Math 558 Lecture #17
Comparison of Treatments after F-test
Comparison of treatments after F-test involves subdivision of the sum of squares for treatments. Recall the eelworm experiment from your Assignment 2. After significant F-test, you may ask the following questions
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Is soil fumigation effective? To respond to this question you will find the average reduction in eelworm numbers for all fumigants.
What are the differences among the effects of different fumigants? The response requires the comparisons of the reduction numbers due to different fumigants.
How effective is the single dose as compared to the double dose?
Comparison of Treatments after F-test
Definition 1
Any linear combination
lm = am1τ1 +am2τ2 +…+amkτk
is called contrast if
am1 + am2 …….amk = 0
The combinations τ1 − τ2 and τ1 +τ2 − τ3 +τ4 are contrasts. 22
Orthogonal Contrasts
Definition 2
Two contrasts l1 and l2 are said to be orthogonal if a11a21 + a12a22 + ……. + a1ka2k = 0
l1 =a11τ1+a12τ2+…+a1kτk l2 =a21τ1+a22τ2+…+a2kτk
Subdivision of SST
If lw is any contrast, then the quantity
S ,where Sw=r(lw1+lw2+….+lwk) w
is a component of SST with 1 degree of freedom. If two contrasts l1 and l2
l2 are said to be orthogonal then S2 is a component of SSTr − l12
This means that first we divide the SST into the contribution from l2
contrast l1 and SST − S1 . Then choose a contrast orthogonal to l1 for
further subdivision.
Subdivision of SSTr
After removing the contribution of l2, remove the contribution of the contrast l3 such that l3 is orthogonal to both l1 and l2. Hence if
l1, l2, ….lt−1 are mutually orthogonal (every pair is orthogonal) then
l2l2 l2 SSTr= 1 + 2 +…..+ t−1 S1 S2 St−1
Hence we can partition the SST with t − 1 degrees of freedom into t − 1 components each with one degree of freedom. Starting from l1 we can find l2, l3, …lt−1 to construct a complete orthogonal set.
Orthogonal Polynomial Contrasts
There are particular sets of orthogonal contrasts that can be used when the independent variable is a quantitative factor. A quantitative factor is a variable whose levels (i.e., groups) differ in quantity. That is, the levels of a quantitative factor can be ordered in regard to a quantitative difference. For example doses in our rat behaviour experiment.
Definition 3
Polynomial contrasts are a set of orthogonal contrasts that reflect linear and nonlinear associations among population means. Each contrast within a set of polynomial contrasts tests a specific trend (e.g., linear, quadratic, cubic, quartic).
Orthogonal Polynomial Contrasts
The number of possible trends and the coefficient weights that reflect the trends vary as a function of the number of levels in the independent variable. As we discussed that we can partition the SST with t − 1 degrees of freedom into t − 1 components each with one degree of freedom. Consequently, there are t − 1 potential trends among treatment means and t − 1 orthogonal polynomials that can independently account for those trends.
Orthogonal Polynomial Contrasts
For example, a study with two quantitative levels can be described only by a linear trend because only a straight line can be fit to two means.A study with three levels can have a linear and quadratic trend because a straight line and/or a curvilinear pattern that has two changes in direction can describe three ordered means.
Being orthogonal, polynomial contrasts account for the entire amount of variation among the treatments. For example, the variation among four treatments that vary quantitatively can be fully accounted for by a linear, quadratic, and cubic trend:
SSTr = SSlinear + SSquadratic + SScubic
contrasts(drugdose)
0.1195229 -0.4780914 0.7171372 -0.4780914 0.1195229
0 -0.6324555 0.5 -0.3162278 1 0.0000000
1.5 0.3162278 2 0.6324555
0.5345225 -0.2672612 -0.5345225 -0.2672612 0.5345225
-3.162278e-01 6.324555e-01 -4.095972e-16 -6.324555e-01 3.162278e-01
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