Differences Among Groups chapter 9 Differences Among Groups
Chapter Outline How statistics test differences Types of t tests Interpreting t Relationship of t and r Analysis of variance Analysis of covariance Experimentwise error rate Understanding multivariate techniques
How Statistics Test Differences Independent and dependent variables Evaluating the null hypothesis Establishing two points: Are the groups different? How meaningful are the differences? Assumptions for F and t distributions Distribution is normal. Samples from population are random. Numerator and denominator are estimates of same thing. Numerator and denominator are independent.
Differences Among Groups: t = M 1 – 2 s / n + d f ( ) Omega squared w 2 = ( t – 1 ) / + n
Dependent t Test t = M p o s – r e 2 + ( ) · / N 1 é ë ê ù û ú
Components of Variance Total variance = True variance + Error variance Test of significance = True variance ÷ Error variance – t and F Test of meaningfulness = True variance ÷ Total variance r2 and omega squared
ANOVA Formulas (continued)
ANOVA Formulas (continued) Summary table Source SS df MS F Between C–B K–1 (C–B)/(K–1) MSB/MSW (true) Within A–C N–K (A–C)/(N–K) (error) Total C–B N–1
Data for ANOVA Group 1 X X2 10 100 11 121 9 81 8 64 48 466 Group 2 10 100 11 121 9 81 8 64 48 466 Group 2 X X2 7 49 8 64 6 36 34 234 Group 3 X X2 3 9 6 36 4 16 5 25 21 95
Analysis of Variance (ANOVA) Summary table for ANOVA Source SS df MS F Between 72.9 2 36.47 29.57* Within (error) 14.8 12 1.23 Total 87.7 14 *p < .01
Follow-Ups to ANOVA Scheffé: Contrast all 3 pairs (1 and 2, 1 and 3, 2 and 3); a difference must exceed calculated Scheffé critical value (CV) to be significant, p < .05. k – 1 ( ) F a; ; N é ë ù û 2(MSW /n 3 . 8 = 2 7 9 2(1 .23 / 5 ) 5 CV =
Model for Factorial ANOVA
Factorial ANOVA IV1: are the 3 levels (rows) different? IV2: are the 2 levels (columns) different? Interaction: does one IV change as a function of the other? Three F ratios to test significance: IV1 IV2 Interaction
Interaction From Factorial ANOVA (continued)
Interaction From Factorial ANOVA (continued)
Interaction From Factorial ANOVA (continued)
Repeated-Measures ANOVA Typical use: do two or more groups change differently over trials (over time)? Example: Two groups (exercise and control) are measured every 2 weeks for 12 weeks. Analysis is a between (group)–by–within (trials) ANOVA with repeated measures on trials (trials are time, every 2 weeks).
Interaction of Groups With Repeated Measures Three age levels and four levels of movement difficulty as repeated measures. Used with permission from Albers, Thomas, & Thomas, 2005.
Experimenterwise Error Bonferroni technique to adjust alpha EW = ÷ number of comparisons If = .05 and three t tests (or ANOVAs) are to be done, then Adjusted = .05/3 = .017
Discriminant Analysis One independent variable with two or more levels and several (two or more) measures (dependent variables). Can a linear combination be made of dependent variables that will identify group membership?
Multivariate Analysis of Variance (MANOVA) Two or more independent variables and two or more dependent variables Independent variables = two age groups (10 & 12 years) and two levels of expertise (experts and novices) Dependent variables = knowledge and performance (continued)
Multivariate Analysis of Variance (MANOVA) (continued) MANOVA F ratios Age Expertise Age expertise Follow-ups
MANOVA With Repeated Measures 2 or more groups with 2 or more repeated measures on 2 or more variables Groups Trials 1 2 3 4 5 1 2 3 Groups = Exp 1, Exp 2, Control (3) Trials = time periods dvs = heart rate and body fat Sphericity assumption – equal r across trials
Analysis of Covariance (ANCOVA) Relationship between one (or more) covariates and dependent variable is removed. ANOVA is calculated on remaining dependent variables after variance due to covariate is removed.
Multivariate Analysis of Covariance (MANCOVA) Relationship between groups of covariates and multiple dependent variables is removed. MANCOVA is done on remaining dependent variables.