Question

the data BaseballTimes contains 4 quantitative variables that might be useful for predicting game "Time".

Runs, Margin, Pitchers, and Attendance

Game	League	Runs	Margin	Pitchers	Attendance	Time
CLE-DET	AL	14	6	6	38774	168
CHI-BAL	AL	11	5	5	15398	164
BOS-NYY	AL	10	4	11	55058	202
TOR-TAM	AL	8	4	10	13478	172
TEX-KC	AL	3	1	4	17004	151
OAK-LAA	AL	6	4	4	37431	133
MIN-SEA	AL	5	1	5	26292	151
CHI-PIT	NL	23	5	14	17929	239
LAD-WAS	NL	3	1	6	26110	156
FLA-ATL	NL	19	1	12	17539	211
CIN-HOU	NL	3	1	4	30395	147
MIL-STL	NL	12	12	9	41121	185
ARI-SD	NL	11	7	10	32104	164
COL-SF	NL	9	5	7	32695	180
NYM-PHI	NL	15	1	16	45204	317

From among the four predictors choose a model for each of the following goals

a. Maximize the coefficient of determination R^2

b. Maximize the adjusted R^2

c. Minimize Mallow's Cp

d. Considering models a-c, whamt model would you choose to predict game Time and why?

e. Using stepwise procedure(forward backwards, elimination process) find the "best" model

Answer 1

Using the above data , We run the Multiple linear Regression ,

We have to predict Game Time using the predictors Runs, Margin, Pitchers, and Attendance

The model is :

Time = B_0 + B_1(Runs) + B_2(Margin) + B_3(Pitchers) + B_4(Attendance)

Where B_0 , B_1 , B_2 , B_3 and B_4 are the regression coefficients.

Hypothesis :

Ho :  B_0 = B_1 = B_2 = B_3 = B_0 = 0

i.e. All variables are insignificant

V/s

H₁ : at leats one coefficient is not Zero.

i.e . the variables are significant.

Calculation table :

coefficients	estimate	t value	p value
B_0	88.0151	4.952	0.00057
B_1	1.5614	0.931	0.3736
B_2	-3.7278	-1.793	0.1032
B_3	8.7322	3.514	0.005594
B_4	0.000726	1.424	0.1848

at 5% level of significance we reject Ho and conclude that the variable Pitchers (B_3) is significant.

therefor the model is ,

Time = B_0 + B_3(Pitchers)

a) Maximize the coefficient of determination (R²) :-

R-squared: 0.8557

i.e. Our model is the good , the all predictors explained the 85.57% variations on dependent variable.

b) Maximize the adjusted R² :-

Adjusted R-squared: 0.798

i.e That means the 79.8% variation is expalined by only those variable which are statistically significant . (here , our Pitchers variable is statistically significant) .

c) Minimize Mallow's C_p :-

where ,SSE = is the error sum of squares for the model with P regressors,

S² = is the residual mean square after regression on the complete set of K regressors and can be estimated by mean square error MSE

N = is the sample size.

and P = no. pf regressors.

Using the above formula we calculate Mallow's Cp

Cp = 3

Here , Mallows' C_p-statistic = 3 is the size of the bias that is introduced into the predicted responses by having an underspecified model.

d) Choosed Model is :-

Time = B_0 + B_3(Pitchers)

Because , The coefficeients B_0 and B_3 are significantally affect to prdict the Time.

and this model have the less AIC i.e. 93.83

e) Using stepwise procedure :-

using the stepwise procedure the best model is,

Time = B_0 + B_3(Pitchers)

because the AIC=93.83 . and have a adjusted R² =0.798

here ,  B_0 = 94.84 and B_3 = 10.71

the model is ,

Time = 94.84 + 10.71(Pitchers)

-----ALL THE BEST-----