Question

It shows graduate program admission decisions (Yes: 1 and No: 2), GRE score and undergraduate GPA (1: ≥ 3.5 and 2: < 3.5) for fifty students. Tasks: Fit a binary logistic regression model with admission decision as the dependent variable, GRE and GPA as the independent variables. Evaluate the goodness of fit of the model. Determine the significance of independent variables. Interpret odds ratios for independent variables. State the binary logistic regression equation. Evaluate the classification accuracy of the model. Check if the residuals are independent.

Tasks:

Fit a binary logistic regression model with admission decision as the dependent variable, GRE and GPA as the independent variables.

• Evaluate the goodness of fit of the model.
• Determine the significance of independent variables.
• Interpret odds ratios for independent variables.
• State the binary logistic regression equation.
• Evaluate the classification accuracy of the model.
• Check if the residuals are independent.

Answer 1

Answer:

Goodness-of-Fit Tests :

H0 : The model is good fit

H1: The model is poor fit

The logistic regression model is statistically significant [Hosmer lemeshow test p> 0.01].that mean the null hypotehsis is fail to reject and the model is good fit  The model explained 22.2% (Nagelkerke R²) of the variance in likelihood of occurrence of admission decision by GRE and GPA.

Minitab output:

Test	DF	Chi-Square	P-Value
Deviance	47	53.77	0.231
Pearson	47	49.91	0.359
Hosmer-Lemeshow	7	5.53	0.596

Analysis:

In our model, the first variable to analyse its impact on admission decision is the GRE score of the student. The statistical results exhibit its positive association with admission decision. With the increase in the GRE score at there is a increase in the likelihood of admission decision this is a significant explanatory variable for likelihood of admission decision in the model (p<0.01).

the second variable to analyse its impact on admission decision is the GPA score of the student, The statistical results exhibit its negative association with admission decision. The likelihood of admission decision in those student where GPA>=3.5 is 4.62 (=1/0.2162) times lesser than those wher GPA <3.5 This implies that more GPA score of the student is, more likely to take decision about admission. Hence, GPA is the significant (p=0.037< 0.05) factor for admission decision.

Output:

Coefficients

Term	Coef	SE Coef	VIF
Constant	-3.40	1.53
GRE	0.00633	0.00252	1.00
GPA
1	-1.532	0.775	1.00

Deviance Table

Source	DF	Adj Dev	Adj Mean	Chi-Square	P-Value
Regression	2	15.225	7.613	15.23	0.000
GRE	1	7.743	7.743	7.74	0.005
GPA	1	4.354	4.354	4.35	0.037
Error	47	53.769	1.144
Total	49	68.994

Odds Ratios for Categorical Predictors

Level A	Level B	Odds Ratio	95% CI
GPA
1	0	0.2162	(0.0473, 0.9879)

Regression Equation

P(2)

=

exp(Y')/(1 + exp(Y'))

LOGIT (Child Mortality) = -3.402 + 0.006335 GRE -1.532 GPA

GPA
0	Y'	=	-3.402 + 0.006335 GRE
1	Y'	=	-4.934 + 0.006335 GRE