This page shows an example of a multinomial logistic regression analysis with footnotes explaining the output. The data were collected on 200 high school students and are scores on various tests, including a video game and a puzzle. The outcome measure in this analysis is the preferred flavor of ice cream – vanilla, chocolate or strawberry- from which we are going to see what relationships exists with video game scores (video), puzzle scores (puzzle) and gender (female). Our response variable, ice_cream, is going to be treated as categorical under the assumption that the levels of ice_cream have no natural ordering, and we are going to allow Stata to choose the referent group. In out example, this will be vanilla. By default, Stata chooses the most frequently occurring group to be the referent group. The first half of this page interprets the coefficients in terms of multinomial log-odds (logits). These will be close to but not equal to the log-odds achieved in a logistic regression with two levels of the outcome variable. The second half interprets the coefficients in terms of relative risk ratios.
use https://stats.idre.ucla.edu/stat/stata/output/mlogit, clear
Before running the regression, obtaining a frequency of the ice cream flavors in the data can inform the selection of a reference group.
tab ice_cream
favorite flavor of ice cream
| Freq. Percent Cum.
------------+-----------------------------------
chocolate | 47 23.50 23.50
vanilla | 95 47.50 71.00
strawberry | 58 29.00 100.00
------------+-----------------------------------
Total | 200 100.00
Vanilla is the most frequently occurring ice cream flavor and will be the reference group in this example.
mlogit ice_cream video puzzle female
Iteration 0: log likelihood = -210.58254
Iteration 1: log likelihood = -194.75041
Iteration 2: log likelihood = -194.03782
Iteration 3: log likelihood = -194.03485
Iteration 4: log likelihood = -194.03485
Multinomial logistic regression Number of obs = 200
LR chi2(6) = 33.10
Prob > chi2 = 0.0000
Log likelihood = -194.03485 Pseudo R2 = 0.0786
------------------------------------------------------------------------------
ice_cream | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
chocolate |
video | -.0235647 .0209747 -1.12 0.261 -.0646744 .017545
puzzle | -.0389243 .0195165 -1.99 0.046 -.0771759 -.0006726
female | .8166202 .3909813 2.09 0.037 .050311 1.582929
_cons | 1.912256 1.127256 1.70 0.090 -.2971258 4.121638
-------------+----------------------------------------------------------------
strawberry |
video | .022922 .0208718 1.10 0.272 -.0179861 .0638301
puzzle | .0430036 .0198894 2.16 0.031 .0040211 .081986
female | -.032862 .3500153 -0.09 0.925 -.7188793 .6531553
_cons | -4.057323 1.222939 -3.32 0.001 -6.45424 -1.660407
------------------------------------------------------------------------------
(ice_cream==vanilla is the base outcome)
Iteration Loga
Iteration 0: log likelihood = -210.58254 Iteration 1: log likelihood = -194.75041 Iteration 2: log likelihood = -194.03782 Iteration 3: log likelihood = -194.03485 Iteration 4: log likelihood = -194.03485
a. Iteration Log – This is a listing of the log likelihoods at each iteration. Remember that multinomial logistic regression, like binary and ordered logistic regression, uses maximum likelihood estimation, 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 stops, and the results are displayed. For more information on this process for binary outcomes, see Regression Models for Categorical and Limited Dependent Variables by J. Scott Long (page 52-61).
Model Summary
Multinomial logistic regression Number of obsc = 200
LR chi2(6)d = 33.10
Prob > chi2e = 0.0000
Log likelihood = -194.03485b Pseudo R2f = 0.0786
b. Log Likelihood – This is the log likelihood of the fitted model. It is used in the Likelihood Ratio Chi-Square test of whether all predictors’ regression coefficients in the model are simultaneously zero and in tests of nested models.
c. Number of obs – This is the number of observations used in the multinomial logistic regression. It may be less than the number of cases in the dataset if there are missing values for some variables in the equation. By default, Stata does a listwise deletion of incomplete cases.
d. LR chi2(6) – This is the Likelihood Ratio (LR) Chi-Square test that for both equations (chocolate relative to vanilla and strawberry relative to vanilla) that at least one of the predictors’ regression coefficient is not equal to zero. The number in the parentheses indicates the degrees of freedom of the Chi-Square distribution used to test the LR Chi-Square statistic and is defined by the number of models estimated (2) times the number of predictors in the model (3). The LR Chi-Square statistic can be calculated by -2*( L(null model) – L(fitted model)) = -2*((-210.583) – (-194.035)) = 33.096, where L(null model) is from the log likelihood with just the response variable in the model (Iteration 0) and L(fitted model) is the log likelihood from the final iteration (assuming the model converged) with all the parameters.
e. Prob > chi2 – This is the probability of getting a LR test statistic as extreme as, or more so, than the observed statistic under the null hypothesis; the null hypothesis is that all of the regression coefficients across both models are simultaneously equal to zero. In other words, this is the probability of obtaining this chi-square statistic (33.10) or one more extreme if there is in fact no effect of the predictor variables. This p-value is compared to a specified alpha level, our willingness to accept a type I error, which is typically set at 0.05 or 0.01. The small p-value from the LR test, <0.00001, would lead us to conclude that at least one of the regression coefficients in the model is not equal to zero. The parameter of the chi-square distribution used to test the null hypothesis is defined by the degrees of freedom in the prior line, chi2(6).
f. Pseudo R2 – This is McFadden’s 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 of the response variable explained by the predictors), we suggest interpreting this statistic with great caution.
Parameter Estimates
------------------------------------------------------------------------------
ice_creamg | Coef.h Std. Err.j zk P>|z|k [95% Conf. Interval]l
-------------+----------------------------------------------------------------
chocolate |
video | -.0235647 .0209747 -1.12 0.261 -.0646744 .017545
puzzle | -.0389243 .0195165 -1.99 0.046 -.0771759 -.0006726
female | .8166202 .3909813 2.09 0.037 .050311 1.582929
_cons | 1.912256 1.127256 1.70 0.090 -.2971258 4.121638
-------------+----------------------------------------------------------------
strawberry |
video | .022922 .0208718 1.10 0.272 -.0179861 .0638301
puzzle | .0430036 .0198894 2.16 0.031 .0040211 .081986
female | -.032862 .3500153 -0.09 0.925 -.7188793 .6531553
_cons | -4.057323 1.222939 -3.32 0.001 -6.45424 -1.660407
------------------------------------------------------------------------------
(ice_cream==vanilla is the base outcome)i
g. ice_cream – This is the response variable in the multinomial logistic regression. Underneath ice_cream are two replicates of the predictor variables, representing the two models that are estimated: chocolate relative to vanilla and strawberry relative to vanilla.
h and i. Coef. and referent group – These are the estimated multinomial logistic regression coefficients and the referent level, respectively, for the model. An important feature of the multinomial logit model is that it estimates k-1 models, where k is the number of levels of the outcome variable. In this instance, Stata, by default, set vanilla as the referent group, and therefore estimated a model for chocolate relative to vanilla and a model for strawberry relative to vanilla. Since the parameter estimates are relative to the referent group, the standard interpretation of the multinomial logit is that for a unit change in the predictor variable, the logit of outcome m relative to the referent group is expected to change by its respective parameter estimate (which is in log-odds units) given the variables in the model are held constant.
chocolate relative to vanilla
video – This is the multinomial logit estimate for a one unit increase in video score for chocolate relative to vanilla, given the other variables in the model are held constant. If a subject were to increase his video score by one point, the multinomial log-odds for preferring chocolate to vanilla would be expected to decrease by 0.024 unit while holding all other variables in the model constant.
puzzle – This is the multinomial logit estimate for a one unit increase in puzzle score for chocolate relative to vanilla, given the other variables in the model are held constant. If a subject were to increase his puzzle score by one point, the multinomial log-odds for preferring chocolate to vanilla would be expected to decrease by 0.039 unit while holding all other variables in the model constant.
female – This is the multinomial logit estimate comparing females to males for chocolate relative to vanilla, given the other variables in the model are held constant. The multinomial logit for females relative to males is 0.817 unit higher for preferring chocolate to vanilla, given all other predictor variables in the model are held constant. In other words, females are more likely than males to prefer chocolate to vanilla.
_cons – This is the multinomial logit estimate for chocolate relative to vanilla when the predictor variables in the model are evaluated at zero. For males (the variable female evaluated at zero) with zero video and puzzle scores, the logit for preferring chocolate to vanilla is 1.912. Note that evaluating video and puzzle at zero is out of the range of plausible scores. If the scores were mean-centered, the intercept would have a natural interpretation: log odds of preferring chocolate to vanilla for a male with average video and puzzle scores.
strawberry relative to vanilla
video – This is the multinomial logit estimate for a one unit increase in video score for strawberry relative to vanilla, given the other variables in the model are held constant. If a subject were to increase his video score by one point, the multinomial log-odds for preferring strawberry to vanilla would be expected to increase by 0.023 unit while holding all other variables in the model constant.
puzzle – This is the multinomial logit estimate for a one unit increase in puzzle score for strawberry relative to vanilla, given the other variables in the model are held constant. If a subject were to increase his puzzle score by one point, the multinomial log-odds for preferring strawberry to vanilla would be expected to increase by 0.043 unit while holding all other variables in the model constant.
female – This is the multinomial logit estimate comparing females to males for strawberry relative to vanilla, given the other variables in the model are held constant. The multinomial logit for females relative to males is 0.033 unit lower for preferring strawberry to vanilla, given all other predictor variables in the model are held constant. In other words, males are more likely than females to prefer strawberry ice cream to vanilla ice cream.
_cons – This is the multinomial logit estimate for strawberry relative to vanilla when the predictor variables in the model are evaluated at zero. For males (the variable female evaluated at zero) with zero video and puzzle scores, the logit for preferring strawberry to vanilla is -4.057.
j. Std. Err. – These are the standard errors of the individual regression coefficients for the two respective models estimated. They are used in both the calculation of the z test statistic, superscript k, and the confidence interval of the regression coefficient, superscript l.
k. z and P>|z| – The test statistic z is the ratio of the Coef. to the Std. Err. of the respective predictor, and the p-value P>|z| is the probability the z test statistic (or a more extreme test statistic) would be observed under the null hypothesis. For a given alpha level, z and P>|z| determine whether or not the null hypothesis that a particular predictor’s regression coefficient is zero, given that the rest of the predictors are in the model, can be rejected. If P>|z| is less than alpha, then the null hypothesis can be rejected and the parameter estimate is considered significant at that alpha level. The z value follows a standard normal distribution which is used to test against a two-sided alternative hypothesis that the Coef. is not equal to zero. In multinomial logistic regression, the interpretation of a parameter estimate’s significance is limited to the model in which the parameter estimate was calculated. For example, the significance of a parameter estimate in the chocolate relative to vanilla model cannot be assumed to hold in the strawberry relative to vanilla model.
chocolate relative to vanilla
For chocolate relative to vanilla, the z test statistic for the
predictor video (-0.024/0.021) is -1.12 with an associated p-value of
0.261. If we set our alpha level to 0.05, we would fail to reject the null
hypothesis and conclude that for chocolate relative to vanilla, the regression
coefficient for video has not been found to be statistically different
from zero given puzzle and female are in the model.
For chocolate relative to vanilla, the z test statistic for
the predictor puzzle (-0.039/0.020) is -1.99 with an associated p-value
of 0.046. If we again set our alpha level to 0.05, we would reject the null
hypothesis and conclude that the regression coefficient for puzzle has
been found to be statistically different from zero for chocolate relative
to vanilla given that video and female are in the model.
For chocolate relative to vanilla, the z test statistic for
the predictor female (0.817/0.391) is 2.09 with an associated p-value of
0.037. If we again set our alpha level to 0.05, we would reject the null
hypothesis and conclude that the difference between males and females has been
found to be statistically different for chocolate relative to vanilla given that
video and female are in the model.
For chocolate relative to vanilla, the z test statistic for
the intercept, _cons (1.912/1.127) is 1.70 with an associated p-value of
0.090. With an alpha level of 0.05, we would fail to reject the null hypothesis
and conclude that a) the multinomial logit for males (the variable female
evaluated at zero) and with zero video and puzzle scores in
chocolate relative to vanilla are found not to be statistically different
from zero; or b) for males with zero video and puzzle scores, you
are statistically uncertain whether they are more likely to be classified as
preferring chocolate or vanilla. We can make the second interpretation
when we view the _cons as a specific covariate profile (males with zero
video and puzzle scores). Based on the direction and significance
of the coefficient, the _cons indicates whether the profile would have a
greater propensity to be classified in one level of the outcome variable than
the other level.
strawberry relative to vanilla
For strawberry relative to vanilla, the z test statistic
for the predictor video (0.023/0.021) is 1.10 with an associated p-value
of 0.272. If we set our alpha level to 0.05, we would fail to reject the null
hypothesis and conclude that for strawberry relative to vanilla, the regression
coefficient for video has not been found to be statistically different
from zero given puzzle and female are in the model.
For strawberry relative to vanilla, the z test statistic for the
predictor puzzle (0.043/0.020) is 2.16 with an associated p-value of
0.031. If we again set our alpha level to 0.05, we would reject the null
hypothesis and conclude that the regression coefficient for puzzle has
been found to be statistically different from zero for strawberry
relative to vanilla given that video and female are in the model.
For strawberry relative to vanilla, the z test statistic for the
predictor female (-0.033/0.350) is -0.09 with an associated p-value of
0.925. If we again set our alpha level to 0.05, we would fail to reject the null
hypothesis and conclude that for strawberry relative to vanilla, the regression
coefficient for female has not been found to be statistically different
from zero given puzzle and video are in the model.
For strawberry relative to vanilla, the z test statistic for the
intercept, _cons (-4.057/1.223) is -3.32 with an associated p-value of
0.001. With an alpha level of 0.05, we would reject the null hypothesis and
conclude that a) the multinomial logit for males (the variable female
evaluated at zero) and with zero video and puzzle scores in
strawberry relative to vanilla are statistically different from zero; or b) for
males with zero video and puzzle scores, there is a statistically
significant difference between the likelihood of being classified as preferring
strawberry or preferring vanilla. Such a male would be more likely to be
classified as preferring vanilla to strawberry. We can make the second
interpretation when we view the _cons as a specific covariate profile
(males with zero video and puzzle scores). Based on the direction
and significance of the coefficient, the _cons indicates whether the
profile would have a greater propensity to be classified in one level of the
outcome variable than the other level.
l. [95% Conf. Interval] – This is the Confidence Interval (CI) for an individual multinomial logit regression coefficient given the other predictors are in the model for outcome m relative to the referent group. For a given predictor with a level of 95% confidence, we’d say that we are 95% confident that the "true" population multinomial logit regression coefficient lies between the lower and upper limit of the interval for outcome m relative to the referent group. It is calculated as the Coef. (zα/2)*(Std.Err.), where zα/2 is a critical value on the standard normal distribution. The CI is equivalent to the z test statistic: if the CI includes zero, we’d fail to reject the null hypothesis that a particular regression coefficient is zero given the other predictors are in the model. An advantage of a CI is that it is illustrative; it provides a range where the "true" parameter may lie.
Relative Risk Ratio Interpretation
The following is the interpretation of the multinomial logistic regression in terms of relative risk ratios and can be obtained by mlogit, rrr after running the multinomial logit model or by specifying the rrr option when the full model is specified. This part of the interpretation applies to the output below.
mlogit ice_cream video puzzle female, rrr
Iteration 0: log likelihood = -210.58254
Iteration 1: log likelihood = -194.75041
Iteration 2: log likelihood = -194.03782
Iteration 3: log likelihood = -194.03485
Iteration 4: log likelihood = -194.03485
Multinomial logistic regression Number of obs = 200
LR chi2(6) = 33.10
Prob > chi2 = 0.0000
Log likelihood = -194.03485 Pseudo R2 = 0.0786
------------------------------------------------------------------------------
ice_cream | RRRa Std. Err. z P>|z| [95% Conf. Interval]b
-------------+----------------------------------------------------------------
chocolate |
video | .9767108 .0204862 -1.12 0.261 .9373726 1.0177
puzzle | .9618236 .0187714 -1.99 0.046 .925727 .9993276
female | 2.262839 .8847276 2.09 0.037 1.051598 4.869199
-------------+----------------------------------------------------------------
strawberry |
video | 1.023187 .0213558 1.10 0.272 .9821747 1.065911
puzzle | 1.043942 .0207633 2.16 0.031 1.004029 1.085441
female | .9676721 .3387 -0.09 0.925 .4872981 1.921595
------------------------------------------------------------------------------
(ice_cream==vanilla is the base outcome)
a. Relative Risk Ratio – These are the relative risk ratios for the multinomial logit model shown earlier. They can be obtained by exponentiating the multinomial logit coefficients, ecoef, or by specifying the rrr option when the mlogit command is issued. Recall that the multinomial logit model estimates k-1 models, where the kth equation is relative to the referent group. The RRR of a coefficient indicates how the risk of the outcome falling in the comparison group compared to the risk of the outcome falling in the referent group changes with the variable in question. An RRR > 1 indicates that the risk of the outcome falling in the comparison group relative to the risk of the outcome falling in the referent group increases as the variable increases. In other words, the comparison outcome is more likely. An RRR < 1 indicates that the risk of the outcome falling in the comparison group relative to the risk of the outcome falling in the referent group decreases as the variable increases. See the interpretations of the relative risk ratios below for examples. In general, if the RRR < 1, the outcome is more likely to be in the referent group.
chocolate relative to vanilla
video – This is the relative risk ratio for a one unit increase in video score for preferring chocolate to vanilla, given that the other variables in the model are held constant. If a subject were to increase her video score by one unit, the relative risk for preferring chocolate to vanilla would be expected to decrease by a factor of 0.977 given the other variables in the model are held constant. So, given a one unit increase in video, the relative risk of being in the chocolate group would be 0.977 times more likely when the other variables in the model are held constant. More generally, we can say that if a subject were to increase her video score, we would expect her to be more likely to prefer vanilla ice cream over chocolate ice cream.
puzzle – This is the relative risk ratio for a one unit increase in puzzle score for preferring chocolate to vanilla, given that the other variables in the model are held constant. If a subject were to increase her puzzle score by one unit, the relative risk for preferring chocolate to vanilla would be expected to decrease by a factor of 0.962 given the other variables in the model are held constant. More generally, we can say that if two subjects have identical video scores and are both female (or both male), the subject with the higher puzzle score is more likely to prefer vanilla ice cream over chocolate ice cream than the subject with the lower puzzle score.
female – This is the relative risk ratio comparing females to males for preferring chocolate to vanilla, given that the other variables in the model are held constant. For females relative to males, the relative risk for preferring chocolate relative to vanilla would be expected to increase by a factor of 2.263 given the other variables in the model are held constant. In other words, females are more likely than males to prefer chocolate ice cream over vanilla ice cream.
strawberry relative to vanilla
video – This is the relative risk ratio for a one unit increase in video score for preferring strawberry to vanilla, given that the other variables in the model are held constant. If a subject were to increase her video score by one unit, the relative risk for strawberry relative to vanilla would be expected to increase by a factor of 1.023 given the other variables in the model are held constant. More generally, we can say that if a subject were to increase her video score, we would expect her to be more likely to prefer strawberry ice cream over vanilla ice cream.
puzzle – This is the relative risk ratio for a one unit increase in puzzle score for preferring strawberry to vanilla, given that the other variables in the model are held constant. If a subject were to increase her puzzle score by one unit, the relative risk for strawberry relative to vanilla would be expected to increase by a factor of 1.043 given the other variables in the model are held constant. More generally, we can say that if two subjects have identical video scores and are both female (or both male), the subject with the higher puzzle score is more likely to prefer strawberry ice cream to vanilla ice cream than the subject with the lower puzzle score.
female – This is the relative risk ratio comparing females to males for strawberry relative to vanilla, given that the other variables in the model are held constant. For females relative to males, the relative risk for preferring strawberry to vanilla would be expected to decrease by a factor of 0.968 given the other variables in the model are held constant. In other words, females are less likely than males to prefer strawberry ice cream to vanilla ice cream.
b. [95% Conf. Interval] – This is the CI for the relative risk ratio given the other predictors are in the model. For a given predictor with a level of 95% confidence, we’d say that we are 95% confident that the "true" population relative risk ratio comparing outcome m to the referent group lies between the lower and upper limit of the interval. An advantage of a CI is that it is illustrative; it provides a range where the "true" relative risk ratio may lie.