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Statistics for the Behavioral Sciences (5th ed.)Gravetter & Wallnau

Chapter 13

Introduction to Analysis of Variance (ANOVA)

University of GuelphPsychology 3320 — Dr. K. Hennig

Winter 2003 Term

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Figure 13-2 (p. 397)A typical situation in which ANOVA would be used. Three separate samples are obtained to evaluate the mean differences among three populations (or treatments) with unknown means.

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Figure 13-3 (p. 399) - mixed model designA research design with two factors. The research study uses two factors: One factor uses two levels of therapy technique (I versus II), and the second factor uses three levels of time (before, after, and 6 months after). Also notice that the therapy factor uses two separate groups (independent measures) and the time factor uses the same group for all three levels (repeated measures).

4Statistics for the Behavioral Sciences, Sixth Edition by Frederick J. Gravetter and Larry B. WallnauCopyright © 2004 by Wadsworth Publishing, a division of Thomson Learning. All rights reserved.

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The test statistics

)(exp errorchancebyecteddifferencemeanssamplebetweendifferencet =

)(exp)(var)(var

errorchancebyectedsdifferenceiancemeanssamplebetweensdifferenceiance

F =

)(

)()(

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2121

MMsMM

t−

−−−=

µµ

n E.g., M1 = 20 M2 = 30; can find the difference.n but if there are three means? Use variance to

measure the size of the difference between groups.

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Logic of ANOVA

n Goal: measure the variability and determine where it comes from.

n Determine the total variability of the datan break into parts (analyze) the variability/variance,

i.e., ANOVA

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Logic (contd.)

n Between-treatment variance (note 50º vs. 70º)nWithin-treatment variance (e.g., 70º - not all the

same)nWhen you see “variance,” think “differences”

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Logic (contd.)

n H0: the differences between treatments are simply due to chance

n H1: the differences are significantly greater than can be explained by chance, i.e., the differences are caused by treatment effects

n Two primary sources of chance differences:n individual differences - people are differentn experimental error - measurement always

involves

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Logic (contd.)

n Compare with chance - how big are the differences when there is no treatment effect?

nWithin-treatments variance:n e.g., in 70º condition individuals were all tested

in same condition, yet different scoresn within-treatment variance is a measure of

difference expected just by chance

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Figure 13-4 (p. 403)The independent-measures analysis of variance partitions, or analyzes, the total variability into two components: variance between treatments and variance within treatments.

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The F-ratio: The test statistic for ANOVA

n An F-ratio near 1.00 means differences are due to chance

n In ANOVA the denominator is the error term; the same as the numerator when the treatment effect is zero

chancetoduesdifferencechancetoduesdifference

chancetoduesdifferencechancetoduesdifferenceeffecttreatment

treatmentswithiniancetreatmentsbetweenianceF

+=

+=

=

0

varvar

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Steps

n 1. Analysis of the SSn SSwithin

n SSbetween

n 2. Analysis of the degrees of freedom (df)n dfwithin

n dfbetween

n 3. Calculation of variances

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1. Analysis of the SS

n Calculate T1,2,3 (?X=) for each of the k (=3) treatment conditions

n Calculate G (grand total) = T1 + T2 + T3

n N =n1 + n2 + n3 Calculate ? X2 for entire N=15

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Table 13-2 (p. 406) Hypothetical data from an experiment examining learning performance under three temperature conditions.

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1a. SS within treatmentsPartitioning the sum of squares (SS) for the independent-measures analysis of variance.

∑ ∑−=NX

XSS2

2 )(

321 SSSSSSSSwithin ++=

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1b. SS between treatments

n But, only works if samples are all same size (ns are equal), thus use a compuationalformula:

NG

nT

SS between

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−= ∑

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Step 2: Analysis of the degrees of freedom (df) Partitioning degrees of freedom (df) for the independent-measures analysis of variance.

dftotal = N - 1 = 15 - 1=14

dfwithin = N - k = 15 - 3 =12dfbetween = k - 1 = 3 - 1 = 2

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Step 3: Analysis of the variances The structure and sequence of calculations for the analysis of variance. (Recall: s2 = SS/df)

within

between

MSMS

=

10

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Results are organized into a table

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Figure 13-8 (p. 413) The distribution of F-ratios with df = 2.12. Of all the values in the distribution, only 5% are larger than F = 3.88, and only 1% are larger than F = 6.93.

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Table 13-3 (p. 414) A portion of the F distribution table. Entries in roman type are critical values for the .05 level of significance, and bold type values are for the .01 level of significance. The critical values for df = 2.12 have been highlighted (see text).

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Figure 13-15 (p. 421)A visual representation of the between-treatments variability and the within-treatments variability that form the numerator and denominator, respectively, of the F-ratio. In (a), the difference between treatments is relatively large and easy to see. In (b), the same 4-point difference between treatments is relatively small and is overwhelmed by the within -treatments variability.

M1=8M2=12

88

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Figure 13-11 (p. 431) The distribution of tstatistics with df = 18 and the corresponding distribution of F-ratios with df = 1.18. Notice that the critical values for α = .05 are t = ±2.101 and that F = 2.1012 = 4.41

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Post hoc tests

n E.g., M1 = 3 M2 = 5 M3 = 10n 2-pt difference M1 and M2

n experimentwise alpha leveln planned vs. unplanned comparisonsn Tukey (HSD)

n Scheffe

nMS

qHSD within=