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Introduction to Analysis of Variance. CJ 526 Statistical Analysis in Criminal Justice. Introduction. An alysis o f Va riance (ANOVA) is an inferential statistical technique. Developer. Developed by Sir Ronald Fisher in the 1920’s Agricultural geneticist.

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introduction to analysis of variance

Introduction to Analysis of Variance

CJ 526 Statistical Analysis in Criminal Justice

introduction
Introduction
  • Analysis of Variance (ANOVA) is an inferential statistical technique
developer
Developer
  • Developed by Sir Ronald Fisher in the 1920’s
      • Agricultural geneticist
relationship between anova and independent t test
Relationship Between ANOVA and Independent t-Test
  • Actually, Independent t-Test is really a special case of ANOVA
similarities with other parametric inferential procedures
Similarities With Other Parametric Inferential Procedures
  • Like all parametric inferential procedures
purpose of anova
Purpose of ANOVA
  • Determine whether differences between the means of the groups are due to chance (sampling error)
anova and research designs
ANOVA and Research Designs
  • Can be used with both experimental and ex post facto research designs
experimental research 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
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
Example of an Experimental Research Design Using ANOVA -- continued
  • Independent Variable
      • Different dosages of new drug
        • 0 mg (placebo)
        • 100 mg
        • 200 mg
ex post facto research designs
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
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
example of an ex post facto research design using anova continued
Example of an Ex Post Facto Research Design Using ANOVA -- continued
  • Independent Variable
      • Political Party Affiliation
        • Democrat
        • Independent
        • Republican
null hypothesis in anova
Null Hypothesis in ANOVA
  • No differences among the population means
alternative hypothesis in anova
Alternative Hypothesis in ANOVA
  • At least one population mean is different from one other population mean
example of pairwise comparisons
Example of Pairwise Comparisons
  • Dr. Mildred wants to determine whether birth order has an effect on number of self-reported delinquent acts
  • Independent Variable
      • Birth Order
        • First Born (or only child)
        • Middle Born (if three or more children)
        • Last Born
example of pairwise comparisons continued
Example of Pairwise Comparisons -- continued
  • Dependent Variable
      • Number of self-reported delinquent acts
  • Possible pairwise comparisons
      • FB ≠ MB
      • FB ≠ LB
      • MB ≠ LB
  • It is possible for this particular analysis that:
      • Any one of the pairwise comparisons could be statistically significant
      • Any two of the pairwise comparisons could be statistically significant
      • All three of the pairwise comparisons could be statistically significant
types of anova
Types of ANOVA
  • One-Way ANOVA
      • One Independent Variable
      • Groups are independent
types of anova continued
Types of ANOVA -- continued
  • Repeated-Measures ANOVA
      • Groups are dependent
      • Measure the dependent variable at more than two points in time
the logic of anova
The Logic of ANOVA
  • Total variability of the DV can be analyzed by dividing it into its component parts
components of total variability
Components of Total Variability
  • Between-Groups
  • Measure of the overall differences between treatment conditions (groups, samples)
within groups variability
Within-Groups Variability
  • Measure of the amount of variability inside of each treatment condition (group, sample)
  • There will always be variability within a group
between group bg variability
Between-Group (BG) Variability
  • Treatment Effect (TE)
within group wg variability
Within-Group (WG) Variability
  • Individual Differences (ID)
  • 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
The F-Ratio
  • Obtained test statistic for ANOVA

Is

the f ratio continued2
The F-Ratio -- continued
  • If H0 is true, TE = 0, F = 1
the f ratio continued4
The F-Ratio -- continued
  • If H0 is false, TE > 0, F > 1
the f ratio continued6
The F-Ratio -- continued
  • F = Systematic Variability
      • divided by
systematic variability
Systematic Variability
  • Due to treatment
unsystematic variability
Unsystematic Variability
  • Uncontrolled or unexplained
factor
Factor
  • Independent variable
level
Level
  • Different values of a factor
notation for anova
Notation for ANOVA
  • k: number of levels of a factor
      • Also the number of different samples
degrees of freedom
Degrees of Freedom
  • Between Groups
      • k - 1
f distribution
F-Distribution
  • Always positive
example
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
Experimental Groups
  • 0 mg (placebo)
  • 50 mg
  • 100 mg
  • 150 mg
  • 200 mg
example continued
Example -- continued
  • Records how many “nonsense” words each police officer recalls after studying a 20-word list for 2 minutes
  • CVC, dif, zup
example of anova
Example of ANOVA
  • Number of Samples: 5
  • Nature of Samples:
  •  Known:
example of anova continued
Example of ANOVA --continued
  • Independent Variable: caffeine
  • Dependent Variable and its Level of Measurement: number of syllables remembered—interval/ratio
example of anova continued1
Example of ANOVA -- continued
  • Target Population:
  • Appropriate Inferential Statistical Technique: one way analysis of variance
  • Null Hypothesis: no differences in memory between the groups
example of anova continued2
Example of ANOVA -- continued
  • Alternative Hypothesis: Caffeine does have an effect on memory and there will be differences among the groups
  • Decision Rule:
      • If the p-value of the obtained test statistic is less than .05, reject the null hypothesis
example of anova continued3
Example of ANOVA -- continued
  • Obtained Test Statistic: F
  • Decision: accept or reject the null hypothesis
results
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
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
assumptions of anova
Assumptions of ANOVA
  • Observations are independent
spss procedure oneway
SPSS Procedure Oneway
  • Analyze, Compare Means, One-Way ANOVA
    • Move DV into Depdent List
    • Move IV into Factor
    • Options
      • Descriptives
      • Homogeniety of Variance
spss procedure one way output
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
SPSS Procedure One-Way Output -- continued
  • Test of Homogeneity of Variance
  • ANOVA Summary Table
    • Sum of Squares
    • df
    • Mean Square
    • F
    • Sig