, rather than lumping everyone together and analyzing with a one-way anova. Sometimes the repeated measures are repeated at different places rather than different times, such as the hip abduction angle measured on the right and left hip of individuals. Repeated measures experiments are often done without replication, although they could be done with replication.
In a repeated measures design, one of main effects is usually uninteresting and the test of its null hypothesis may not be reported. If the goal is to determine whether a particular exercise program affects running speed, there would be little point in testing whether individuals differed from each other in their average running speed; only the change in running speed over time would be of interest.
Randomized blocks: Another experimental design that is analyzed by a two-way anova is randomized blocks. This often occurs in agriculture, where you may want to test different treatments on small plots within larger blocks of land. Because the larger blocks may differ in some way that may affect the measurement variable, the data are analyzed with a two-way anova, with the block as one of the nominal variables. Each treatment is applied to one or more plot within the larger block, and the positions of the treatments are assigned at random. This is most commonly done without replication (one plot per block), but it can be done with replication as well.
Examples
Shimoji and Miyatake (2002) raised the West Indian sweetpotato weevil for 14 generations on an artificial diet. They compared these artificial diet weevils (AD strain) with weevils raised on sweet potato roots (SP strain), the weevil's natural food. Multiple females of each strain were placed on either the artificial diet or sweet potato root, and the number of eggs each female laid over a 28-day period was counted. There are two nominal variables, the strain of weevil (AD or SP) and the oviposition test food (artificial diet or sweet potato), and one measurement variable (the number of eggs laid).
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| Mean total numbers of eggs of females from SP and AD strains on sweet-potato pieces (mesh bars) and artificial-diet pieces (hollow bars). Values are mean ± SEM. (Fig. 4 from Shimoji and Miyatake [2002]). |
The results of the two-way anova with replication are a significant effect for weevil strain (F1, 117=8.82, P=0.0036) and oviposition test food (F=1, 117=345.92, P=9 x 10-37). However, there is also a significant interaction term (F1, 117=17.02, P=7 x 10-5). Both strains lay more eggs on sweet potato root than on artificial diet, but the difference is smaller for the AD strain.
I assayed the activity of the enzyme mannose-6-phosphate isomerase (MPI) in the amphipod crustacean Platorchestia platensis (McDonald, unpublished data). There are three genotypes at the locus for MPI, Mpiff, Mpifs, and Mphss, and I wanted to know whether the genotypes had different activity. Because I didn't know whether sex would affect activity, I also recorded the sex. Each amphipod was lyophilized, weighed, and homogenized; then MPI activity of the soluble portion was assayed. The data (in δO.D. units/sec/mg dry weight) are shown below as part of the SAS example. The results indicate that the interaction term, the effect of sex and the effect of genotype are all non-significant.
Graphing the results
Sometimes the results of a two-way anova are plotted on a 3-D graph, with the measurement variable on the Y-axis, one nominal variable on the X-axis, and the other nominal variable on the Z-axis (going into the paper). This makes it difficult to visually compare the heights of the bars in the front and back rows, so I don't recommend this. Instead, I suggest you plot a bar graph with the bars clustered by one nominal variable, with the other nominal variable identified using the color or pattern of the bars.
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| Mannose-6-phosphate isomerase activity in three MPI genotypes in the amphipod crustacean Platorchestia platensis. Black bars are Mpiff, gray bars are Mpifs, and white bars are Mpiss. |
If one of the nominal variables is the interesting one, and the other is just a possible confounder, I'd group the bars by the possible confounder and use different patterns for the interesting variable. For the amphipod data described above, I was interested in seeing whether MPI phenotype affected enzyme activity, with any difference between males and females as an annoying confounder, so I group the bars by sex.
Similar tests
A two-way anova without replication and only two values for the interesting nominal variable may be analyzed using a paired t-test. The results of a paired t-test are mathematically identical to those of a two-way anova, but the paired t-test is easier to do. Data sets with one measurement variable and two nominal variables, with one nominal variable nested under the other, are analyzed with a nested anova.
Data in which the measurement variable is severely non-normal or heteroscedastic may be analyzed using the non-parametric Friedman's method (for a two-way design without replication) or the Scheirer–Ray–Hare technique (for a two-way design with replication). See Sokal and Rohlf (1995), pp. 440-447.
Three-way and higher order anovas are possible, as are anovas combining aspects of a nested and a two-way or higher order anova. The number of interaction terms increases rapidly as designs get more complicated, and the interpretation of any significant interactions can be quite difficult. It is better, when possible, to design your experiments so that as many factors as possible are controlled, rather than collecting a hodgepodge of data and hoping that a sophisticated statistical analysis can make some sense of it.
How to do the test
Spreadsheet
I haven't put together a spreadsheet to do two-way anovas.
Web pages
Web pages are available to perform: a 2x2 two-way anova with replication, up to a 6x4 two-way anova without replication or with up to 4 replicates.
Rweb lets you do two-way anovas with or without replication. To use it, choose "ANOVA" from the Analysis Menu and choose "External Data: Use an option below" from the Data Set Menu, then either select a file to analyze or enter your data in the box. On the next page (after clicking on "Submit"), select the two nominal variables under "Choose the Factors" and select the measurement variable under "Choose the response."
SAS
Use PROC GLM for a two-way anova. Here is an example using the MPI activity data described above:
data amphipods; input ID $ sex $ genotype $ activity; cards; 1 male ff 1.884
====See the web page for the full data set====
49 male ss 3.110 ; proc glm data=amphipods; class sex genotype; model activity=sex genotype sex*genotype; run;
The results indicate that the interaction term is not significant (P=0.60), the effect of genotype is not significant (P=0.84), and the effect of sex concentration not significant (P=0.77).
Source DF Type I SS Mean Square F Value Pr > F sex 1 0.06808050 0.06808050 0.09 0.7712 genotype 2 0.27724017 0.13862008 0.18 0.8400 sex*genotype 2 0.81464133 0.40732067 0.52 0.6025
Further reading
Sokal and Rohlf, pp. 321-342.
Zar, pp. 231-271.
References
Shimoji, Y., and T. Miyatake. 2002. Adaptation to artificial rearing during successive generations in the West Indian sweetpotato weevil, Euscepes postfasciatus (Coleoptera: Curuculionidae). Annals of the Entomological Society of America 95: 735-739.
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This page was last revised August 10, 2008. Its address is http://udel.edu/~mcdonald/stattwoway.html.
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