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STAT 350: Lecture 27

Variable Selection Methods

PROBLEM: Find a set of predictor variables which gives a good fit, predicts the dependent value well and is as small as possible.

Thus far we have used F and t tests to compare 2 models at a time. We have followed a sequence of tests to try to find a good set of variables but our method has been informal; other statisticians using the same method might select a different final model. Here we investigate 4 mechanical (more or less) variable selection methods: Forward, Backward, Stepwise and All Subsets.

FORWARD

Start with a model with no predictors.
Add variable with largest F-statistic (provided P less than some cut-off).
Refit with this variable. Recompute all F statistics for adding one of the remaining variables and add variable with largest F statistic.
Continue until no variable is significant at cut-off level.

BACKWARD

Start with model with all predictors.
Delete variable with smallest F-statistic (provided P more than some cut-off).
Refit with this variable deleted. Recompute all F statistics for deleting one of the remaining variables and delete variable with smallest F statistic.
Continue until every remaining variable is significant at cut-off level.

STEPWISE

Start with model with no predictors.
Add variable with largest F-statistic (provided P more than some cut-off).
Refit with this variable added. Recompute all F statistics for adding one of the remaining variables and delete variable with smallest F statistic.
At each step after adding a variable try to eliminate any variable not significant at some level (that is, do BACKWARD elimination till that stops).
After doing the backwards steps take another FORWARD step.
Continue until every remaining variable is significant at cut-off level and every excluded variable is insignificant OR until variable to be added is same as last deleted variable.

ALL SUBSETS

For each subset of the set of predictors fit the model and compute some summary statistic of the quality of the fit. Pick the model which makes this summary as large (or sometimes as small) as possible.
With k predictors fit models; impractical for k too large. Special Best subsets algorithms work without looking at all models.
Possible summary statistics:
- : but NOTE -- adding a variable increases so this is most useful for comparing models of the same size.
- Adjusted : This method adjusts to try to compensate for the fact that more variables produces larger even when the extra variables are irrelevant.
- : Like Adjusted but based on a trade off of bias and variance.
- PRESS: The sum of squares of the PRESS residuals (See Lecture 13 and the class notes for that lecture.)

Example

FORWARD Selection

data scenic;
 infile 'scenic.dat' firstobs=2;
 input Stay  Age Risk Culture Chest Beds 
    School Region Census Nurses Facil;
 Nratio = Nurses / Census  ;
proc reg  data=scenic;
  model Risk = Culture Stay Nurses Nratio 
     Chest Beds Census Facil / 
     selection=forward;
run ;

EDITED SAS OUTPUT (Complete output)

          Forward Selection Procedure for Dependent Variable RISK    
Step 1   Variable CULTURE Entered   R-square = 0.31265864   C(p) = 47.47794976
                 DF         Sum of Squares      Mean Square          F   Prob>F
 Regression       1            62.96314170      62.96314170      50.49   0.0001
 Error          111           138.41668131       1.24699713
 Total          112           201.37982301
                 Parameter        Standard          Type II
 Variable         Estimate           Error   Sum of Squares          F   Prob>F
 INTERCEP       3.19789965      0.19376813     339.64905575     272.37   0.0001
 CULTURE        0.07325862      0.01030975      62.96314170      50.49   0.0001
--------------------------------------------------------------------------------
Step 2   Variable STAY Entered      R-square = 0.45040256   C(p) = 18.11960703
                 DF         Sum of Squares      Mean Square          F   Prob>F
 Regression       2            90.70198757      45.35099379      45.07   0.0001
 Error          110           110.67783543       1.00616214
 Total          112           201.37982301
                 Parameter        Standard          Type II
 Variable         Estimate           Error   Sum of Squares          F   Prob>F
 INTERCEP       0.80549102      0.48775579       2.74400250       2.73   0.1015
 CULTURE        0.05645147      0.00979843      33.39687778      33.19   0.0001
 STAY           0.27547211      0.05246473      27.73884588      27.57   0.0001
--------------------------------------------------------------------------------
Step 3   Variable FACIL Entered     R-square = 0.49340010   C(p) = 10.33092385
                 DF         Sum of Squares      Mean Square          F   Prob>F
 Regression       3            99.36082444      33.12027481      35.39   0.0001
 Error          109           102.01899857       0.93595412
 Total          112           201.37982301
                 Parameter        Standard          Type II
 Variable         Estimate           Error   Sum of Squares          F   Prob>F
 INTERCEP       0.49133226      0.48163614       0.97401801       1.04   0.3099
 CULTURE        0.05419997      0.00947933      30.59827862      32.69   0.0001
 STAY           0.22390748      0.05336561      16.47664606      17.60   0.0001
 FACIL          0.01963027      0.00645392       8.65883687       9.25   0.0029
--------------------------------------------------------------------------------
Step 4   Variable NRATIO Entered    R-square = 0.52547952   C(p) =  5.02782551
                 DF         Sum of Squares      Mean Square          F   Prob>F
 Regression       4           105.82097194      26.45524298      29.90   0.0001
 Error          108            95.55885107       0.88480418
 Total          112           201.37982301
                 Parameter        Standard          Type II
 Variable         Estimate           Error   Sum of Squares          F   Prob>F
 INTERCEP      -0.49505513      0.59376426       0.61507231       0.70   0.4063
 CULTURE        0.04818092      0.00948204      22.84513509      25.82   0.0001
 STAY           0.26758404      0.05434637      21.44995791      24.24   0.0001
 NRATIO         0.79262357      0.29333869       6.46014750       7.30   0.0080
 FACIL          0.01747585      0.00632554       6.75349077       7.63   0.0067
--------------------------------------------------------------------------------
Step 5   Variable CHEST Entered     R-square = 0.53792463   C(p) =  4.19461013
                 DF         Sum of Squares      Mean Square          F   Prob>F
 Regression       5           108.32716704      21.66543341      24.91   0.0001
 Error          107            93.05265597       0.86965099
 Total          112           201.37982301
                 Parameter        Standard          Type II
 Variable         Estimate           Error   Sum of Squares          F   Prob>F
 INTERCEP      -0.76804342      0.61022741       1.37763165       1.58   0.2109
 CULTURE        0.04318856      0.00984976      16.71979631      19.23   0.0001
 STAY           0.23392650      0.05741114      14.43814950      16.60   0.0001
 NRATIO         0.67240318      0.29931440       4.38883521       5.05   0.0267
 CHEST          0.00917860      0.00540681       2.50619510       2.88   0.0925
 FACIL          0.01843860      0.00629673       7.45710068       8.57   0.0042
--------------------------------------------------------------------------------
Step 6   Variable CENSUS Entered    R-square = 0.54146833   C(p) =  5.38786192
                 DF         Sum of Squares      Mean Square          F   Prob>F
 Regression       6           109.04079737      18.17346623      20.86   0.0001
 Error          106            92.33902564       0.87112288
 Total          112           201.37982301
                 Parameter        Standard          Type II
 Variable         Estimate           Error   Sum of Squares          F   Prob>F
 INTERCEP      -0.60982957      0.63526660       0.80275743       0.92   0.3393
 CULTURE        0.04327612      0.00985857      16.78604192      19.27   0.0001
 STAY           0.21806849      0.06007156      11.47961755      13.18   0.0004
 NRATIO         0.74253899      0.30942748       5.01649265       5.76   0.0182
 CHEST          0.00967176      0.00543875       2.75481205       3.16   0.0782
 CENSUS         0.00092337      0.00102018       0.71363033       0.82   0.3675
 FACIL          0.01171496      0.00974167       1.25977812       1.45   0.2318
--------------------------------------------------------------------------------
No other variable met the 0.5000 significance level for entry into the model.
     Summary of Forward Selection Procedure for Dependent Variable RISK    
         Variable   Number   Partial     Model
 Step    Entered        In      R**2      R**2        C(p)           F   Prob>F
    1    CULTURE         1    0.3127    0.3127     47.4779     50.4918   0.0001
    2    STAY            2    0.1377    0.4504     18.1196     27.5690   0.0001
    3    FACIL           3    0.0430    0.4934     10.3309      9.2513   0.0029
    4    NRATIO          4    0.0321    0.5255      5.0278      7.3012   0.0080
    5    CHEST           5    0.0124    0.5379      4.1946      2.8818   0.0925
    6    CENSUS          6    0.0035    0.5415      5.3879      0.8192   0.3675

BACKWARD Selection

data scenic;
 infile 'scenic.dat' firstobs=2;
 input Stay  Age Risk Culture Chest Beds 
    School Region Census Nurses Facil;
 Nratio = Nurses / Census  ;
proc reg  data=scenic;
  model Risk = Culture Stay Nurses Nratio 
     Chest Beds Census Facil / 
     selection=backward;
run ;

EDITED SAS OUTPUT (Complete output)

         Backward Elimination Procedure for Dependent Variable RISK    
Step 0    All Variables Entered     R-square = 0.54317205   C(p) =  9.00000000
                 DF         Sum of Squares      Mean Square          F   Prob>F
 Regression       8           109.38389081      13.67298635      15.46   0.0001
 Error          104            91.99593220       0.88457627
 Total          112           201.37982301
                 Parameter        Standard          Type II
 Variable         Estimate           Error   Sum of Squares          F   Prob>F
 INTERCEP      -0.61544324      0.66644759       0.75436112       0.85   0.3579
 CULTURE        0.04410334      0.01004539      17.05081019      19.28   0.0001
 STAY           0.20541554      0.06405141       9.09797087      10.29   0.0018
 NURSES        -0.00087592      0.00216138       0.14527948       0.16   0.6861
 NRATIO         0.85012850      0.39334446       4.13198139       4.67   0.0330
 CHEST          0.00946882      0.00549665       2.62501031       2.97   0.0879
 BEDS          -0.00106503      0.00265225       0.14263677       0.16   0.6888
 CENSUS         0.00295314      0.00357645       0.60311345       0.68   0.4109
 FACIL          0.01312502      0.01010817       1.49138778       1.69   0.1970
--------------------------------------------------------------------------------
Step 1   Variable BEDS Removed      R-square = 0.54246375   C(p) =  7.16124870
                 DF         Sum of Squares      Mean Square          F   Prob>F
 Regression       7           109.24125404      15.60589343      17.78   0.0001
 Error          105            92.13856897       0.87751018
 Total          112           201.37982301
                 Parameter        Standard          Type II
 Variable         Estimate           Error   Sum of Squares          F   Prob>F
 INTERCEP      -0.66993458      0.64987468       0.93251908       1.06   0.3050
 CULTURE        0.04396605      0.00999939      16.96447009      19.33   0.0001
 STAY           0.21222554      0.06151830      10.44332482      11.90   0.0008
 NURSES        -0.00101551      0.00212470       0.20045667       0.23   0.6337
 NRATIO         0.85643956      0.39145742       4.20026358       4.79   0.0309
 CHEST          0.00947189      0.00547465       2.62672060       2.99   0.0865
 CENSUS         0.00178564      0.00207440       0.65020801       0.74   0.3913
 FACIL          0.01228514      0.00984983       1.36506916       1.56   0.2151
--------------------------------------------------------------------------------
Step 2   Variable NURSES Removed    R-square = 0.54146833   C(p) =  5.38786192
                 DF         Sum of Squares      Mean Square          F   Prob>F
 Regression       6           109.04079737      18.17346623      20.86   0.0001
 Error          106            92.33902564       0.87112288
 Total          112           201.37982301
                 Parameter        Standard          Type II
 Variable         Estimate           Error   Sum of Squares          F   Prob>F
 INTERCEP      -0.60982957      0.63526660       0.80275743       0.92   0.3393
 CULTURE        0.04327612      0.00985857      16.78604192      19.27   0.0001
 STAY           0.21806849      0.06007156      11.47961755      13.18   0.0004
 NRATIO         0.74253899      0.30942748       5.01649265       5.76   0.0182
 CHEST          0.00967176      0.00543875       2.75481205       3.16   0.0782
 CENSUS         0.00092337      0.00102018       0.71363033       0.82   0.3675
 FACIL          0.01171496      0.00974167       1.25977812       1.45   0.2318
--------------------------------------------------------------------------------
Step 3   Variable CENSUS Removed    R-square = 0.53792463   C(p) =  4.19461013
                 DF         Sum of Squares      Mean Square          F   Prob>F
 Regression       5           108.32716704      21.66543341      24.91   0.0001
 Error          107            93.05265597       0.86965099
 Total          112           201.37982301
                 Parameter        Standard          Type II
 Variable         Estimate           Error   Sum of Squares          F   Prob>F
 INTERCEP      -0.76804342      0.61022741       1.37763165       1.58   0.2109
 CULTURE        0.04318856      0.00984976      16.71979631      19.23   0.0001
 STAY           0.23392650      0.05741114      14.43814950      16.60   0.0001
 NRATIO         0.67240318      0.29931440       4.38883521       5.05   0.0267
 CHEST          0.00917860      0.00540681       2.50619510       2.88   0.0925
 FACIL          0.01843860      0.00629673       7.45710068       8.57   0.0042
--------------------------------------------------------------------------------
All variables left in the model are significant at the 0.1000 level.
   Summary of Backward Elimination Procedure for Dependent Variable RISK    
         Variable   Number   Partial     Model
 Step    Removed        In      R**2      R**2        C(p)           F   Prob>F
    1    BEDS            7    0.0007    0.5425      7.1612      0.1612   0.6888
    2    NURSES          6    0.0010    0.5415      5.3879      0.2284   0.6337
    3    CENSUS          5    0.0035    0.5379      4.1946      0.8192   0.3675

STEPWISE Selection

data scenic;
 infile 'scenic.dat' firstobs=2;
 input Stay  Age Risk Culture Chest Beds 
     School Region Census Nurses Facil;
 Nratio = Nurses/Census;
proc reg  data=scenic;
  model Risk = Culture Stay Nurses Nratio 
     Chest Beds Census Facil / 
     selection=stepwise sle=0.20 sls=0.05;
run ;

EDITED SAS OUTPUT (Complete output)

               Stepwise Procedure for Dependent Variable RISK    
Step 1   Variable CULTURE Entered   R-square = 0.31265864   C(p) = 47.47794976
                 DF         Sum of Squares      Mean Square          F   Prob>F
 Regression       1            62.96314170      62.96314170      50.49   0.0001
 Error          111           138.41668131       1.24699713
 Total          112           201.37982301
                 Parameter        Standard          Type II
 Variable         Estimate           Error   Sum of Squares          F   Prob>F
 INTERCEP       3.19789965      0.19376813     339.64905575     272.37   0.0001
 CULTURE        0.07325862      0.01030975      62.96314170      50.49   0.0001
--------------------------------------------------------------------------------
Step 2   Variable STAY Entered      R-square = 0.45040256   C(p) = 18.11960703
                 DF         Sum of Squares      Mean Square          F   Prob>F
 Regression       2            90.70198757      45.35099379      45.07   0.0001
 Error          110           110.67783543       1.00616214
 Total          112           201.37982301
                 Parameter        Standard          Type II
 Variable         Estimate           Error   Sum of Squares          F   Prob>F
 INTERCEP       0.80549102      0.48775579       2.74400250       2.73   0.1015
 CULTURE        0.05645147      0.00979843      33.39687778      33.19   0.0001
 STAY           0.27547211      0.05246473      27.73884588      27.57   0.0001
--------------------------------------------------------------------------------
Step 3   Variable FACIL Entered     R-square = 0.49340010   C(p) = 10.33092385
                 DF         Sum of Squares      Mean Square          F   Prob>F
 Regression       3            99.36082444      33.12027481      35.39   0.0001
 Error          109           102.01899857       0.93595412
 Total          112           201.37982301
                 Parameter        Standard          Type II
 Variable         Estimate           Error   Sum of Squares          F   Prob>F
 INTERCEP       0.49133226      0.48163614       0.97401801       1.04   0.3099
 CULTURE        0.05419997      0.00947933      30.59827862      32.69   0.0001
 STAY           0.22390748      0.05336561      16.47664606      17.60   0.0001
 FACIL          0.01963027      0.00645392       8.65883687       9.25   0.0029
--------------------------------------------------------------------------------
Step 4   Variable NRATIO Entered    R-square = 0.52547952   C(p) =  5.02782551
                 DF         Sum of Squares      Mean Square          F   Prob>F
 Regression       4           105.82097194      26.45524298      29.90   0.0001
 Error          108            95.55885107       0.88480418
 Total          112           201.37982301
                 Parameter        Standard          Type II
 Variable         Estimate           Error   Sum of Squares          F   Prob>F
 INTERCEP      -0.49505513      0.59376426       0.61507231       0.70   0.4063
 CULTURE        0.04818092      0.00948204      22.84513509      25.82   0.0001
 STAY           0.26758404      0.05434637      21.44995791      24.24   0.0001
 NRATIO         0.79262357      0.29333869       6.46014750       7.30   0.0080
 FACIL          0.01747585      0.00632554       6.75349077       7.63   0.0067
--------------------------------------------------------------------------------
Step 5   Variable CHEST Entered     R-square = 0.53792463   C(p) =  4.19461013
                 DF         Sum of Squares      Mean Square          F   Prob>F
 Regression       5           108.32716704      21.66543341      24.91   0.0001
 Error          107            93.05265597       0.86965099
 Total          112           201.37982301
                 Parameter        Standard          Type II
 Variable         Estimate           Error   Sum of Squares          F   Prob>F
 INTERCEP      -0.76804342      0.61022741       1.37763165       1.58   0.2109
 CULTURE        0.04318856      0.00984976      16.71979631      19.23   0.0001
 STAY           0.23392650      0.05741114      14.43814950      16.60   0.0001
 NRATIO         0.67240318      0.29931440       4.38883521       5.05   0.0267
 CHEST          0.00917860      0.00540681       2.50619510       2.88   0.0925
 FACIL          0.01843860      0.00629673       7.45710068       8.57   0.0042
--------------------------------------------------------------------------------
Step 6   Variable CHEST Removed     R-square = 0.52547952   C(p) =  5.02782551
                 DF         Sum of Squares      Mean Square          F   Prob>F
 Regression       4           105.82097194      26.45524298      29.90   0.0001
 Error          108            95.55885107       0.88480418
 Total          112           201.37982301
                 Parameter        Standard          Type II
 Variable         Estimate           Error   Sum of Squares          F   Prob>F
 INTERCEP      -0.49505513      0.59376426       0.61507231       0.70   0.4063
 CULTURE        0.04818092      0.00948204      22.84513509      25.82   0.0001
 STAY           0.26758404      0.05434637      21.44995791      24.24   0.0001
 NRATIO         0.79262357      0.29333869       6.46014750       7.30   0.0080
 FACIL          0.01747585      0.00632554       6.75349077       7.63   0.0067
--------------------------------------------------------------------------------
All variables left in the model are significant at the 0.0500 level.
The stepwise method terminated because the next variable to be entered was just 
removed.
         Summary of Stepwise Procedure for Dependent Variable RISK    
        Variable        Number   Partial    Model
 Step   Entered Removed     In      R**2     R**2      C(p)          F   Prob>F
    1   CULTURE              1    0.3127   0.3127   47.4779    50.4918   0.0001
    2   STAY                 2    0.1377   0.4504   18.1196    27.5690   0.0001
    3   FACIL                3    0.0430   0.4934   10.3309     9.2513   0.0029
    4   NRATIO               4    0.0321   0.5255    5.0278     7.3012   0.0080
    5   CHEST                5    0.0124   0.5379    4.1946     2.8818   0.0925
    6           CHEST        4    0.0124   0.5255    5.0278     2.8818   0.0925

Comments on code and results

Notice the option selection on the model lines.
Forward adds variables until the smallest P-value is more than 0.5.
Backward removes variables until all remaining are significant at 0.1 level.
Final models for backward, forward and stepwise are virtually the same here: variables retained are Culture, Stay, Nratio, Chest and Facil. Exception is tthat forward retains Census at P=0.36.
Significance levels to add or delete variables are controlled by sle= andsls= options.

ALL SUBSETS

data scenic;
 infile 'scenic.dat' firstobs=2;
 input Stay  Age Risk Culture Chest Beds 
     School Region Census Nurses Facil;
 Nratio = Nurses / Census  ;
proc reg  data=scenic;
  model Risk = Culture Stay Nurses Nratio  
   Chest Beds Census Facil / selection=cp ;
run ;

EDITED SAS OUTPUT (Complete output)

N = 113     Regression Models for Dependent Variable: RISK    
      C(p)    R-square      Variables in Model
                        In   
   4.19461  0.53792463   5  CULTURE STAY NRATIO CHEST FACIL 
   4.81202  0.53521260   5  CULTURE STAY NRATIO CHEST CENSUS 
   5.02783  0.52547952   4  CULTURE STAY NRATIO FACIL 
   5.33543  0.53291351   5  CULTURE STAY NRATIO CHEST BEDS 
   5.38786  0.54146833   6  CULTURE STAY NRATIO CHEST CENSUS FACIL 
   5.69350  0.54012581   6  CULTURE STAY NRATIO CHEST BEDS FACIL 
   5.89630  0.53923499   6  CULTURE STAY NURSES NRATIO CHEST FACIL 
   6.00546  0.52118519   4  CULTURE STAY NRATIO CENSUS 
   6.23202  0.52897517   5  CULTURE STAY NURSES NRATIO CHEST 
   6.47628  0.51911707   4  CULTURE STAY NRATIO BEDS 
   6.50213  0.52778865   5  CULTURE STAY NRATIO CENSUS FACIL 
   6.70444  0.53568517   6  CULTURE STAY NURSES NRATIO CHEST CENSUS 
   6.73959  0.52674562   5  CULTURE STAY NRATIO BEDS FACIL 
   6.77459  0.53537702   6  CULTURE STAY NRATIO CHEST BEDS CENSUS 
   6.91746  0.52596429   5  CULTURE STAY NURSES NRATIO FACIL

  81.27048  0.17300751   2  BEDS FACIL 
  83.31964  0.15522130   1  NURSES 
  83.60929  0.17151925   3  NURSES BEDS CENSUS 
  84.59092  0.15842223   2  NURSES CENSUS 
  85.31844  0.15522654   2  NURSES BEDS 
  85.53858  0.14547441   1  CENSUS 
  86.28567  0.15097790   2  BEDS CENSUS 
  89.19019  0.12943445   1  BEDS 
 111.09898  0.03319840   1  NRATIO

Comments on code and results

Every one of the models was tried.
Good possible models have small and not too far from p because when the model in question is correct.

First listed model has

a bit over 4; this is ok since only values larger than p can indicate a bias (resulting from a missing variable in the model. This method selects CULTURE, STAY, NRATIO, CHEST and FACIL as did BACKWARD and STEPWISE. (FORWARD also included CENSUS with a quite large P-value.

INCLUDING CATEGORICAL COVARIATES

Conceptually it is easy to do variable selection in the same way when some of the variables are categorical. In SAS you have to use proc reg and that procedure has no facility for categorical variables. You create columns of the design matrix yourself and group together the columns which correspond to the categorical variable as follows:

options pagesize=60 linesize=80;
data scenic;
 infile 'scenic.dat' firstobs=2;
 input Stay  Age Risk Culture Chest Beds School 
      Region Census Nurses Facil;
 Nratio = Nurses / Census  ;
 R1 = -(Region-4)*(Region-3)*(Region-2)/6;
 R2 = (Region-4)*(Region-3)*(Region-1)/2;
 R3 = -(Region-4)*(Region-2)*(Region-1)/2;
 S1 = School-1;
proc reg  data=scenic;
  model Risk = S1 Culture Stay Nurses Nratio { R1 R2 R3 }
  Chest Beds Census Facil / selection=stepwise 
  groupnames = 'School' 'Culture' 'Stay' 'Nurses' 'Nratio' 
  'Region' 'Chest' 'Beds' 'Census' 'Facil';
run ;

COMMENTS ON CODE

Variable R1 is 1 for cases in Region 1 and 0 for other cases. R2 is 1 for Region 2, R3 is 1 for Region 3. These 3 columns are the columns for the factor REGION using the corner point coding , in the notation of previous lectures.
S1 is 1 for hospitals not attached to medical schools.
Variables R1, R2 and R3 are grouped together by braces so that the selection method must put them all in or all out.
The data step can be used more simply to compute R1, R2 and R3. See the HELP facility in SAS.
groupnames names groups of variables so that, e.g., R1, R2 and R3 have a name, Region.

EDITED SAS OUTPUT (Complete output)

               Stepwise Procedure for Dependent Variable RISK    
Step 1   Group Culture  Entered     R-square = 0.31265864   C(p) = 58.36413224
                 Parameter        Standard          Type II
 Variable         Estimate           Error   Sum of Squares          F   Prob>F
 INTERCEP       3.19789965      0.19376813     339.64905575     272.37   0.0001
 --- Group Culture  ---                         62.96314170      50.49   0.0001
 CULTURE        0.07325862      0.01030975      62.96314170      50.49   0.0001
--------------------------------------------------------------------------------
Step 2   Group Stay     Entered     R-square = 0.45040256   C(p) = 26.82418731
                 Parameter        Standard          Type II
 Variable         Estimate           Error   Sum of Squares          F   Prob>F
 INTERCEP       0.80549102      0.48775579       2.74400250       2.73   0.1015
 --- Group Culture  ---                         33.39687778      33.19   0.0001
 CULTURE        0.05645147      0.00979843      33.39687778      33.19   0.0001
 --- Group Stay     ---                         27.73884588      27.57   0.0001
 STAY           0.27547211      0.05246473      27.73884588      27.57   0.0001
--------------------------------------------------------------------------------
Step 3   Group Facil    Entered     R-square = 0.49340010   C(p) = 18.35450472
                 Parameter        Standard          Type II
 Variable         Estimate           Error   Sum of Squares          F   Prob>F
 INTERCEP       0.49133226      0.48163614       0.97401801       1.04   0.3099
 --- Group Culture  ---                         30.59827862      32.69   0.0001
 CULTURE        0.05419997      0.00947933      30.59827862      32.69   0.0001
 --- Group Stay     ---                         16.47664606      17.60   0.0001
 STAY           0.22390748      0.05336561      16.47664606      17.60   0.0001
 --- Group Facil    ---                          8.65883687       9.25   0.0029
 FACIL          0.01963027      0.00645392       8.65883687       9.25   0.0029
--------------------------------------------------------------------------------
Step 4   Group Nratio   Entered     R-square = 0.52547952   C(p) = 12.54332929
                 Parameter        Standard          Type II
 Variable         Estimate           Error   Sum of Squares          F   Prob>F
 INTERCEP      -0.49505513      0.59376426       0.61507231       0.70   0.4063
 --- Group Culture  ---                         22.84513509      25.82   0.0001
 CULTURE        0.04818092      0.00948204      22.84513509      25.82   0.0001
 --- Group Stay     ---                         21.44995791      24.24   0.0001
 STAY           0.26758404      0.05434637      21.44995791      24.24   0.0001
 --- Group Nratio   ---                          6.46014750       7.30   0.0080
 NRATIO         0.79262357      0.29333869       6.46014750       7.30   0.0080
 --- Group Facil    ---                          6.75349077       7.63   0.0067
 FACIL          0.01747585      0.00632554       6.75349077       7.63   0.0067
--------------------------------------------------------------------------------
Step 5   Group Chest    Entered     R-square = 0.53792463   C(p) = 11.51300690
                 Parameter        Standard          Type II
 Variable         Estimate           Error   Sum of Squares          F   Prob>F
 INTERCEP      -0.76804342      0.61022741       1.37763165       1.58   0.2109
 --- Group Culture  ---                         16.71979631      19.23   0.0001
 CULTURE        0.04318856      0.00984976      16.71979631      19.23   0.0001
 --- Group Stay     ---                         14.43814950      16.60   0.0001
 STAY           0.23392650      0.05741114      14.43814950      16.60   0.0001
 --- Group Nratio   ---                          4.38883521       5.05   0.0267
 NRATIO         0.67240318      0.29931440       4.38883521       5.05   0.0267
 --- Group Chest    ---                          2.50619510       2.88   0.0925
 CHEST          0.00917860      0.00540681       2.50619510       2.88   0.0925
 --- Group Facil    ---                          7.45710068       8.57   0.0042
 FACIL          0.01843860      0.00629673       7.45710068       8.57   0.0042
--------------------------------------------------------------------------------
Step 6   Group Region   Entered     R-square = 0.56825843   C(p) = 10.12688089
                 Parameter        Standard          Type II
 Variable         Estimate           Error   Sum of Squares          F   Prob>F
 INTERCEP      -0.66156855      0.68931767       0.77004723       0.92   0.3394
 --- Group Culture  ---                         19.41848300      23.23   0.0001
 CULTURE        0.04717749      0.00978882      19.41848300      23.23   0.0001
 --- Group Stay     ---                         18.64724032      22.31   0.0001
 STAY           0.28408192      0.06015054      18.64724032      22.31   0.0001
 --- Group Nratio   ---                          1.86769604       2.23   0.1380
 NRATIO         0.47735146      0.31936579       1.86769604       2.23   0.1380
 --- Group Region   ---                          6.10861501       2.44   0.0689
 R1            -0.91152625      0.33831556       6.06877293       7.26   0.0082
 R2            -0.61170886      0.30630883       3.33408744       3.99   0.0484
 R3            -0.54005754      0.30531855       2.61565335       3.13   0.0799
 --- Group Chest    ---                          3.10587423       3.72   0.0566
 CHEST          0.01029102      0.00533912       3.10587423       3.72   0.0566
 --- Group Facil    ---                          7.66252029       9.17   0.0031
 FACIL          0.01883340      0.00622080       7.66252029       9.17   0.0031
--------------------------------------------------------------------------------
Step 7   Group School   Entered     R-square = 0.57830628   C(p) =  9.68027972
                 Parameter        Standard          Type II
 Variable         Estimate           Error   Sum of Squares          F   Prob>F
 INTERCEP      -1.29313397      0.79443852       2.18445103       2.65   0.1066
 --- Group School   ---                          2.02343484       2.45   0.1203
 S1             0.45874175      0.29282732       2.02343484       2.45   0.1203
 --- Group Culture  ---                         21.14238169      25.64   0.0001
 CULTURE        0.05016596      0.00990650      21.14238169      25.64   0.0001
 --- Group Stay     ---                         19.90843811      24.15   0.0001
 STAY           0.29583936      0.06020399      19.90843811      24.15   0.0001
 --- Group Nratio   ---                          1.42881407       1.73   0.1909
 NRATIO         0.42026288      0.31924279       1.42881407       1.73   0.1909
 --- Group Region   ---                          7.09035688       2.87   0.0402
 R1            -0.99737538      0.34041455       7.07745167       8.58   0.0042
 R2            -0.64425716      0.30489819       3.68115979       4.46   0.0370
 R3            -0.59950685      0.30557155       3.17349874       3.85   0.0525
 --- Group Chest    ---                          2.85453005       3.46   0.0656
 CHEST          0.00987802      0.00530873       2.85453005       3.46   0.0656
 --- Group Facil    ---                          9.68526975      11.75   0.0009
 FACIL          0.02391008      0.00697611       9.68526975      11.75   0.0009
--------------------------------------------------------------------------------
Step 8   Group Nratio   Removed     R-square = 0.57121116   C(p) =  9.40790549
                 Parameter        Standard          Type II
 Variable         Estimate           Error   Sum of Squares          F   Prob>F
 INTERCEP      -0.83240584      0.71570292       1.12313185       1.35   0.2475
 --- Group School   ---                          2.46231681       2.97   0.0880
 S1             0.50274483      0.29193670       2.46231681       2.97   0.0880
 --- Group Culture  ---                         23.66688888      28.50   0.0001
 CULTURE        0.05233635      0.00980270      23.66688888      28.50   0.0001
 --- Group Stay     ---                         18.47964968      22.26   0.0001
 STAY           0.27469386      0.05822575      18.47964968      22.26   0.0001
 --- Group Region   ---                          9.68716458       3.89   0.0111
 R1            -1.10696516      0.33123989       9.27275385      11.17   0.0012
 R2            -0.76673818      0.29137725       5.74922078       6.92   0.0098
 R3            -0.75936643      0.28139304       6.04647398       7.28   0.0081
 --- Group Chest    ---                          3.92124933       4.72   0.0320
 CHEST          0.01132621      0.00521177       3.92124933       4.72   0.0320
 --- Group Facil    ---                         11.30278424      13.61   0.0004
 FACIL          0.02545939      0.00690031      11.30278424      13.61   0.0004
--------------------------------------------------------------------------------
All groups of variables left in the model are significant at the 0.1500 level.
No other group of variables met the 0.1500 significance level for entry into 
the model.
         Summary of Stepwise Procedure for Dependent Variable RISK    
        Group           Number   Partial    Model
 Step   Entered Removed     In      R**2     R**2      C(p)          F   Prob>F
    1   Culture              1    0.3127   0.3127   58.3641    50.4918   0.0000
    2   Stay                 2    0.1377   0.4504   26.8242    27.5690   0.0000
    3   Facil                3    0.0430   0.4934   18.3545     9.2513   0.0029
    4   Nratio               4    0.0321   0.5255   12.5433     7.3012   0.0080
    5   Chest                5    0.0124   0.5379   11.5130     2.8818   0.0925
    6   Region               8    0.0303   0.5683   10.1269     2.4357   0.0689
    7   School               9    0.0100   0.5783    9.6803     2.4542   0.1203
    8           Nratio       8    0.0071   0.5712    9.4079     1.7330   0.1909

COMMENTS ON OUTPUT

Final model selected has variables SCHOOL, CULTURE, STAY, REGION, CHEST and FACIL.
Variable NRATIO included at step 4 was eliminated at step 8.
It is possible to name the groups so that the printout is more easily read.

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Richard Lockhart
Mon Mar 10 11:03:54 PST 1997