ANOVA: One-Way Analysis of Variance

Transcription

1 ANOVA: One-Way Analysis of Variance This is the kind of statistical analysis we do when we want to talk about more than two means at once (i.e., with 2 means, we use a form of the t test). The simplest kind of design is where we have 1 independent variable with 3 or more levels. For example, we might want to do a study on the effects of caffeine on test performance where: Level 1: One group gets no caffeine (the Control group), Level 2: A second group gets a low dose (Mild Buzz group) and Level 3: A third group gets a heavy dose (the Jolt group). Thus, we would use an ANOVA to analyze the results to decide if there were statistically significant differences between the groups. Assumptions: 1. Must have independent random samples. 2. Each population needs to be normal. Do histogram for each group. If you have one group that is skewed, may want to pursue the ANOVA s non-parametric counterpart: Kruskal-Wallis The population needs to have a common variance σ Boxplots should show about the same spreads. Hypotheses: H 0 : All populations are the same H a : At least one population is different from the others

2 Example: Effects of Caffeine on Test Performance Group 1: Control Group 2:Mild Group 3: Jolt Test Scores Individual Group s = 79 SD = 3.16 SD = 3.16 SD = 3.16 Computations SQUARED Individual SQUARED Individual SQUARED Scores SS tot scores Individual SS w Group SS b Group control m= sd= Group Mild m= sd=

3 Computations SQUARED Individual SQUARED Individual SQUARED Scores SS tot scores Individual SS w Group SS b G Jolt m= st= Sum ANOVA Summary Table Source SS df MS F Between Groups 250 k-1=2 SS/df 250/2=125 MS b F=MS b /MS w 125/ Within Groups 120 N-k 15-3=12 SS/df 120/12=10 Total 370 N-1=14 Note: Every time we estimate something, we lose a degree of freedom (df). df are also the numbers you divide by to estimate a population variance. squares (MS) are average (mean) sums of square deviations. That is, they are variance estimates. The variance is the mean-square-deviation from the mean. The standard deviation is the root-mean-square deviation from the mean. F is a ratio of two mean squares. MS w is the variance within groups. This is the yardstick we use to judge how large the between groups variance is. If there is a treatment effect, then MS b will be larger than MS w, and the F ratio will be larger than 1.0. MS w

4 F has a sampling distribution that is used to compute significance tests. The sampling distribution of F is not normal. However, the form of the distribution is known and can be looked up in the F Distribution table in our textbook. Unlike the other distributions we have studied, F demands that we supply 2 quantities, in addition to alpha, before its form is fully specified. The quantities we supply are the df for the numerator and the df for the denominator (i.e., df b and df w ). Fortunately, there is no decision about one- and two-tailed tests; F is unidirectional. Numerator df: df b Denominator df: df w 1---> 5% > 5% > 5% > 5% > 5% > 5% To find the critical value of the F distribution in this instance with df = 2 for the numerator and df = 12 for the denominator, we find at the.05 level the intersection of these two values to be Our calculated F (12.50) is greater than the critical F (3.88) and, therefore, in the critical region, so we reject the H o.

5 Also, the effect size, eta square =.676, informs us percentage-wise regarding the amount of real difference present in the sample between the null hypothesis and the alternative hypothesis. In general, effect sizes derived from ANOVA methods with values of.02,.15, and.35 are considered to represent small, medium, and large effects. So, we have a very large difference between the null and the alternative. We conclude that these data provide evidence of statistically significant differences among the three populations of caffeinated drinks, but we do not know where. We will perform post-hoc comparisons to determine where the difference exists between the groups. Post Hoc Comparisons: There are many comparison tests from which to choose. Assuming equality of variance, here are 4 tests that are often used in educational research. 1. Scheffe Method: Most conservative and least likely to pick-up on a difference. However, if there is a difference, we are quite sure about the difference because of this test s conservative nature. It takes into account the number of groups we are looking at. 2. Bonferroni Method: Not as conservative as Scheffe. 3. Least Significant Difference (LSD) Test: A more liberal test than the previous two in terms of finding mean differences between groups. 4. Tukey s HSD (Honestly Significant Difference)