Logistic Regression

Version info: Code for this page was tested in Stata 12.

Logistic regression, also called a logit model, is used to model dichotomous
outcome variables. In the logit model the log odds of the outcome is modeled as a linear
combination of the predictor variables.

Please note: The purpose of this page is to show how to use various data analysis commands.
It does not cover all aspects of the research process which researchers are expected to do. In
particular, it does not cover data cleaning and checking, verification of assumptions, model
diagnostics and potential follow-up analyses.

Examples of logistic regression

Example 1:  Suppose that we are interested in the factors

that influence whether a political candidate wins an election.  The

outcome (response) variable is binary (0/1);  win or lose.

The predictor variables of interest are the amount of money spent on the campaign, the

amount of time spent campaigning negatively and whether or not the candidate is an

incumbent.

Example 2:  A researcher is interested in how variables, such as GRE (Graduate Record Exam scores),

GPA (grade

point average) and prestige of the undergraduate institution, effect admission into graduate

school. The response variable, admit/don’t admit, is a binary variable.

Description of the data

For our data analysis below, we are going to expand on Example 2 about getting
into graduate school.  We have generated hypothetical data, which can be
obtained from our website.

use https://stats.idre.ucla.edu/stat/stata/dae/binary.dta, clear

This data set has a binary response (outcome, dependent) variable called admit.

There are three predictor

variables:  gre, gpa and rank. We will treat the
variables gre and gpa as continuous. The variable rank takes on the
values 1 through 4. Institutions with a rank of 1 have the highest prestige,
while those with a rank of 4 have the lowest.

summarize gre gpa

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
         gre |       400       587.7    115.5165        220        800
         gpa |       400      3.3899    .3805668       2.26          4

tab rank 

       rank |      Freq.     Percent        Cum.
------------+-----------------------------------
          1 |         61       15.25       15.25
          2 |        151       37.75       53.00
          3 |        121       30.25       83.25
          4 |         67       16.75      100.00
------------+-----------------------------------
      Total |        400      100.00

tab admit

      admit |      Freq.     Percent        Cum.
------------+-----------------------------------
          0 |        273       68.25       68.25
          1 |        127       31.75      100.00
------------+-----------------------------------
      Total |        400      100.00

tab admit rank

           |                    rank
     admit |         1          2          3          4 |     Total
-----------+--------------------------------------------+----------
         0 |        28         97         93         55 |       273 
         1 |        33         54         28         12 |       127 
-----------+--------------------------------------------+----------
     Total |        61        151        121         67 |       400

Analysis methods you might consider

Below is a list of some analysis methods you may have encountered.
Some of the methods listed are quite reasonable while others have either
fallen out of favor or have limitations.

Logistic regression

Below we use the logit command to estimate a logistic regression
model. The i. before rank indicates that rank is a factor
variable (i.e.,
categorical variable), and that it should be included in the model as a series
of indicator variables. Note that this syntax was introduced in Stata 11.

logit admit gre gpa i.rank 

Iteration 0:   log likelihood = -249.98826  
Iteration 1:   log likelihood = -229.66446  
Iteration 2:   log likelihood = -229.25955  
Iteration 3:   log likelihood = -229.25875  
Iteration 4:   log likelihood = -229.25875  

Logistic regression                               Number of obs   =        400
                                                  LR chi2(5)      =      41.46
                                                  Prob > chi2     =     0.0000
Log likelihood = -229.25875                       Pseudo R2       =     0.0829

------------------------------------------------------------------------------
       admit |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         gre |   .0022644    .001094     2.07   0.038     .0001202    .0044086
         gpa |   .8040377   .3318193     2.42   0.015     .1536838    1.454392
             |
        rank |
          2  |  -.6754429   .3164897    -2.13   0.033    -1.295751   -.0551346
          3  |  -1.340204   .3453064    -3.88   0.000    -2.016992   -.6634158
          4  |  -1.551464   .4178316    -3.71   0.000    -2.370399   -.7325287
             |
       _cons |  -3.989979   1.139951    -3.50   0.000    -6.224242   -1.755717
------------------------------------------------------------------------------

We can test for an overall effect of rank
using the test command. Below we see that the overall effect of rank is
statistically significant.

test 2.rank 3.rank 4.rank

 ( 1)  [admit]2.rank = 0
 ( 2)  [admit]3.rank = 0
 ( 3)  [admit]4.rank = 0

           chi2(  3) =   20.90
         Prob > chi2 =    0.0001

We can also test additional hypotheses about the differences in the
coefficients for different levels of rank. Below we
test that the coefficient for rank=2 is equal to the coefficient for rank=3.
(Note that if we wanted to estimate this difference, we could do so using the
lincom command.)

test 2.rank = 3.rank

 ( 1)  [admit]2.rank - [admit]3.rank = 0

           chi2(  1) =    5.51
         Prob > chi2 =    0.0190

You can also exponentiate

the coefficients and interpret them as odds-ratios. Stata will do this

computation for you

if you use the or option, illustrated below.  You could also use the
logistic command.

logit , or

Logistic regression                               Number of obs   =        400
                                                  LR chi2(5)      =      41.46
                                                  Prob > chi2     =     0.0000
Log likelihood = -229.25875                       Pseudo R2       =     0.0829

------------------------------------------------------------------------------
       admit | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         gre |   1.002267   .0010965     2.07   0.038      1.00012    1.004418
         gpa |   2.234545   .7414652     2.42   0.015     1.166122    4.281877
             |
        rank |
          2  |   .5089309   .1610714    -2.13   0.033     .2736922    .9463578
          3  |   .2617923   .0903986    -3.88   0.000     .1330551    .5150889
          4  |   .2119375   .0885542    -3.71   0.000     .0934435    .4806919
------------------------------------------------------------------------------

Now we can say that for a one unit increase in gpa, the odds of being
admitted to graduate school (versus not being admitted) increase by a factor of
2.23. For more information on interpreting odds ratios see our FAQ page
How do I interpret odds ratios in logistic regression?
.

You can also use predicted probabilities to help you understand the model.
You can calculate predicted probabilities using the margins command,
which was
introduced in Stata 11. Below we use the margins command to calculate the
predicted probability of admission at each level of rank, holding all
other variables in the model at their means. For more information on using the margins
command to calculate predicted probabilities, see our page
Using margins for predicted probabilities.

margins rank, atmeans

Adjusted predictions                              Number of obs   =        400
Model VCE    : OIM

Expression   : Pr(admit), predict()
at           : gre             =       587.7 (mean)
               gpa             =      3.3899 (mean)
               1.rank          =       .1525 (mean)
               2.rank          =       .3775 (mean)
               3.rank          =       .3025 (mean)
               4.rank          =       .1675 (mean)

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        rank |
          1  |   .5166016   .0663153     7.79   0.000     .3866261    .6465771
          2  |   .3522846   .0397848     8.85   0.000     .2743078    .4302614
          3  |    .218612   .0382506     5.72   0.000     .1436422    .2935819
          4  |   .1846684   .0486362     3.80   0.000     .0893432    .2799937
------------------------------------------------------------------------------

In the above output we see that the predicted probability of being accepted
into a graduate program is 0.51 for the highest prestige undergraduate
institutions (rank=1), and 0.18 for the lowest ranked institutions (rank=4),
holding gre and gpa at their means.

Below we generate the predicted probabilities for values of gre from
200 to 800 in increments of 100. Because we have not specified either atmeans
or used at(…) to specify values at with the other predictor
variables are held, the values in the table are average predicted probabilities
calculated using the sample values of the other
predictor variables. For example, to calculate the average predicted probability
when gre = 200, the predicted probability was calculated for each case,
using that case’s values of rank and gpa,
with gre set to 200.

margins , at(gre=(200(100)800))  vsquish

Predictive margins                                Number of obs   =        400
Model VCE    : OIM

Expression   : Pr(admit), predict()
1._at        : gre             =         200
2._at        : gre             =         300
3._at        : gre             =         400
4._at        : gre             =         500
5._at        : gre             =         600
6._at        : gre             =         700
7._at        : gre             =         800

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         _at |
          1  |   .1667471   .0604432     2.76   0.006     .0482807    .2852135
          2  |    .198515   .0528947     3.75   0.000     .0948434    .3021867
          3  |   .2343805   .0421354     5.56   0.000     .1517966    .3169643
          4  |   .2742515   .0296657     9.24   0.000     .2161078    .3323951
          5  |   .3178483    .022704    14.00   0.000     .2733493    .3623473
          6  |   .3646908   .0334029    10.92   0.000     .2992224    .4301592
          7  |   .4141038   .0549909     7.53   0.000     .3063237    .5218839
------------------------------------------------------------------------------

In the table above we can see that the mean predicted probability of being
accepted is only 0.167 if one’s GRE score is 200 and increases to 0.414 if one’s GRE score is 800 (averaging
across the sample values of gpa and rank).

It can also be helpful to use graphs of predicted probabilities to understand and/or present
the model.

We may also wish to see measures of how well our model fits. This can be particularly useful when comparing
competing models. The user-written command fitstat produces a
variety of fit statistics. You can find more information on fitstat by typing
search fitstat (see
How can I use the search command to search for programs and get additional help?
for more information about using search).

fitstat

Measures of Fit for logit of admit

Log-Lik Intercept Only:       -249.988   Log-Lik Full Model:           -229.259
D(393):                        458.517   LR(5):                          41.459
                                         Prob > LR:                       0.000
McFadden's R2:                   0.083   McFadden's Adj R2:               0.055
ML (Cox-Snell) R2:               0.098   Cragg-Uhler(Nagelkerke) R2:      0.138
McKelvey & Zavoina's R2:         0.142   Efron's R2:                      0.101
Variance of y*:                  3.834   Variance of error:               3.290
Count R2:                        0.710   Adj Count R2:                    0.087
AIC:                             1.181   AIC*n:                         472.517
BIC:                         -1896.128   BIC':                          -11.502
BIC used by Stata:             494.466   AIC used by Stata:             470.517

Things to consider

See also

References