Question

1. Consider the National Football League data in Table B.1.

   1. Use the forward selection algorithm to select a subset regression model.

   2. Use the backward elimination algorithm to select a subset regression

      model.

   3. Use stepwise regression to select a subset regression model.

   4. Comment on the final model chosen by these three procedures.

y	x1	x2	x3	x4	x5	x6	x7	x8	x9
10	2113	1985	38.9	64.7	4	868	59.7	2205	1917
11	2003	2855	38.8	61.3	3	615	55	2096	1575
11	2957	1737	40.1	60	14	914	65.6	1847	2175
13	2285	2905	41.6	45.3	-4	957	61.4	1903	2476
10	2971	1666	39.2	53.8	15	836	66.1	1457	1866
11	2309	2927	39.7	74.1	8	786	61	1848	2339
10	2528	2341	38.1	65.4	12	754	66.1	1564	2092
11	2147	2737	37	78.3	-1	761	58	1821	1909
4	1689	1414	42.1	47.6	-3	714	57	2577	2001
2	2566	1838	42.3	54.2	-1	797	58.9	2476	2254
7	2363	1480	37.3	48	19	984	67.5	1984	2217
10	2109	2191	39.5	51.9	6	700	57.2	1917	1758
9	2295	2229	37.4	53.6	-5	1037	58.8	1761	2032
9	1932	2204	35.1	71.4	3	986	58.6	1709	2025
6	2213	2140	38.8	58.3	6	819	59.2	1901	1686
5	1722	1730	36.6	52.6	-19	791	54.4	2288	1835
5	1498	2072	35.3	59.3	-5	776	49.6	2072	1914
5	1873	2929	41.1	55.3	10	789	54.3	2861	2496
6	2118	2268	38.2	69.6	6	582	58.7	2411	2670
4	1775	1983	39.3	78.3	7	901	51.7	2289	2202
3	1904	1792	39.7	38.1	-9	734	61.9	2203	1988
3	1929	1606	39.7	68.8	-21	627	52.7	2592	2324
4	2080	1492	35.5	68.8	-8	722	57.8	2053	2550
10	2301	2835	35.3	74.1	2	683	59.7	1979	2110
6	2040	2416	38.7	50	0	576	54.9	2048	2628
8	2447	1638	39.9	57.1	-8	848	65.3	1786	1776
2	1416	2649	37.4	56.3	-22	684	43.8	2876	2524
0	1503	1503	39.3	47	-9	875	53.5	2560	2241

Answer 1

I have done forward selection, backward elimination and stepwise regression using minitaM software. Steps to perform this procedure is given below,

Open the worksheet containing the data file.

Steps to run stepwise regression in Minitab:

1. Click Stat → Regression → Regression → Fit Regression Model
2. A new window named “Regression” appears.
3. Select “y” as the “Responses” and select all the other variables into the “Continuous Predictors” box.

1. Click the “Stepwise” button and a new window named “Regression: Stepwise” pops up.
2. Select the method of stepwise regression and enter the alphas to enter/remove. In this example, we use the “Forward selection” method and the alpha to enter is 0.25.

Here, we apply forward selection method for given data. Minitab uses t Statistics for decision making regarding variable selection. In this example we will use alpha = 0.25

From the table of analysis of variance, we see that the regressor most highly correlated with y is , X2, X7,X8 & X9 .( Since p value regarding x2,X7, x8 & x9 is less than alpha=0.25).

So the forward selection procedure terminates with,

Yhat = -1.82 + 0.003819X2 + 0.2169 x7 -0.00401 x8 -0.00163 x9

Also R sq value is 80.12% which implies good fit to the model.

imilarlilyyou can done for backward ellimination procedure.

In this run we have selected the cutoff value by using alpha=0.10 , the default in Minitab .

Minitab uses p value for removing variables. Thus a regressor is dropped if the p value is greater than aplha=0.10 . In the above analysis of variance table, the regressors with corresponding p value is less the alpha = 0.10 . So we include all the regressor variables in the regression model ,

The fitted regression model is,

Y hat = -1.81 +0.003598 X2 +0.1940X7 -0.00482 X8

for stepwise regression

Stepwise regression model requires two cutoff values, one for entering valriabva and one for removing them. In the given model we select only those regressors variables which has p value is less than alpha = 0.15

The fitted regression model is,

Yhat = -1.81+0.003598X2 + 0.1940 X7 -0.00482 X8.