1. Linear Contrasts and Multiple Comparisons (Chapter 9)
One-way classified design AOV example.
Develop the concept of multiple comparisons and linear contrasts.
Multiple comparisons methods needed due to potentially large number of comparisons that may be made if Ho rejected in the one-way AOV test.
Terms:
Linear Contrasts
Multiple comparisons
Data dredging
Mutually orthogonal contrasts
Experimentwise error rate
Comparisonwise error rate
MCPs:
Fisher’s Protected LSD
Tukey’s W (HSD)
Studentized range distribution
Student-Newman-Keuls procedure
Scheffe’s Method
Dunnett’s procedure
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2. One-Way Layout Example
A study was performed to examine the effect of a new sleep inducing drug on a population of insomniacs. Three (3) treatments were used:
Standard Drug
New Drug
Placebo (as a control)
What is the role of the placebo in this study?
What is a control in an experimental study?
18 individuals were drawn (at random) from a list of known insomniacs maintained by local physicians. Each individual was randomly assigned to one of three groups. Each group was assigned a treatment. Neither the patient nor the physician knew, until the end of the study, which treatment they were on (double-blinded).
Why double-blind?
A proper experiment should be:
randomized, controlled, and double-blinded.
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3. Average number of hours of sleep per night.
Response:
Average number of hours of sleep per night.
Placebo: 5.6, 5.7, 5.1, 3.8, 4.6, 5.1
Standard Drug: 8.4, 8.2, 8.8, 7.1, 7.2, 8.0
New Drug: 10.6, 6.6, 8.0, 8.0, 6.8, 6.6
yij = response for the j-th individual on the i-th treatment.
Hartley’s test for equal variances:
Fmax = 4.77 < Fmax_critical = 10.8
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4. Excell Analysis Tool Output
What do we conclude here?
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5. Linear Contrasts and Multiple Comparisons
If we reject H0 of no differences in treatment means in favor of HA, we conclude that at least one of the t population means differs from the other t-1.
Which means differ from each other?
Multiple comparison procedures have been developed to help determine which means are significantly different from each other.
Many different approaches - not all produce the same result.
Data dredging and data snooping - analyzing only those comparisons which look interesting after looking at the data – affects the error rate!
Problems with the confidence assumed for the comparisons:
1-a for a particular pre-specified comparison?
1-a for all unplanned comparisons as a group?
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6. Where the ai are constants satisfying the constraint:
Linear Comparisons
Any linear comparison among t population means, m1, m2, ...., mt can be written as:
Where the ai are constants satisfying the constraint:
Example: To compare m1 to m2 we use the equation:
with coefficients
Note constraint is met!
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7. Variance of a linear contrast:
A linear comparison estimated by using group means is called a linear contrast.
Variance of a linear contrast:
Test of significance
Ho: l = 0 vs. Ha: l  0
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8. These two contrasts are said to be orthogonal if:
Orthogonal Contrasts
These two contrasts are said to be orthogonal if:
in which case l1 conveys no information about l2 and vice-versa.
A set of three or more contrasts are said to be mutually orthogonal if all pairs of linear contrasts are orthogonal.
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9. Orthogonal Non-orthogonal
Compare average of drugs (2,3) to placebo (1).
Contrast drugs (2,3).
Orthogonal
Non-orthogonal
Contrast Standard drug (2) to placebo (1).
Contrast New drug (3) to placebo (1).
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10. Drug Comparisons
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11. Importance of Mutual Orthogonality
Assume t treatment groups, each group having n individuals (units).
t-1 mutually orthogonal contrasts can be formed from the t means. (Remember t-1 degrees of freedom.)
Treatment sums of squares (SSB) can be computed as the sum of the sums of squares associated with the t-1 orthogonal contrasts. (i.e. the treatment sums of squares can be partitioned into t-1 parts associated with t-1 mutually orthogonal contrasts).
t-1 independent pieces of information about the variability in the treatment means.
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12. Example of Linear Contrasts
Objective: Test the wear quality of a new paint.
Treatments: Weather and wood combinations.
Treatment Code Combination
A m1 hardwood, dry climate
B m2 hardwood, wet climate
C m3 softwood, dry climate
D m4 softwood, wet climate
(Obvious) Questions:
Q1: Is the average life on hardwood the same as average life on softwood?
Q2: Is the average life in dry climate the same as average life in wet climate?
Q3: Does the difference in paint life between wet and dry climates depend upon whether the wood is hard or soft?
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13. Q1
MSE= 5 t= 4 n t
-t= 8
Q1: Is the average life on hardwood the same as average life on softwood?
Comparison:
Estimated Contrast
Test H0: l1 = 0 versus HA: l1  0
What is MSl1 ?
Test Statistic:
Rejection Region: Reject H0 if
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14. Conclusion: Since F=29. 4 > 5
Conclusion: Since F=29.4 > 5.32 we reject H0 and conclude that there is a significant difference in average life on hard versus soft woods.
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15. Q2
MSE= 5 t= 4 n t
-t= 8
Q2: Is the average life in dry climate the same as average life in wet climate?
Comparison:
Estimated Contrast
Test H0: l2 = 0 versus HA: l2  0
Test Statistic:
Rejection Region: Reject H0 if
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16. Conclusion: Since F=0. 6 < 5
Conclusion: Since F=0.6 < 5.32 we do not reject H0 and conclude that there is not a significant difference in average life in wet versus dry climates.
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17. Q3
MSE= 5 t= 4 n t
-t= 8
Q3: Does the difference in paint life between wet and dry climates depend upon whether the wood is hard or soft?
Comparison:
Estimated Contrast
Test H0: l3 = 0 versus HA: l3  0
Test Statistic:
Rejection Region: Reject H0 if
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18. Conclusion: Since F=0 < 5
Conclusion: Since F=0 < 5.32 we do not reject H0 and conclude that the difference between average paint life between wet and dry climates does not depend on wood type. Likewise, the difference between average paint life for the wood types does not depend on climate type (i.e. there is no interaction).
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19. Mutual Orthogonality The three are mutually orthogonal.
Total Error SS = dferror x MSE = 8 x 5 = 40
SSl1 = MSl1 = 147
SSl2 = MSl2 = 3
SSl3 = MSl3 = 0
Treatment SS = 150
The three mutually orthogonal contrasts add up to the Treatment Sums of Squares.
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20. (Type I) Error Rate
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21. p-value<0.05, even though there is no effect!
If Ho is true, and α=0.05, we can expect to make a Type I error 5% of the time…
1 out of every 20 will yield
p-value<0.05, even though there is no effect!
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22. Types of Error Rates
Compairsonwise Error Rate - the probability of making a Type I error in a single test that involves the comparison of two means. (Our usual definition of Type I error thus far…)
Question: How should we define Type I error in an experiment (test) that involves doing several tests? What is the “overall” Type I error?
The following definition seems sensible:
Experimentwise Error Rate - the probability of observing an experiment in which one or more of the pairwise comparisons are incorrectly declared significantly different. This is the probability of making at least one Type I error.
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23. Error Rates: Problems
Suppose we make c mutually orthogonal (independent) comparisons, each with Type I comparisonwise error rate of a. The experimentwise error rate, e, is then:
If the comparisons are not orthogonal, then the experimentwise error rate is smaller.
Thus in most situations we actually have:
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24. The Bonferroni Solution
Solution: set e=0.05 and solve for :
But there’s a problem…
E.g. if c=8, we get =0.0064!
Very conservative…, thus type II error is large.
Bonferroni’s inequality provides an approximate solution to this that guarantees:
We set:
E.g. if c=8, we get =0.05/8=
Still conservative!
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25. Multiple Comparison Procedures (MCPs): Overview
Terms:
If the MCP requires a significant overall F test, then the procedure is called a protected method.
Not all procedures produce the same results. (An optimal procedure could be devised if the degree of dependence, and other factors, among the comparisons were known…)
The major differences among all of the different MCPs is in the calculation of the yardstick used to determine if two means are significantly different. The yardstick can generically be referred to as the least significant difference. Any two means greater than this difference are declared significantly different.
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26. Multiple Comparison Procedures: Overview
Yardsticks are composed of a standard error term and a critical value from some tabulated statistic.
Some procedures have “fixed” yardsticks, some have “variable” yardsticks. The variable yardsticks will depend on how far apart two observed means are in a rank ordered list of the mean values.
Some procedures control Comparisonwise Error, other Experimentwise Error, and some attempt to control both. Some are even more specialized, e.g. Dunnett’s applies only to comparisons of treatments to a control.
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27. Fisher’s Least Significant Difference - Protected
Mean of group i (mi) is significantly different from the mean of group j (mj) if
if all groups have same size n.
Type I (comparisonwise) error rate = a
Controls Comparisonwise Error. Experimentwise error control comes from requiring a significant overall F test prior to performing any comparisons, and from applying the method only to pre-planned comparisons.
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28. Tukey’s W (Honestly Significant Difference) Procedure
Primarily suited for all pairwise comparisons among t means.
Means are different if:
{Table 10 - critical values of the studentized range.}
Experimentwise error rate = a
This MCP controls experimentwise error rate! Comparisonwise error rates is thus very low.
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29. Student Newman Keul Procedure
A modified Tukey’s MCP. Rank the t sample means from smallest to largest. For two means that are r “steps” apart in the ranked list, we declare the population means different if:
{Table 10 - critical values of the studentized range. Depends on which mean pair is being considered!}
varying yardstick
r=6
r=5
r=2
r=3
r=4
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30. Duncan’s New Multiple Range Test (Passe)
Neither an experimentwise or comparisonwise error rate control alone.
Based on a ranking of the observed means.
Introduces the concept of a “protection level” (1-a)r-1
Number of steps Protection Level Probability of
Apart, r (0.95)r-1 Falsely Rejecting H0
{Table A -11 (later) in these notes}
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31. Dunnett’s Procedure
A MCP that is used for comparing treatments to a control. It aims to control the experimentwise error rate.
Compares each treatment mean (i) to the mean for the control group (c).
dα(k,v) is obtained from Table A-11 (in the book) and is based on:
α = the desired experimentwise error rate
k = t-1, number of noncontrol treatments
v = error degrees of freedom.
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32. Scheffé’s S Method For any linear contrast: Estimated by:
With estimated variance:
To test H0: l = 0 versus Ha: l ¹ 0
For a specified value of a, reject H0 if:
where:
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33. Adjustment for unequal sample sizes: The Harmonic Mean
If the sample sizes are not equal in all t groups, the value of n in the equations for Tukey and SNK can be replaced with the harmonic mean of the sample sizes:
E.g. Tukey’s W becomes:
Or can also use Tukey-Cramer method:
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34. MCP Confidence Intervals
In some MCPs we can also form simultaneous confidence intervals (CI’s) for any pair of means, μi - μj.
Fisher’s LSD:
Tukey’s W:
Scheffe’s for a contrast I:
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35. A Nonparametric MCP (§9.9)
The (parametric) MCPs just discussed all assume the data are random samples from normal distributions with equal variances.
In many situations this assumption is not plausible, e.g. incomes, proportions, survival times.
Let τi be the shift parameter (e.g. median) for population i, i=1,…,t. Want to determine if populations differ with respect to their shift parameters.
Combine all samples into one, rank obs from smallest to largest. Denote mean of ranks for group i by:
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36. Nonparametric Kruskal-Wallis MCP
This MCP controls experimentwise error rate.
Perform the Kruskall-Wallis test of equality of shift parameters (null hypothesis).
If this test yields an insignificant p-value, declare no differences in the shift parameters and stop.
If not, declare populations i and j to be different if
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37. Comparisonwise error rates for different MCP
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38. Experimentwise error rates for different MCP
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