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Yesterday Correlation -Definition -Deviation Score Formula, Z score formula -Hypothesis Test Regression -Intercept and Slope -Unstandardized Regression Line -Standardized Regression Line -Hypothesis Tests Slide 2 Summary Correlation: Pearsons r Unstandardized Regression Line Standardized Regression Line Slide 3 Some issues with r Outliers have strong effects Restriction of range can suppress or augment r Correlation is not causation No linear correlation does not mean no association Slide 4 Outliers Child 19 is lowering r Child 18 is increasing r Slide 5 The restricted range problem The relationship you see between X and Y may depend on the range of X For example, the size of a childs vocabulary has a strong positive association with the childs age But if all of the children in your data set are in the same grade in school, you may not see much association Slide 6 Common causes, confounds Two variables might be associated because they share a common cause. There is a positive correlation between ice cream sales and drownings. Also, in many cases, there is the question of reverse causality Slide 7 Non-linearity Some variables are not linearly related, though a relationship obviously exists For monotonic relationships that are not linear we use Spearmans r Slide 8 How well does the regression line describe the data? Assessing fit relies on analysis of residuals Are the residuals randomly distributed? (If no, perhaps a linear model is inappropriate) How large are the residuals? Too big? (low correlation means big residuals) Regression: Analyzing the Fit Slide 9 The residuals have mean of 0 and variance of s resid 2 The residuals are uncorrelated with X The residuals are homoscedastic (similarly sized across the range of x) Assumptions of Regression Slide 10 Residual Diagnostics I: Graphing r =.96 r 2 =.92 Slide 11 Residual Plot resid Problem: curvilinearity Residual Diagnostics I: Graphing Slide 12 Agreeableness Time 2 Slide 13 Problem: heteroscedasticity Residual Diagnostics I: Graphing Residuals Residual Plot Slide 14 How well does the regression line describe the data? Assessing fit relies on analysis of residuals Are the residuals randomly distributed? (If no, perhaps a linear model is inappropriate) How large are the residuals? Too big? (low correlation means big residuals) Regression: Analyzing the Fit Residual plotsANOVA Slide 15 Regression ANOVA SS Y SS model SS resid Y Y Slide 16 Regression ANOVA SourceSSdfs2s2 Model Error Total the amount of variance in Y explained by our model F=t 2 Slide 17 Exercise Fill in the ANOVA table XY 13 14 54 56 96 97 Slide 18 Exercise XYY Predicted value (Y-Y) 2 Residual (Unpredicted deviation) (Predicted Deviation) 13 14 54 56 96 97 SS resid = SS model = Slide 19 Exercise XYY Predicted value (Y-Y) 2 Residual (Unpredicted deviation) (Predicted Deviation) 133.5(-0.5) 2 (-1.5) 2 143.5(0.5) 2 (-1.5) 2 545(-1) 2 (0) 2 565(1) 2 (0) 2 966.5(-0.5) 2 (1.5) 2 976.5(0.5) 2 (1.5) 2 SS resid = SS model = 39 Slide 20 Regression ANOVA SourceSSdfs2s2 F Model Error Total Slide 21 Regression ANOVA SourceSSdfs2s2 F Model91912 Error34.75 Total125