L8.pdf - APPLIED STATISTICS Variable Selection Dr Tao Zou Research School of Finance Actuarial Studies Statistics The Australian National University

L8.pdf - APPLIED STATISTICS Variable Selection Dr Tao Zou...

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APPLIED STATISTICS Variable Selection Dr Tao Zou Research School of Finance, Actuarial Studies & Statistics The Australian National University Last Updated: Mon Sep 18 18:16:03 2017 1 / 56
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Overview Motivation Sequential Variable Selection Variable Selection Among All Subsets Cross Validation for Variable Selection Results Multicollinearity 2 / 56
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References 1. F.L. Ramsey and D.W. Schafer (2012) Chapter 12 of The Statistical Sleuth 2. The slides are made by R Markdown . 3 / 56
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Motivation There are two prime reasons for variable selection: 1. Simple models with less variables are preferable to complex models with more variables. 2. Including unnecessary variables in a model results in a loss of precision overfitting. Variable selection involves choosing a subset of explanatory variables to construct the multiple linear regression model. Because if the exlanatory variables selected in MLR are determined, then the MLR model with those exlanatory variables is given. Hence, sometimes we also call model selection. Different subsets of explanatory variables determine different models. We call those models candidate models. 4 / 56
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Motivation Example: Significance Depends on Other Explanatory Variables in the Model (Con’d) Suppose we are interested in predicting ANU students’ 2nd year GPA ( Y ) given their 1st year GPA ( X 1 ) and UAC score ( X 2 ). The following regression line is fit: μ { Y | X 1 , X 2 } = β 0 + β 1 X 1 + β 2 X 2 . (1) Based on the data, the p -values for the t -tests of whether β j = 0 versus β j = 0 for j = 1 , 2 are 0.15 and 0.20, respectively. Does this mean that we do not need to select both X 1 and X 2 in the model? NO! The test for β 2 tells us whether X 2 is needed in the model that already contains X 1 , i.e., does X 2 offer any information about mean GPA over and above that of X 1 ? The meaning of the coefficient of an explanatory variable depends on what other explanatory variables have been included in the regression. 5 / 56
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Motivation Example: Significance Depends on Other Explanatory Variables in the Model (Con’d) If we fit the following two models: μ { Y | X 1 } = α 0 + α 1 X 1 and μ { Y | X 2 } = γ 0 + γ 2 X 2 . For both models, the p -values for the t -tests of α 1 = 0 versus α 1 = 0 and γ 2 = 0 versus γ 2 = 0 can be computed. Based on the data, the results of the p -values are 0.01 and 0.02, respectively. Hence at least one of X 1 and X 2 is needed in the model. In this example X 1 and X 2 are probably highly correlated so we might expect this to be the case. The following F -test of model (1) avoids this problem. H 0 : none of X 1 and X 2 is needed in the model H a : at least one of X 1 and X 2 is needed in the model . However, the F -test does not answer: which of X 1 and X 2 should be selected in the model. 6 / 56
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Sequential Variable Selection - Backward Elimination Steps Backward Elimination Step for j Explanatory Variables 7 / 56
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Sequential Variable Selection - Forward Selection Steps Forward Selection Step for j Explanatory Variables 8 / 56
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Sequential Variable Selection The idea behind sequential techniques is a sequential search through all
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