topic10 - Statistics 512: Applied Linear Models Topic 10...

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Statistics 512: Applied Linear Models Topic 10 Topic Overview This topic will cover One-way Analysis of Covariance ANCOVA with more than one factor / covariate One-way Analysis of Covariance ANCOVA is really “ANOVA with covariates” or, more simply, a combination of ANOVA and regression used when you have some categorical factors and some quantitative predictors. The predictors ( X variables on which to perform regression) are called “covariates” in this context. The idea is that often these covariates are not necessarily of primary interest, but still their inclusion in the model will help explain more of the response, and hence reduce the error variance. Example: An Illustration of why ANCOVA can be important Our response Y is the number of months a patient lives after being placed on one of three different treatments available to treat an aggressive form of cancer. We could analyze these treatments with a one-way ANOVA as follows: At first glance, the treatment variable would appear to be important. In fact if we run the one-way analysis of variance we get: Dependent Variable: y 1
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Sum of Source DF Squares Mean Square F Value Pr > F Model 2 1122.666667 561.333333 14.43 0.0051 Error 6 233.333333 38.888889 Corrected Total 8 1356.000000 Mean N trt A 39.333 3 1 B 24.667 3 2 C 12.000 3 3 The analysis tells us that there is a big difference between the treatments. Treatment 1 is clearly the best as people live longer. Suppose we put a large group of people on Treatment 1 expecting them to live 30+ months only to find that over half of them die prior to 25 months. What did we do wrong???? It turns out that we have neglected an important variable. We need to consider X ,th e stage to which the cancer has progressed at the time treatment begins. We can see its effect in the following plot: There is clearly a linear relationship between X and Y , and we notice that the group assigned to the first treatment were all in a lower stage of the disease, those assigned to treatment 2 were all in a mid-stage, and those assigned to treatment 3 were all in a late stage of the disease. We would suspect looking at this plot to find the treatments are not all that different. The following ANCOVA output leads to the same conclusion: Sum of Source DF Squares Mean Square F Value Pr > F x 1 1297.234815 1297.234815 192.86 <.0001 trt 2 25.134378 12.567189 1.87 0.2478 Error 5 33.630807 6.726161 Total 8 1356.000000 LSMEAN 2
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trt y LSMEAN Number 1 20.3039393 1 2 23.6762893 2 3 32.0197715 3 Least Squares Means for Effect trt t for H0: LSMean(i)=LSMean(j) / Pr > |t| i/j 1 2 3 1 -0.85813 -1.56781 0.4300 0.1777 2 0.858127 -1.89665 0.4300 0.1164 3 1.567807 1.89665 0.1777 0.1164 So the stage of the cancer was what actually was affecting the lifetime - it really didn’t have anything to do with the choice of treatment. It just happened that everyone on treatment 1 was in an earlier stage of the disease and so that made it look like there was a treatment effect. And notice that if there was to be a difference, treatment 3 actually would have been the best. So to give everyone treatment 1 on the basis of our original analysis could have been a deadly mistake.
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topic10 - Statistics 512: Applied Linear Models Topic 10...

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