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Class_16 and test review

# Class_16 and test review - Analysis of Covariance ANOVA is...

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Analysis of Covariance

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ANOVA is a class of statistics developed to evaluate controlled experiments. Experimental control, random selection of subjects, and random assignment of subjects to subgroups are devices to control or hold constant all the other (UNMEASURED) influences on the dependent (Y ij ) variable so that the effects of the independent (X ij ) variable can be assessed. Without experimental control, random selection, and random assignment, other (non-random) differences besides the treatment variable enter the picture. Remember: Inferential statistics only assess the likelihood that chance could have affected the sample results; they do not take into account non-random factors.
For example, without randomly selecting students and compelling them to take PPD 404, then randomly assigning them to an instructor, plus controlling their lives for an entire semester (e.g., forbidding them to work), differences that are not random creep in. To some extent, this problem of uncontrolled, non- random differences can be compensated for by introducing covariates as statistical controls . Covariates are continuous variables that hold constant non-random differences. For example, by asking students how many hours per week they were working , we could add this variable to our ANOVA model. Let's look briefly at the analysis of covariance with one of the classic examples in the statistical literature.

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The data are from an experiment involving the use of two drugs for treating leprosy. Drug A and Drug B were experimental drugs; Drug C was a placebo. Subjects were children in the Philippines suffering from leprosy. Thirty children who were taken to a clinic were given either Drug A, B, or C (the treatments) in order of their arrival. Thus each subgroup consisted of 10 children. The outcome measure, Y ij , was a microscopic count of leprosy bacilli in samples taken from six body sites on each child at the end of the experiment. Data are in the following table.
——————————————————————————————————————————————————————— Group A Group B Group C Y Y 2 X X 2 XY Y Y 2 X X 2 XY Y Y 2 X X 2 XY ——————————————————————————————————————————————————————— 6 36 11 121 66 0 0 6 36 0 13 169 16 256 208 0 0 8 64 0 2 4 6 36 12 10 100 13 169 130 2 4 5 25 10 3 9 7 49 21 18 324 11 121 198 8 64 14 196 112 1 1 8 64 8 5 25 9 81 45 11 121 19 361 209 18 324 18 324 324 23 529 21 441 483 4 16 6 36 24 4 16 8 64 32 12 144 16 256 192 13 169 10 100 130 14 196 19 361 266 5 25 12 144 60 1 1 6 36 6 9 81 8 64 72 16 256 12 144 192 8 64 11 121 88 1 1 5 25 5 1 1 7 49 7 0 0 3 9 0 9 81 15 225 135 20 400 12 144 240

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