Model II ANOVA: Variance Components Model II MS A = s 2 + ns 2 A MS A MS W = ns 2 A (MS A MS W )/n = ns 2 A /n = s2 A Usually Expressed: s 2 A /(s2 A + s2 W ) x 100
Assumptions of ANOVA Random Sampling Independence of experimental units Homogeneity of variances Normal distribution of residual error
Meeting ANOVA Assumptions Through Data Transformation Log transformations Power transformations (e.g., square root) Arcsine transformation
Nonparametric Alternatives to 1-way ANOVA Wilcoxon or Mann-Whitney U test for two-sample comparisons (analogous to a t-test) Kruskall-Wallis test (analogous to 1- way ANOVA)
More Complex ANOVA Designs Proper experimental designs are essential in order to: Meet ANOVA assumptions Increase the power of an experiment of a specified size Accommodate simultaneous consideration of multiple null hypotheses
Completely Randomized Design Simplest of all experimental designs Treatments are assigned to experimental units in an entirely random fashion. Advantages: Flexibility Missing data is not a big problem Analysis is simple
Example of Completely Randomized Design Scores Before Scores After 1.5 1.5 1.5 2.0 1.9 1.4 1.5 1.4 1.8 1.7 1.2 1.5 1.6 1.7 1.4 1.6 1.6 1.7 1.9 2.1 1.7 1.7 1.7 2.0 1.6 1.2 1.6 1.6 1.9 1.6
H0: Supplemental instruction does not improve reading scores HA: Supplemental instruction improves reading scores Y 1 = 1.57 Y 2 = 1.70 s 2 1 = 0.045 n 1 = 15 Is this a 1- or 2- tailed test? How many degrees of freedom? What is the critical value of t 0.05? s 2 2 = 0.046 n 2 = 15 1-tailed n 1 + n 2 2 = 28 t critical = 1.701
H0: Supplemental instruction does not improve reading scores HA: Supplemental instruction improves reading scores Y 1 = 1.57 s 2 1 = 0.045 n 1 = 15 Y 2 = 1.71 s 2 2 = 0.047 n 2 = 15 a = 0.05 t critical = 1.701 df = 28 t = 1.669 Accept H0 t = Y 2 -Y 1 / 1/n(s 2 1 + s2 2 )
Power of our test The power of a given test is determined by 3 things: 1) sample size 2) variability of the populations sampled 3) the difference between means The power of a performed t-test can be calculated in the following manner: 1) Calculate f = (nd 2 2s 2 p )/ 4s2 p 2) Find the calculated value of f on the x-axis of the power curve 3) Find the corresponding power (1 b) on the y- axis
Example of Paired Design 0.2 1.6 1.4 15 0.2 1.9 1.7 14 0.0 1.6 1.6 13 0.1 1.6 1.5 12 0.0 1.2 1.2 11-0.1 1.6 1.7 10 0.2 2.0 1.8 9 0.3 1.7 1.4 8 0.2 1.7 1.5 7 0.3 1.7 1.4 6 0.2 2.1 1.9 5-0.1 1.9 2.0 4 0.2 1.7 1.5 3 0.1 1.6 1.5 2 0.1 1.6 1.5 1 Improvement (D) Score After Score Before Pupil
Paired t-test H0: The difference between before and after = 0 HA: The difference between before and after > 1 b = 15 D = 0.127 s 2 D = 0.0164 t = D/ s 2 D /b df = b - 1 1-tailed test t critical = 1.761 t = 3.84 Reject H0
What is the power of our paired t-test? 1) Calculate t b,n : t b,n = d s 2 D /b- t a,n 2) Compute the probability of t b,n using SAS probt function or by looking up in a t-table and calculating 1-p.
Why has the power of our test increased? Pairing data in a logical fashion can lower the (within group) experimental error. Members of each pair have something in common that affects the experimental units of the pair in the same way. This something in common is a variance component due to the paring criterion. Recognizing the pairing has the effect of removing this variance component from the within-group MS.
Randomized Complete Block Design The more generalized form of the paired design for a 2 groups. Each block represents a set of experimental units, each randomly assigned to a different group (e.g., treatment) and all with something in common. The experiment is analyzed by a 2-way ANOVA.
RCBD Example Block 1 Trt 4 Trt 1 Trt 3 Trt 2 Environmental Gradient Block 2 Trt 2 Trt 1 Trt 3 Trt 4 Block 3 Block 4 Trt 3 Trt 2 Trt 4 Trt 1 Trt 2 Trt 3 Trt 1 Trt 4 4 Treatments 6 Blocks 24 Exp. Units Block 5 Trt 1 Trt 2 Trt 4 Trt 3 Block 6 Trt 1 Trt 3 Trt 4 Trt 2
RCB ANOVA If we had a completely randomized design, our ANOVA model would have been: Y ij = m + a i + e ij With a Randomized Block design, the model expands to: Y ij = m + a i + B j + [(ab) ij ] + e ij
Model I or Model II? The decision as to whether group (treatment) effects are fixed or random is made exactly as before. Block effects, because they are usually trying to control for random environmental variation, are usually random (Model II) effects. The entire ANOVA model can therefore be either a Model II (if the group effects are random) or a Mixed Model ANOVA if the group effects are fixed.
Completely Randomized Design (Model I) Source Treatment Error Total df a - 1 a(n-1) ab-1 SS ns(y-y) 2 SS(Y-Y) 2 SS(Y-Y) 2 RCBD ANOVA Table (Mixed Model) Source Block Treatment Error Total df b - 1 a - 1 (a-1)(b-1) ab 1 Expected MS as(y B -Y) 2 bs(y T -Y) 2 SS(Y-Y T -Y B + Y) 2 SS(Y-Y) 2
RCBD ANOVA Table (Mixed Model) Source df Expected MS Block b - 1 s 2 + as 2 B Treatment a - 1 s 2 + [s 2 AB ] + b/(a-1)sa2 Error (a 1)(b 1) s 2 + [s 2 AB]
2-way ANOVA: Factorial Design Two treaments applied to each experimental unit. The experimenter is usually interested in how the treatments act individually and in concert (interaction).
Factorial Design Crosspollination Selfpollination Species 1 Species 2
Factorial Design Two fixed-effect treatments: species and pollination type Two levels of each treatment: species (M. guttatus vs. M. micranthus); pollination (self vs. cross) Equal replication within each treatment combination
Factorial ANOVA (Model I) Source Species df r - 1 EMS s 2 + nc/(r-1)sa 2 Pollination treatment Interaction (species X poll. trt.) Error c - 1 (r - 1)(c - 1) rc(n-1) s 2 + nr/(c-1)sb 2 s 2 + n/[(r - 1)(c - 1)]S(ab) s 2 Total rcn - 1
Main Effects and Interactions Main Effect of Species and Pollination Treatment No Interaction 60 50 M. guttatus M. micranthus Main Effect of Species and Pollination Treatment Significant Interaction 100 M. guttatus M. micranthus Number of Flowers 40 30 20 Number of Flowers 10 10 0 Self Outcross 1 Self Outcross Pollination Treatment Pollination Treatment Main Effect of Species and Pollination Treatment Significant Interaction 60 50 M. guttatus M. micranthus 60 50 No Main Effects Significant Interaction Number of Flowers 40 30 20 Number of Flowers 40 30 20 10 10 M. guttatus M. micranthus Self Outcross Self Outcross Pollination Treatment Pollination Treatment
Factorial ANOVA (Model I) Source EMS (Model I) EMS (Model II) EMS (Mixed Model) Species s 2 + nc/(r-1)sa 2 s 2 + ns 2 SP + ncs2 S s 2 + ns 2 SP +nc/(r-1)sa2 Pollination treatment s 2 + nr/(c-1)sb 2 s 2 + ns 2 SP + nrs2 P s 2 + nrs 2 P Interaction (species X poll. trt.) s 2 + n/[(r - 1)(c - 1)]S(ab) s 2 + ns 2 SP s 2 + ns 2 SP Error s 2 s 2 s 2
Unbalanced Factorial Designs Adjustments for unequal sample size in factorial designs (2- and multiway ANOVAs in general) are more complicated than with 1-way ANOVAs. When doing factorial designs and sample sizes are unequal, ignore Type I SS and use only Type III SS.