This page shows an example of logistic regression regression analysis with footnotes explaining the output. These data were collected on 200 high schools students and are scores on various tests, including science, math, reading and social studies (socst). The variable female is a dichotomous variable coded 1 if the student was female and 0 if male.
Because we do not have a suitable dichotomous variable to use as our dependent variable, we will create one (which we will call honcomp, for honors composition) based on the continuous variable write. We do not advocate making dichotomous variables out of continuous variables; rather, we do this here only for purposes of this illustration.
use http://www.ats.ucla.edu/stat/data/hsb2, clear generate honcomp = (write >=60) logit honcomp female read science
Iteration 0: log likelihood = -115.64441
Iteration 1: log likelihood = -84.558481
Iteration 2: log likelihood = -80.491449
Iteration 3: log likelihood = -80.123052
Iteration 4: log likelihood = -80.118181
Iteration 5: log likelihood = -80.11818
Logit estimates Number of obs = 200
LR chi2(3) = 71.05
Prob > chi2 = 0.0000
Log likelihood = -80.11818 Pseudo R2 = 0.3072
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honcomp | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-----------+-------------------------------------------------------------
female | 1.482498 .4473993 3.31 0.001 .6056111 2.359384
read | .1035361 .0257662 4.02 0.000 .0530354 .1540369
science | .0947902 .0304537 3.11 0.002 .035102 .1544784
_cons | -12.7772 1.97586 -6.47 0.000 -16.64982 -8.904589
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Iteration Log
Iteration 0: log likelihood = -115.64441 Iteration 1: log likelihood = -84.558481 Iteration 2: log likelihood = -80.491449 Iteration 3: log likelihood = -80.123052 Iteration 4: log likelihood = -80.118181 Iteration 5:a log likelihood = -80.11818
a. This is a listing of the log likelihoods at each iteration. (Remember that logistic regression uses maximum likelihood, which is an iterative procedure.) The first iteration (called iteration 0) is the log likelihood of the "null" or "empty" model; that is, a model with no predictors. At the next iteration, the predictor(s) are included in the model. At each iteration, the log likelihood increases because the goal is to maximize the log likelihood. When the difference between successive iterations is very small, the model is said to have "converged", the iterating is stopped and the results are displayed. For more information on this process, see Regression Models for Categorical and Limited Dependent Variables by J. Scott Long.
Model Summary
Logit estimates Number of obsc = 200
LR chi2(3)d = 71.05
Prob > chi2e = 0.0000
Log likelihood = -80.11818b Pseudo R2f = 0.3072
b. Log likelihood - This is the log likelihood of the final model. The value -80.11818 has no meaning in and of itself; rather, this number can be used to help compare nested models.
c. Number of obs - This is the number of observations that were used in the analysis. This number may be smaller than the total number of observations in your data set if you have missing values for any of the variables used in the logistic regression. Stata uses a listwise deletion by default, which means that if there is a missing value for any variable in the logistic regression, the entire case will be excluded from the analysis.
d. LR chi2(3) - This is the likelihood ratio (LR) chi-square test. The likelihood chi-square test statistic can be calculated by hand as 2*(115.64441 - 80.11818) = 71.05. This is minus two (i.e., -2) times the difference between the starting and ending log likelihood. The number in the parenthesis indicates the number of degrees of freedom. In this model, there are three predictors, so there are three degrees of freedom.
e. Prob > chi2 - This is the probability of obtaining the chi-square statistic given that the null hypothesis is true. In other words, this is the probability of obtaining this chi-square statistic (71.05) if there is in fact no effect of the independent variables, taken together, on the dependent variable. This is, of course, the p-value, which is compared to a critical value, perhaps .05 or .01 to determine if the overall model is statistically significant. In this case, the model is statistically significant because the p-value is less than .000.
f. Pseudo R2 - This is the pseudo R-squared. Logistic regression does not have an equivalent to the R-squared that is found in OLS regression; however, many people have tried to come up with one. There are a wide variety of pseudo-R-square statistics. Because this statistic does not mean what R-square means in OLS regression (the proportion of variance explained by the predictors), we suggest interpreting this statistic with great caution.
Parameter Estimates
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honcompg| Coef.h Std. Err.i zj P>|z|j [95% Conf. Interval]k
-----------+-------------------------------------------------------------
female | 1.482498 .4473993 3.31 0.001 .6056111 2.359384
read | .1035361 .0257662 4.02 0.000 .0530354 .1540369
science | .0947902 .0304537 3.11 0.002 .035102 .1544784
_cons | -12.7772 1.97586 -6.47 0.000 -16.64982 -8.90459
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g. honcomp - This is the dependent variable in our logistic regression. The variables listed below it are the independent variables.
h. Coef. - These are the values for the logistic regression equation for predicting the dependent variable from the independent variable. They are in log-odds units. Similar to OLS regression, the prediction equation is
log(p/1-p) = b0 + b1*female + b2*read + b3*science
where p is the probability of being in honors composition. Expressed in terms of the variables used in this example, the logistic regression equation is
log(p/1-p) = -12.7772 + 1.482498*female + .1035361*read + 0947902*science
These estimates tell you about the relationship between the independent variables and the dependent variable, where the dependent variable is on the logit scale. These estimates tell the amount of increase in the predicted log odds of honcomp = 1 that would be predicted by a 1 unit increase in the predictor, holding all other predictors constant. Note: For the independent variables which are not significant, the coefficients are not significantly different from 0, which should be taken into account when interpreting the coefficients. (See the columns with the z-values and p-values regarding testing whether the coefficients are statistically significant). Because these coefficients are in log-odds units, they are often difficult to interpret, so they are often converted into odds ratios. You can do this by hand by exponentiating the coefficient, or by using the or option with logit command, or by using the logistic command.
female - The coefficient (or parameter estimate) for the variable female is 1.482498. This means that for a one-unit increase in female (in other words, going from male to female), we expect a 1.482498 increase in the log-odds of the dependent variable honcomp, holding all other independent variables constant.
read - For every one-unit increase in reading score (so, for every additional point on the reading test), we expect a .1035361 increase in the log-odds ofhoncomp, holding all other independent variables constant.
science - For every one-unit increase in science score, we expect a .0947902 increase in the log-odds of honcomp, holding all other independent variables constant.
constant - This is the expected value of the log-odds of honcomp when all of the predictor variables equal zero. In most cases, this is not interesting. Also, oftentimes zero is not a realistic value for a variable to take.
read - For every one-unit increase in reading score (so, for every additional point on the reading test), we expect a .1035361 increase in the log-odds ofhoncomp, holding all other independent variables constant.
science - For every one-unit increase in science score, we expect a .0947902 increase in the log-odds of honcomp, holding all other independent variables constant.
constant - This is the expected value of the log-odds of honcomp when all of the predictor variables equal zero. In most cases, this is not interesting. Also, oftentimes zero is not a realistic value for a variable to take.
i. Std. Err. - These are the standard errors associated with the coefficients. The standard error is used for testing whether the parameter is significantly different from 0; by dividing the parameter estimate by the standard error you obtain a z-value (see the column with z-values and p-values). The standard errors can also be used to form a confidence interval for the parameter, as shown in the last two columns of this table.
j. z and P>|z| - These columns provide the z-value and 2-tailed p-value used in testing the null hypothesis that the coefficient (parameter) is 0. If you use a 2-tailed test, then you would compare each p-value to your preselected value of alpha. Coefficients having p-values less than alpha are statistically significant. For example, if you chose alpha to be 0.05, coefficients having a p-value of 0.05 or less would be statistically significant (i.e., you can reject the null hypothesis and say that the coefficient is significantly different from 0). If you use a 1-tailed test (i.e., you predict that the parameter will go in a particular direction), then you can divide the p-value by 2 before comparing it to your preselected alpha level. With a 2-tailed test and alpha of 0.05, you may reject the null hypothesis that the coefficient for female is equal to 0. The coefficient of 1.482498 is significantly greater than 0.
The coefficient for read is .1035361 significantly different from 0 using alpha of 0.05 because its p-value is 0.000, which is smaller than 0.05.
The coefficient for science is .0947902 significantly different from 0 using alpha of 0.05 because its p-value is 0.000, which is smaller than 0.05.
The coefficient for read is .1035361 significantly different from 0 using alpha of 0.05 because its p-value is 0.000, which is smaller than 0.05.
The coefficient for science is .0947902 significantly different from 0 using alpha of 0.05 because its p-value is 0.000, which is smaller than 0.05.
k. [95% Conf. Interval] - This shows a 95% confidence interval for the coefficient. This is very useful as it helps you understand how high and how low the actual population value of the parameter might be. The confidence intervals are related to the p-values such that the coefficient will not be statistically significant if the confidence interval includes 0.
Odds Ratios
In this next example, we will illustrate the interpretation of odds ratios. We will use the logistic command so that we see the odds ratios instead of the coefficients. In this example, we will simplify our model so that we have only one predictor, the binary variable female. Before we run the logistic regression, we will use the tab command to obtain a crosstab of the two variables.
tab female honcomp
| honcomp
female | 0 1 | Total
-----------+----------------------+----------
male | 73 18 | 91
female | 74 35 | 109
-----------+----------------------+----------
Total | 147 53 | 200
If we divide the number of males who are in honors composition, 18, by the number of males who are not in honors composition, 73, we get the odds of being in honors composition for males, 18/73 = .24657534. If we do the same thing for females, we get 35/74 = .47297297. To get the odds ratio, which is the ratio of the two odds that we have just calculated, we get .47297297/.24657534 = 1.9181682. As we can see in the output below, this is exactly the odds ratio we obtain from the logistic command. The thing to remember here is that you want the group coded as 1 over the group coded as 0, so honcomp=1/honcomp=0 for both males and females, and then the odds for females/odds for males, because the females are coded as 1.
With regard to the 95% confidence interval, we do not want this to include the value of 1. When we were considering the coefficients, we did not want the confidence interval to include 0. If we exponentiate 0, we get 1 (exp(0) = 1). Hence, this is two ways of saying the same thing. As you can see, the 95% confidence interval includes 1; hence, the odds ratio is not statistically significant. Because the lower bound of the 95% confidence interval is so close to 1, the p-value is very close to .05.
There are a few other things to note about the output below. The first is that although we have only one predictor variable, the test for the odds ratio does not match with the overall test of the model. This is because the z statistic is actually the result of a Wald chi-square test, while the test of the overall model is a likelihood ratio chi-square. While these two types of chi-square tests are asymptotically equivalent, in small samples they can differ, as they do here. Also, we have the unfortunate situation in which the results of the two tests give different conclusions. This does not happen very often. In a situation like this, it is difficult to know what to conclude. One might consider the power, or one might decide if an odds ratio of this magnitude is important from a clinical or practical standpoint.
logistic honcomp female
Logistic regression Number of obs = 200
LR chi2(1) = 3.94
Prob > chi2 = 0.0473
Log likelihood = -113.6769 Pseudo R2 = 0.0170
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honcomp | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval]
-----------+-------------------------------------------------------------
female | 1.918168 .6400451 1.95 0.051 .9973827 3.689024
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