Study Case: Logistic Regression – Model Interpretation

A.Field Goal PercentageRef: Introduction to Data Analysis and Statistical Inference (Morris, C. N. and Rolph, J. E.). Prentice Hall, 1981.


Let:

p = probability of success in field goal attempt by a NFL kicker

X = distance in yards from the goal.

Predicting equation: logit(p) = 3.80 – 0.108 X

Questions

(1)Interpret the coefficient of the distance, X.

(2)What is the probability of a successful 30-yard field goal?

(3)Compute p for various values of X and plot against X. Do you think that the shape of the curve looks reasonable?


B. Predicting bankruptcy

Ref.: Chaterjee and Price. Regression Analysis by Examples (John Wiley, 1991)

Detecting ailing financial and business establishments is an important

function of audit and control. Systematic failure to do so can lead to grave

consequences, such as the savings and loan fiasco of the 1980s. Table 5.9

gives the operating financial ratios of 33 firms that went bankrupt after 2

years and 33 that remained solvent during the same period. The data were

provided by Edward Altman of New York University. Five financial ratios

were available for each firm:

X1 = (working capital)/(total assets)

X2 = (retained earnings)/(total assets)

X3 = (earnings before interest and taxes)/(total assets)

X4 = (market-value equity)/(book value of total liabilities)

X5 = sales/(total assets)

Y = / 0 if bankrupt after 2 years
1 if solvent after 2 years

A multiple logistic regression model was fitted using variables X2 and X3.

The other three variables did not substantially add to the explanatory

power of the model. The fitted logistic model is

g ( X ) = ln () = -0.550 + 0.157 X2 + 0.194X3.

The predicted probabilities (X) for remaining solvent is given by

and that for bankruptcy is given by 1-(X).

If we classify any firm with(X) greater than 0.5 to be a solvent firm, the model given in misclassifies only two firms, one from the bankrupt and one from the

solvent category.

Appendix: Interpretation of the Coefficients

Let:

logit (p) =

Interpretation of : proportional change of odds given one unit change of X.

= logit(p){x+l}- logit(p){x} = log

Therefore:

so that:

From calculus, recall:

Therefore:

for

That is:

1