29 Homework Regression - M4L12HW2 – Bayesian Statistics

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--- title: "Homework Regression - M4L12HW2" subtitle: "Bayesian Statistics: From Concept to Data Analysis" categories: - Bayesian Statistics keywords: - Exponential Distribution - Homework - Honnors --- ::::: {.content-visible unless-profile="HC"} ::: {.callout-caution} Section omitted to comply with the Honor Code ::: ::::: ::::: {.content-hidden unless-profile="HC"} ::: {#exr-hregression-golf-1} [**Golf**]{.column-margin} \index{dataset!pgalpga2008} The data are found at [pgalpga2008.dat](http://www.stat.ufl.edu/~winner/data/pgalpga2008.dat) and consist of season statistics for individual golfers on the United States LPGA and PGA tours. The first column reports each player's average driving distance in yards. The second column reports the percentage of the player's drives that finish in the fairway, measuring their accuracy. The third and final column has a 1 to denote a female golfer (on the LPGA tour), and a 2 to denote male golfer (on the PGA tour). Now consider a multiple regression on the full data set, including both female and male golfers. Modify the third variable to be a 0 if the golfer is female and 1 if the golfer is male and fit the following regression: $$ \mathbb{E}[y] = b_0 + b_1x_1 + b_2x_2 $$ where - $x_1$ is the average driving distance and - $x_2$ is the indicator that the golfer is male. What is the posterior mean estimate of $b_0$ ? Round your answer to the nearest whole number. ::: ::: {.solution .callout-tip collapse="true"} #### Solution: ```{r} #| label: C1-L12-Ex2-1 dat=read.table("data/pgalpga2008.dat.txt", header=T) # Note that attaching this masks T which is originally TRUE attach(dat) colnames(dat) <- c('distance','accuracy','gender') dat$gender = dat$gender -1 mod2 <- lm(accuracy ~ distance + gender, data=dat) summary(mod2) ``` ::: ::: {#exr-hregression-golf-2} [**Golf**]{.column-margin} The posterior mean estimates of the other two coefficients are $\hat{b}_1=−0.323$, and $\hat{b}_2=8.94$. What is the interpretation of $\hat{b}_1$? ::: ::: {.solution .callout-tip collapse="true"} #### Solution: - [ ] Holding all else constant, being male is associated with a 0.323 increase in drive accuracy percentage. - [ ] Holding all else constant, each additional yard of distance is associated with a 0.323 increase in drive accuracy percentage. - [x] Holding all else constant, each additional yard of distance is associated with a 0.323 decrease in drive accuracy percentage. - [ ] Holding all else constant, being male is associated with a 0.323 decrease in drive accuracy percentage. ::: ::: {#exr-hregression-golf-3} [**Golf**]{.column-margin} The standard error for $b_1$ (which we can think of as marginal posterior standard deviation in this case) is roughly 1/10 times the magnitude of the posterior mean estimate $\hat{b}_1=−0.323$. In other words, the posterior mean is more than 10 posterior standard deviations from 0. What does this suggest? ::: ::: {.solution .callout-tip collapse="true"} #### Solution: - [ ] The posterior probability that $b_1<0$ is very low, suggesting a negative relationship between driving distance and accuracy. - [ ] The posterior probability that $b_1<0$ is about 0.5, suggesting no evidence for an association between driving distance and accuracy. - [x] The posterior probability that $b_1<0$ is very high, suggesting a negative relationship between driving distance and accuracy. ::: ::: {#exr-hregression-golf-4} [**Golf**]{.column-margin} The estimated value of $b_2$ would typically be interpreted to mean that holding all else constant (for a fixed driving distance), golfers on the PGA tour are about 9% more accurate with their drives on average than golfers on the LPGA tour. However, if you explore the data, you will find that the PGA tour golfers' average drives are 40+ yards longer than LPGA tour golfers' average drives, and that the LPGA tour golfers are actually more accurate on average. Thus $b_2$ , while a vital component of the model, is actually a correction for the discrepancy in driving distances. Although model fitting can be easy (especially with software), interpreting the results requires a thoughtful approach. It would also be prudent to check that the model fits the data well. One of the primary tools in regression analysis is the residual plot. Residuals are defined as the observed values y minus their predicted values $\hat{y}$ . Patterns in the plot of $\hat{y}$ versus residuals, for example, can indicate an inadequacy in the model. These plots are easy to produce. ```{r} #| label: C1-L12-Ex2-2 plot(fitted(mod2), residuals(mod2)) abline(lm(residuals(mod2)~fitted(mod2)), col="red") # regression line (y~x) ``` where "mod" is the model object fitted with the lm() command. ::: ::: {.solution .callout-tip collapse="true"} #### Solution: - [ ] The residuals appear to be random and lack any patterns or trends. There are no outliers (extreme observations). - [X] The residuals appear to be random and lack any patterns or trends. However, there is at least one outlier (extreme observation) that we may want to investigate. - [ ] The residuals appear to be more spread apart for smaller predicted values $\hat{y}$ . There are no outliers (extreme observations). - [ ] The residuals appear to exhibit a curved trend. There is at least one outlier (extreme observation) that we may want to investigate. ::: :::::