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Fully replicated factorial ANOVAPrinciples Fixed effects model Formulae Mixed effects model Factorial design in randomized blocks Nested crossfactored design AssumptionsPrinciplesWe define a factorial design as when you have fully replicated measures on two or more crossed factors. Note that there are several other definitions of a factorial design in the literature. For example Sokal & Rohlf (1995) only use the term when there are more than two treatment factors. You can also find the (very odd) term 'one way factorial ANOVA' used quite widely in the literature, presumably to mean one factor ANOVA. We only use the term factorial design to describe a design with replicated measures on two or more crossed factors. In a factorial design multiple independent effects are tested simultaneously. Each level of one factor is tested in combination with each level of the other(s) so the design is orthogonal. The analysis of variance aims to investigate both the independent and combined effect of each factor on the response variable. The combined effect is investigated by assessing whether there is a significant interaction between the factors. If there is no interaction, the effect on the response variable of certain levels of two factors acting together will equal the sum of the responses to the same levels of the two factors in isolation. If interaction is present, the effect of the two factors together may be greater or less than the sum of the two in isolation. Use of interaction plots In the analysis one first calculates sums of squares for the main effects. These are the effect of each independent variable on the response variable, ignoring the effects of all other independent variables. Only then do we consider the interactions which are the variablity leftover after main effects have been considered. Since the sums of squares are accounting for components in sequence they are known as sequential or Type I SS. This is important because (as we shall see below) there are other ways to calculate sums of squares that we will encounter when we look at unbalanced designs. When it comes to interpreting those interactions the process works in the opposite direction. One first examines the interaction terms. If an interaction is significant, one compares levels of one factor at each level of the other factor. These are (rather confusingly) known as the simple main effects. Generally the main effect (the average level irrespective of the other factor) is only of interest if the interaction is not significant. However, if there is only slight interaction one may still be interested in average level. One should always bear in mind that any factorial ANOVA has less power to detect an interaction than it has to detect a main effect. For example for a 2 × 2 factorial one would need four times as many experimental units to have the same power to demonstrate the interaction as a main effect. This is important if the main interest is in detecting an interaction (see our example on detecting the interaction between temperature and carbon dioxide on plant Factorial treatment structures can either be used in a completely randomized design or as part of a variety of other designs. We provide here the mathematical model and computational details for the designs we covered in the core text (the completely randomized and randomized complete block designs). We also consider the nested crossfactored design and (in a related topic) the thorny issue of nonorthogonal (unbalanced) factorial designs.
Factorial in completely randomized designFixed effects modelWe first consider an experiment where we have two (or more) fixed factors. Treatment combinations are assigned at random to experimental units.
The model for analysis of variance of this design is given below.
The Fratio for factor A is obtained by dividing MS_{A} by MS_{Error}. The Pvalue for this Fratio is obtained for The Fratio for factor B is obtained by dividing MS_{B} by MS_{Error}. The Pvalue for this Fratio is obtained for ab − 1 and ab(n − 1) degrees of freedom. The Fratio for the A × B interaction is obtained by dividing MS_{A×B} by MS_{Error}. The Pvalue for this Fratio is obtained for ab − 1 and ab(n − 1) degrees of freedom. It is not uncommon to find (usually after a certain amount of introspection) that one has only one independent observation of each combination rather than the anticipated 'n' replicated observations (see nested cross factored design below). One can analyze such designs if one assumes one or more of the interaction terms is zero, and then use the interaction term as the error. This is equivalent to the randomized complete block design which is a twoway factorial anova with only one replicate per cell. Computational formulaeWe will take a balanced experiment with 'a' levels of Factor A, 'b' levels of factor B, with 'n' replicates of each factor combination. Factor A totals are denoted as TA_{1} to TA_{a}, Factor B totals as TB_{1} to TB_{b}, subtotals (AB combinations) as T(AB)_{1} to T(AB)_{ab} and the grand total as G. The factor A, factor B, A × B interaction, error and total sums of squares are calculated as follows:
Mixed effects modelIn field work a more common application of factorial ANOVA is the mixed effects model. This is sometimes used to analyze data from a generalized randomized block desig, for example when completely randomized one factor experiments are repeated in multiple locations (= blocks). This approach is recommended by Underwood There appears to be no simple answer to this matter. We would tend to support the treatment nested within block approach, but have to admit that most recent texts seem quite happy to treat it as a fully replicated factorial design. (Obviously if one has only one replicate of each treatment in each block the issue does not arise  you have to analyze it as a randomized complete block with all the necessary assumptions.)
The Fratio for (fixed) factor A is obtained by dividing MS_{A} by MS_{A×B}. We have highlighted the MS_{A×B} term as this is the main difference between the fixed and mixed effects models. The Pvalue for this Fratio is obtained for The Fratio for (random) factor B is obtained by dividing MS_{B} by MS_{Error}. The Pvalue for this Fratio is obtained for b − 1 and ab(n − 1) degrees of freedom. The Fratio for the A × B interaction is obtained by dividing MS_{A×B} by MS_{Error}. The Pvalue for this Fratio is obtained for (a− 1)(b− 1) and ab(n − 1) degrees of freedom. If there is little evidence for interaction (in other words if P > 0.25) , then it is permissible pool the interaction mean square with the error mean square to enable more powerful main effects tests of both A and B with mean square error as the denominator. But if there is any indication of interaction, the Fratio test for factor A over the A × B interaction is a quasi Fratio and provides only an approximate test of the main effect.
Factorial in randomized blocks
The Fratio for (fixed) factor A is obtained by dividing MS_{A} by MS_{S × A}. The Fratio for (fixed) factor B is obtained by dividing MS_{B} by MS_{S × B}. The Fratio for the A × B interaction is obtained by dividing MS_{A×B} by MS_{Error}. Note we do not have to assume there is no blocks × treatment interaction, but we do have to assume that there is no threeway interaction. If replication (number of blocks) is inadequate, the model will have low power to identify treatment effects.
The Fratio for (fixed) factor A is obtained by dividing MS_{A} by MS_{Error}. The Fratio for (fixed) factor B is obtained by dividing MS_{B} by MS_{Error}. The Fratio for the A × B interaction is obtained by dividing MS_{A×B} by MS_{Error}. Note we now have to assume there are no blocks × treatment interactions, even though we can estimate those interactions. You will have more power to identify treatment effects, but that power is bought at the cost of adopting the Nelson approach! Nested crossfactored design
The Fratios for factors A and B and the A × B interaction are obtained by dividing their respective mean squares by MS_{C(A ×B)}. This design is a common cause of massive pseudoreplication because the evaluation units are taken as the experimental (or sampling) unit. In other words the main effects and interaction are wrongly tested over the error mean square rather than the C(A × B) mean square. Unfortunately some authorities argue in favour of pooling the error mean square and the C(A × B) mean square, if the Pvalue for C (A × B) is greater than some prespecified value, say > 0.25. The effect of this recommendation is to make respectable those experiments where there is little or no genuine (independent) replication  simply because one has very low power for the test of C(A × B) if the number of true replicates (c) is very small. Hence we follow Hurlbert's point of view that this is simply pseudoreplication in another guise.
AssumptionsThe same assumptions as for a onefactor ANOVA must also hold for factorial ANOVA, namely:
