C532: ANOVA & Regression Modeling Nov 29, 2011 Y. Cao 1Topics: learn aboutfactorial design and two-way ANOVA model. Datasets: np423aug2007.sav, C530_clbp.sav Two wayANOVA, we mean that we have 2 explanatory variablesof categorical type, x1and x2. A factorial design, by definition, we mean that it’s an experiment designed to explore the effects of several (2 or more) treatment factors on a single outcome (or dependent as in SPSS) variable. Each factor (because it affects the outcome so we call it a factor) takes a small number of different values (typically 2, 3 or 4), which may or may not be quantitative and if it is NOT quantitative we call the values of the factor (variable) levels of the factor. The main effects(because it is the primary factors in the study) compare the mean values of the observations at the different levels of a factor, while the interactioncombines effects of two variables. The equation may look like The sum of separate effects of the two variables in an interaction isnotequivalent to that of the combined effects of them. The main effect is the mean difference at different levels for the outcome. In some experiments, the difference in outcome between the levels of one factor is not the same at all levels of the other factor(s). When this happens, we say that there is an interaction between the factors. A significant interaction (with p value < 0.05 for the term) may mask the significance of main effects if there is one. Therefore, if there is an interaction, the main effects of the factors involved in the interaction may not so meaningful. We cannot report it with clear explanations, that is, we do not report the main effects under this situation! Based on these definitions (you can see that mathematics is something that consists of a series of definitions!), if the interaction has significant affect on the outcome then it makes the explanation of individual main effects harder or even impossible. That is, the two factors (variables) only affect the outcome “simultaneously” but not separable! Usually, we use two-way interaction like x1*x2for our design but theoretically we can have three way interaction like x1*x2*x3or even high order interactions.
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