1. The American Statistician (1990) Vol. 44, pp. 204-209.
The Alignment Method for Displaying and Analyzing Treatments in Blocking Designs
Richard F. Fawcett
The American Statistician (1990) Vol. 44, pp
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2. Outline Introduction Examples Final Remarks
Responses to one of the treatments in a Graeco-Latin Square
Treatments in a randomized complete block design
Treatments in a balanced incomplete block design
Final Remarks
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3. Introduction
Students taking the elementary course, upon reaching the chapter on the one-way analysis of variance, are taught to graph their data --- to construct dot diagrams --- and then to draw tentative conclusions from their graphs.
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4. Students are encouraged to graph their data and then speculate, from their graph, the answers to such questions as the following:
Can the responses to the treatments be reasonably viewed as random samples from normal populations having equal variances?
Will the null hypothesis of equal treatment means be accepted or reject?
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5. The responses can be viewed as observations from normal populations having equal variances.
The null hypothesis of equal treatment means will be rejected at any sensible level of significance.
If paired comparisons of the treatment means are performed, the results will be
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6. When some of these students take a course in design and analysis of experiments a bit later on, they rely almost exclusively on the formal analyses of their data.
The dot diagrams, suitably constructed, can provide useful information about the responses to treatments, even when observations come from relatively complex experimental designs.
The aligned responses to the treatments have the appearance of observations from a completely randomized design.
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7. computed from the aligned responses.
We can calculate the quantity from the aligned responses and then adjust this quantity to produce the value of the variance ratio associated with the design in question.
When the design under study is a balanced and complete blocking design, a particularly simple and interesting relationship exists between the variance ratio computed from the original data [MS(tr)/MS(E)] and the quantity
computed from the aligned responses.
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8. m = 3 --- Graeco-Latin square design. m = 2 --- Latin square design.
The relationship is
where k = the number of treatment levels and m = the number of nuisance variables (blocking variables) in the design.
m = Graeco-Latin square design.
m = Latin square design.
m = RCBD.
m = CRD.
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9. Graeco-Latin Square
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10. Aligned Responses and Corresponding Analysis
Let i be the subscript associated with the level of the Latin treatment, and let denote the jth aligned responses to the ith level of the Latin treatment.
Then
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11. When computed from the aligned responses (AL),
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12. Dot Diagrams of the Original and Aligned Responses
The design adjustment factor , serves to remind us that our obtained F ratio is only one-fourth as large as it appears to be from the graph.
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13. The graph of the aligned responses yields other interesting characteristics of the data.
The residuals associated with the treatments appear to have come from populations having equal variance, but those populations are not normal.
Every residual in this example has one of the values -.5 or +.5 (recall that the residuals associated with a treatment are centered at the treatment mean).
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14. When data contains outliers or when observations come from populations that are very thick-tailed or long-tailed, the effects on the power of the F ratio can be devastating, but when the error-term population is thin-tailed or short-tailed, as it is in this example, the power of the F test is relatively unaffected (e.g., see Groggel 1987).
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15. RCBD
The preceding discussion serves to remind us of how sensitive the power of the classical F test can be to particular types of violations of the basic assumptions.
This example illustrates the points.
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16. The Data and the Classical Analysis
The data shown in Figure 5 contain an outlier (Y34), and, not coincidentally, we do not even come close to rejecting the null hypothesis of equal treatment means.
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17. This is frustrating, because a closer look at the data reveals that, within every block, treatment 1 is less than treatment 2, which in turn is less than treatment 3.
The large sum of square (SS) due to blocks dispels any notation that blocking may have been a mistake and that the cost of blocking was too high in this instance.
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18. The Aligned Responses and Analysis
The jth aligned responses to the ith treatment is defined as follows:
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19. The data are flagrantly nonnormal and contain at least one outlier.
Treatment 3 is producing responses at a different level than those produced by treatment 1 and 2.
The data are flagrantly nonnormal and contain at least one outlier.
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20. The grapgs also reveal something else --- every aligned response to treatment 3 exceeds every aligned response to treatment 1 and every aligned response to treatment 2.
At this point, we are tempted to overrule the formal test results and reject the null hypothesis of equal treatment means on visual evidence alone and conclude that
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21. A more acceptable suggestion is to perform an analysis that is not so dependent on the classical assumptions.
My personal choice, not surprisingly, is the aligned ranks transformation test (see Fawcett and Salter 1984), in which one merely replaces the aligned responses in Figure 6 with rank numbers and then proceeds with the usual F test computed from the ranks.
When this is done, a partial summary of results is as follows: SS(tr)=108.5, SS(E)=28.5, and F0=9.61, (.01<p<.025).
This seems to provide a more accurate reflection of the experimental situation.
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22. BIBD
The pioneering work on balanced incomplete block designs was by Yates (1936, 1940).
b: the numbers of blocks
t: treatments
k: treatments per block
r: replications of each treatment
N ( = tr = bk): the total number of experimental units
: the number of times any two treatments occur together in the same block.
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23. The Data to Be Analyzed and the Classical Analysis
In the example in Figure 8, b = t =4, k = r =3, N=12 and .
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25. The Aligned Responses and Analysis
Alignment 1 ( ) is illustrated in Figure 9.
The principal advantages of this alignment scheme are the comparatively simple calculations required to produce the aligned responses.
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26. Alignment 2 ( ) is illustrated in Figure 10.
is unbiased estimators of the block means.
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27. The dot diagrams suggest that the assumptions of normality and equal variances are not being flagrantly violated.
The different treatments seem to be producing responses at very different levels and would no doubt lead one to speculate that H0 will be rejected at any sensible level of significance.
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28. However, we recall that the computed F ratio was 4
However, we recall that the computed F ratio was 4.61, but the 5% critical value was 5.41.
The smaller-than-expected F ratio leads to speculate that a severe cost of blocking is at work here.
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29. Final Remarks
The alignment method is a dimension-reducing procedure in which one first aligns the data by removing the effects of nuisance parameters (blocks) and then analyzes the aligned responses to the treatments.
Aligned responses resemble observations from completely randomized designs and may be displayed with simple dot diagrams that allow rough visual checks on the tenability of the assumptions of normality and homogeneity, on the effectiveness of blocking, and whether the treatments are producing responses at different levels.
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30. One may test using classical formulas developed from the particular design under study, or with equivalent formulas from the aligned responses.
The concept of alignment, subtracting the effects of nuisance parameters and then examining the resulting aligned responses to the treatments of interest, is so general that it can be applied to a wide variety of experimental situations.
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31. In this article, attention has been restricted to balanced (complete and incomplete) blocking designs with fixed-effects factors, in which nuisance parameters are blocks.
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32. Thanks for your attention!