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Introduction
1. Analysis of Variance (ANOVA) is an inferential statistical technique
2. Developed by Sir Ronald Fisher, an agricultural geneticist, in the 1920s.
Relationship Between ANOVA and Independent t-Test
1. Actually, Independent t-Test is really a special case of ANOVA
2. It is like other parametric inferential procedures such as t test, but there are more than two groups
Purpose of ANOVA
1. Determine whether differences between the means of the groups are due to chance (sampling error)
2. Can be used with both experimental and ex post facto designs
Experimental Research Designs
Researcher manipulates levels of Independent Variable to determine its effect on a Dependent Variable
Example of an Experimental Research Design Using ANOVA
Dr. Sophie studies the effect of different dosages of a new drug on impulsivity among children at-risk of becoming delinquent
Example of an Experimental Research Design Using ANOVA -- continued
1. Independent Variable1. Different dosages of new drug
1. 0 mg (placebo)
2. 100 mg
3. 200 mg
4. Measure impulsivity in each group, compare groups
Ex Post Facto Research Designs
Researcher investigates effects of pre-existing levels of an Independent Variable on a Dependent Variable
Example of an Ex Post Facto
Research Design Using ANOVA Dr. Horace wants to determine whether
political party affiliation has an effect on attitudes toward the death penalty
using a scale assessing attitudes
Example of an Ex Post Facto Research Design Using ANOVA -- continued
1. Independent Variable1. Political Party Affiliation
1. Democrat
2. Independent
3. Republican
4. Measure attitudes toward the death penalty in each group
5. No manipulation
Null and Alternative Hypothesis in ANOVA
1. No differences among the group means
2. Alternative: at least one group differs from at least one other group
Example of Pairwise Comparisons
1. Dr. Mildred wants to determine whether birth order has an effect on number of self-reported delinquent acts
2. Independent Variable1. Birth Order
1. First Born (or only child)2. Middle Born (if three or more children)3. Last Born
Example of Pairwise Comparisons -- continued
3. Dependent Variable1. Number of self-reported delinquent acts
4. Possible pairwise comparisons1. FB ≠ MB2. FB ≠ LB3. MB ≠ LB
5. It is possible for this particular analysis that:1. Any one of the pairwise comparisons could be statistically
significant2. Any two of the pairwise comparisons could be statistically
significant3. All three of the pairwise comparisons could be statistically
significant
Types of ANOVA -- continued
Repeated-Measures ANOVA1. Groups are dependent
2. Measure the dependent variable at more than two points in time
ANOVA and Multiple t-Tests
1. Testwise alpha
2. If multiple t tests are run, there is error each time. If p < .05, one in twenty will be significant, which could just be error
3. Better to conduct ANOVA rather than multiple t tests
The Logic of ANOVA
Total variability of the DV can be analyzed by dividing it into its component parts
Components of Total Variability
1. Between-Groups
2. Measure of the overall differences between treatment conditions (groups, samples)
Within-Groups Variability
1. Measure of the amount of variability inside of each treatment condition (group, sample)
2. There will always be variability within a group
Within-Group (WG) Variability
1. Individual Differences (ID)
2. Example: for race, there is more within group variability than between group variability (more genetic variation among white, or Asians, etc, than between the races
ANOVA Vocabulary
1. Factor, (an IV)
2. Levels are different values of a factor
3. k, number of levels of a factor (also the number of samples)
Degrees of Freedom
1. Between Groups1. k – 1 (number of samples-1)
2. Within groups: n – k (total number of subjects minus number of samples)
3. Total degrees of freedom: n - 1
F-Distribution
1. Always positive
2. See p. 727, p < .05, p. 728, p < .01
3. n1 refers to within degrees of freedom, n2 to between degrees of freedom
Example
A police psychologist wants to determine whether caffeine has an effect on learning and memory
Randomly assigns 120 police officers to one of five groups:
Example -- continued
Records how many “nonsense” words each police officer recalls after studying a 20-word list for 2 minutes
(for example, CVC, dif, zup)
ANOVA Summary Table
Sum ofSquares df
MeanSquares F
BetweenGroups 82.72 4 20.68 5.14WithinGroups 462.3 115 4.02Total 545.02 119
Example of ANOVA --continued
3. Independent Variable: caffeine
4. Dependent Variable and its Level of Measurement: number of syllables remembered—interval/ratio
Example of ANOVA -- continued
5. Appropriate Inferential Statistical Technique: one way analysis of variance
6. Null Hypothesis: no differences in memory (DV) between the groups who are administered differing amount of caffeine (IV)
Example of ANOVA -- continued
7. Decision Rule:1. If the p-value of the obtained test statistic is
less than .05, reject the null hypothesis
Example of ANOVA -- continued
8. Obtained Test Statistic: F
9. Decision: accept or reject the null hypothesis
Results
The results of the One-way ANOVA involving caffeine as the independent variable and number of nonsense words recalled as the dependent variable were statistically significant,
F = (4, 115) = 5.14, p < .01. The means and standard deviations for the five groups are contained in Table 1.
Discussion
It appears that the ingesting small to moderate amounts of caffeine results in better retention of nonsense syllables, but that ingesting moderate to large amounts of caffeine interferes with the ability to retain nonsense syllables
SPSS Procedure Oneway
Analyze, Compare Means, One-Way ANOVAMove DV into Depdent ListMove IV into FactorOptions
DescriptivesHomogeniety of Variance
Sample Printout: ANOVADescriptives
Score on Drug Index
7 9.43 12.541 4.740 -2.17 21.03 0 30
4 7.75 9.032 4.516 -6.62 22.12 0 20
9 18.33 15.969 5.323 6.06 30.61 0 50
20 13.10 13.924 3.114 6.58 19.62 0 50
Catholic
Jewish
Protestant
Total
N Mean Std. Deviation Std. Error Lower Bound Upper Bound
95% Confidence Interval forMean
Minimum Maximum
Test of Homogeneity of Variances
Score on Drug Index
.831 2 17 .452
LeveneStatistic df1 df2 Sig.
ANOVA
Score on Drug Index
455.336 2 227.668 1.199 .326
3228.464 17 189.910
3683.800 19
Between Groups
Within Groups
Total
Sum ofSquares df Mean Square F Sig.
Sample Printout: Post Hoc Tests
Multiple Comparisons
Dependent Variable: Score on Drug Index
Bonferroni
1.68 8.638 1.000 -21.25 24.61
-8.90 6.945 .651 -27.34 9.53
-1.68 8.638 1.000 -24.61 21.25
-10.58 8.281 .655 -32.57 11.40
8.90 6.945 .651 -9.53 27.34
10.58 8.281 .655 -11.40 32.57
(J) Religious Affiliationof RespondentJewish
Protestant
Catholic
Protestant
Catholic
Jewish
(I) Religious Affiliationof RespondentCatholic
Jewish
Protestant
MeanDifference
(I-J) Std. Error Sig. Lower Bound Upper Bound
95% Confidence Interval
SPSS Procedure One-Way Output
Descriptives Levels of IV N Mean Standard Deviation Standard Error of the Mean 95% Confidence Interval
Lower Bound Upper Bound
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