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
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 Variable
1. 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 preexisting 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 Variable
1. 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 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
One-Way ANOVA
1. One Independent Variable
2. Groups are independent
Types of ANOVA -continued
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
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
F
BG
WG
The F-Ratio -- continued
F
TE  ID  EE
ID  EE
The F-Ratio -- continued
1. If H0 is true, TE = 0, F = 1
The F-Ratio -- continued
F
0  ID  EE
ID  EE
The F-Ratio -- continued
1. If H0 is false, TE > 0, F > 1
The F-Ratio -- continued
F
TE  ID  EE
ID  EE
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 Groups
1. 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
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.
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
ANOVA
Move DV into Depdent List
 Move IV into Factor
 Options

Descriptives
 Homogeniety of Variance

Sample Printout: ANOVA
Descriptives
Score on Drug Ind ex
95% Con fidence Interval for
Mean
N
Mean
Std. Deviation
Std. Error
Lower Bou nd
Upper Bou nd
Min im um
Maxim um
Catholic
7
9.43
12.541
4.740
-2 .17
21.03
0
30
Jewish
4
7.75
9.032
4.516
-6 .62
22.12
0
20
Pro testant
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
Total
ANOVA
Score on Drug Index
Test of Homogeneity of Variances
Sum of
Squares
Score on Drug Index
Levene
Statistic
.831
df1
df2
2
Sig.
17
.452
Between Groups
df
Mean Square
455.336
2
227.668
Within Groups
3228.464
17
189.910
Total
3683.800
19
F
1.199
Sig.
.326
Sample Printout: Post Hoc
Tests
Multiple Comparisons
Dependent Variable: Score on Drug Index
Bonferroni
(I) Religious Affiliation
of Res pondent
Catholic
Jewish
Mean
Difference
(I-J)
1.68
Std. Error
8.638
Sig.
1.000
Lower Bound
-21.25
Upper Bound
24.61
Protes tant
-8.90
6.945
.651
-27.34
9.53
Catholic
-1.68
8.638
1.000
-24.61
21.25
-10.58
8.281
.655
-32.57
11.40
Catholic
8.90
6.945
.651
-9.53
27.34
Jewish
10.58
8.281
.655
-11.40
32.57
(J) Religious Affiliation
of Res pondent
Jewish
Protes tant
Protes tant
95% Confidence Interval
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
