Analysis of Variance (One Factor)
ANOVA Analysis 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 independent
Each subject contributes just one score to the overall analysis
Sources of variance Treatment effect
The existence of at least one difference between the population means defined by the independent variable.
Between groups
Within groups (random error)
Similar to pooled variance estimate s2p
Understanding variability Progress 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 test F ratio
variability between groups
F = variability within groups
random error
If Ho is true then F = random error
random error + treatment effect
If Ho is false then F = random error
F test An 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 Estimates Sum of squares is the variance estimate
Sample variance (s2) is the mean of the variance
Mean square (MS) is the synonymous with s2
SS
MS = df
SS computationsT2 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 Terms Dftotal = 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 variability
Total 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 nondirectional Since 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 F Since 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 comparisons Use Tukey’s HSD test to find differences between pairs
of means.
Tukey’s “honestly significant difference” test
MSwithin
HSD = q√ n
Where q (studentized range statistic) comes from Table G, Appendix C, page 529
Tukey’s HSD Create 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 Size Once 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)