Estimate a Multinomial logistic regression (MNL) for classification
To estimate a Multinomial logistic regression (MNL) we require a categorical response variable with two or more levels and one or more explanatory variables. We also need to specify the level of the response variable to be used as the base for comparison. In the example data file,
ketchup, we could assign
heinz28 as the base level by selecting it from the
Choose level dropdown in the Summary tab.
To access the
ketchup dataset go to Data > Manage, select
examples from the
Load data of type dropdown, and press the
Load button. Then select the
In the Summary tab we can test if two or more variables together improve the fit of a model by selecting them in the
Variables to test dropdown. This functionality can be very useful to evaluate the overall influence of a variable of type
factor with three or more levels.
Additional output that requires re-estimation:
Additional output that does not require re-estimation:
As an example we will use a dataset on choice behavior for 300 individuals in a panel of households in Springfield, Missouri (USA). The data captures information on 2,798 purchase occasions over a period of approximately 2 years and includes the follow variables:
The screenshot of the Data > Pivot tab shown below indicates that
heinz32 is the most popular choice option, followed by
hunts32 are much less common choices among the household panel members.
Suppose we want to investigate how prices of the different products influence the choice of ketchup brand and package size. In the Model > Multinomial logistic regression (MNL) > Summary tab select
choice as the response variable and
heinz28 from the Choose base level dropdown menu. Select
price.hunts32 as the explanatory variables. In the screenshot below we see that most, but not all, of the coefficients have very small p.values and that the model has some predictive power (p.value for the chi-squared statistic < .001). The left-most output column shows which product a coefficient applies to. For example, the 2nd row of coefficients and statistics captures the effect of changes in
price.heinz28 on the choice of
heinz32 relative to the base product (i.e.,
heinz28). If consumers see
heinz32 as substitutes, which seems likely, we would expect that an increase in
price.heinz28 would lead to an increase in the odds that a consumer chooses
heinz32 rather than
Unfortunately the coefficients from a multinomial logistic regression model are difficult to interpret directly. The
RRR column, however, provides estimates of Relative-Risk-Ratios (or odds) that are easier to work with. The
RRR values are the exponentiated coefficients from the regression (i.e., $exp(1.099) = 3.000). We see that the
risk (or odds) of buying
heinz28 compared to
heinz32 is 3 times higher after a $1 increase in
price.heinz28, keeping all other variables in the model constant.
For each of the explanatory variables the following null and alternate hypotheses can be formulated:
A selected set of relative risk ratios from the multinomial logistic regression can be interpreted as follows:
price.heinz32on the relative odds or purchasing
heinz28is 0.101. If the price for
heinz32increased by $1, the odds of purchasing
heinz28would decrease by a factor of 0.101, or decrease by 89.9%, while holding all other variables in the model constant.
price.heinz28on the relative odds or purchasing
heinz28is 3.602. If the price for
heinz28increased by $1, the odds of purchasing
heinz28would increase by a factor of 3.602, or increase by 260.2%, while holding all other variables in the model constant.
price.hunts32on the relative odds or purchasing
heinz28is 0.070. If the price for
hunts32increased by $1, the odds of purchasing
heinz28would decrease by a factor of 0.070, or decrease by 93%, while holding all other variables in the model constant.
RRRs estimated in the model can be interpreted similarly.
In addition to the numerical output provided in the Summary tab we can also evaluate the link between
choice and the prices of each of the four products visually (see Plot tab). In the screenshot below we see a coefficient (or rather an RRR) plot with confidence intervals. We see the following patterns:
price.heinz28increases by $1 the relative purchase odds for
price.heinz32increases, the odds of purchase for
heinz28decrease significantly. We see the same pattern for
hunts32when their prices increase
hunts32is the only product to see a significant improvement in purchase odds relative to
heinz28from an increase in
Probabilities, are often more convenient for interpretation than coefficients or RRRs from a multinomial logistic regression model. We can use the Predict tab to predict probabilities for each of the different response variable levels given specific values for the selected explanatory variable(s). First, select the type of input for prediction using the
Prediction input type dropdown. Choose either an existing dataset for prediction (“Data”) or specify a command (“Command”) to generate the prediction inputs. If you choose to enter a command, you must specify at least one variable and one value in the Prediction command box to get a prediction. If you do not specify a value for each of the variables in the model either the mean value or the most frequently observed level will be used. It is only possible to predict probabilities based on variables used in the model. For example,
price.heinz32 must be one of the selected explanatory variables to predict the probability of choosing to buy
heinz32 when priced at $3.80.
hunts32is available in stores type
disp.hunts32 = "yes"as the command and press enter
heinz41is (not) on display and (not) featured type
disp.heinz41 = c("yes", "no"), feat.heinz41 = c("yes", "no")and press enter
price.heinz28 = seq(3.40, 5.20, 0.1)and press enter. See screenshot below.
The figure above shows that the probability of purchase drops sharply for
heinz32, the most popular option in the data, is predicted to see a large increase in purchase probability following an increase in
price.heinz28. Although the predicted increase in purchase probability for
hunts32 does not look as impressive in the graph compared to the effect on
heinz32, the relative predicted increase is larger (i.e., 3.2% to 8.4% for
hunts32 versus 39.3% to 72.8% for
For a more comprehensive assessment of the impact of price changes for each of the four products on purchase probabilities we can generate a full table of predictions by selecting
Data from the
Prediction input type dropdown in the Predict tab and selecting
ketchup from the
Predict data dropdown. You can also create a dataset for input in Data > Transform using
Expand grid or in a spreadsheet and then paste it into Radiant using the Data > Manage tab.
Once the desired predictions have been generated they can be saved to a CSV file by clicking the download icon on the top right of the prediction table. To add predictions to the dataset used for estimation, click the
Note that MNL models generate as many columns of probabilities as there are levels in the categorical response variable (i.e., four in the ketchup data). If you want to add only the predictions for the first level (i.e.,
heinz28) to the dataset used for estimation, provide only one name in the
Store predictions input. If you want to store predictions for all ketchup products, enter four variable names, separated by a comma.
Note: We ignored endogeneity concerns in the above discussion. Suppose, for example, that
price.heinz28changes due to changes in the quality of
heinz28. Changes in quality effect the price and, likely, also demand for the product. Unless we control in some way for these changes in quality, the estimated effects of price changes are likely to be incorrect (i.e., biased).
Add code to Report > Rmd to (re)create the analysis by clicking the icon on the bottom left of your screen or by pressing
ALT-enter on your keyboard.
If a plot was created, it can be customized using
ggplot2 commands or with
gridExtra. See example below and Data > Visualize for details.
For an overview of related R-functions used by Radiant to estimate a multinomial logistic regression model see Model > Multinomial logistic regression.
The key functions used in the
mnl tool are
multinom from the
nnet package and
linearHypothesis from the