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ANOVA • Be able to explain in general terms and using an example what a one-way ANOVA is (370). Know the purpose of the one-way ANOVA (371, 2). Carefully read page 372 in which the logic of the one-way ANOVA is described. You should be able to explain this back to me on the exam (372). • One way refers to only 1 independent variable, with 2 or more independent levels (usually three or more!). • Used to test whether difference among means of multiple groups is due to chance only, or to treatment as well.
Logic of F test Logic of the F test F(observed) = Variance Betwn Groups = var of scores w/in groups + treatment Variance Within Groups = var of scores w/in groups If treatment = 0? Smaller F… If treatment = 10? Bigger F… If F hovers around 1 = no treatment effect. (The higher the number generated from the F, the greater the likelihood of a treatment effect.)
F Distribution 13-2.Be able to describe the F distribution and how it is created (e.g. many samples in which the F is calculated and put into a frequency distribution (373). Know the five characteristics of the F distribution. You should be able to explain the difference between dfB and dfW (374). The F Distribution A. F(observed) = Variance Between Groups Variance within groups B. Calculate F(observed) a bunch of times; create a frequency distribution of F’s C. Use this distribution to determine the probability of obtaining a particular F by chance alone. If likely = “no treatment effect” If unlikely = “treatment effect” = “statistically significant”
Five characteristics of the F distribution 1. The mean of the distribution approaches 1 as the sample size increases. 2. The F distribution is unimodal. 3. The sampling distribution is positively skewed. 4. A different distribution exists for each DF (thus, a family of distributions) 5. When comparing only two groups will get the same result as the t-test (two tailed only)
F-test: degrees of freedom Degrees of freedom for the F: Two different numbers of degrees of freedom. • A. One set of degrees of freedom depends on the number of sample group means being compared, and • B. The other set depends on the number of subjects in each group. • dfB = number of groups-1 = k-1 • dfW = group 1 (n-1) + group 2 (n-1) b + group 3….
Shavelson Chapter 13 S13-4A. Know the following regarding the one way ANOVA:A. The design requirements (377) as well as the assumptions (378).B. Be able to create and recognize correct examples of hypotheses (Ho and H1) (378)C. Know the decision rules for rejecting or not the null hypothesis. Given data, you should be able to generate both the Fobserved and Fcritical and determine whether or not to reject the null. I will not ask you to calculate the sum of squares. Given a particular study, you should be able to say what rejecting (or not) the null means. (384) Assumptions of one-way ANOVA • Scores are independent of each other • The scores of the population from which the sample was drawn should be normally distributed (but little problem with this when levels of IV are fixed) • The variances of the populations from which the samples were drawn should be equal: homogeneity of variance (little problem with this when groups are all the same size (equal Ns)
Shavelson Chapter 13 S13-4A continued Design requirements for the one-way ANOVA: • One IV with 2 or more levels • The levels of the Iv can differ quantitatively or qualitatively • Participant may only appear in one group (that is one level of IV) and was randomly selected from the population
Shavelson Chapter 13 S13-4B Null Hypotheses used: Ho: 1 = 2 = 3 (All means are equal) H1: i ≠ i' (At least one of the pairs of means differ from one another) Only two-tailed tests! e.g. directional would be goofy: 1> 2< 3> 4 etc.
Strength of Association S13-6. Be able to explain how one would obtain information about the size of a treatment effect (in general terms). Given the value of the omega-square (e.g. .71) explain what that number means. (387-388) Strength of Association (size of treatment effects) Omega-square: indicates the amount of variability in the DV accounted for by the IV Thus, a larger number (e.g. .71) indicates a larger effect by the IV (more DV variability is accounted for by knowing the IV) Said another way, with an omega-square of .71, there is a strong relationship between the IV and DV.
Post hocs S13-7. Why would one conduct post hoc comparisons? (389). Be able to describe the general steps taken in conducting a post hoc comparison. This includes creating the null and alternative hypotheses; writing the comparisons as a set of weighted means; and finally when this comparison would be run. (390-394) Post hoc comparisons: When a significant F is found, used to determine which means caused the significant F (or said another way, determine which pair, or combination of pairs of means have a significant difference between them)
Post hocs S13-7. Why would one conduct post hoc comparisons? (389). Be able to describe the general steps taken in conducting a post hoc comparison. This includes creating the null and alternative hypotheses; writing the comparisons as a set of weighted means; and finally when this comparison would be run. (390-394) Post hoc comparisons: When a significant F is found, used to determine which means caused the significant F (or said another way, determine which pair, or combination of pairs of means have a significant difference between them) Scheffe Post hoc comparison • Used to detect significant differences between pairs of means and combinations of means. Tukey’s HSD test • Used to test all pairs (only pairs, no complex combinations) • More powerful than the Scheffe for this
