You may discuss homework problems with other students, but you have to prepare the written assignments yourself.

Please combine all your answers, the computer code and the figures into one file.

Grading scheme: 10 points per question, total of 30.

Due date: March 7, 2017, 11:59PM.

Question 1

The data set http://stats191.stanford.edu/data/asthma.table contains data on the number of admittances, Y to an emergency room for asthma-related problems in a hospital for several days. On each day, researchers also recorded the daily high temperature T, and the level of some atmospheric pollutants P.

  1. Fit a linear regression model to the observed counts as a linear function of T and P.

  2. Looking at the usual diagnostics plots, does the constant variance assumption seem justified?

  3. The outcomes are counts, for which a common model is the so-called Poisson model which says that $\text{Var}(Y) = E(Y)$. In words, this says that the variance of the outcome is equal to the expected value of the outcome. Using a two-stage procedure, fit a weighted least squares regression to Y as a function of T and P with weights being inversely proportional to the fitted values of the initial model in 1.

  4. Looking at the usual diagnostics plots of this model (which takes the weights into account), does the constant variance assumption seem more reasonable? (The change may not be astonishing -- the point of the problem is to try using weighted least squares.)

  5. Using the weighted least squares fit, test the hypotheses at level $\alpha = 0.05$ that

    • the number of asthma cases is uncorrelated to the temperature allowing for pollutants;
    • the number of asthma cases is uncorrelated to the atmospheric pollutants allowing for temperature.

    Use a Bonferroni correction.

Question 2 (Based on RABE 8.4-8.6)

The file

http://www1.aucegypt.edu/faculty/hadi/RABE5/Data5/P229-30.txt

contains the values of the daily DJIA (Dow Jones Industrial Average) for all the trading days in 1996. The variable Time denotes the trading day of the year. There were 262 trading days in 1996.

  1. Fit a linear regression model connecting DJIA with `Time using all 262 trading days in 1996. Is the linear trend model adequate? Examine the residuals for time dependencies, including a plot of the autocorrelation function.

  2. Regress DJIA[t] against its lagged by one version DJIA[t-1]. Is this an adequate model? Are there any evidences of autocorrelation in the residuals?

  3. The variability (volatility) of the daily DJIA is large, and to accomodate this phenomenon the analysis is crried out on the logarithm of the DJIA. Repeat 2. above using log(DJIA) instead of DJIA.

  4. A simplified version of the random walk model of stock prices states that the best prediction of the stock index at Time=t is the value of the index at Time=t − 1. Show that this corresponds to a simple linear regression model for 2. with an intercept of 0 and a slope of 1.

  5. Carry out the the appropriate tests of significance at level α = 0.05 for 4. Test each coefficient separately ($t$-tests) , then test both simultaneously (i.e. an F test).

  6. The random walk theory implies that the first differences of the index (the difference between successive values) should be independently normally distributed with mean zero and constant variance. What kind of plot can be used to visually assess this hypothesis? Provide the plot.

Question 3

In this question we will look at inference ($t$-tests and confidence intervals) after model selection. We will use the prostate data from ElemStatLearn.

In [ ]:
library(ElemStatLearn)
data(prostate)
  1. Fit the full model lcavol + lweight + age + lbph + svi + lcp + pgg45 with response lpsa.

  2. Write a function in R that takes a regression model output by lm and creates a new design matrix by adding some number k columns to the model's design matrix whose entries are filled with rnorm. The functions cbind, matrix, rnorm and as.data.frame will be helpful here. The function should return a new data.frame with all the original variables as well as these k new ones.

  3. Try adding k=20 columns to the design matrix. After adding some additional rnorm noise to lpsa of variance about half the estimated variance from the model in step 1., run step in a forward direction with the largest model including these 20 new columns. A simple way to do this is along the lines of full.lm = lm(lpsa.noisy ~ ., data=prostate) (substituting your data.frame for prostate below. Then, you can put list(upper=full.lm) as the scope argument to step where lpsa.noisy = lpsa + rnorm(length(lpsa), sd=?) and an appropriate choice of sd.

  4. We know that all of the new variables have nothing to do with the real data so they must have coefficient 0 in these regression models we've created. Do the $p$-values you see after running step look as if they come from regressions with 0 (i.e. null) coefficients?

  5. Write a function that repeats steps 3. & 4. returning a list with one entry pvalues for the null $p$-values and the other entry intervals for 95% confidence intervals for these null variables. Repeatedly call this function, storing the $p$-values until you have 1000 $p$-values and plot a histogram. Do they look as you'd expect?

  6. Repeat 5., this time checking which of the confidence intervals cover 0 (the true coefficient in this case). Is it roughly 95%?