ANOVA
Chapter 12
Review
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Independent Variable: the variable manipulated by the experimenter in an effort to determine its effect
*ie. The “cause”
*note: there may be 2 or more levels of this variable
-an IV can be a condition of the subject’s external environment usually created by the experimenter
EX: exposure to violent tv, time of day, a drug, etc.
-an IV can be a characteristic of the subject
EX: gender, IQ, degree of attractiveness, etc.
**in this case the variable is not literally manipulated
**Instead, the experimenter assigns subjects to conditions based on the subject’s characteristics
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Dependent Variable: the behavior that may be affected by the independent variable.
*ie. The “effect”
*note: this is the variable that is measured
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Extraneous & Confounding Variables: influence the DV but not what the experimenter is studying.
-these must be controlled
-they cause error when they still influence the DV
**main source of this error is individual differences
**we would like all the subjects in our experiment to be exactly the same but that’s impossible
-therefore, we must have a measure of this error
EX: Standard Deviation measures error because it measures the differences between subjects
Multiple Comparisons
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t-test: used to test the effects of one IV with two levels to determine if the two groups (levels) are significantly
different.
EXAMPLE: Are people affected in their judgments of guilt & innocence by how physically attractive the
accused is?
M1= Attractive Defendant
M2=Unattractive Defendant
-the measure of error is the SED in the t-test
What if we wanted to include a third level of this one IV?
M1= Attractive
M2= Unattractive
M3= No information/Control Group
What if we wanted to test two IV’s?
Attractiveness of Defendant
Attractive
Unattractive
Control
Male
Gender of Defendant
Female
The Problem With Multiple Comparisons
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The Type I error rate increases drastically with multiple comparisons
The more statistical tests we perform, the more likely it is that some will be statistically significant purely by
chance
The following table illustrates how the alpha error originally set at .05 inflates as the number of decisions to
reject increases:
In doing research, we can not afford to take this kind of chance of committing alpha error.
-To correct for this, there is the Bonferroni Test and Analysis of Variance (ANOVA)
Bonferroni Test
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Corrects for multiple t-tests (ie. multiple comparisons)
Used to even out the level of significance among those t comparisons
-only used when comparisons are planned from the outset of the study
Sets a lower alpha level for all the t tests combined
Take the original alpha and divide it by the number of t tests performed
EX: If the original alpha is set at .05 and there are 3 t tests planned then the
alpha is calculated:
α=.05/3=.016 or .02
In this example, you would make three decisions about whether or not to
reject Ho, each at the .02 level
The theory behind the Bonferroni is that the probability of all the t-tests
combined can not be greater than the sum of each individual t-test
EX: α=.016 + .016 + .016=.048 or .05
One-Way ANOVA
• Compares more than two groups using only one statistical decision
• Uses the F distribution so it’s also called “The F test”
• “One-Way” when there is only one IV with various levels
Ho: μ1 = μ2 = μ3 =……μk
Ha: μ1 ≠ μ2 ≠ μ3 ≠……μk
k=the number of levels of the independent variable
Rejection of Ho tells us there is at least one significant difference but it doesn’t
tell us where it is
Developer of ANOVA
Logic of ANOVA
Attractive
x
x
x
x
x
m1
Unattractive
x
x
x
x
x
m2
No Information
x
x
x
x
x
m3
Overall Mean: M
Investigates Three Sources of Deviation
SSt
1) Deviation of each score from the overall mean
SSb
2) Deviation of each group mean from the overall mean
3) Deviation of each score form its own group mean (measure of error)
SSw
Ronald Fisher
Sum of Squares: sums of squared deviations from the mean
-we square & sum the scores from each of the 3 sources of deviation to end up with 3 types of sums of squares
**NOTE:SSb + SSw = SSt
One-Way ANOVA
Calculation of the Sums of Squares (SS):
SSt
SSb
SSw
One-Way ANOVA
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Variance Estimates: estimates of the population variance
-the sums of squares are just descriptive statistics
-In order to evaluate our data we must attempt to make inferences from the sample to population
-in order to make inferences, we calculate mean squares
Mean Squares: mean of the squared deviations
-ie. An independent estimate of the population variance
-SS is affected by sample size so we need to determine the mean of each source of deviation
-we use degrees of freedom to arrive at mean squares
Total
dft = N-1
MSb = SSb/dfb
Between dfb = k-1
Within
dfw = N-k
MSw = SSw/dfw
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F-ratio: the ratio of the difference between groups to the difference within groups
- Calculate:
Evaluation ANOVA
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Use Table G: Critical values of F
-Use the df for the numerator and the df for the denominator to arrive at
the critical values at the .05 or .01 levels
*df for the numerator = dfb
*df for the denominator = dfw
-On Table G, the lightface type is α=.05 and the boldface type is α=.01
-The obtained F is significant at a given level if it is equal to or greater
than the critical value (table value)
**Note: If you have one IV with two levels t = √F and t2 = F
One-Way ANOVA Worksheet
One-Way ANOVA Homework due Next Class
Tukey’s HSD
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Tukey’s HSD (Honestly Significant Difference) is a post-hoc multiple comparison test that can be applied
when the null hypothesis is rejected.
-determines exactly where the significant differences are between groups
-we need critical values to evaluate the differences in the comparisons of an ANOVA
-Tukey’s HSD calculates the critical values
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Calculation:
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Step 1: Determine the Tukey table value (Table H) at either α.05 or α.01
-use k and dfw (for the mean square within groups)
Step 2: plug the table value into the formula (in the place of α.05)
-plug all other values into the formula
Step 3: Find the differences between each pair of means (subtract them)
A-B or M1 – M2
A-C or M1 – M3
B-C or M2 – M3
Interpreting HSD
-the difference between any pair of means must be equal to or greater than Tukey’s HSD to reject Ho
Tukey’s HSD Worksheet
Tukey’s HSD Homework due Next Class
Factorial ANOVA
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When there is only one IV with 3 or more levels, the ANOVA is a One-Way ANOVA
When there is more than one IV, the ANOVA is a Factorial ANOVA
There are still 3 sources of deviation (with one change):
1) TOTAL: Deviation of each score from the overall mean
2) BETWEEN:
-Difference between Male/Female: MAIN EFFECT A
-Difference between Attractive/Unattractive/No Info: MAIN EFFECT B
-Interaction of the 2 variables: A x B Interaction
3) WITHIN: Deviation of each score form its own group mean (measure of error)
Factorial ANOVA
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Example of a Three-Way ANOVA (2 x 2 x 3)
-3 Independent Variables
-7 F-Ratios will be evaluated
MAIN EFFECT A: Male vs. Female
MAIN EFFECT B: Property vs. Person Crime
MAIN EFFECT C: Attractive vs. Unattractive vs. No Info
A x B Interaction
A x C Interaction
B x C Interaction
A x B x C Interaction
Factorial ANOVA: Calculation
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Before Calculation: Draw the block of cells
-One IV is set up in columns and the other IV is set up in rows
-a cell is where the row intersects the column
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Step 1: Calculate SSt and Identify C
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Step 2: Calculate SSb
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Step 3: Calculate SSw
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Step 4: Calculate SSA
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Step 5: Calculate SSB
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Step 6: Calculate SSAxB
Two-Way ANOVA Worksheet
C
Factorial ANOVA
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Step 7: Set up Summary Table, fill in SS & calculate df, MS, & F
SS
SSA
SSB
SSAxB
SSw
Degrees of Freedom (df)
dfA= # of columns – 1
dfB = # of rows – 1
dfAxB = (dfA) (dfB)
dfw = N-k
Mean Squares (MS)
MSA= SSA/dfA
MSB = SSB/dfB
MSAxB = SSAxB/dfAxB
MSw = SSw/dfw
F
MSA/MSw
MSB/MSw
MSAxB = MSw
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Step 8: Graph the Interaction
-start by calculating the means for each cell and the interactions
- IV #1 (or A or columns) is plotted on the X-axis in words
- DV is plotted on the Y-axis in numbers
- IV #2 (or B or rows) is represented by the lines that connected the plotted points
**use solid lines for one condition and a dashed line for the other
NOTE: If lines cross or they differ significantly from being parallel there is a significant interaction
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Step 9: Evaluate the 3 F-ratios for significance (Table G)
-same as with a one-way ANOVA but now you have three F-ratios instead of one
Factorial ANOVA: 2x3
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If in a two-way ANOVA an IV has more than 2 levels then calculation remains the
same but evaluation changes slightly.
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The two main effects will have different df’s if the IV’s have a different number of
levels
-in this case, you need to look up two sets of critical values in Table G
**Main effect A at the .01 and .05 levels
**Main effect B at the .01 and .05 levels
-You do not need to look anything up for the interaction
**if the F for the interaction is equal to or greater than ANY of the critical values
you looked up for the main effects then you can reject the null
Factorial ANOVA Homework due Next Class