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Linear Regression Models Statistics

创建日期：2024-07-20 14:52:04

所属分类：统计学statistics

标签： Statistics 4205

Linear Regression Models Statistics

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Linear Regression Models Statistics 4205/5205

By Monday, September 28, read Chapters 1–3 and Appendices A.1–A.8.3 of Applied Linear Regression, fourth edition;

Assignment 3

Reading:

By Monday, September 28, read Chapters 1–3 and Appendices A.1–A.8.3 of Applied Linear Regression, fourth edition; by Sanford Weisberg; and Chapters 0–3 of Weisberg’s Computing Primer for Applied Linear Regression Using R.

For Monday, October 5, read Chapter 4 of the textbook and the R primer.

Homework 3:

The following problems are due by the end of the day on Wednesday, October 7.

This problem uses the United Nations example (data file UN11) to demonstrate many of the properties of added-variable plots. This problem is based on the mean function fertility ~ log(ppgdp) +pctUrban.

(a)Examine the scatterplot matrix for (fertility, log(ppgdp), pctUrban), and com- ment on the marginal relationships.

(b)Fit the two simple regressions, fertility ~ log(ppgdp)and fertility ~ pctUrban, and verify that the slope coefficients are significantly differ- ent from 0 at any conventional level of significance.

(c)Obtain the added-variable plots for both predictors. Based on the added-variable plots, is log(ppgdp) useful after adjusting for pctUrban, and similarly, is pctUrban useful after adjusting for log(ppgdp)? Compute the estimated mean function with both predictors included as regressors, and verify the findings of the added-variable plots.

(d)Show that the estimated coefficient for pctUrban is the same as the estimated slopein the added-variable plot for pctUrban after log(ppgdp). This correctly suggests that all the estimates in a multiple linear regression model are adjusted for all the other regressors in the mean function.

(e)Showthat the residuals in the added-variable plot are identical to the residuals from the mean function with both predictors.

2.The Berkeley Guidance Study (data file BGSgirls) enrolled children born in Berkeley, California, between January 1928 and June 1929, and then measured them periodically until age 18.

The data we use include heights in centimeters at ages 2, 9, and 18 (HT2, HT9,and HT18), weights in kilogram (WT2, WT9, and WT18), and strength in kilogram (ST2, ST9, and ST18). An additional measure is body mass index BMI18, computed as weight in kilogram divided by the square of mass in meters. For this problem we use only the data on the girls.

(a)Draw the scatterplot matrix of (HT2, HT9, WT2, WT9, ST9, BMI18). Write a summary ofthe information in this scatterplot

(b)Starting with the mean function E(BMI18|WT9) = β0+β1WT9, use added-variable plots to explore adding ST9 to get the mean function E(BMI18|WT9, ST9) = β0 + β1WT9 + β2ST9.

Obtain the marginal plots of BMI18 versus each of WT9 and ST9, the plot of ST9 versus WT9, and then the added-variable plot for ST9. Summarize your results.

(c)Fit the multiple linear regression model with the mean function E(BMI18|X= x) = β0 + β1HT2 + β2WT2 + β3HT9 + β4WT9 + β5ST9

Find σˆ and R2. Compute the t-statistics used to test β2 and β4 to be 0 against two- sided alternatives. Explicitly state the hypotheses being tested, and the conclusions.

3.The following questions all refer to the meanfunction

E(Y |X1 = x1, X2 = x2) = β0 + β1x1 + β2x2 (1)

(a)Suppose we fit (1) to data for which x1= 2.2x2, with no error (for example, if x1 were a weight in pounds, and x2 were the weight of the same object in kilogram). Describe the appearance of the added-variable plot for X2 after X1.

(b)Again referring to (1), suppose now that Y = 3X1without error, but X1 and X2 are not perfectly correlated. Describe the appearance of the added-variable plot for X2 after X1.

(c)Under what conditions will the added-variable plot for X2after X1 have exactly the same shape as the marginal plot of Y versus X2?线性回归模型统计作业代写

(d)True or false: The vertical variation of the points in an added-variable plot for X2after X1 is always less than or equal to the vertical variation in a plot of Y versus X2.

4.Suppose we have a regression in which we want to fit the mean function (1),

given in the previous problem, to a data set for which the two terms X1and X2 have sample correlation equal to 0. This means that

(a)Givethe formula for the estimated slope in the regression of Y on X1, and for Y on

X2. Give the value of estimated slope in the regression of X2 on X1.

(b)Give formulas for the residuals for the regressions of Y on X1and for X2 on X1. The plot of these two sets of residuals corresponds to the added-variable plot for X2.

(c)Compute the slope of the regression corresponding to the added-variable plot for the regression of Y on X2after X1, and show that this slope is exactly the same as the slope for the simple regression of Y on X2, ignoring X1.

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