st: Hosmer Lemeshow table incorrectly reported probabilities?

["Matthew Baldwin, MD" <mrb45@columbia.edu>]

Dear Statlisters,

I created a logistic regression model that predicts mortality.  I am  
examining the model's calibration, and the predicted probabilities for  
quintiles of risk seem to be incorrectly calculated using STATA.

Using the code "gof, group(5), table" gives me the 5 groups used for  
the Hosmer-Lemeshow Chi squared test used to test model calibration  
and gives predicted probabilities (in this case, probability of  
death), observed and expected counts for both outcomes (deaths in this  
case), and totals for each group.

However, the expected probabilities that the "estat gof, group(5)  
table" code in STATA gives are different from the expected  
probabilities I calculate from the expected counts in the table  
itself. I am confused about this:

 Logistic model for death2, goodness-of-fit test

  (Table collapsed on quantiles of estimated probabilities)
  +--------------------------------------------------------+
  | Group |   Prob | Obs_1 | Exp_1 | Obs_0 | Exp_0 | Total |
  |-------+--------+-------+-------+-------+-------+-------|
  |     1 | 0.0958 |    15 |  20.1 |   296 | 290.9 |   311 |
  |     2 | 0.1757 |    47 |  41.7 |   264 | 269.3 |   311 |
  |     3 | 0.3006 |    66 |  73.3 |   245 | 237.7 |   311 |
  |     4 | 0.4494 |   119 | 114.8 |   192 | 196.2 |   311 |
  |     5 | 0.9328 |   183 | 180.2 |   127 | 129.8 |   310 |
  +--------------------------------------------------------+

       number of observations =      1554
             number of groups =         5
      Hosmer-Lemeshow chi2(3) =         3.46
                  Prob > chi2 =         0.3265


Here are my calculations for expected probabilities based on the table:

x (expected mortality)     y (observed mortality)
20.1/311 = 0.065           15/311 = 0.048
41.7/311 = 0.13            47/311 = 0.15
73.3/311 = 0.24            66/311 = 0.21
114.8/311 = 0.37           119/311 = 0.38
180.2/310 = 0.58           183/310 = 0.58


Can anyone explain how the expected probabilities in the table were derived?

Is the "gof, group(x) table" function in STATA reporting incorrect  
expected probabilities?

Thanks,

Matthew Baldwin, MD
Department of Pulmonary, Allergy, and Critical Care Medicine
New York Presbyterian Hospital
Columbia University College of Physicians and Surgeons


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