Analysis of Variance (ANOVA) is a powerful test for statistical significance, and it is used when we need to compare the difference of means of more than two samples. The basic purpose of ANOVA is to test the homogeneity of several means.

N-way Analysis of Variance

In ANOVA, the variation in the observed values is assumed to be due to the different levels of the independent variables, with random error accounting for the remaining variation.

Data categorized according to multiple independent variables can be examined using an N-way (or factorial) Analysis of Variance. For instance, a two-way ANOVA (two-factor ANOVA) can assess differences in participant gender and treatment effects simultaneously. An ANOVA allows for using more than two independent variables (e.g., three-way, four-way). Suppose there are significant main effects of the independent variables and interaction effects between independent variables in a data set. In that case, an N-way factorial ANOVA can demonstrate these relationships. When the effect of one independent variable depends on the value of the other independent variable, this is known as an interaction effect.

Hypothesis Testing

In order to determine whether the experiment s data are sufficient to reject the null hypothesis and declare the strategy factor statistically significant, an analysis of variance (ANOVA) procedure is provided. The null hypothesis that the population means corresponding to the various strategies are all the same can be tested using an ANOVA. Think about an experiment with only one factor (independent variable), and it has, let us say, four levels. Assume that the learning strategy is the factor and that the levels of the factor correspond to various learning strategies. The scores at each level would be the number of items correctly remembered by participants in a memory experiment. Each learning technique could be thought of as having a hypothetical population of scores representing all the scores that have been or could be obtained using the technique if the experiment were repeated. The scores obtained in the four groups can be viewed as random samples from the populations associated with the various strategies if the participants in the current experiment are appropriately selected and assigned to the learning groups.

ANOVA can take multiple factors effects into account in one analysis. ANOVA allows us to determine whether each factor in a design with two factors is significant. Additionally, we can test whether there is a significant interaction between the variables, indicating whether the two variables have a combined effect that cannot be determined by looking at each variable separately. An ANOVA tests very broad null hypotheses. The null hypothesis for tests of the main effect is that a factor s population means are all equal. The null hypothesis is that the joint effects, which cannot be obtained by adding the main effects of the factors in question, are all zero when tests of the interactions of two or more factors are being conducted.

Assumptions of ANOVA

The assumptions for two-way ANOVA are the same as for t-tests and one-way ANOVA: variance homogeneity and normally distributed data. You may put these assumptions to the test using the processes we have previously described. Remember that ANOVA is extremely robust and can thus manage all but the most egregious breaches of these assumptions. If you are certain that the assumptions are broken, you can use several non-parametric two-way ANOVA analogs. However, many of these non-parametric processes have yet to be completely developed or are not generally available in computer systems. You should also be aware that two-way (and n-way) ANOVAs are classified into three types

Fixed-effects Model I ANOVA − The researcher determines the factor levels in this two-way ANOVA. As a result, as previously stated, the components are said to be "crossed." Model, I ANOVA is illustrated in the example we just finished. This is the same Model I ANOVA we studied in the one-way ANOVA section. The interaction term is another feature that distinguishes two-way ANOVA from one-way ANOVA. We discovered that the interaction impact was not substantial in our scenario. As a result, we could test for inspanidual factor impacts. We could only conclude something meaningful about factor impacts if the interaction were substantial since the difference between levels of one factor is not consistent at all levels of the other factor.

Random-effects Model II ANOVA − This uncommon two-way ANOVA occurs when the factor levels are chosen randomly. The F-statistics are computed differently than the Model I ANOVA, and this is the same Model II ANOVA discussed in the one-way ANOVA section.

Unbalanced Two-way ANOVA (Unequal Reppcation)

ANOVA is most effective when reppcation is equal for all component levels. If you have uneven reppcation, you may still run an ANOVA. Continuing from our last example, suppose we had uneven reppcation. The data would then be entered the same manner as previously, with certain levels of the components being uneven. SAS will handle these data correctly, but you must utipze "Type III" sums of squares to obtain the required F-statistics for hypothesis testing. Examine the result from our balanced design example (in the pdf file). The SAS output has "Type I" and "Type III" sums of squares. I do not want to get into SAS “details,” but you need to know that, most simply stated, the Type III SS are the ones to use in most cases. You will always use the Type III SS when the design is unbalanced. Type I and Type III SS will be the same for balanced designs, so you can use either one in this case.

Conclusion

ANOVA designs can take a lot of different forms. A pure between-subjects design is one in which each subject provides a single score at only one combination of levels of the design s factors. A pure within-subjects or repeated-measures design is one in which each subject provides a score for each combination of levels of the factors in the design. The use of mixed designs, where a given subject provides scores at all levels of one or more within-subjects factors but only at one level of one or more between-subjects factors, is quite common. ANOVA is frequently used to examine experimental data. Because ANOVA treats all factors as categorical and uncorrelated, it is less suitable for data from observational studies