Question

data file:

"STATE" "MALE" "BIRTH" "DIVO" "BEDS" "EDUC" "INCO" "LIFE"
AK 119.1 24.8 5.6 603.3 14.1 4638 69.31
AL 93.3 19.4 4.4 840.9 7.8 2892 69.05
AR 94.1 18.5 4.8 569.6 6.7 2791 70.66
AZ 96.8 21.2 7.2 536.0 12.6 3614 70.55
CA 96.8 18.2 5.7 649.5 13.4 4423 71.71
CO 97.5 18.8 4.7 717.7 14.9 3838 72.06
CT 94.2 16.7 1.9 791.6 13.7 4871 72.48
DC 86.8 20.1 3.0 1859.4 17.8 4644 65.71
DE 95.2 19.2 3.2 926.8 13.1 4468 70.06
FL 93.2 16.9 5.5 668.2 10.3 3698 70.66
GA 94.6 21.1 4.1 705.4 9.2 3300 68.54
HW 108.1 21.3 3.4 794.3 14.0 4599 73.60
IA 94.6 17.1 2.5 773.9 9.1 3643 72.56
ID 99.7 20.3 5.1 541.5 10.0 3243 71.87
IL 94.2 18.5 3.3 871.0 10.3 4446 70.14
IN 95.1 19.1 2.9 736.1 8.3 3709 70.88
KS 96.2 17.0 3.9 854.6 11.4 3725 72.58
KY 96.3 18.7 3.3 661.9 7.2 3076 70.10
LA 94.7 20.4 1.4 724.0 9.0 3023 68.76
MA 91.6 16.6 1.9 1103.8 12.6 4276 71.83
MD 95.5 17.5 2.4 841.3 13.9 4267 70.22
ME 94.8 17.9 3.9 919.5 8.4 3250 70.93
MI 96.1 19.4 3.4 754.7 9.4 4041 70.63
MN 96.0 18.0 2.2 905.4 11.1 3819 72.96
MO 93.2 17.3 3.8 801.6 9.0 3654 70.69
MS 94.0 22.1 3.7 763.1 8.1 2547 68.09
MT 99.9 18.2 4.4 668.7 11.0 3395 70.56
NC 95.9 19.3 2.7 658.8 8.5 3200 69.21
ND 101.8 17.6 1.6 959.9 8.4 3077 72.79
NE 95.4 17.3 2.5 866.1 9.6 3657 72.60
NH 95.7 17.9 3.3 878.2 10.9 3720 71.23
NJ 93.7 16.8 1.5 713.1 11.8 4684 70.93
NM 97.2 21.7 4.3 560.9 12.7 3045 70.32
NV 102.8 19.6 18.7 560.7 10.8 4583 69.03
NY 91.5 17.4 1.4 1056.2 11.9 4605 70.55
OH 94.1 18.7 3.7 751.0 9.3 3949 70.82
OK 94.9 17.5 6.6 664.6 10.0 3341 71.42
OR 95.9 16.8 4.6 607.1 11.8 3677 72.13
PA 92.4 16.3 1.9 948.9 8.7 3879 70.43
RI 96.2 16.5 1.8 960.5 9.4 3878 71.90
SC 96.5 20.1 2.2 739.9 9.0 2951 67.96
SD 98.4 17.6 2.0 984.7 8.6 3108 72.08
TN 93.7 18.4 4.2 831.6 7.9 3079 70.11
TX 95.9 20.6 4.6 674.0 10.9 3507 70.90
UT 97.6 25.5 3.7 470.5 14.0 3169 72.90
VA 97.7 18.6 2.6 835.8 12.3 3677 70.08
VT 95.6 18.8 2.3 1026.1 11.5 3447 71.64
WA 98.7 17.8 5.2 556.4 12.7 3997 71.72
WI 96.3 17.6 2.0 814.7 9.8 3712 72.48
WV 93.9 17.8 3.2 950.4 6.8 3038 69.48
WY 100.7 19.6 5.4 925.9 11.8 3672 70.29

use all R commands you need, including lm() function, to answer the following questions.

1. (a) Fit the MLR model with LIFE (y) as the response variable, and MALE (x1), BIRTH (x2), DIVO (x3), BEDS (x4), EDUC (x5), and INCO (x6), as predictors.

2. (b) At level α = 5%, conduct the F-test for the overall fit of the regression. Comment on the results.

3. (c) At level α = 1%, test each of the individual regression coefficients. Do the results indicate that any of the explanatory variables should be removed from the model?

4. (d) Determine the regression model with the explanatory variable(s) identified in part (c) removed. Write down the estimated regression equation.

5. (e) Perform a partial F-test at level α = 1% to determine whether the variables associated with MALE and INCO can be removed from the model

6. (f) Compute and report the F test statistic for comparing the two models E(Yi|xi) = β0 + β1xi1,

   E(Yi|xi) = β0 + β1xi1 + β2xi2 + β3xi3 + β4xi4 + β5xi5 + β6xi6,

7. (g) Perform a partial F-test at level α = 1% for comparing the two models E(Yi|xi) = β0,

   E(Yi|xi) = β0 + β1xi1 + β2xi2,

8. (h) Compute and report the terms in the decomposition

   SSreg(β1, β2, β3|β0) = SSreg(β3|β0) + SSreg(β2|β0, β3) + SSreg(β1|β0, β3, β2)

9. (i) Suppose we are interested in fitting a regression model using LIFE as the response variable

   and some subset of the variables (MALE, BIRTH, DIVO, and INCO) as predictor.

   (i.1) Perform variable selection by finding the subset model that minimizes the AIC criteria. State the ’best model’.

   2. (i.2) Perform variable selection using forward selection. State the ’best model’.

   3. (i.3) Perform variable selection using backward selection. State the ’best model’.

Answer 1

Here I attach the R code along with the solution :

x=read.csv("C:\\Users\\PC-10\\Desktop\\s.csv")
head(x)
library(car)
m=lm(x$LIFE~x$MALE+x$BIRTH+x$DIVO+x$BEDS+x$EDUC+x$INCO)
vif(m)
summary(m)

Residuals: -2. 5563 -0.6629 0.0755 0.6983 3.3215 Coefficients: Min 1Q Median 3Q Max Estimate std. Error t value Pr(ltI) (Intercept) 70.5577813 4.2897471 16.448 <2e-16 * xSMALE X$BIRTH x$DIVO x$BEDS X $EDUC XSINCO 0.1261019 0.0472318 2.670 0.01059 * -0. 5160558 0.1172775 -4.400 6.78e-05 -0.1965375 0. 0739533 -2. 658 0.01093 * -0. 0033392 0. 0009795 -3.409 0.00141 ** 0. 2368223 0.1110225 2.133 0.03853 * -0. 0003612 0.0004598 -0.786 0.43633 signif. codes : 0 ‘ , 0.001 ‘**, 0.01 ‘*, 0.05 ‘., 0.1 ‘ ’ 1 Residual standard error: 1.176 on 44 degrees of freedom Multiple R-squared 0.4685, Adjusted R-squared 0. 396 F-statistic: 6.464 on 6 and 44 DF, p-value: 6.112e-05

Here at 1% level male, divo, educ and income are not significant. we continue our regression by removing these varaibles.

x=read.csv("C:\\Users\\PC-10\\Desktop\\s.csv")
head(x)
library(car)
m=lm(x$LIFE~x$BIRTH+x$BEDS)
vif(m)
summary(m)
r=residuals(m)
shapiro.test(r)
Box.test(r,type = "Ljung-Box")

shapiro-wilk normality test data: r w = 0.97064 , p-value 0.235 > Box.test (r,type"Ljung-Box") Box-Ljung test data: r x-squared 0.70332, df 1, p-value0.4017

The model satisfies the assumptions of normality and autocorrelation of residuals.

The estimated regression model is:

Residuals -2. 5627 -0. 8180 -0. 0819 0. 9261 3.6202 Coefficients: Min 1Q Median 3Q Max Estimate std. Error t value Pr(ltI) (Intercept) 79.1473186 2.2717401 34. 840 < 2e-16 X$BIRTH x$BEDS -0. 3281679 0.1026214 -3.198 0.00245 ** -0.0027415 0.0009388 -2. 920 0.00531** signif. codes : 0 ‘ , 0.001 ‘**, 0.01 ‘*’ 0.05 ‘., 0.1 ‘ , 1 Residual standard error: 1.352 on 48 degrees of freedom Multiple R-squared 0.2329, Adjusted R-squared 0.2009 F-statistic: 7.286 on 2 and 48 DF, p-value: 0.001725

Life=79.147-0.328*Birth-0.0027*Beds