# Two-factor analysis of variance, Structural models of two-factor...

## Two-way variance analysis

So, we saw that the two-factor plan allows you to explore three effects of independent variables: the two main effects of factors and one interaction between these two independent variables. In addition, there is a statistical (experimental) error in the two-factor intergroup plan that determines the intra-group differences between subjects in the same group. These differences are not related to the experimental effects studied.

Thus, the entire variance observed in experiments of this type can be decomposed into two additive parts: 1) the variance that determines the variability of the data between the groups of subjects The variance of data between groups of subjects, in turn, can be decomposed into three more additive parts: variance, the source of which is each of the experimental factors in separate spine, and the variance, defined by their interaction. This reasoning is illustrated graphically in Fig. 5.1. Fig. 5.1. Dispersion sources in the intergroup two-factor plan

Let Then, keeping the conventions about the mean values ​​adopted in paragraph 5.1 on each level of independent variables and the general average, the variances (average squares) of the basic effects of independent variables can obviously be estimated as follows: The average square of the interaction of the factors A and In is estimated using the following formula: Finally, the statistical error effect, reflecting the intra-group variations of the data, can be estimated as follows: As usual, in order to assess the statistical reliability of the experimental effects, it is necessary to build the F -relations correctly. However, in the case of factorial plans, the construction of such statistics proves to be ambiguous. The fact is that the rules for constructing F -relations differ depending on the chosen structural model, and it, among other things, is determined by the way in which independent variables - as random or as fixed.

## Structural models of two-way variance analysis

Recall that fixed refers to such independent variables that in the experiment take all possible values. This is the most common case, and, as a rule, the statistical packages used for data processing are configured by default to process such data. It happens, however, that the researcher can not in the experiment grasp all possible values ​​of the factors. For example, researchers may be interested in word identification processes, and the words themselves are treated as levels of an independent variable. It is clear that the researcher in the experiment is always limited in the choice of words of natural language due to their extreme multiplicity. In this case, he randomly selects from the dictionary a limited set of words, which will represent a random sample of the levels of the corresponding factor. Such an independent variable will be random.

Depending on what factors of the two-factorial experimental plan will be considered as fixed or random, we can distinguish three possible structural models (Table 5.2):

• a completely fixed model in which both independent variables are fixed (model I),

• A mixed model in which one of the experimental factors turns out to be fixed, and the second one is random (model II),

• completely random model, where both factors are considered by the researcher as random (model III).

Since both experimental factors, whose effects on the dependent variable are investigated in the experiment, are equivalent from the point of view of organizing and planning this experiment, in the future we will consider that in the mixed model the factor A is random, an