Multi-Factor ANOVA Online: Main Effects and Interactions
What Is Multi-Factor ANOVA?
Multi-factor ANOVA examines the influence of two or more factors on a response variable simultaneously, testing not only each factor's main effect but also the interactions between factors, that is, whether the effect of one factor changes with the level of another. For example, when studying the effect of temperature and pressure on strength, a temperature x pressure interaction exists if the temperature effect differs between high and low pressure. The sum-of-squares decomposition splits the total variation into factor and interaction terms, each tested with an F-test.
Fixed Versus Random Effects
A factor is fixed when its levels are chosen deliberately by the experimenter and conclusions apply only to those levels (for example, three materials or two temperature settings). A factor is random when its levels are drawn at random from a larger population and conclusions are meant to generalize to that population (for example, five randomly selected machines or three randomly chosen raw-material batches). Random-effect factors have different expected mean squares and different F-test denominators than fixed effects; the tool gives the correct tests based on your selection, and choosing the wrong type directly affects the validity of the F and p values.
How to Read Interactions
When an interaction is significant, interpreting main effects alone can mislead, so look at the interaction plot: parallel lines indicate no interaction, while crossed or clearly non-parallel lines indicate a significant interaction. The tool outputs interaction plots and a table of means at each factor level to help you judge the practical direction of the effects. With a significant interaction, examine simple effects by splitting one factor's levels and evaluating the other factor within each level, and assess whether the interaction has engineering meaning. Always keep the interaction term in the analysis.
How to Use It (Step by Step)
Log in and organize the data in a factor x factor x replicate structure (one row per observation with factor levels and the response), choose the factor type (fixed or random) and whether to include interactions, and the tool outputs the ANOVA table (SS, df, MS, F and p for main effects, interactions and error) plus effect plots. Check residual normality and variance homogeneity first, transform the response if needed, and use the DOE tool before the experiment to plan factor levels and replication.