PS102Lecture22

PS102Lecture22 - + Evaluating Regression Models Political...

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Unformatted text preview: + Evaluating Regression Models Political Science 102 Introduction to Political Inquiry Lecture 22 + Good Models and “Explaining” Variance What do we mean when we say a model is “good” or “explains” the dependent variable? Explanation exists in our theory, not in any data we might observe. Thus “explained” variance cannot be measured with a statistic (or with data). What do we mean when we say we have a “good model” of our data? Changes in X have a large impact on changes in Y ( That is, b is large) Remaining error terms are small (That is, σ u 2 is small) + A Composite Measure of Model Quality Developed an overall measure of these concepts that is insensitive to the scale measuring X & Y Measure is based on fundamental goal of the OLS estimator – minimize squared errors. Take the ratio of “Explained”Sum of Squares (ESS) to Total Sum of Squares (TSS) This ratio is known as the “coefficient of determination” or R 2 R 2 is literally just the correlation between Y and Yhat, squared TSS USS TSS ESS R- = = 1 2 + What IS R 2 Really? The R 2 Stew Want a measure of model quality to compare across samples Like correlation coefficients – R 2 cannot generalize It depends on the variance of X and the variance of the errors R2 for bivariate regression can be written as: Which is: 2 2 2 2 2 2 * * u x x b b R σ σ σ + = ( causal _ strength ) 2 * s x 2 ( causal _ strength ) 2 * s x 2 + good _ fit USS ESS ESS TSS ESS R + = = 2 + Don’t Drink the R 2 Kool Aid Thus R 2 conflates our two aspects of a “good model,” combines them with σ x 2 , and places them on a dimensionless scale The resulting value is nearly uninterpretable Basically measures the shape of the cloud of observations around our regression line. R 2 tells us little we want to know – pay it no heed Also avoid standardized regression coefficients We must make substantive evaluations of our models: Size of Coefficients (hypothesis tests) Size of Substantive Effects Ability to forecast out-of-sample + Standards of Model Evaluation Statistical Significance of Coefficients T-tests Substantive Size of Effects Generate predictions from the model Size of Residuals Compare to substantive effects Forecasting Out-of-Sample + Statistical Significance of Regression Coefficients In general, our theories give us hypotheses that B>0 or B<0 We can estimate b, but we need a way to assess the validity of statements that B is positive or negative We can rely on our estimate of b and its variance to use probability theory to test such statements. T-tests (or Z-scores) give us confidence that relationships we observe generalize to the population Don’t get too focused on .05 as a “magical” threshold for significance + Z – Scores & Hypothesis Tests Recall that b ~ N( B , σ b ) Subtracting B from both sides, we can see that...
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This note was uploaded on 10/21/2011 for the course POL SCI 102 taught by Professor Gelpi during the Spring '11 term at Duke.

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PS102Lecture22 - + Evaluating Regression Models Political...

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