Orthogonal contrasts

As the standard test matrix has been followed, the effects diagrams in Figs 18.3 to 18.5 above will give the same message as half normal plots. The advantage of half normal plots is that they show which effects are statistically significant. The numerical values of the factor effects for the half normal plot need to be found from orthogonal contrasts. 

The orthogonal contrasts have the property that they completely partition the total variation in the results. The sum of squared factor effects is equal to the total sum of squares (see, for example, Cochran and Cox, 1957). 

If a set of non-orthogonal contrasts is applied to the results, then the constituent parts will not sum to the total variation and they will not be independent of each other. The experimental design used here can be analysed in a number of ways corresponding to different sets of orthogonal contrasts. This is explored further in Appendix II. 

Allowing for the full effect of the three level factors, a useful set of orthogonal contrasts leads to an assessment of 

Because these contrasts are orthogonal, each effect can be evaluated independently of any other. Table 18.4 represents the different levels of the factors. Temp and Qtemp stand for the linear and quadratic effects of upper 

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Significantly more phosphorus was excreted in the feces when 1.2% calcium and 1.2% phosphorus diets were fed with either egg white (317 mg) or soy (303 mg), than when any of the other rations were fed. Orthogonal contrast analysis indicated that statistically significant sources of these differences included ration phosphorus level (P <0.001), calcium level (P< 0.0001), and calcium level x protein source (P< 0.0461). The degree of effectiveness of increasing phosphorus excretion with increased ration level of calcium was greater in egg white fed animals than in soy fed animals. 

Mean fecal calcium loss at high levels of calcium intake was 75.0 mg and at low calcium intake was 16.0 mg. Mean calcium and phosphorus levels are illustrated in Figure 6. Orthogonal contrast analysis indicated that the level of calcium in the ration was the only significant source of difference in the data. 

The sequence of meal consumption was determined by random assignment of diets to subjects. Statistical analysis was performed by a General Linear Models Procedure (20) using split-plot in time analysis with the following non-orthogonal contrasts ... 

Note that the sum of the squares of the coefficients of r/j, and (j>2 is not unity, since these atom-centered functions are not orthogonal (contrast the simple Hiickel method, Section 4.3.4). 

These tests included analysis of variance and, if the design of the study warranted, Duncan s Multiple Range Test and/or orthogonal contrast. 

Effects of treatment (sib, non-sib, or blank water), clutch (sibship), and their interaction were evaluated by analyses of variance (General Linear Models Procedure, SAS Release 82.4). Treatment effects were further analyzed by orthogonal contrasts (sib vs non-sib, experimental vs control). 

The set of orthogonal contrasts in Table 18.4 applied to the results for mean time to spall (see Appendix I for data) were used to produce the half normal plot in Fig. 18.6. For example, the standardised contrast value for hot is 119. This is the vector product of the contrast for hot and the results column... 

Note that the orthogonal contrast for Qdwell assumes that the levels are equally spaced whereas the factor levels are actually 4,8 and 20 h which are not equally spaced. This will tend to bias the contrast value so that Qdwell is larger than it should be. The half normal plot should be looked at in conjunction with the main effects plot. The main effects plot for time to spall shows that the effect of upper dwell time is approximately linear which supports the lack of significance in the half normal plot in Fig. 18.6. 

The variation in the nine sets of results can be broken up into constituent parts. These can correspond to the factors and interactions above. The constituent parts are obtained by calculating orthogonal contrasts of the results. There are several alternative sets of orthogonal contrasts. Not all sets correspond to the design, for example the following set is not meaningful for the experiment carried out here (8, -1, -1, -1, -1, -1, -1, -1, -1), (0, 7, -1, -1, -1, -1, -1, -1, -1), and so on to (0, 0, 0, 0, 0, 0, 0, 1, -1). The analysis in this chapter is for main effects of all factors only, i.e. the first six effects listed above. The contrasts used are orthogonal to each other. 

Overall, there are a number of alternative ways of sub-dividing the variation in the nine trial results but the set of orthogonal contrasts used in the analysis in this chapter are the most reliable. This is because they correspond to the experimental design and provide the most nearly balanced assessment of the effects of the cycling parameters. 

Table 18.70 Alternative set of orthogonal contrasts including mixed level interactions...

Table 18.12 Set of orthogonal contrasts with centre point ...

Two orthogonal contrasts were performed to compare the three AA treatments 1) Contrast EAA (Glu vs. Ideal and 4EAA) and 2) and contrast Ideal (4EAA vs. Ideal). 

Data were analyzed by ANOVA and treatments means were compared by orthogonal contrasts using SAS. 

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