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9.3: One-Way ANOVA

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  • The purpose of a one-way ANOVA test is to determine the existence of a statistically significant difference among several group means. The test actually uses variances to help determine if the means are equal or not. In order to perform a one-way ANOVA test, there are five basic assumptions to be fulfilled:

    1. Each population from which a sample is taken is assumed to be normal.
    2. All samples are randomly selected and independent.
    3. The populations are assumed to have equal standard deviations (or variances).
    4. The factor is a categorical variable.
    5. The response is a numerical variable.

    The Null and Alternative Hypotheses

    The null hypothesis is simply that all the group population means are the same. The alternative hypothesis is that at least one pair of means is different. For example, if there are k groups:

    \(H_{0} : \mu_{1}=\mu_{2}=\mu_{3}=\ldots \mu_{k}\)

    \(H_a\): At least two of the group means \(\mu_{1}, \mu_{2}, \mu_{3}, \dots, \mu_{k}\) are not equal. That is, \(\mu_{i} \neq \mu_{j}\) for some \(i \neq j\).

    The graphs, a set of box plots representing the distribution of values with the group means indicated by a horizontal line through the box, help in the understanding of the hypothesis test. In the first graph (red box plots), \(H_{0} : \mu_{1}=\mu_{2}=\mu_{3}\) and the three populations have the same distribution if the null hypothesis is true. The variance of the combined data is approximately the same as the variance of each of the populations.

    If the null hypothesis is false, then the variance of the combined data is larger which is caused by the different means as shown in the second graph (green box plots).

    The first illustration shows three vertical boxplots with equal means. The second illustration shows three vertical boxplots with unequal means.

    Figure 9.3 (a) \(H_0\) is true. All means are the same; the differences are due to random variation. (b) H0 is not true. All means are not the same; the differences are too large to be due to random variation.