class: center, middle, inverse, title-slide # Interpretations ### Dr. D’Agostino McGowan --- layout: true <div class="my-footer"> <span> Dr. Lucy D'Agostino McGowan </span> </div> --- class: middle ## `\(\hat\beta\)` interpretation in multiple linear regression -- The coefficient for `\(x\)` is `\(\hat\beta\)` (95% CI: `\(LB_\hat\beta, UB_\hat\beta\)`). A one-unit increase in `\(x\)` yields an expected increase in y of `\(\hat\beta\)`, **holding all other variables constant**. --- ## Hitters data ``` ## ## Call: ## lm(formula = Salary ~ AtBat + Hits, data = Hitters) ## ## Residuals: ## Min 1Q Median 3Q Max ## -1006.05 -247.38 -79.15 179.41 2002.17 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 141.2720 76.5526 1.845 0.066113 ## AtBat -1.2160 0.6372 -1.908 0.057430 ## Hits 8.2119 2.0799 3.948 0.000101 ## ## Residual standard error: 404.1 on 260 degrees of freedom ## (59 observations deleted due to missingness) ## Multiple R-squared: 0.2036, Adjusted R-squared: 0.1975 ## F-statistic: 33.23 on 2 and 260 DF, p-value: 1.405e-13 ``` --- class: middle # standard error How much we expect the sample slope to vary from one random sample to another. --- ## Hitters data ``` ## ## Call: ## lm(formula = Salary ~ AtBat + Hits, data = Hitters) ## ## Residuals: ## Min 1Q Median 3Q Max ## -1006.05 -247.38 -79.15 179.41 2002.17 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 141.2720 76.5526 1.845 0.066113 ## AtBat -1.2160 0.6372 -1.908 0.057430 ## Hits 8.2119 2.0799 3.948 0.000101 ## ## Residual standard error: 404.1 on 260 degrees of freedom ## (59 observations deleted due to missingness) ## Multiple R-squared: 0.2036, Adjusted R-squared: 0.1975 ## F-statistic: 33.23 on 2 and 260 DF, p-value: 1.405e-13 ``` --- class: middle # p-value The probability of getting a statistic as extreme or more extreme than the observed test statistic **given the null hypothesis is true** --- ## Hitters data ``` ## ## Call: ## lm(formula = Salary ~ AtBat + Hits, data = Hitters) ## ## Residuals: ## Min 1Q Median 3Q Max ## -1006.05 -247.38 -79.15 179.41 2002.17 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 141.2720 76.5526 1.845 0.066113 ## AtBat -1.2160 0.6372 -1.908 0.057430 ## Hits 8.2119 2.0799 3.948 0.000101 ## ## Residual standard error: 404.1 on 260 degrees of freedom ## (59 observations deleted due to missingness) ## Multiple R-squared: 0.2036, Adjusted R-squared: 0.1975 ## F-statistic: 33.23 on 2 and 260 DF, p-value: 1.405e-13 ```