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.

Dispersion sources in the intergroup two-factor plan

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, and the factor In is fixed.

Structural models for two-way ANOVA for an intergroup plan

Factors

A is fixed

A is random

In is fixed

Model I

Model II

In is random

Model II

Model III

We will assume that every value of the dependent variable includes five additive parameters:

• population mean μ, constant for the whole population;

• the effect of the factor A - α;

• the effect of the factor B - β;

• the effect of the interaction of the factors A and In - αβ;

• the effect of the experimental error is ε.

In other words, we will assume that the randomly chosen result of the subject k in the group ij can be decomposed into the following components:

As we can see, we again meet with the application in the experimental work of general linear models, now even more complex.

Denote by the capital Latin letter the number of factor levels in the general population, and the lowercase in the sample. Thus, P will denote the number of levels of the factor A in an endless experiment, or, what is the same, the general of the aggregate. Accordingly, the letter Q will denote the number of levels of the factor In in the same infinite experiment where the researcher can go over the maximum possible number of levels of this factor. Then the lowercase letters p and q will denote the number of factor levels A and B, which is explored in a real experiment, i.e. selective values ​​of independent variable levels.

Theoretically, it is proved that the mean squares, variances, of the basic experimental effects of the A and B factors should be described by the following relationships:

When A and In are fixed , i.e. when p = p and q = Q, these relations must obviously be rewritten as follows:

If the values ​​of the factors A and B, that will be investigated in the experiment, are randomly selected from an infinite the number of their possible values ​​in the general population, it is logical to assert that the selective number of levels of the independent is much less than an infinitely large number of their values ​​in the general population. Then we can conclude that the ratio of selective values ​​of the number of levels of independent variables to their population values ​​is negligibly small from zero. Thus, the average squares of the main effects in the case of randomness independent variables should theoretically be described by the following equations:

As for the variance of the interaction, its estimated value in the experiment should be described by the following relation:

Finally, the expected value of the experimental error will be expressed as:

Table 5.3

Theoretically expected mean square values ​​of the main effects of independent variables, their interaction and experimental error in the two-factor intergroup experiment

Average square

Model

fixed (model I)

mixed (model II)

random (model II)

In Table. 5.3 reflects the expected values ​​of the estimated variance of the main effects, their interaction, and the experimental error for the three possible structural models of two-factor analysis of variances in relation to the intergroup plan. Again, in this case, the mixed model implies chance of the A strong> and fixed factors B.

Now we can identify statistical hypotheses that are tested in a two-factor experiment. It is clear that they concern the main effects of factors A and B and their interactions. Assuming the equality of these effects at all levels of independent variables, the null hypotheses will look like this:

Obviously, each of these hypotheses assumes a zero value of the effect of each of the experimental factors and their interaction, although the values ​​of α, β and αβ themselves may differ from zero ones. In other words,

Alternative hypotheses affirming the inequality of the main effects of independent variables and their interactions at different levels of these variables will look like this:

Then, to assess the statistical reliability of the effects found in the experiment, guided by the theoretical assumptions of the structural models reflected in Table. 5.3, we can formulate the following rules for constructing F -relations.

Fixed model (model I). All effects (average squares) are estimated relative to the intra-group variance, which is nothing more than the variance of the experimental error

Mixed model (model 11). The effect of a random factor is evaluated relative to its interaction with a fixed factor, and the effect of a fixed factor and the interaction of independent variables are estimated relative to the intra-group variance.

• The random model (model III). The main effects are evaluated relative to their interaction, and the interaction of the independent variables is evaluated with respect to the intra-group variance.

These rules can also be formulated in a different way: all effects in the two-factor intergroup plan need to be evaluated with respect to the intra-group variance, except for the case when the effect of a random variable is evaluated; In this exceptional case, statistical reliability is determined relative to the interaction of a random variable with another independent variable.

It should be specially noted that the evaluation of the effects of random variables relative to the mean square for the interaction is meaningful only in the case when the effect of the interaction itself is sufficiently pronounced. If it is poorly expressed, it is necessary to first assess the statistical reliability of this effect. If this estimate gives a negative result, in evaluating the effect of a random variable, you must act in the same way as when evaluating the effects of a fixed variable.

As in the case of single-factor plans, the constructed F-ratio in the case of the validity of the null hypothesis should theoretically be equal to unity. And the statistics itself should be described by the law F -distributions. If the alternative hypothesis is correct, the value F -statistics should be greater than one. As far as statistically reliable, this excess in the situation of practical data processing, as usual, can be estimated using the F -distribution function. With manual The statistical tables can be used for this purpose (see Appendix 4). Once again, we pay attention to the fact that no structural model assumes a value F -statistics that is less than one. If in practice this happens, it means automatic and unconditional accepting the null hypothesis. The statistical reliability of such a result is not investigated and is not stated in the experimental report. Such actions will be meaningless from the point of view of the investigated structural models.

thematic pictures

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