Introduction to Analysis of Variance

CJ 526 Statistical Analysis in Criminal Justice

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

One-Way ANOVA 1. One Independent Variable

2. Groups are independent

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

Between-Group (BG) Variability

1. Treatment Effect (TE)

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

The F-Ratio

WG

BGF

The F-Ratio -- continued

EEID

EEIDTEF

The F-Ratio -- continued

1. If H0 is true, TE = 0, F = 1

The F-Ratio -- continued

EEID

EEIDF

0

The F-Ratio -- continued

1. If H0 is false, TE > 0, F > 1

The F-Ratio -- continued

EEID

EEIDTEF

Systematic Variability

1. Due to treatment

2. Unsystematic variability: uncontrolled or unexplained

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:

Experimental Groups

1. 0 mg (placebo)

2. 50 mg

3. 100 mg

4. 150 mg

5. 200 mg

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

1. Number of Samples: 5

2. Nature of Samples:1. independent

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

SPSS Procedure One-Way Output -- continued

Test of Homogeneity of Variance

ANOVA Summary TableSum of SquaresdfMean SquareFSig