Fall 2013
STATISTICS 479
Assignment #7 (45 points)
Instructions: Turn in the programs, the plots and the written answers (when required) for each problem.
The data supplied in the file air pollution.txt are on air pollution in 41 U.S. cities. The type of air
pollution under study is the annual mean concentration of sulfur dioxide. It is desired to develop a regression
model to predict air pollution using 6 explanatory variables x1 − x6 measured as described on the back of this
page. A SAS procedure was used to fit a multiple regression model:
y = β0 + β1 x1 + β2 x2 + β3 x3 + β4 x4 + β5 x5 + β6 x6 + using data for all 41 cities.
1. Use a scatterplot matrix and correlation matrices to carry out a preliminary assessment of the pairwise
relationships between y and x1 , x2 , x3 , x4 , x5 , and x6 . On this basis alone, select a few explanatory
variables that may good predictors in a multiple regression model. Using the plot and the correlation
matrix, find the four variables that are most strongly correlated among the explanatory variables. Based
on above analysis alone suggest the explanatory variables that are most strongly involved in multicollinearity
when fitting the full model.
2. Use a SAS procedure to fit the above multiple regression model to all 41 cities. Discuss the fit of this
model using the Anova table, R2 , and the estimates table. Use other diagnostic tools including output
statistics and plots of residuals to examine the adequacy of the model (use the diagnostic panel of plots).
Comment on these cases, identifying any possible problems with model assumptions indicated by them and
cases that may be outliers and/or influential.
3. Identify any least squares estimates of the regression coefficients (β̂’s) from the fit of the model in Prob
2.), that have a sign (positive or negative) that is different from what you would expect for the parameter
– an indication of multicollinearity? Use the standard errors of the parameter estimates to show that these
are poorly estimated. Do the variance inflation factors (VIF’s) identify these parameters?
4. Remove the explanatory variables x3 and x6 from the five-variable model and use a multiple regression
model to relate y to x1 , x2 , x4 and x5 only. What can you observe about the multicollinearity in the new
model? Is there improvement in the accuracy of estimation of parmeters of this model (for e.g., decreases
standard errors, more t-statisticas are significant etc.)? Justify your answers.
5. Perform a residual analysis using diagnostic statistics including relevant graphics for the above 4-variable
model. Specifically, determine whether a single city does not appear to fit the model very well and provide
evidence from your analysis for making this conclusion.
6. Remove the case you determined above to be a possible outlier from the data and use all 6 variables for
the analyses described in the following three parts:
(i) Use a SAS procedure to do all possible regressions containing no less than 2 and no more than 4
explanatory variables. Print statistics for only the 4 best models in each case. Construct a plot of
the Cp statistic for all models with ‘reasonable’ Cp values. Select a single model, each with 2, 3,
and 4 explanatory variables, respectively, for the purpose of predicting annual mean concentration of
sulfur dioxide in a city, indicating your reasons for selection of each model. There may be
several possible choices, i.e., there may be many ‘good’ models but give arguments for each of your
choices. Primarily, use s2 , R2 , and Cp in your arguments. Select one of these models as your final
model and provide arguments supporting your choice.
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(ii) Use the backward elimination subset selection procedure with significance level of 0.05 for deleting
variables to select a possible model. State the model selected and report estimates of parameters and
the analysis of variance table for this model.
(iii) Use the stepwise subset selection procedure, with significance levels of 0.10 for entry and 0.05 for
deletion of variables, respectively, to select a possible model. State the model selected and report
estimates of parameters and the analysis of variance table for this model.
Data: Note that the first column of the data set is a city number. Data values for the variables appear in the
order they are described below. A link to a text file is available on the assignment page
city=
y=
x1=
x2=
x3=
x4=
x5=
x6=
city number (input as character data)
the annual mean concentration of sulfur dioxide (mcg/meter3 )
average annual temperature ◦ F
number of manufacturing enterprises employing 20 or more workers
population size (1970 census) in thousands
average annual wind speed (mph)
average annual precipitation (inches)
average number of days with precipitation per year
1
10 70.3 213 582 6.0 7.05 36
2
13 61.0
91 132 8.2 48.52 100
4
17 51.9 454 515 9.0 12.95 86
5
56 49.1 412 158 9.0 43.37 127
6
36 54.0
80
80 9.0 40.25 114
7
29 57.3 434 757 9.3 38.89 111
8
14 68.4 136 529 8.8 54.47 116
9
10 75.5 207 335 9.0 59.80 128
10 24 61.5 368 497 9.1 48.34 115
11 110 50.6 3344 3369 10.4 34.44 122
12 28 52.3 361 746 9.7 38.74 121
13 17 49.0 104 201 11.2 30.85 103
14
8 56.6 125 277 12.7 30.58 82
15 30 55.6 291 593 8.3 43.11 123
16
9 68.3 204 361 8.4 56.77 113
17 47 55.0 625 905 9.6 41.31 111
18 35 49.9 1064 1513 10.1 30.96 129
19 29 43.5 699 744 10.6 25.94 137
20 14 54.5 381 507 10.0 37.00 99
21 56 55.9 775 622 9.5 35.89 105
22 14 51.5 181 347 10.9 30.18 98
23 11 56.8
46 244 8.9 7.77 58
24 46 47.6
44 116 8.8 33.36 135
25 11 47.1 391 463 12.4 36.11 166
26 23 54.0 462 453 7.1 39.04 132
27 65 49.7 1007 751 10.9 34.99 155
28 26 51.5 266 540 8.6 37.01 134
29 69 54.6 1692 1950 9.6 39.93 115
30 61 50.4 347 520 9.4 36.22 147
31 94 50.0 343 179 10.6 42.75 125
32 10 61.6 337 624 9.2 49.10 105
33 18 59.4 275 448 7.9 46.00 119
34
9 66.2 641 844 10.9 35.94 78
35 10 68.9 721 1233 10.8 48.19 103
36 28 51.0 137 176 8.7 15.17 89
37 31 59.3
96 308 10.6 44.68 116
38 26 57.8 197 299 7.6 42.59 115
39 29 51.1 379 531 9.4 38.79 164
40 31 55.2
35
71 6.5 40.75 148
41 16 45.7 569 717 11.8 29.07 123
Note: It may take several computer runs to complete the required analyses. You may cut-and-paste parts of
computer printout into your answer sheets.
Due Thursday November 21, 2013
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