1. Chapter 12
Introduction to Analysis of Variance
PowerPoint Lecture Slides
Essentials of Statistics for the
Behavioral Sciences
Eighth Edition
by Frederick J. Gravetter and Larry B. Wallnau
2. Chapter 12 Learning Outcomes
• Explain purpose and logic of Analysis of
Variance1
• Perform Analysis of Variance on data from
single-factor study2
• Know when and why to use post hoc tests
(posttests)3
• Compute Tukey’s HSD and Scheffé test post
hoc tests4
• Compute η2 to measure effect size5
3. Tools You Will Need
• Variability (Chapter 4)
– Sum of squares
– Sample variance
– Degrees of freedom
• Introduction to hypothesis testing (Chapter 8)
– The logic of hypothesis testing
• Independent-measures t statistic (Chapter 10)
4. 12.1 Introduction to Analysis
of Variance
• Analysis of variance
– Used to evaluate mean differences between two
or more treatments
– Uses sample data as basis for drawing general
conclusions about populations
• Clear advantage over a t test: it can be used
to compare more than two treatments at the
same time
6. Terminology
• Factor
– The independent (or quasi-independent) variable
that designates the groups being compared
• Levels
– Individual conditions or values that make up
a factor
• Factorial design
– A study that combines two or more factors
8. Statistical Hypotheses for ANOVA
• Null hypothesis: the level or value on the
factor does not affect the dependent variable
– In the population, this is equivalent to saying that
the means of the groups do not differ from each
other
•
3210 : H
9. Alternate Hypothesis for ANOVA
• H1: There is at least one mean difference
among the populations (Acceptable
shorthand is “Not H0”)
• Issue: how many ways can H0 be wrong?
– All means are different from every other mean
– Some means are not different from some others,
but other means do differ from some means
10. Test statistic for ANOVA
• F-ratio is based on variance instead of sample
mean differences
effecttreatmentnowithexpectedes)(differencvariance
meanssamplebetweenes)(differencvariance
F
11. Test statistic for ANOVA
• Not possible to compute a sample mean
difference between more than two samples
• F-ratio based on variance instead of sample
mean difference
– Variance used to define and measure the size of
differences among sample means (numerator)
– Variance in the denominator measures the mean
differences that would be expected if there is no
treatment effect
12. Type I Errors and
Multiple-Hypothesis tests
• Why ANOVA (if t can compare two means)?
– Experiments often require multiple hypothesis
tests—each with Type I error (testwise alpha)
– Type I error for a set of tests accumulates testwise
alpha experimentwise alpha > testwise alpha
• ANOVA evaluates all mean differences
simultaneously with one test—regardless of
the number of means—and thereby avoids
the problem of inflated experimentwise alpha
13. 12.2 Analysis of Variance Logic
• Between-treatments variance
– Variability results from general differences
between the treatment conditions
– Variance between treatments measures
differences among sample means
• Within-treatments variance
– Variability within each sample
– Individual scores are not the same within each
sample
14. Sources of Variability
Between Treatments
• Systematic differences caused
by treatments
• Random, unsystematic differences
– Individual differences
– Experimental (measurement) error
15. Sources of Variability
Within Treatments
• No systematic differences related to treatment
groups occur within each group
• Random, unsystematic differences
– Individual differences
– Experimental (measurement) error
effectstreatmentnowithsdifference
effectstreatmentanyincludingsdifference
F
17. F-ratio
• If H0 is true:
– Size of treatment effect is near zero
– F is near 1.00
• If H1 is true:
– Size of treatment effect is more than 0.
– F is noticeably larger than 1.00
• Denominator of the F-ratio is called the
error term
18. Learning Check
• Decide if each of the following statements
is True or False
• ANOVA allows researchers to compare
several treatment conditions without
conducting several hypothesis tests
T/F
• If the null hypothesis is true, the
F-ratio for ANOVA is expected (on
average) to have a value of 0
T/F
19. Learning Check - Answers
• Several conditions can be
compared in one testTrue
• If the null hypothesis is true, the
F-ratio will have a value near 1.00
False
20. 12.3 ANOVA Notation
and Formulas
• Number of treatment conditions: k
• Number of scores in each treatment: n1, n2…
• Total number of scores: N
– When all samples are same size, N = kn
• Sum of scores (ΣX) for each treatment: T
• Grand total of all scores in study: G = ΣT
• No universally accepted notation for ANOVA;
Other sources may use other symbols
24. Degrees of Freedom Analysis
• Total degrees of freedom
dftotal= N – 1
• Within-treatments degrees of freedom
dfwithin= N – k
• Between-treatments degrees of freedom
dfbetween= k – 1
26. Mean Squares and F-ratio
within
within
withinwithin
df
SS
sMS 2
between
between
betweenbetween
df
SS
sMS 2
within
between
within
between
MS
MS
s
s
F 2
2
27. ANOVA Summary Table
Source SS df MS F
Between Treatments 40 2 20 10
Within Treatments 20 10 2
Total 60 12
•Concise method for presenting ANOVA results
•Helps organize and direct the analysis process
•Convenient for checking computations
•“Standard” statistical analysis program output
28. Learning Check
• An analysis of variance produces SStotal = 80
and SSwithin = 30. For this analysis, what is
SSbetween?
• 50A
• 110B
• 2400C
• More information is neededD
29. Learning Check - Answer
• An analysis of variance produces SStotal = 80
and SSwithin = 30. For this analysis, what is
SSbetween?
• 50A
• 110B
• 2400C
• More information is neededD
30. 12.4 Distribution of F-ratios
• If the null hypothesis is true, the value of F will
be around 1.00
• Because F-ratios are computed from two
variances, they are always positive numbers
• Table of F values is organized by two df
– df numerator (between) shown in table columns
– df denominator (within) shown in table rows
32. 12.5 Examples of Hypothesis
Testing and Effect Size
• Hypothesis tests use the same four steps that
have been used in earlier hypothesis tests.
• Computation of the test statistic F is done
in stages
– Compute SStotal, SSbetween, SSwithin
– Compute MStotal, MSbetween, MSwithin
– Compute F
34. Measuring Effect size for
ANOVA
• Compute percentage of variance accounted
for by the treatment conditions
• In published reports of ANOVA, effect size is
usually called η2 (“eta squared”)
– r2 concept (proportion of variance explained)
total
treatmentsbetween
SS
SS
2
35. In the Literature
• Treatment means and standard deviations are
presented in text, table or graph
• Results of ANOVA are summarized, including
– F and df
– p-value
– η2
• E.g., F(3,20) = 6.45, p<.01, η2 = 0.492
37. MSwithin and Pooled Variance
• In the t-statistic and in the F-ratio, the
variances from the separate samples are
pooled together to create one average value
for the sample variance
• Numerator of F-ratio measures how much
difference exists between treatment means.
• Denominator measures the variance of the
scores inside each treatment
38. 12.6 post hoc Tests
• ANOVA compares all individual mean
differences simultaneously, in one test
• A significant F-ratio indicates that at least one
difference in means is statistically significant
– Does not indicate which means differ significantly
from each other!
• post hoc tests are follow up tests done to
determine exactly which mean differences are
significant, and which are not
39. Experimentwise Alpha
• post hoc tests compare two individual means
at a time (pairwise comparison)
– Each comparison includes risk of a Type I error
– Risk of Type I error accumulates and is called the
experimentwise alpha level.
• Increasing the number of hypothesis tests
increases the total probability of a Type I error
• post hoc (“posttests”) use special methods to
try to control experimentwise Type I error rate
40. Tukey’s Honestly Significant
Difference
• A single value that determines the minimum
difference between treatment means that is
necessary to claim statistical significance–a
difference large enough that p < αexperimentwise
– Honestly Significant Difference (HSD)
n
MS
qHSD within
41. The Scheffé Test
• The Scheffé test is one of the safest of all
possible post hoc tests
– Uses an F-ratio to evaluate significance of the
difference between two treatment conditions
groupstwoofSSwithcalculatedBA versus
within
between
MS
MS
F
42. Learning Check
• Which combination of factors is most likely to
produce a large value for the F-ratio?
• large mean differences
and large sample variancesA
• large mean differences
and small sample variancesB
• small mean differences
and large sample variancesC
• small mean differences
and small sample variancesD
43. Learning Check - Answer
• Which combination of factors is most likely to
produce a large value for the F-ratio?
• large mean differences
and large sample variancesA
• large mean differences
and small sample variancesB
• small mean differences
and large sample variancesC
• small mean differences
and small sample variancesD
44. Learning Check
• Decide if each of the following statements
is True or False
• Post tests are needed if the decision from
an analysis of variance is “fail to reject the
null hypothesis”
T/F
• A report shows ANOVA results: F(2, 27) =
5.36, p < .05. You can conclude that the
study used a total of 30 participants
T/F
45. Learning Check - Answers
• post hoc tests are needed only if
you reject H0 (indicating at least
one mean difference is significant)
False
• Because dftotal = N-1 and
• Because dftotal = dfbetween + dfwithin
True
46. 12.7 Relationship between
ANOVA and t tests
• For two independent samples, either t or F
can be used
– Always result in same decision
– F = t2
• For any value of α, (tcritical)2 = Fcritical
48. Independent Measures
ANOVA Assumptions
• The observations within each sample must be
independent
• The population from which the samples are
selected must be normal
• The populations from which the samples are
selected must have equal variances
(homogeneity of variance)
• Violating the assumption of homogeneity of
variance risks invalid test results
FIGURE 12.1 A typical situation in which ANOVA would be used. Three separate samples are obtained to evaluate the mean differences among three populations (or treatments) with unknown means.
FIGURE 12.2 A research design with two factors. The research study uses two factors: One factor uses two levels of therapy technique (I versus II), and the second factor uses three levels of time (before, after and 6 months after). Also notice that the therapy factor uses two separate groups (independent measures) and the time factor uses the same group for all three levels (repeated measures).
Some instructors may want to spend some time listing some of the ways that the null hypothesis cold be wrong that do NOT include μ1 ≠ μ2 ≠ μ3 because many students make the logic error of assuming that if null is wrong then the ONLY way it can be wrong is if all the groups are significantly different from each other.
Testwise Type I error rate and Experimentwise Type I error rate are sufficiently complex concepts that some instructors may want to expand the lecture materials on this topic.
FIGURE 12.3 The independent-measures ANOVA partitions, or analyzes, the total variability into two components: variance between treatments and variance within treatments.
FIGURE 12.4 The structure and sequence of calculations for the ANOVA.
FIGURE 12.5 Partitioning the sum of squares (SS) for the independent-measures ANOVA.
FIGURE 12.6 Partitioning degrees of freedom (df) for the independent-measures ANOVA.
Some instructors may want to begin the discussion of the computational procedures with this slide. Students sometimes get lost in all the formulas, but if they can see that the summary table is the “goal” of the process, they can break it down into the small steps required to fill in each part of the table.
FIGURE 12.7 The distribution of F-ratios with df = 2, 12. Of all the values in the distribution, only 5% are larger than F = 3.88, and only 1% are larger than F = 6.93.
FIGURE 12.8 The distribution of F-ratios with df = 3, 20. The critical values for α = .01 is 4.94.
A visual representation of the between-treatments variability and the within0-treatments variability that form the numerator and denominator, respectively, of the F-ratio. In (a), the difference between treatments is relatively large and easy to see. In (b), the same 4-point difference between treatments is relatively small and is overwhelmed by the within-treatments variability.
FIGURE 12.10 The distribution of t statistics with df = 18 and the corresponding distribution of F-ratios with df = 1, 18. Notice that the critical values for α = .05 are t = ±2.101 and F = 2.1012 = 4.41.
FIGURE 12.10 Formulas for ANOVA.
FIGURE 12.12 SPSS Output of the ANOVA for the studying strategy experiment in Example 12.1.
