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Is ANOVA a model or test?

Analysis of variance (ANOVA) models apply to data that occur in groups. The fundamental ANOVA model is the one-way model that specifies a common mean value for the observations in a group. The analysis of variance associated with the one-way model is presented.

sciencedirect.com - Analysis of Variance - an overview | ScienceDirect Topics

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Analysis of variance (ANOVA)

All of the previous tests have concerned two groups of observations. When there are three or more groups, one tests the null hypothesis that there is no difference in the group means by examining the variances. Consider the data, shown in Table 9.7, from three groups of four patients. There are several sources of variation in this sample. First, there is the total variation of the whole sample. To calculate this value one calculates the sample variance, ignoring the group to which each measurement belongs. This is sometimes called the total mean squares (MS total ) or total variance. Second, there is the variation of the group means about a grand mean of all observations. This is sometimes called the between-groups variance (MS between ) or sometimes MS treat . Finally, there is the variation between the group measurements and their individual group means. This is sometimes called the within-groups mean squares or residual mean squares (MS residual ) or within-groups variance. The relationship between these sources of variation is: SS total = SS between + SS residual ( MS = SS df ) If each of the groups is drawn from the same population with equal population means, the between-groups variance will be comparable to the within-groups variation (MS residual ). Alternatively, if the three groups have different population means, the between-groups variation will be large compared with the within-groups variance. In order to test which one of these situations is more likely, we use the test statistic F. F = Variation between samples Variation within samples = MS between MS residual In the example above the total variance is calculated from the whole sample (as if they were not in groups). The between groups sum of squares is the sum of squared deviations between the group means and the overall mean multiplied by the number of observations in each group. The residual sum of squares is calculated usually by subtraction. Mean squares are the sum of squares (SS) divided by the appropriate degrees of freedom. Most statistical packages when performing an ANOVA will produce an output similar to that shown in Table 9.8. The sum of squares is just the sum of the squared difference between each value and its corresponding mean. By dividing by the degrees of freedom, we can calculate the within-groups (MS residual ), between-groups (MS between ) and total variance. In the above example, F = 11.9, which is significant, being less than 0.05. We can therefore reject the null hypothesis that there is no difference between the groups.

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ANOVA can also be extended to the analysis of data which can be classified in a number of ways. For example, in an observational study measuring memory performance scores, patients may be classified by diagnosis, sex and treatment. If one wished to compare memory score by diagnosis, a one-way ANOVA (as above) could be conducted. However, if gender and treatment also affected memory score, the difference might not be due solely to the effect of diagnosis alone. To avoid this potential pitfall, a factorial ANOVA can include any number of the factors in a single experiment. The resulting analysis could give the effect of each factor independently, but can also provide information about interactions between factors. For example, a factorial ANOVA could detect that memory scores may be impaired in males with schizophrenia but not females, whereas a one-way ANOVA might fail to detect any differences. ANOVA • Is a parametric statistical test • Tests the null hypothesis that the mean values of three or more independent groups are equal • The test statistic F is the ratio of the between-groups to within-groups variance • The value of F and the two degrees of freedom should always be stated • ANOVA has a non-parametric equivalent called the Kruskal–Wallis test An analysis of variance can be extended to include paired values from the same samples when it is called a repeated measures ANOVA. Where data can be classified in several ways (e.g. by group and gender) the appropriate statistical test is the factorial ANOVA. An analysis of covariance (ANCOVA) is used when we wish to see if the mean of a variable differs across three or more groups, while taking into account a possible confounder. If, for example, one examined cognition in the three groups of subjects above, their performance may be confounded by their premorbid general intellectual ability or their age. In order to take into account these factors we can either use a regression analysis (see later) or use ANCOVA, where IQ or age or both would be covariates.

Table 9.9 summarises tests for differences, for various data types.

sciencedirect.com - Analysis of Variance - an overview | ScienceDirect Topics

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