# PSYC 3430 Lecture Notes - Lecture 5: The Instructor, Statistical Hypothesis Testing, Dependent And Independent Variables

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**preview**shows pages 1-3. to view the full**16 pages of the document.**Instructor Notes - Chapter 12 - page 169

Chapter 12: Introduction to Analysis of Variance

Chapter Outline

12.1 Introduction

Terminology in Analysis of Variance

Statistical Hypotheses for ANOVA

The Test Statistic for ANOVA

Type I Errors and Multiple Hypothesis Tests

12.2 The Logic of Analysis of Variance

Between-Treatments Variance

Within-Treatments Variance

The F-Ratio: The Test Statistic for ANOVA

12.3 ANOVA Notation and Formulas

ANOVA Formulas

Analysis of Sum of Squares (SS)

Analysis of Degrees of Freedom (df)

Calculation of Variances (MS) and the F-Ratio

12.4 The Distribution of F-Ratios

The F Distribution Table

12.5 Examples of Hypothesis Testing and Effect Size with ANOVA

Measuring Effect Size for ANOVA

In the Literature - Reporting the Results of ANOVA

A Conceptual View of ANOVA

MSwithin and Pooled Variance

An Example with Unequal Sample Sizes

Assumptions for the Independent-Measures ANOVA

12.6 Post Hoc Tests

Post Tests and Type I Errors

Tukeyβs Honestly Significant Difference (HSD) Test

The ScheffΓ© Test

12.7 The Relationship between ANOVA and t Tests

Assumptions for the Independent-Measures ANOVA

Learning Objectives and Chapter Summary

1. Students should understand the basic purpose for analysis of variance and the general logic

that underlies this statistical procedure.

Analysis of variance is a hypothesis test that evaluates the significance of mean

differences. That is, the goal is to determine whether the mean differences that are found

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Instructor Notes - Chapter 12 - page 170

in sample data are greater than can be reasonable explained by chance alone.

The size of the mean differences is measured by computing a variance between

treatments (MSbetween); large mean differences produce a large variance and small mean

differences produce a small variance. The amount of variance expected simply by chance

(without any treatment effect) is measured by computing the variance within treatments

(MSwithin). The logic behind this second variance is that all of the individuals inside a

treatment condition are treated exactly the same. Therefore, any differences that exist

(variance) cannot be caused by the treatment and must be due to chance alone. Finally,

the two variances are compared in an F-ratio to determine whether the mean differences

(MSbetween) are significantly bigger than chance (MSwithin).

2. Students should be able to perform an analysis of variance to evaluate the data from a single-

factor, independent-measures research study.

Although an analysis of variance requires a relatively long sequence of calculations, all of

the calculations are logically connected in a fairly simple pattern:

The goal is to obtain an F-ratio which is the ratio of two variances. Each variance is

computed as SS/df. Thus, we must obtain SSbetween and dfbetween for the variance in the

numerator of the F-ratio and we must obtain SSwithin and dfwithin for the variance in the

denominator. The two SS values are obtained by analyzing the total SS into two

components, and the two df values are obtained by analyzing the total df into two

components. All of the formulas for the ANOVA are shown in Figure 12.12.

3. Students should understand when post tests are necessary and the purpose that they serve.

Students should be familiar with post test techniques such as Tukeyβs HSD and the ScheffΓ¨ test.

Whenever the null hypothesis is rejected for an analysis of variance comparing more than

two treatments, we conclude that at least one of the means is significantly different from

another mean. The problem is that we have not determined exactly which means are

different and which are not. This is the purpose for post tests. Typically, a post test will

compare the means two at a time to determine exactly which pairs are significantly

different.

4. Students should be able to compute Ξ·2 (the percentage of variance accounted for) to measure

effect size for the sample means in an analysis of variance.

Just as r2 measures how much variability is explained by the treatment effects in a t test,

Ξ·2 (eta squared) measures the percentage of explained variability in an analysis of

variance. In ANOVA, the definition and calculation of Ξ·2 are very straightforward: you

simply determine how much of the total variability (SStotal) is explained by the variability

between treatments (SSbetween).

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Instructor Notes - Chapter 12 - page 171

Other Lecture Suggestions

1. The terms total, between, and within can help students learn and understand the calculations

in ANOVA. Total refers to the entire set of scores. Thus, SStotal is the sum of squares for the

entire set of scores and dftotal is simply N β 1 for the entire set. Within refers to the SS and df

values inside the individual treatments. Thus, SSwithin is determined by the SS values inside each

treatment. Finally, between refers to the mean differences between treatments. Thus, with 3

treatments, there are 3 β 1 = 2 degress of freedom between treatments (dfbetween = 2).

2. Although ANOVA requires extensive calculations, there are actually very few formulas that

students need to know. Instead of formulas, students should focus on the relationships among

the pieces. The three basic relationships are:

1. For both SS and df values, the basic relationship is that total = between + within. If

two of the values are known, the third can be computed easily.

2. For each variance or MS value, MS = SS/df. Again, there is a relationship among

three values (MS, SS, and df). If any two are known, the third can be computed

easily.

3. The final F-ratio also forms a relationship among 3 values (F, MSbetween , and MSwithin).

Again, if two values are known, the third can be computed.

Knowing and using these relationships can be demonstrated and tested with problems that

require students to fill in the missing values in an ANOVA summary table (see 16-18 in the end

of chapter problems).

3. Point out that the variance within treatments (MSwithin) in the bottom of the F-ratio is another

example of the pooled variance that was used in the bottom of the t statistic formula. The

calculation of MSwithin simply extends the formula by adding extra SS and df values.

SS1 + SS2 SS1 + SS2 + SS3 + . . .

pooled variance = βββββββ MSwithin = ββββββββββββββ

df1 + df2 df1 + df2 + df3 + . . .

4. Problems that ask students to fill in the missing values in a summary table are easy to

construct, easy to grade, and provide a good test of student knowledge. To construct one of these

problems, start with an empty summary table:

Source SS df MS

ββββββββββββββββββββββββββββββββββ

Between Treatments ___ ___ ___ F = ___

Within Treatments ___ ___ ___

Total ___ ___

a. Pick any values for the number of treatments and the number of participants in each

treatment. For example, 3 treatments with n = 8 in each treatment.

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