## Sunday, March 18, 2018

### Event Probabilities from Logit Models

Event Probabilities from Logit Models

Question:   The table below contains coefficients of a logit regression model for whether people age 58 to 66 are employed or not employed.   Use the estimated coefficients to obtain the employment probabilities for three people.

All three people are 58-year males with a college degree.  The first person does not have diabetes.   The second person has diabetes but does not have complications from diabetes.    The third person has both diabetes and some diabetic related complication.

Data:

 A Logit Model for Work Status -- Age 58 to 66 working Coef. z male 0.4130 5.59 age5859 0.5112 3.82 age6061 0.2282 1.69 age6364 -0.2251 -1.64 age6566 -0.8567 -6.11 College Degree 0.6273 4.15 no_college -0.0071 -0.06 diabetes -0.5045 -5.1 complications -1.1857 -5.15 _cons 0.0645 0.43

Formula for event probabilities from logit models:

Define XB as the dot product of the explanatory variables with the coefficients for the explanatory variables.   The formula for the event probability (in this case the probability of being employed) is:

P= Exp(XB)/(1+Exp(XB))

Calculations:  The first step is to set up a table with the logit coefficient next to the variable vectors for the three people.

 Logit Coefficients and Variable Values for Three People working Coef. Person 1 Person 2 Person 3 male 0.4130 1 1 1 age5859 0.5112 1 1 1 age6061 0.2282 0 0 0 age6364 -0.2251 0 0 0 age6566 -0.8567 0 0 0 college_deg 0.6273 1 1 1 no_college -0.0071 0 0 0 diabetes -0.5045 0 1 1 complications -1.1857 0 0 1 _cons 0.0645 0 0 0

The next steps are to calculate XB with the SUMPRODUCT function in Excel and then plug this value into the formula to get p.

 Calculation of Employment Probabilities for Three People Person 1 Person 2 Person 3 SUMPRODUCT 1.552 1.047 -0.139 EXP SUMPRODUCT 4.719 2.849 0.871 P 0.825 0.740 0.465

Concluding Remarks:  The person without diabetes has an 82.5 percent likelihood of being employed.   The person with diabetes but no complications has a 74.0 percent chance of being employed.   The person with diabetes and diabetes-related complications has a 46.5 percent chance of being employed.

Other exercises for students of statistics.   Use the data presented here to calculate employment likelihoods for a female age 66 with a college degree and with and without diabetes or diabetes related complications.