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Unformatted text preview: PubH 7405: EGRESSION ANALYSIS MLR: INFERENCES, Part I TESTING HYPOTHESES Once we have fitted a multiple linear regression model and obtained estimates for the various parameters of interest, we want to answer questions about the contributions of factor or factors to the prediction of the dependent variable Y. There are three types of tests : (1) An overall test (2) Test for the value of a single factor (3) Test for contribution of a group of factors OVERALL TEST The question is: Taken collectively, does the entire set of explanatory or independent variables contribute significantly to the prediction of the Dependent Variable Y?. The Null Hypothesis for this test may stated as: All k independent variables, considered together, do not explain the variation in the values of Y ". In other words, we can write: ... : 2 1 k o H Logically, we could look for some individual effect even without the overall test if its the primary aim of the research project. For example, this factor represents treatment assignment ina clinical trial (say, =1 if treated & 0 if placebo) TEST FOR SINGLE FACTOR The question is: Does the addition of one particular factor of interest add significantly to the prediction of Dependent Variable over and above that achieved by other factors in the model ?. The Null Hypothesis for this test may stated as: " Factor X i does not have any value added to the explain the variation in Yvalues over and above that achieved by other factors ". In other words, we can write: : i H TEST FOR A GROUP OF VARABLES The question is : Does the addition of a group of factors add significantly to the prediction of Y over and above that achieved by other factors ? The Null Hypothesis for this test may stated as: "Factors {X i+1 , X i+2 ,, X i+m }, considered together as a group , do not have any value added to the prediction of the Mean of Y over and above that achieved by other factors ". In other words, ... : 2 1 m i i i H TEST FOR A GROUP OF VARABLES This multiple contribution test is often used to test whether a similar group of variables , such as demographic characteristics , is important for the prediction of the mean of Y; these variables have some trait in common . Other groupings: Psychological and Social aspects of health in QofL research Another application: collection of powers and/or product terms . It is of interest to assess powers & interaction effects collectively before considering individual interaction terms in a model. It reduces the total number of tests & helps to provide better control of overall Type I error rates which may be inflated due to multiple testing . This lecture is aimed to provide the details for these tests of significance. Since the primary focus is the regression approach , well go through the decomposition of the sums of squares the basic approach of ANOVA (Analysis of Variance ). ANOVA IN REGRESSION...
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 Fall '08
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