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One-Way Analysis of Variance (ANOVA)

The basic underlying idea is to compare the variability of the observations among groups to that within groups:

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Here is the analysis of variance:

In some cases, you can choose the values of the predictivevariables, because they just describe the conditions inwhich the experiment was conducted.

Actually, this is merely a linear regression of each value of thequalitative variable.

The sum of squares for the variationis either given by the symbol SSW (sum of square within)
or SSE (sum of square for error).

Complete the following analysis of variance table: Source S.S.

Notice that in the first example, where the null hypothesis is true, the variability of Lifesavers (or anxiety levels) each group was the same as the variability of Lifesavers the groups. The extent to which there were different kinds of Lifesavers within each group roughly corresponds to the within-group estimate of the population variance. The extent to which the composition of Lifesavers in each group differed from the composition of the other groups corresponds to the between-group variance. (Technically, it is variance several groups and should be called, but traditionally the word "between" has been used.) When the null hypothesis is true, the variance equals the variance This is so, because the means of the groups differ by chance (due to sampling error) much like the individual scores within the groups differ by chance (due to the same sources of sampling error). Notice that in the second example, where the null hypothesis was false, the groups differed from each other more so than did the Lifesavers within each group. Since students who said "red" tended to have mostly red Lifesavers, the variability of Lifesavers within the "red" group was very small. Since students who said "blue" tended to have mostly blue Lifesavers and did not have blue, green, or yellow ones, the variability within the "blue" group was small. The variability among the groups (between-group variance) is much larger, because "mostly red" is very different from "mostly blue." If you think of the different Lifesaver colors as different levels of interpersonal anxiety, then anxiety scores were similar within each attachment group, and then were different across the groups. Thus, a group with mostly low anxiety scores would have a low mean score, whereas a group with mostly high anxiety scores would have a high mean score, and the difference (variance) group means would be larger than the variance among individual scores the groups.

This example illustrates a case where the F-statistic is large enough to be significant, but where the null hypothesis of sigificant differences in group variances can not be rejected.

This is precisely the null hypothesis of the F-test in this analysis.

Example 4 shows how (relative to Example 3) the presence of larger within-group variances here reduces the apparent significance of the F statistic (to the point of non-significance). It is harder to demonstrate differences among means when the variability within groups is larger.

When you have several quanlitative variables, interactionscan start playing an important role. For instance, it ispossible that the mean of Y does not depend on X1 nor on X2,but on the pair (X1,X2). For instance

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  • While the analysis of variance reached fruition in ..

    Source of Variation

  • Hypothesis Testing - Analysis of Variance (ANOVA)

    Instead of a univariate value, we would obtain amultivariate value (' λ λ

  • Analysis of variance (ANOVA) - Hypothesis Testing - …

    Remember that the error variance is computed (SS error) byadding up the sums of squares within each group.

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Introduction to Analysis of Variance - Free Statistics Book

What separates ANOVA from other statistical techniques is that it is used to make multiple comparisons. This is common throughout statistics, as there are many times where we want to compare more than just two groups. Typically an overall test suggests that there is some sort of difference between the parameters we are studying. We then follow this test with some other analysis to decide which parameter differs.

Analysis of Variance (ANOVA) - SlideShare

This last example demonstrates a situation that often arises in practice–the variabilty of two groups of data may differ more than the means do. (This could be more scientifically meaningful that obsrving a simple difference among means.)

Analysis of Variance (ANOVA) - Definition - ThoughtCo

In contrast, the p-value for the homogeneity of variance test, 0.1397, is large enough (i.e. greater than 0.05) to suggest that there is little support for rejecting the null hypothesis that the variances of the data in the individual groups is equal.

Analysis Of Variance (ANOVA) - Statistics Solutions

In practice, one would proceed by discussing the analysis of variance anyway (even though its assumptions are violated), because that’s what people do!

ANOVA 2 | Analysis Of Variance | Null Hypothesis

Know how to construct an ANOVA Table.The various statistics computed from the analysis of variance abovecan be summarized in an ANOVA Table
as shown below: These summaries are then used to draw inference aboutthe various samples or treatments of which we are studying.

Multivariate Analysis of Variance (MANOVA)

The sum of squares for the variation is either given by the symbol SSB (sum of squares between)
or SSTR (sum of squares for treatments) and is the explainedvariation.

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