Question

Applications 16. A study provided data on variables that may be related to the number of weeks a manufac- turing worker has been jobless. The dependent variable in the study (Weeks) was defined as the number of weeks a worker has been jobless due to a layoff. The following indepen- dent variables were used in the study. The age of the worker Age DATA file Educ The number of years of education 2 Layoffs the tre A dummy variable; 1 if married, 0 otherwise A dummy variable; 1 if the head of household, 0 otherwise SATAG Married Head TRC Tenure The number of years on the previous job A dummy variable; 1 if management occupation, 0 otherwise A dummy variable; 1 if sales occupation, 0 otherwise ses Manager Sales The data are available in the file named Layoffs. Develop the best one-variable estimated regression equation. b. Use the stepwise procedure to develop the best estimated regression equation. Use values of .05 for Alpha to enter and Alpha to remove. c. Use the forward selection procedure to develop the best estimated regression equation. Use a value of .05 for Alpha to enter. d. Use the backward elimination procedure to develop the best estimated regression equation. Use a value of .05 for Alpha to remove. Use the best-subsets regression procedure to develop the best estimated regression equation. a. EEL nding noitua oizangn olgilum bel e.

Answer 1

SOLUTION:

(a) Let us consider the sample correlation coefficients between each pair of variables.

The following figure is the correlation matrix obtained using Minitab.

Correlations: Weeks. Age, Educ, Married. Head. Tenure. Manager. Sales

Weeks Age Educ Married Head Tenure Manager

Age 0.577

Educ 0.007 0.100

Married -0.130 -0.209 -0.151

Head -0.205 0.027 -0.158 -0.449

Tenure 0.398 0.459 0.174 -0.057 -0.046

Manager -0.198 0.097 0.160 0.073 -0.200 -0.113

Sales -0.134 0.137 0.124 -0.148 -0.013 0.097 -0.156
Looking at the sample correlation coefficients between Weeks and each of the independent variables can give us a quick indication of which independent variables are. by themselves good predictors. We see that the single best predictor of Weeks is Age. because it has the highest sample correlation coefficient.
Age can explain (0.577)² (000)= 33.29 %. of the variability in weeks.

So we construct an estimated regression equation using the variable Age:

Regression Analysis: Weeks versus Age

The regression equation is

Weeks = - 8.0 + 1.51 Age

Predictor Coef SE Coef T P

Constant -8.8611.01 -0.80 0.425

Age 1.5092 0.3080 4.90 0.000

S = 19.5342 R-Sq = 33.3% R-Sq (adj) = 32.0%

Analysis of Variance

Source DF SS MS F P

Regression 1 9161.4 9161.4 24.01 0.000

Residual Error 48 18316.1 381.6

Total 49 27477.5

The estimated regression equation used to predict the number of weeks a worker has been jobless due to a layoff given the age of the worker is:
_{y = -8.9+1.51.0}

(b)

The following figure shows the results obtained by using the minitab stepwise regression procedure for the given data using values of 0.05 for Alpha to remove and 0.05 for Alpha to enter.

Step wise Regression : Weeks versus Age , Educ, Married, Head , Tenure, Manager , Sales

Alpha - to - Enter : 0.05 Alpha - to - Removes : 0.05

Response is Weeks on 7 predictors. with N = 50

Step 1 2 3 4

Constant -8.86002 -9.09741 -0.10922 -0.06890

Age 1.51 1.57 1.61 1.73

T-Value 4.90 5.30 5.74 6.51

P-Value 0.000 0.000 0.000 0.000

Manager -20.1 -24.6 -28.7

T-Value -2.26 -2.88 -3.53

P-Value 0.029 0.006 0.001

Head -14.3 -15.1

T-Value -2.61 -2.95

P-Value 0.012 0.005

Sales -17.4

T-Value -2.79

P-Value 0.008

S 19.5 18.7 17.7 16.5

R-Sq 33.34 39.87 47.64 55.38

R-Sq(ad) 31.95 37.31 44.22 51.41

Mallows C-p 22.5 17.8 11.8 5.9

The stepwise procedure terminated after four steps. The estimated regression equation identified by the Minitab stepwise regression procedure is:

Weeks = -0.06890+1.73Age-28.7Manger-15.1Head-17.4Sales

The value of R-sq has been increased from 33.34% to 55.38% and the recommended estimated regression equation has an R-Sq (adj) value of 51.41%

(c)

The following figure shows the results obtained by using the Minitab stepwise regression procedure for the given data using values of am for Alpha to remove and 0.05 for Alpha to enter.

Stepwise Regression: Weeks versus Age, Educ, Married, Head, Tenure, Manager, Sales Forward selection: Alpha-to-Enter. 0.5

Response is Weeks on 7 predictors. with N = 50

Step 1 2 3 4

Constant -8.86002 -9.09741 -0.10922 -0.06890

Age 1.51 1.57 1.61 1.73

T-Value 4.90 5.30 5.74 6.51

P-Value 0.000 0.000 0.000 0.000

Manager -20.1 -24.6 -28.7

T-Value -2.26 -2.88 -3.53

P-Value 0 029 0.006 0.001

Head -14.3 -15.1

T-Value -2.61 -2.95

P-Value 0.012 0.005

Sales -17.4

T-Value -2.79

P-Value 0.008

S 19.5 18.7 17.7 16.5

R-Sq 33.34 39.87 47.64 55.38

R-Sq(adj) 31.95 37.31 44.22 51.41

Mallows C-p 22.5 17.8 11.8 5.9

The forward selection procedure terminated after four steps. The estimated regression equation identified by the Minitab forward selection procedure. similar to stepwise regression procedure is:

Weeks = -0.06890+1.73 Age - 28.7 Manager – 15.1 Head – 17.4 Sales

The value of R-sq has been increased from 33.34% to 55.38% and the recommended estimated regression equation has an R-Sq (adj) value of 51.41%

(d)

Stepwise Regression: Weeks versus Age, Educ,

Backward elimination: Alpha-to-Remove: 0.05

Response isiAteks on 7 predictors. with N = 50

Step 1234

Constant 22.85070 13.62308 13.06817 -0.06890

Age 1.51 1.52 1.64 1.73

T-Value 4.96 5.04 6.18 6.51

P-Value 0.000 0.000 0.000 0.000

Educ -0.61

T-Value -0.66

P-Value 0.516

Married -10.7 -9.9 -9.8

T-Value -1.79 -1.69 -1.69

P-Value 0.081 0.098 0099

Head -19.8 -19.0 -19.4 -15.1

T-Value -3.39 -3.35 -3.44 -2.95

P-value 0.002 0.002 0.001 0.005

Tenure 0.43 0.37
T-Value 0.91 0.82

P-Value 0.366 0.418

Manager -26.7 -27.7 -29.0 -28.7

T-Value -3.21 -3.40-3.64 -3.53

P-Value 0.003 0.001 0.001 0.001

Sales -18.6 -19.0 -19.0 -17.4

T-Value -2.96 -3.06-3.07 -2.79

P-Value 0.005 0.004 0.004 0.008

S 16.3 16.2 16.2 16.5

R-Sq 59.14 58.72 58.08 55.38

R-Sq (adj) 52.33 52.96 53.32 51.41

Mallows C-p 8.06.4 5.1 5.9

The backward selection procedure terminated after four steps. The estimated regression equation identified by the Minitab forward selection procedure, similar to stepwise regression procedure is:

Weeks = -0.06890+1.73 Age - 28.7 Manager – 15.1 Head – 17.4 Sales

The value of R-sq has been decreased from 59.14% to 55.38% and the recommended estimated regression equation has an R-Sq (adj) value of 51.41%

(e)

Best-subsets regression enables the user to find the best regression model given a specified number of independent variables.

The following figure is a portion of the computer output obtained by using the best-subsets procedures for the Layoffs data set.

Results ton LAYOFFS.MTW

Best Subsets Regression: Weeks versus Age. Educ....

Response is weeks

MM Vars R-Sq R-Sq(adj) 1 33.3 32.0 1 15.8 14.0 2 39.9 37.3 2 38.2 35.6 3 47.6 44.2 3 46.8 43.3 4 55.4 51.4 4 49.1 44.6 5 58.1 53.3 5 56.0 51.0 IS ETH dieud1 Mallows gue a C-P Secdd 22.5 19.534 X 40.6 21.954 17.9 10.749 X 19.5 19.005 X X 11.8 17.686 X X 12.7 17.831 X 5.9 16.507 X X 12.3 17.628 X X X 5.1 16.179 X X X X X 7.3 16.582 X XXXX

This output identifies the two best one-variable estimated regression equations, the two best two-variable equations. the two three-variable equations, and so on. The criterion used in determining which estimated regression equations are best for any number of predictors is the value of the coefficient of determination (R-Sq.).

For instance. Age with an R-Sq = 33.3%. provides the best estimated regression equation using only one independent variable: Age and Manager. with an R-Sq = 39.9% provides the best estimated regression equation using two independent variables: Age. Head and Manager with an R-Sq = 47.6%. provides the best estimated regression equation with three independent variables.
The adjusted coefficient of determination (Adj. R-Sq = 53.3%) is largest for the model with five independent variables: Age. Marred. Head. Manager and Sales

The best-subsets procedure shows that the best five-variable model contains the independent variables Age. Married, Manage. Head and Sales.

The best estimated regression equation is obtained by using the regression routine of Minitab: Regression Analysis: Weeks versus Age, Married, Head, Manager, Sales

The regression equation is

Weeks = 13.1 + 1.64 Age - 9.76 Married - 19 .4 Head - 29.0 Manager - 19.0 Sales

Predictor Coef SE Coef T P

Constant 13.07 12.40 1.05 0.298

Age 1.6369 0.2651 6.18 0.000

Married -9.764 5.794 -1.69 0.099

Head -19.405 5.636 -3.44 0.001

Manager -28.986 7.958 -3.64 0.001

Sales -18.967 6.181 -307 0.004

S = 16.1794 R-Sq = 58.1% R-Sq (adj) = 53.3%

Analysis of Variance Source OF SS MS F P

Regression 5 15959.5 3191.9 12.19 0.000

Residual Error 44 115180 261.8

Total 49 27477.5

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