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Analysis of Variance (ANOVA)
A procedure used to evaluate the differences between two or moretreatments (groups) – a ttest compares only two groups.

Statistical Hypotheses for ANOVA
 Assuming there are 3 groups – more μ values if there are more groups
 H_{0}: μ_{1} = μ_{2} = μ_{3}
 H_{1}: not H_{0}


Why variance?
When there is more than 2 sample means it is impossible to computemean differences (therefore need to use “variance”)

What is variance?
Variance is simply a method for measuring how big the differences arefor a set of numbers (variance = differences)

The Logic of ANOVA
 (1) Not all the scores are the same – there is variability (variance)
 (2) There are 3 types of variability –
 (a) total
 (b) betweentreatments
 (c) withintreatments

Total Variability
 (1) The deviation of each score from the Grand Mean (viz. the average of all the scores)
 (2) This variability can be partitioned or divided into 2 parts:
 (a) the amount that score deviates from its group mean
 (i.e., withintreatment variation)
 (b) the amount that score’s group mean differs from the grand mean
 (i.e., betweentreatment variation)

BetweenTreatments Variability
 (1) Measures the differences between sample means
 (2) What causes those differences between means?
 (a) Treatment Effects: Systematic differences between groups due to the effect of the IV
 (b) Chance or Experimental Error
 (3) Sources of chance or experimental error
 (a) Individual Differences (e.g., sleep, past drug use, IQ, etc.)
 (b) Measurement Error (e.g., not calibrating a machine, misreading a dial, etc.)

Treatment Effects
Systematic differences between groups due to the effect of the IV

WithinTreatments Variability Logic
If all Ss were the same (no experimental error), all the subjects who received thesame treatment (e.g., 10 mg of a drug) would have the same score on the DV

WithinTreatments Variability Conclusion
The variability of subjects in the same condition gives us an estimate ofchance or experimental error. As such, withintreatments variability is solelycaused by chance or experimental error.

Goal of ANOVA
To test for treatment effects (did the IV impact the DV). The Fratio allows youto accomplish this goal.


If H_{o} is true, then there are no treatment effects and the “conceptual formula” reduces to:
 This equals 1

If Ho is false, then there are treatment effects and the numerator or the “conceptual formula” gets
 bigger and bigger (making the overall fraction large).
 If the fraction (i.e.,value of this test statistic) gets large enough, we will feel that we haveadequate evidence to Reject H_{o} and conclude that we have treatment effects

ANOVA Notation and Formulas
 k = the number of treatment conditions (groups)
 n = the number of scores in each treatment
 N = the total number of scores in the entire study
 T = the total for each treatment condition (ΣX) [needed for formulas so memorize]
 G ("Grand Total") = the sum of all the scores in the study G = ΣT [needed for formulas so memorize]

Total Sums of Squares (SS_{Withintreatments}) [Total Variability]

WithinTreatments Sums of Squares (SS_{Withintreatments}) [WithinTreatments Variability]
 (add up all SS values for each group)

Degrees of freedom for ANOVA
 (1) df_{Total} = N – 1
 (2) df_{Withintreatments} = N – k
 (3) df_{Betweentreatments} = k – 1

Why do we Need Mean Squares?
Need to convert Sums of Squares (SS) into Variance estimates [i.e., Mean Square (MS)values] to use them in Analysis of Variance

What is a Mean Square?
MS or Variance = SS/df

MS Calculations
 (1) MS_{Betweentreatments} = SS_{Betweentreatments} / df_{Betweentreatments}
 (2) MS_{Withintreatments} = SS_{Withintreatments} / df_{Withintreatments}

F_{ratio} Calculation
F = MS_{Betweentreatments} / MS_{Withintreatments}

Characteristics of the Fdistribution
 (1) Fvalues will always be positive (there is never a negative variance)
 (2) When H_{0} is true, F is close to 1.00. As such, the distribution piles up around 1.00.

Fdistribution Table (Table B.4)
Purpose: Used to obtain critical value for boundary of your Critical Region

Hypothesis Testing Example
 * Research Question: Are these drugs equally effective as pain relievers?*
 DV: # of seconds that a painful stimulus is endured
 (A) Step #1 – State the hypotheses
 (1) H_{0}: μ_{1} = μ_{2} = μ_{3} = μ_{4}
 (2) H_{1}: not H_{0}
 (B) Step #2 – Set the criteria for a decision
 (1) α = .05
 (C)Step #3 Determine and draw boundary of the critical region(from Table B.4)
 ** Draw graph with the critical regionsPage #5
 (D) Step #4 – Collect data and compute sample statistics
 (1) Part 1 = calculate SS values
 (2) Part 2 = calculate MS values
 (3) Part 3 = calculate F statistic
 (4) Part 4 = complete the ANOVA Summary Table
 (E) Step #5 – Make a decision
 (1) Draw obtained score (“test statistic”) on the graph
 (2) Decision Rule:
 (a) If obtained value is in the critical region, you “Reject H_{0}”
 (b) If obtained value is not in the critical region, you “Fail to Reject H0”
 (3) Decision = Reject H_{0} (conclude that these drugs effective are not equally effective as painrelievers  that is there are differences among the 4 groups)
Note: Still need to do posthocs to determine “what differs from what”

Assumptions of the 1way Between subjects ANOVA
 (A) Independent scores
 (B) Normality (the populations from which the samples are selected must be normal)
 (C) Homogeneity of Variance (the populations from which the

