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Contrasts
When testing hypotheses and ANOVA indicates a significant effect, we seek more specific information about our data Which group means are different? Contrasts, or comparisons, let us examine differences
Independent variable: brand of coffee Dependent variable: employee productivity (number of cream puffs
produced)What are the relevant comparisons you would want to make?
Contrasts
One procedure for making individual comparisons among sample means is called the method of planned (independent) comparisons
After overall F test (based on dividing MSA by MSS/A) is found to be significant, follow up with a series of individual F tests, each with one degree of freedom for the numerator
Contrasts
Planned comparisons (also called pairwise comparisons or single-df tests) give more information about what is happening in the data
Complex comparisons allow testing average of 2 or more groups with another group
Unplanned contrasts limited to (k-1) tests of significance
Contrasts
Suppose we are interested in the effects of coffee on employees’ cream-puff production
Starbucks, Peet’s, Maxwell House, Folgers, Sanka There are a possible total of a(a – 1)/2 comparisons In our coffee example: (5*4)/2 = 10 possible pairwise
comparisons Starbucks vs. each of the 4 other brands, Peet’s vs. each
of the other 3 brands, Maxwell House vs. 2 other brands, Folgers vs. Sanka
But not all of these makes sense
Coffee Contrasts
H01: X1 – X2 = 0
H02: X1 – X3 = 0
H03: X1 – X4 = 0
H04: X1 – X5 = 0
H05: X2 – X3 = 0
H06: X2 – X4 = 0
H07: X2 – X5 = 0
H08: X3 – X4 = 0
H09: X3 – X5 = 0
H010: X4 – X5 = 0
Coefficients for pairwise contrasts {1, -1}
Ψj = C1X1 + C2X2 + C3X3 + …+ CkXk where Σ Cj = 0 j=1
k
Coffee Contrasts
A comparison of the Sanka (decaf) with the other brands
Comparisons among the café brands and the store brands
Comparisons between the café brands (Starbucks and Peet’s)
Comparisons between the store brands (Maxwell House and Folgers)
(+1)X1 + (+1)X2 + (+1)X3 + (+1)X4 + (-4)X5
(+1)X1 + (-1)X2 + (0)X3 + (0)X4 + (0)X5
(+1)X1 + (+1)X2 + (-1)X3 + (-1)X4 + (0)X5
(0)X1 + (0)X2 + (+1)X3 + (-1)X4 + (0)X5
Contrasts
Convert contrast into sums of squares
nΨ 2
SS =
Σc2
Sum of squared coefficients
Difference between pair of means
Contrasts
Contrasts that flow from the omnibus F test and are followed up with pairwise comparisons allow “continuity”
When overall F test is not significant, t tests of theoretically important mean comparisons may be appropriate
Experimentwise Error Rate
Type I error rate accumulates over a family of tests
If each test is evaluated at α significance level, probability of avoiding Type I error is 1- α
Probability of making no familywise errors with c tests = (1- α)c
Pairwise Corrections
Bonferroni Sidak Dunnett Tukey’s HSD Fisher-Hayter Newman-Keuls Scheffe (for post-hoc error correction)
Trend Analysis
If an experiment contains a quantitative IV then the shape of the function relating the levels of this quantitative IV to the DV is often of interest Trend analysis can be
used to test different aspects of the shape of the function relating the IV (advertising repetitions) and the DV (sales)
0
200
400
600
800
1000
2 4 6 8
Sales
Repetitions
Trend Analysis
The linear component of trend is used to test whether there is an overall increase (or decrease) in the DV as the IV increases A test of the linear
component of trend is a test of whether this increase in sales is significant
0
200
400
600
800
1000
2 4 6 8
Sales
Repetitions
Trend Analysis
If there were a perfectly linear relationship between repetitions and sales, then no components of trend other than the linear would be present
The quadratic component of trend is used to test whether the slope increases (or decreases) as the independent variable increases 0
200
400
600
800
1000
2 4 6 8
Sales
Repetitions
Trend Analysis
Trend analysis is computed as a set of orthogonal comparisons using a particular set of coefficients
Each set of comparisons is tested for significance Linear Quadratic Cubic
Trend Analysis
Examples of theoretically motivated trend analysis Effect of ad repetition on persuasion Moderate schema incongruity Law of diminishing returns Losses loom larger than gains
Effect Sizes
Effect size is a name given to a family of indices that measure the magnitude of a treatment effect Unlike significance tests, effect size indices are independent of sample size Measures are the common currency of meta-analyses
Two ways to think about effect sizes The standardized difference between means
d = (m1 – m2)/s Appropriate when comparing two groups
How much of variability can be attributed to treatments Effect size is ratio of variability explained to total variability or the SSEffect/SSTotal
Expressed as R2
Appropriate for > 2 groups
Check out effect-size calculators at http://davidmlane.com/hyperstat/effect_size.html
Effect Sizes
Effect sizes expressed as d and R2 are descriptive statistics
In the populationω2 = (σTotal
2 - σError2)/ σTotal
2
ω2 = SSA – (a-1) MSS/A
SStotal + MSS/A
ω2 = (a - 1) (F -1)
(a - 1) (F -1) + (an)
Effect Sizes
Properties of ω2: Varies between 0 and 1 (except in a fluke occurrence
when F < 1, (negative ω2) then treat it like zero) Unlikely to get very high estimates of ω2 because
behavioral research has so much error variance Cohen’s guidelines
.01 = small .5 = moderate .8 = large
Effect Sizes
0.2 effect size corresponds to the difference between the heights of 15 and 16 year old girls in the US
0.5 effect size corresponds to the difference between the heights of 14 and 18 year old girls
0.8 effect size equates to the difference between the heights of 13 and 18 year old girls