Analysis of Variance (One Factor)

ANOVAAnalysis of Variance

Tests whether differences exist among population means categorized by only one factor or independent variable.

Such as: hours of sleep deprivation.Assumptions:

All scores are independentEach subject contributes just one score to the

overall analysis

Sources of varianceTreatment effect

The existence of at least one difference between the population means defined by the independent variable.

Between groupsWithin groups (random error)

Similar to pooled variance estimate s2p

Understanding variabilityProgress check 16.1, page 335

Hypothesis Test SummaryOne-Factor F Test (Sleep Deprivation Experiment, Outcome B pg 331-2)

Research Problem

On average, are subjects’ aggression scores in a controlled social situation affected by sleep deprivation periods of 0, 24, or 48 hours?

Statistical Hypothesis

Ho: µo = µ24 = µ48 H1: Ho is false

Decision Rule

Reject Ho at .05 level of significance if F ≥ 5.14 (from Table C, Appendix C, given dfbetween = 2 and dfwithin = 6)

Calculations

F = 7.36 (See Tables 16.3 (p. 342) and 16.6 (p. 348)for details )

Decision

Reject Ho at the .05 level of significance because F = 7.36 exceeds 5.14

Interpretation

Hours of sleep deprivation affect the subjects’ mean aggression scores in a controlled social situation.

F testF ratio

variability between groupsF = variability within groups

random errorIf Ho is true then F = random error

random error + treatment effect

If Ho is false then F = random error

F testAn F test of the null hypothesis,

if Ho is true, then numerator and denominator will be about the same.

If Ho is false, then the numerator will tend to be larger then the denominator. Suggesting true differences between the groups as a result of the treatment.

Variance EstimatesSum of squares is the variance estimateSample variance (s2) is the mean of the

varianceMean square (MS) is the synonymous with s2

SSMS = df

SS computations T2 G2

SSbetween = Σ n N

T2

SSwithin = Σ X2 – Σ n

G2

SStotal = Σ X2 – N

T = total group n = group sample size G = Grand total N = grand (combined) sample size

Formulas for df TermsDftotal = N - 1, number of all scores – 1

Dfbetween = k – 1, number of groups – 1

Dfwithin = N – k, number of all scores – number of groups

Sources of variabilityTotal Variability

Variability between groups

Variability within groups

MSbetween = SSbetween

dfbetween

MSwithin = SSwithin

dfwithin

F = MSbetween

MSwithin

Progress check 16.3 page 347

Progress check 16.4 page 347

Progress check 16.5 page 348

F test is nondirectionalSince all the variations in F are squared, this

test is by nature a nondirectional test, even though only the upper tail of the sampling distribution contains the rejection area.

Effect size for FSince F only indicates that the null is

probably false, the effect size allows the test to have a certain level of confidence.

Effect size for F is called “eta squared” (η2) SSbetween

η2 = SStotal

Guidelines for η2 η2 Effect

.01 Small

.09 Medium

.25 Large

Multiple comparisonsUse Tukey’s HSD test to find differences

between pairs of means.

Tukey’s “honestly significant difference” testMSwithin

HSD = q√ n

Where q (studentized range statistic) comes from Table G, Appendix C, page 529

Tukey’s HSDCreate a grid containing all possible

combinations of differences between means for all groups.

The absolute mean difference is compared the value of HSD.

Any absolute mean difference values greater than HSD can be considered significant at the critical probability level chosen.

Table 16.8Absolute differences between means (for

sleep deprivation experiment)X0 = 2 X24 = 5 X48 = 8

X0 = 2 ---- 3 6*

X24 = 5 ---- 3

X48 = 8 ----

* Significant at the .05 level. HSD = 4.77 (page 354)

Estimating Effect SizeOnce a pair of means is determined to have

an effect based on Tukey’s HSD, you can determine the effect size using Cohen’s d

X1 – X2

D = √ MSwithin

SPSS Output for One-way ANOVA - TukeyHSD

Sessions NSubset for alpha = .05

1 2 3

0 8 1.6250

1 8 3.1250 3.1250

2 8 5.0000 5.0000

3 8 7.5000

Sig. .523 .332 .123

Means for groups in homogeneous subsets are displayed. a. Uses Harmonic Mean Sample Size = 8.000.

Final Interpretation of Sleep Deprivation Experiment

Aggression scores for subjects deprived of sleep for zero hours (X = 5, s = 1.73), and those deprived for 48 hours (X = 8, s = 2.00) differ significantly [F(2,6) = 7.36, MSE = 3.67; p < .05; η2 = .71]. According to Tukey’s HSD test, however, only the difference of 6 between mean aggression scores for the zero and 48-hour groups is significant (HSD = 4.77, p < .05, d = 3.13).

Flow chart for one-factor ANOVAF-TEST

Nonsignificant F (ns) Significant F (p < .05)

ESTIMATE TEST

EFFECT SIZE (η2) MULTIPLE COMPARISONS (HSD)

Nonsignificant HSD (ns) Significant HSD (p <.05)

ESTIMATE EFFECT SIZE (d)