Numeric versus Categorical Attributes in Choice Models

The difference between a numeric and categorical price attribute

The chart below illustrates the implications of treating price as being categorical versus numeric. When price is treated as a numeric attribute, the model assumes that there is a linear relationship between price and utility, as shown by the orange line. When price is modeled as being categorical, non-linear relationships can be found. In this example, which is from the home delivery market, the categorical price attribute leads to the conclusion the drop in utility that goes with changing price from $15 to $20 is particularly large, being much larger than the drop from $10 to $15 and even from $20 to $30. By contrast, when a numeric price attribute is used, the model assumes that the effect is constant across all price points; each extra dollar of price leads to a constant drop-off in utility.

How to interpret numeric price coefficients

The output below shows the estimated distribution of a numeric attribute relating to the price of home-delivered food. It would be very easy to look at this and conclude that, relative to Cuisine, Price per person is both not very important and that there is very little variation in terms of its importance in the population. However, this is not the right way to read this output.

To illustrate this point, the output below shows a model that I have re-estimated, but rather than using prices of 10, 12, 15, 20, and 30, I've divided them by 10 and have used prices of 1, 1.2, 1.5, 2. and 3. Price appears to be much more important below, but in reality, it is equally important in both models. The thing to keep in mind is that the values shown for Cuisine are utilities, whereas the distribution shown for Price per person is instead a coefficient. For categorical data the ideas of a utility and a coefficient are interchangeable, but with numeric attributes, they are not. In order to compute utility, we need to multiply the coefficient of the numeric attribute by the values. In the output above, it shows that the coefficient for Price per person is 0.2; this is rounded, and with an extra decimal the value is -0.17. The utility for $10 is then 10*-.17 = -1.7 and the utility for $30 is 30*-.17 = -5.1.

Choosing an appropriate coding for numeric variables

As seen in the previous section, we have various options when choosing how to code a numeric variable. For example, we can encode prices of $10, $12, $15, $20, and $30 as:

• 10, 12, 15, 20, and 30.
• 1, 1.2, 1.5, 2. and 3.
• 1, 2, 3, 4, and 5.

There is no single correct answer. However:

• It is usually advisable to use linear coding if possible (i.e., the first two, not the third coding). That is, a coding where you multiply the values by some consistent number (e.g., 1, 10, 0.1) to get the coding.  Where the categories are ordered, and there is no way to create a linear coding, using consecutive integers (e.g., 1, 2, 3, 4, and 5) has been found to work well.
• Ideally, the values will be in the range of -5 to 5. This is because this will mean they are in a similar range to the other coefficients that are estimated, and this will mean there is less of the algorithm getting confused (some algorithms implicitly assume that all coefficients are on a similar scale).
•  It's advisable to use a scale that makes it easy for you to see the distribution in the outputs from the model (e.g., as above) so that you can gain insights and spot problems.

The benefit of treating price as categorical

The key benefit of treating price as being categorical is that it is more consistent with what we know about consumer behavior. Study after study has found interesting non-linear relationships between price and consumer preferences, and in marketing, it is the norm to view these as indicating interesting "psychological" pricing findings. For example, in most markets, there are believed to be price thresholds (e.g., keep a phone under $1,000), and various other interesting pricing points (e.g., prices ending in 99). Identifying such psychological price points and using them as a basis for setting price seems like good strategy.

In marketing and market research, it is the norm to treat price as being a categorical attribute in choice-based conjoint analysis. 

The benefits of treating price as numeric

Treating price as being numeric has quite a few advantages:

• Consistency with economic theory
• More parsimonious models
• Draw conclusions about price points not tested in the research
• Fit models in situations where it is impractical to estimate the utility of separate price points
• Simplicity of analysis
• Compute average willingness-to-pay

Consistency with economic theory

Economic theory suggests that the relationship between price and utility should be linear. To use the jargon: the price coefficient is seen as being the marginal utility of income. Although it is routine for studies to find statistical evidence that the price is nonlinear, it is possible that psychological price points are just research artifacts. Perhaps surveys show non-linearity, but in the real world with real money people behave more rationally.

More parsimonious models

A basic principle of modeling is that the fewer parameters (e.g., coefficients) that are estimated, all else being equal, the better. When we use numeric attributes we make models more parsimonious. This benefit is most pronounced when fitting models with small or noisy data sets.

Draw conclusions about price points not tested in the research

When we treat price as being a numeric attribute, it allows us to use interpolation and extrapolation to make more precise conclusions about price. In the example above, where we have treated price as being categorical, we can only safely draw conclusions at the specific price points tested in the research. In this case, these are $10, $12, $15, $20, and $30 (i.e., the end-points and the places where the lines join). By contrast, with the numeric variable we can price at any point along the line, and if we are brave can extrapolate beyond the line. 

Fit models in situations where it is impractical to estimate the utility of separate price points

Another benefit of treating price as being numeric is that it allows you to use choice modeling with data that has too many different price points to make it practical to estimate a separate utility for each price point. Such data is widespread in economics and transportation research, where the prices that are shown to each respondent are customized to their specific circumstances (e.g., if an attribute is the price of traveling by car, each person's fuel costs and depreciation will be slightly different).

Simplicity of analysis

There is a further benefit of treating price as being numeric: it makes it a lot simpler to make price-related conclusions from the research.

Compute average willingness-to-pay

When we have a price coefficient, we can easily scale all other utilities by dividing by this coefficient. For example, where the coefficient for price is 0.17, using the data shown above, we end up computing the average mean utility for the different cuisines as follows:

• Chicken: $0
• Chinese: $1.18
• Hamburgers: $0.59
• Indian:-$20.00
• Italian: $1.18
• Mexican: $-$1.18
• Pizza:$7.06
• Thai: -$13.53

These are variously known as dollar-metric utilities and as willingness-to-pay (WTP). Comparing, for example, Chinese with Hamburgers, Chinese has a $0.59 higher WTP ($1.18 - $0.59), and an interpretation of this is that, on average, people are prepared to pay $0.59 more for a Chinese meal than a hamburger meal.