Transcript Introduction to Analysis of Variance

Introduction to Analysis
of Variance
CJ 526 Statistical Analysis in
Criminal Justice
Introduction
1. Analysis of Variance (ANOVA) is an
inferential statistical technique
Developer
1. Developed by Sir Ronald Fisher in the
1920’s
1. Agricultural geneticist
Relationship Between ANOVA
and Independent t-Test
1. Actually, Independent t-Test is really a
special case of ANOVA
Similarities With Other
Parametric Inferential Procedures
1. Like all parametric inferential
procedures
Purpose of ANOVA
1. Determine whether differences
between the means of the groups are
due to chance (sampling error)
ANOVA and Research
Designs
1. Can be used with both experimental
and ex post facto research designs
Experimental Research
Designs
1. Researcher manipulates levels of
Independent Variable to determine its
effect on a Dependent Variable
Example of an Experimental
Research Design Using ANOVA
1. 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 Variable
1. Different dosages of new drug
1. 0 mg (placebo)
2. 100 mg
3. 200 mg
Ex Post Facto Research
Designs
1. 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
1. Dr. Horace wants to determine
whether political party affiliation has an
effect on attitudes toward the death
penalty
Example of an Ex Post Facto Research
Design Using ANOVA -- continued
1. Independent Variable
1. Political Party Affiliation
1. Democrat
2. Independent
3. Republican
Null Hypothesis in
ANOVA
1. No differences among the population
means
Alternative Hypothesis
in ANOVA
1. At least one population mean is
different from one other population
mean
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 Variable
1. 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 Variable
1. Number of self-reported delinquent acts
4. Possible pairwise comparisons
1. FB ≠ MB
2. FB ≠ LB
3. MB ≠ LB
5. It is possible for this particular analysis that:
1. Any one of the pairwise comparisons could be statistically
significant
2. Any two of the pairwise comparisons could be statistically
significant
3. All three of the pairwise comparisons could be statistically
significant
Types of ANOVA
1. One-Way ANOVA
1. One Independent Variable
2. Groups are independent
Types of ANOVA -continued
3. Repeated-Measures ANOVA
1. Groups are dependent
2. Measure the dependent variable at more than
two points in time
ANOVA and Multiple tTests
1. Testwise alpha
The Logic of ANOVA
1. 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
1. Obtained test statistic for ANOVA
Is
The F-Ratio -- continued
BG
F
WG
The F-Ratio -- continued
TE  ID  EE
F
ID  EE
The F-Ratio -- continued
1. If H0 is true, TE = 0, F = 1
The F-Ratio -- continued
0  ID  EE
F
ID  EE
The F-Ratio -- continued
1. If H0 is false, TE > 0, F > 1
The F-Ratio -- continued
TE  ID  EE
F
ID  EE
The F-Ratio -- continued
1. F = Systematic Variability
1. divided by
Systematic Variability
1. Due to treatment
Unsystematic Variability
1. Uncontrolled or unexplained
ANOVA Vocabulary
1. Factor
Factor
1. Independent variable
Level
1. Different values of a factor
Notation for ANOVA
1. k: number of levels of a factor
1. Also the number of different samples
Degrees of Freedom
1. Between Groups
1. k - 1
F-Distribution
1. Always positive
Example
1. A police psychologist wants to
determine whether caffeine has an
effect on learning and memory
2. 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
3. Records how many “nonsense” words
each police officer recalls after
studying a 20-word list for 2 minutes
4. CVC, dif, zup
ANOVA Summary Table
Between
Groups
Within
Groups
Total
Sum of
Squares
df
Mean
Squares
F
82.72
4
20.68
5.14
462.3
545.02
115
119
4.02
Example of ANOVA
1. Number of Samples: 5
2. Nature of Samples:
1.
3.  Known:
Example of ANOVA -continued
4. Independent Variable: caffeine
5. Dependent Variable and its Level of
Measurement: number of syllables
remembered—interval/ratio
Example of ANOVA -continued
6. Target Population:
7. Appropriate Inferential Statistical
Technique: one way analysis of
variance
8. Null Hypothesis: no differences in
memory between the groups
Example of ANOVA -continued
9. Alternative Hypothesis: Caffeine does
have an effect on memory and there
will be differences among the groups
10. Decision Rule:
1. If the p-value of the obtained test statistic is
less than .05, reject the null hypothesis
Example of ANOVA -continued
11. Obtained Test Statistic: F
12. Decision: accept or reject the null
hypothesis
Results
1. 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
1. 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
Assumptions of ANOVA
1. Observations are independent
SPSS Procedure Oneway
Analyze, Compare Means, One-Way
ANOVA
Move DV into Depdent List
 Move IV into Factor
 Options

Descriptives
 Homogeniety of Variance

Sample Printout: ANOVA
Descriptives
Score on Drug Index
N
Cathol ic
Jewi sh
Protestant
Total
7
4
9
20
Mean
9.43
7.75
18.33
13.10
Std. Deviation
12.541
9.032
15.969
13.924
Std. Error
4.740
4.516
5.323
3.114
95% Confidence Interval for
Mean
Lower Bound
Upper Bound
-2.17
21.03
-6.62
22.12
6.06
30.61
6.58
19.62
Mini mum
0
0
0
0
Maximum
30
20
50
50
ANOVA
Score on Drug Index
Test of Homogeneity of Variances
Score on Drug Index
Levene
Statistic
.831
df1
df2
2
17
Sig.
.452
Between Groups
Within Groups
Total
Sum of
Squares
455.336
3228.464
3683.800
df
2
17
19
Mean Square
227.668
189.910
F
1.199
Sig.
.326
Sample Printout: Post Hoc
Tests
Multiple Comparisons
Dependent Variable: Score on Drug Index
Bonferroni
(I) Religious Affiliation
of Respondent
Catholic
Jewish
Protestant
(J) Religious Affiliation
of Respondent
Jewish
Protestant
Catholic
Protestant
Catholic
Jewish
Mean
Difference
(I-J)
1.68
-8.90
-1.68
-10.58
8.90
10.58
Std. Error
8.638
6.945
8.638
8.281
6.945
8.281
Sig.
1.000
.651
1.000
.655
.651
.655
95% Confidence Interval
Lower Bound
Upper Bound
-21.25
24.61
-27.34
9.53
-24.61
21.25
-32.57
11.40
-9.53
27.34
-11.40
32.57
SPSS Procedure OneWay Output
Descriptives






Levels of IV
N
Mean
Standard Deviation
Standard Error of the Mean
95% Confidence Interval


Lower Bound
Upper Bound
SPSS Procedure OneWay Output -- continued
Test of Homogeneity of Variance
ANOVA Summary Table
Sum of Squares
 df
 Mean Square
 F
 Sig
