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st: Sparse Data Problem


From   john metcalfe <johnzmetcalfe@gmail.com>
To   statalist@hsphsun2.harvard.edu
Subject   st: Sparse Data Problem
Date   Fri, 6 Mar 2009 19:12:18 -0800

Dear Statalist,
I am analyzing a small data set with outcome of interest 'clstr', with
the primary goal of the analysis to determine if the variables 's315t'
and 'east' have independent associations with the outcome.  However,
2315t is highly deterministic for the outcome clstr, as below. I am
concerned that exact logistic regression is not fully accounting for
the small cell bias. I would like to employ a hierarchical logistic
regression, but it seems that the stata command 'hireg' is only for
linear linear regressions??
It may be that I simply am unable to make any valid inferences with
this dataset, but I just want to make sure I have explored the
appropriate possible remedies.
Thanks,
John

John Metcalfe, M.D., M.P.H.
University of California, San Francisco


. tab s315 clstr,e

           |         clstr
     s315t |         0          1 |     Total
-----------+----------------------+----------
         0 |        22          1 |        23
         1 |        58         32 |        90
-----------+----------------------+----------
     Total |        80         33 |       113

           Fisher's exact =                 0.002
   1-sided Fisher's exact =                 0.002




. logit clstr ageat s315t east emb sm num,or

Iteration 0:   log likelihood = -62.686946
Iteration 1:   log likelihood = -51.860098
Iteration 2:   log likelihood = -50.754342
Iteration 3:   log likelihood = -50.661741
Iteration 4:   log likelihood = -50.660257
Iteration 5:   log likelihood = -50.660256

Logistic regression                               Number of obs   =        100
                                                  LR chi2(6)      =      24.05
                                                  Prob > chi2     =     0.0005
Log likelihood = -50.660256                       Pseudo R2       =     0.1919

------------------------------------------------------------------------------
       clstr | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
   ageatrept |   .9908837   .0139884    -0.65   0.517     .9638428    1.018683
       s315t |   9.238959   10.28939     2.00   0.046     1.041462    81.96011
  east_asian |   4.219755   2.215279     2.74   0.006     1.508083    11.80727
         emb |   .9964845   .6599534    -0.01   0.996     .2721043    3.649268
          sm |   2.138175   1.696319     0.96   0.338      .451589    10.12379
  num_resist |   1.064089   .2385192     0.28   0.782     .6857694    1.651116
------------------------------------------------------------------------------



Strategy 1: Two-way contingency tables

. tab clstr s315t if east==1,e

           |         s315t
     clstr |         0          1 |     Total
-----------+----------------------+----------
         0 |         6         19 |        25
         1 |         1         24 |        25
-----------+----------------------+----------
     Total |         7         43 |        50

           Fisher's exact =                 0.098
   1-sided Fisher's exact =                 0.049

. tab clstr s315t if east==0,e

           |         s315t
     clstr |         0          1 |     Total
-----------+----------------------+----------
         0 |        12         33 |        45
         1 |         0          8 |         8
-----------+----------------------+----------
     Total |        12         41 |        53

           Fisher's exact =                 0.175
   1-sided Fisher's exact =                 0.108



Strategy 2: Exact Logistic Regression

observation 102: enumerations =       1128
observation 103: enumerations =        574

Exact logistic regression                        Number of obs =       103
                                                 Model score   =  19.78112
                                                 Pr >= score   =    0.0000
---------------------------------------------------------------------------
       clstr | Odds Ratio       Suff.  2*Pr(Suff.)     [95% Conf. Interval]
-------------+-------------------------------------------------------------
       s315t |   10.44218          32      0.0135      1.391627    474.4786
  east_asian |   5.414021          25      0.0006      1.933718    16.65417




(output omitted)
observation 103: enumerations =        574

Exact logistic regression                        Number of obs =       103
                                                 Model score   =  19.78112
                                                 Pr >= score   =    0.0000
---------------------------------------------------------------------------
       clstr |      Coef.       Score    Pr>=Score     [95% Conf. Interval]
-------------+-------------------------------------------------------------
       s315t |   2.345854    6.763266      0.0129      .3304732    6.162216
  east_asian |   1.688992    12.98631      0.0004      .6594448    2.812661
---------------------------------------------------------------------------


Strategy 3: Hierarchical Regression

. hireg clstr (s315t) (east)(ageat emb sm)

Model 1:
   Variables in Model:
   Adding            : s315t

      Source |       SS       df       MS              Number of obs =     113
-------------+------------------------------           F(  1,   111) =    9.18
       Model |   1.7840879     1   1.7840879           Prob > F      =  0.0030
    Residual |   21.578744   111  .194403099           R-squared     =  0.0764
-------------+------------------------------           Adj R-squared =  0.0680
       Total |  23.3628319   112  .208596713           Root MSE      =  .44091

------------------------------------------------------------------------------
       clstr |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       s315t |   .3120773   .1030162     3.03   0.003     .1079438    .5162108
       _cons |   .0434783   .0919364     0.47   0.637    -.1386999    .2256565
------------------------------------------------------------------------------

Model 2:
   Variables in Model: s315t
   Adding            : east

      Source |       SS       df       MS              Number of obs =     103
-------------+------------------------------           F(  2,   100) =   12.03
       Model |  4.34936038     2  2.17468019           Prob > F      =  0.0000
    Residual |  18.0778241   100  .180778241           R-squared     =  0.1939
-------------+------------------------------           Adj R-squared =  0.1778
       Total |  22.4271845   102  .219874358           Root MSE      =  .42518

------------------------------------------------------------------------------
       clstr |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       s315t |   .2817301   .1086887     2.59   0.011     .0660947    .4973654
  east_asian |   .3247109   .0843486     3.85   0.000     .1573656    .4920561
       _cons |  -.0669987   .1023736    -0.65   0.514     -.270105    .1361075
------------------------------------------------------------------------------
R-Square Diff. Model 2 - Model 1 = 0.118   F(1,100) = 14.190  p = 0.000

Model 3:
   Variables in Model: s315t  east
   Adding            : ageat emb sm

      Source |       SS       df       MS              Number of obs =     100
-------------+------------------------------           F(  5,    94) =    4.72
       Model |  4.36538233     5  .873076466           Prob > F      =  0.0007
    Residual |  17.3946177    94  .185049124           R-squared     =  0.2006
-------------+------------------------------           Adj R-squared =  0.1581
       Total |       21.76    99   .21979798           Root MSE      =  .43017

------------------------------------------------------------------------------
       clstr |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       s315t |   .2335983   .1163422     2.01   0.048     .0025981    .4645984
  east_asian |   .2694912   .0945411     2.85   0.005     .0817777    .4572048
   ageatrept |  -.0012444   .0024199    -0.51   0.608    -.0060491    .0035603
         emb |   .0396897   .0989203     0.40   0.689    -.1567189    .2360984
          sm |   .1063985   .1087626     0.98   0.330    -.1095522    .3223492
       _cons |  -.0454117   .1512602    -0.30   0.765    -.3457423     .254919
------------------------------------------------------------------------------
R-Square Diff. Model 3 - Model 2 = 0.007   F(3,94) =  0.029  p = 0.993


Model  R2      F(df)              p         R2 change  F(df) change       p
   1:  0.076   9.177(1,111)       0.003
   2:  0.194  12.030(2,100)       0.000     0.118     14.190(1,100)       0.000
   3:  0.201   4.718(5,94)        0.001     0.007      0.029(3,94)        0.993
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