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Lecture 5

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

Department
Psychology
Course Code
PSYC 3430
Professor
Lecture
5

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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
Tukeyβs Honestly Significant Difference (HSD) 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 Tukeyβs 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 = βββββββ MSwithin = ββββββββββββββ
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
Source SS df MS
ββββββββββββββββββββββββββββββββββ
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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