The F value could be written as F (2, 359) = 11.67. The F-value is our F-test statistic, which in this case is 11.67 with a p-value of 0.000. For a one-way ANOVA, nothing is being adjusted. Error is our within-groups variation so the degrees of freedom are n - K or 361-2 = 359. The source of the Award is our between-groups variation so the DF K -1 or 3-1 = 2. The analysis of variance table is also known as our ANOVA source table. The factor information tells us that our factor is the Award. The largest standard deviation is 151.3 and the smallest is 114.1 for a ratio of 1.33 which is less than 2. Again we can use Minitab to look at the standard deviations across the groups. These plots show that the distributions are all approximately normal.Īssumption: The population variances are equal across responses for the group levels (if the largest sample standard deviation divided by the smallest sample standard deviation is not greater than two, then assume that the population variances are equal). This assumption is met.Īssumption: The response variable is approximately normally distributed for each group or all group sample sizes are at least 30: To check this we can construct a histogram with groups in Minitab. Each student is in only one group and those groups are in no way matched or paired. The assumptions for a one-way between-groups ANOVA are:Īssumption: Samples are independent: Each student selected either Olympic, Academy, or Nobel. Tukey Simultaneous Tests for Differences of Meansġ0.5 - Example: SAT-Math Scores by Award Preference 10.5 - Example: SAT-Math Scores by Award Preferenceġ. Means that do not share a letter are significantly different. Pooled StDev = 2.75386 Grouping Information Using the Tukey Method and 95% Confidence Factor Next, you will also learn how to obtain these results using Minitab.įor each pairwise comparison, \(H_0: \mu_i - \mu_j=0\) and \(H_a: \mu_i - \mu_j \ne 0\).ġ0.4 - Minitab: One-Way ANOVA 10.4 - Minitab: One-Way ANOVAĮqual variances were assumed for the analysis Factor Information Factor In the examples later in this lesson you will see a number of Tukey post-hoc tests. This analysis takes into account the fact that multiple tests are being performed and makes the necessary adjustments to ensure that Type I error is not inflated. In this class we will be relying on statistical software to perform these analyses, if you are interested in seeing how the calculations are performed, this information is contained in the notes for STAT 502: Analysis of Variance and Design of Experiments. This specific post-hoc test makes all possible pairwise comparisons. Most statistical software, including Minitab, will compute Tukey's pairwise comparisons for you. Here, we will learn about one of the most common tests known as Tukey's Honestly Significant Differences (HSD) Test. There are many different post-hoc analyses that could be performed following an ANOVA. Post-hoc tests are conducted after an ANOVA to determine which groups differ from one another. In order to determine which groups are different from one another, a post-hoc test is needed. While the results of a one-way between groups ANOVA will tell you if there is what is known as a main effect of the explanatory variable, the initial results will not tell you which groups are different from one another. For \(k\) independent groups there are \(\frac\). This procedure is known as a one-way between groups analysis of variance, or more often as a "one-way ANOVA." Why not multiple independent t-tests?Ī frequently asked question is, "why not just perform multiple two independent samples \(t\) tests?" If you were to perform multiple independent \(t\) tests instead of a one-way between groups ANOVA you would need to perform more tests. In this lesson, we will learn how to compare the means of more than two independent groups. In previous lessons you learned how to compare the means of two independent groups.
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