# Why is there such a mismatch between the Model's predicted probability and theoretical probability in logistic regression?

I am trying to do Logistic Regression using SAS Enterprise Miner. My Independent variables are

CPR/Inc (Categorical 1 to 7)
OD/Inc (Categorical 1 to 4)
Insurance (Binary 0 or 1)
Income Loss (Binary 0 or 1)
Living Arrangement (Categorical 1 to 7)
Employment Status (categorical 1 to 8)


My Dependent Variable is Default (Binary 0 or 1)

The following is the output from running Regression Model.

Analysis of Maximum Likelihood Estimates

                                  Standard          Wald
Parameter       DF    Estimate       Error    Chi-Square    Pr > ChiSq    Exp(Est)

Intercept        1     -0.4148      0.0645         41.30        <.0001       0.660
CPR___Inc  1     1     -0.8022      0.1051         58.26        <.0001       0.448
CPR___Inc  2     1     -0.4380      0.0966         20.57        <.0001       0.645
CPR___Inc  3     1      0.3100      0.0871         12.68        0.0004       1.363
CPR___Inc  4     1    -0.00304      0.0898          0.00        0.9730       0.997
CPR___Inc  5     1      0.1331      0.0885          2.26        0.1324       1.142
CPR___Inc  6     1      0.1694      0.0881          3.70        0.0546       1.185
Emp_Status 1     1     -0.2289      0.1006          5.18        0.0229       0.795
Emp_Status 2     1      0.4061      0.0940         18.66        <.0001       1.501
Emp_Status 3     1     -0.2119      0.1004          4.46        0.0347       0.809
Emp_Status 4     1      0.1100      0.0963          1.30        0.2534       1.116
Emp_Status 5     1     -0.2280      0.1007          5.12        0.0236       0.796
Emp_Status 6     1      0.3761      0.0943         15.91        <.0001       1.457
Emp_Status 7     1     -0.3337      0.1026         10.59        0.0011       0.716
Inc_Loss   0     1     -0.1996      0.0449         19.76        <.0001       0.819
Insurance  0     1      0.1256      0.0559          5.05        0.0246       1.134
Liv_Arran  1     1     -0.1128      0.0916          1.52        0.2178       0.893
Liv_Arran  2     1      0.2576      0.0880          8.57        0.0034       1.294
Liv_Arran  3     1      0.0235      0.0904          0.07        0.7950       1.024
Liv_Arran  4     1      0.0953      0.0887          1.16        0.2825       1.100
Liv_Arran  5     1     -0.0493      0.0907          0.29        0.5871       0.952
Liv_Arran  6     1     -0.3732      0.0966         14.93        0.0001       0.689
OD___Inc   1     1     -0.2136      0.0557         14.72        0.0001       0.808
OD___Inc   2     1     -0.0279      0.0792          0.12        0.7248       0.973
OD___Inc   3     1     -0.0249      0.0793          0.10        0.7534       0.975


Now I used this Model to Score a new set of data. An example row of my new data is

CPR - 7
OD - 4
Living Arrangement - 4
Employment Status - 4
Insurance - 0
Income Loss - 1


For this sample row, the model predicted output (Probability of default = 1) as 0.7335 To check this manually, I added the estimates

Intercept + Emp Status 4 + Liv Arran 4 + Insurance 0
-0.4148   + 0.1100  +   0.0953   +   0.1256    =   -0.0839


Odds ratio = Exponential(-0.0839) = 0.9195

Hence probability = 0.9195 / (1 + 0.9195) = 0.4790

I am unable to understand why there is such a mismatch between the Model's predicted probability and theoretical probability.

Any help would be much appreciated . Thanks

• I don't know much about SAS, but your math is clearly right. Perhaps it's a bug in how you called the model predicted output? – Ben Jun 19 '14 at 17:49
• Ben, thanks for the reply. I called the Scoring node directly from SAS Enterprise Miner, not much options we can change there. I compared the scoring output with what R predicted, and they perfectly matched. Then used the estimates manually , R's estimates theoretically matched with the output. It's just the problem with what SAS is giving as output, I am unable to understand that. – Shrikant Soni Jun 20 '14 at 18:45
• Well I'm glad you were able to get a reasonable result with R! Hopefully someone who knows why this issue exists in SAS can answer the question! – Ben Jun 22 '14 at 13:04