Chapter 13 - ANOVA

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 testLogic 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 Distribution13-2.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 GroupsVariance 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 distribution1. 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….

F Distribution

Shavelson Chapter 13S13-4A. 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

1. Scores are independent of each other

2. 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)

3. 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)

S13-4A continued

Design requirements for the one-way ANOVA:

1. One IV with 2 or more levels

2. The levels of the Iv can differ quantitatively or qualitatively

3. Participant may only appear in one group (that is one level of IV) and was randomly selected from the population

Shavelson Chapter 13

S13-4BNull 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.

Shavelson Chapter 13

Shavelson Chapter 13

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 hocsS13-7. 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 hocsS13-7. 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 comparison1. Used to detect significant differences

between pairs of means and combinations of means.

Tukey’s HSD test1. Used to test all pairs (only pairs, no complex

combinations)2. More powerful than the Scheffe for this

Post hocs

Post hocs