{"id":595,"date":"2021-08-14T13:08:01","date_gmt":"2021-08-14T17:08:01","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/statspsych\/?post_type=chapter&p=595"},"modified":"2022-06-04T03:29:52","modified_gmt":"2022-06-04T07:29:52","slug":"chapter-9","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/statspsych\/chapter\/chapter-9\/","title":{"raw":"9. Factorial ANOVA and Interaction Effects","rendered":"9. Factorial ANOVA and Interaction Effects"},"content":{"raw":"

9a. Factorial Analysis<\/h1>\r\n\"\"\r\n\r\nIn factorial analysis, just like the fractals we see in nature, we can add multiple branchings to every experimental group, thus exploring combinations of factors and their contribution to the meaningful patterns we see in the data.\r\n\r\nIn this chapter we will tackle two-way Analysis of Variance and explore conceptually how factorial analysis works. To understand when you need two-way ANOVA and how to set up the analyses, you need to understand the matching research design terminology. We will also need to define and interpret [pb_glossary id=\"635\"]main effects[\/pb_glossary]<\/strong> and [pb_glossary id=\"636\"]interaction[\/pb_glossary]<\/strong> effects, both of which can be analyzed in a factorial research design. Later we will approach the detection and interpretation of interaction<\/strong> effects, specifically, which will really help you see the extraordinary complexity of information factorial analyses can offer.\r\n\r\n\"\"Factorial analyses such as a two-way ANOVA are required when we analyze data from a more complex experimental design than we have seen up until now. Specifically, when an experiment (or quasi-experiment) includes two or more independent variables (or [pb_glossary id=\"624\"]participant variables[\/pb_glossary]<\/strong>), we need factorial analysis. \"\"Examples of designs requiring two-way ANOVA (in which there are two factors) might include the following: a drug trial with three doses as well as the sex of the participant, or a memory test using four different colours of stimuli and also three different lengths of word lists.\r\n\r\nAs we saw in the chapter on Analysis of Variance, the total variability among scores in a dataset can be separated out, or partitioned, into two buckets. The first bucket, often called between-groups variance or treatment effect, refers to the systematic differences caused by treatments or associated with known characteristics. \"\"These are the differences among scores we are hoping to see \u2014 the explained differences \u2014 and thus I casually refer to this as the \u201cgood\u201d bucket of variance and colour code it in green. The other bucket, often called within-groups variance or error, refers to the random, unsystematic differences that cannot be explained by the research design. These are the unexplained individual differences that represent the noise in the data, obscuring the signal or pattern we are looking for, and thus I casually refer to it as the \u201cbad\u201d bucket of variance and colour code it in red.\r\n\r\nWe can revisit our visual example from before, in which the goal is to separate colour swatches according to some factor, such that the colours within each grouping (or level) is more uniform. If we first sort the colours according to the factor of hue, let\u2019s say into green or blue hues, then we explain some of the overall variability. But if we add a second factor, brightness, then we can explain even more of the differences among the colour swatches, making each grouping a little more uniform.\r\n\r\n\"\"\r\n\r\nClearly there is still some work to be done, and if in factor A we could have included a third level of \u201cred\u201d, the uniformity would have been much improved. And with factorial analysis, there is technically no limit to the number of factors or the number of levels we can employ to explain away the variability in the data. The more variance we can explain, through multiple factors and\/or multiple levels, the better! This is what we will be able to do with two-way ANOVA and factorial designs.\r\n\r\nYou will recall the jargon of ANOVA, including factors and levels. To grasp factorial research designs, it becomes even more important to develop comfort with these concepts, so that you can identify and describe the design and thus the requisite analysis setup. Let us suppose that we have a research study that measures the effect of a placebo, a low dose and a high dose of the drug, and also takes into account whether the participants were male or female. The first factor could be succinctly identified as \u201cdrug dose\u201d, and the second factor as \u201csex\u201d. In another example, perhaps we show participants words in black, red, blue or green, and we also take into account whether the word list presented is long, medium, or short. What would you call each of those two factors?\r\n\r\nWhat if, in a drug study, you notice that men seem to react differently than women? If you have that information (male\/female), you can use it in your ANOVA and see if you can put more variance in your \u201cgood\u201d bucket.\r\n\r\n\"\"\r\n\r\nIn the design illustrated here, we see that it is a 3 x 2 ANOVA. There are three levels in the first factor (drug dose), and there are two levels in the second factor (sex). This notation, that identifies the number of levels in each factor with a multiplier between, helps us see clearly how many samples are needed to realize the research design. In this example, we would need six samples in total, each of which would need to have a good enough sample size to allow for the central limit theorem to justify the normality assumption (N=30+). That is a lot of participants! However, we could learn much more by including both factors, if indeed the sex of the participant is associated with a different response to the drug.\r\n\r\n\"\"We can see an example of a 4x3 two-way ANOVA here, with our example of word colour and length of list. Altogether, this design would require 12 samples.\r\n\r\nAnd just for the sake of showing you the potential of factorial analyses, you could also impose a third factor on the design: the age of the participants. In this case, you have a 4x3x2 design, requiring 12 samples. At 30 participants each, that would be 30x12=360 people! You can appreciate how each factor exponentially increases the practical demands (costs) of the research study. For this reason, a cost-benefit analysis must be carefully applied in factorial research design, such that the minimum complexity is used to answer the key research questions sufficiently.\r\n\r\nIn a two-way ANOVA, just as in a one-way ANOVA, we calculate various flavours of Sums of Squares (SS).\"\" The SS total is broken down into SS between and SS within. However, with a two-way ANOVA, the SS between must be further broken down, because there are now two different factors that can have a main effect<\/strong> (i.e., can explain some of the total variance). Also, with more than one factor, there can be an interaction<\/strong> between the two that itself uniquely accounts for some of the variance. So now, we can SS row (the first factor), SS column (the second factor) and SS interaction<\/strong>. For each SS, you can also see the matching degrees of freedom.\r\n\r\nHere is the full ANOVA table expanded to accommodate the three subtypes of between-groups variability.\r\n\r\n\"\"\r\n\r\nNote that all of the Sums of Squares and degrees of freedom still should add up to the total. As you can see, there will now be three F-test results from this one omnibus analysis, one for each of the between-groups terms. Each can be compared to the appropriate degrees of freedom to determine the statistical significance of the degree to which that factor (or interaction) accounts for variance in the dependent variable that was measured in the study.\r\n\r\nTo help you interpret the formulas as they reference row means, column means, and cell means, I have added a diagram here to help you see how to locate these numbers in a 2x2 two-way ANOVA scenario.\r\n\r\n\"\"\r\n\r\nThe row and column means, the averages of cell means going across or down this matrix, are often referred to as marginal means (because they are noted at the margins of the data matrix).\r\n\r\nWhen it comes to hypothesis testing, a two-way ANOVA can best be thought of as three hypothesis tests in one. For each factor, and also for the interaction<\/strong> of the two, you need to identify populations and hypotheses, cutoffs, calculate the SS between, degrees of freedom, variance between, and F-test results. All three will share the same error terms, the SS, degrees of freedom, and variance within groups. As you can imagine, the complexity of calculating such an analysis could be daunting, but a systematic, organized approach and the use of the ANOVA table keeps it well under control.\r\n\r\nAs with one-way ANOVA, if any factor has more than two levels, you may need to calculate pairwise contrasts for that factor to determine where exactly a significant difference among group means lies. Even with a 2x2 ANOVA, the interaction<\/strong> effect has four possible pairwise comparisons to investigate, and that would require a planned contrast or post-hoc test. The same rules apply to such analyses as before: they may only be conducted if there is a significant overall ANOVA result, and the experimentwise risk of Type I error must be controlled.\r\n\r\nWe can continue building our statistical decision tree to help us decide which test to use when we examine a research question\/design. If we have two independent variables (factors) in the experimental design, then we need to use a two-way ANOVA to analyze the data.\r\n\r\n\"\"\r\n\r\nBefore we move on to detecting and interpreting main effects<\/strong> and interactions<\/strong>, I would like to bring in two cautions about factorial designs. Many researchers new to the trade are keen to include as many factors as possible in their research design, and to include lots of levels just in case it is informative. This is an understandable impulse, given how much effort and expense can go into designing and conducting a research study. We want to gather as much information as possible from that effort! However, as we saw before, the more factors we add in, the more participants we need to ensure a decent sample size in each cell of our data matrix. There is another important element to consider, as well. For each factor we add in, we add interaction<\/strong> terms. If we were ambitious enough to include three factors in our research design, we would have the potential for interaction<\/strong> effects among each pair of the factors, but we would also potentially see a three-way interaction<\/strong> effect.\r\n
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Interactions for a three-way ANOVA<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nIn a three-way ANOVA involving factors A, B, and C, one must analyze the following interactions:\r\n