# st: Sparse Data Problem

 From john metcalfe 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:

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
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

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
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

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
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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