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 s 2 p

Understanding variability Progress check 16.1, page 335

Hypothesis Test Summary One-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 H o : µ o = µ 24 = µ 48 H 1 : H o is false Decision Rule Reject H o at.05 level of significance if F ≥ 5.14 (from Table C, Appendix C, given df between = 2 and df within = 6) Calculations F = 7.36 (See Tables 16.3 (p. 342) and 16.6 (p. 348)for details ) Decision Reject H o 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 H o is true then F = random error random error + treatment effect If H o is false then F = random error

F test An F test of the null hypothesis, if H o is true, then numerator and denominator will be about the same. If H o 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 (s 2 ) is the mean of the variance Mean square (MS) is the synonymous with s 2 SS MS = df

SS computations T 2 G 2 SS between = Σ n N T 2 SS within = Σ X 2 – Σ n G 2 SS total = Σ X 2 – N T = total group n = group sample size G = Grand total N = grand (combined) sample size

Formulas for df Terms Df total = N - 1, number of all scores – 1 Df between = k – 1, number of groups – 1 Df within = N – k, number of all scores – number of groups

Sources of variability Total Variability Variability between groupsVariability within groups MS between = SS between df between MS within = SS within df within F = MS between MS within

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” (η 2 ) SS between η 2 = SS total

Guidelines for η 2 η2η2 Effect.01Small.09Medium.25Large

Multiple comparisons Use Tukey’s HSD test to find differences between pairs of means. Tukey’s “honestly significant difference” test MS within 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 16.8 Absolute differences between means (for sleep deprivation experiment) X 0 = 2X 24 = 5X 48 = 8 X 0 = *6* X 24 = X 48 = * 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 X 1 – X 2 D = √ MS within

SPSS Output for One-way ANOVA - TukeyHSD SessionsN Subset for alpha = Sig Means for groups in homogeneous subsets are displayed. a. Uses Harmonic Mean Sample Size =

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; η 2 =.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 ANOVA F-TEST Nonsignificant F (ns)Significant F (p <.05) ESTIMATETEST EFFECT SIZE (η 2 )MULTIPLE COMPARISONS (HSD) Nonsignificant HSD (ns)Significant HSD (p <.05) ESTIMATE EFFECT SIZE (d)