A digression: logit and probit choice models

The concepts behind logistic regression and the logit function have also been proposed as a tool to model brand choice in marketing applications. Since choice models are a good way to see integrated use of decision and statistical models, we outline the approach in this section. Consider an individual who chooses between two brands. Ideally, we could model her choice in terms of a multiattribute utility function u(x) depending on the features of each brand. If such features are collected into vectors x₁ and x₂, the individual would choose brand 1 if

Of course, we do not know the utility function of the individual; furthermore, there may be factors at play that are not included in the vector x; indeed, the choice may also depend on factors that are not really related to the product itself, but to price, type of display at a supermarket, etc. Last but not least, we are actually interested in modeling the choice of a “typical” individual, so that we observe different individuals, resulting in some randomness in the choice. A simple approach is to split utility into a term that can be related to product attributes and a term that is completely random:

where i = 1, 2 refers to brand and t to time or observation number. The simplest model for attributes is v_it = β^Tx_it, where x_it is the vector of attributes of brand i when offered at time t. The individual chooses brand 1 at time t if

or, in other words, when the random effect is not larger than the difference in utilities linked to attributes:

If we introduce the random variable , the probability of choosing brand 1 is given by its CDF

where we also assume that the distribution of η does not depend on time. Choice models differ in the assumed probability distribution for η.

• If a normal distribution is assumed, a probit model is obtained. A possible disadvantage of this choice is that, since the CDF of a normal variable is not available in closed form, we do not find an explicit functional form for the probability of choice.
• An alternative choice, which defines a logit model, is the logistic distribution, which is characterized by a CDF of the form (16.11). Taking the derivative of the logistic function, we see that its PDF is given by

This PDF, as shown in Fig. 16.2, has a shape similar to a normal distribution, but its tails are a bit different (and fatter).

Fig. 16.2 The PDF of the logistic distribution.

If we assume a logit choice model, the probability of choosing brand 1 over brand 2 at time t is given by

Probit and logit models can be extended to cope with the more complex setting of a choice between m > 2 alternatives. In such a case, the probability of choosing brand k is just the probability that the random variable U_kt is the largest in the set of variables U_jt, j = 1,…, m. Here we must specify a joint distribution of a vector  collecting the random terms associated with each possible choice. Different assumptions lead to different choice models, like multinomial probit and multinomial logit models. Each model has advantages and disadvantages in terms of readability and ease of estimation, whose discussion is beyond. One point worth emphasizing, from a managerial perspective, is the sensibility of model predictions. It turns out that the multinomial logit model implies a property called independence of irrelevant alternatives (IIA). In fact, using simplifying assumptions about the errors, the predicted probability of choosing brand k is given by

where v_j is a linear function of attributes. The denominator in this ratio is a normalizing factor, which does not influence the ratio of choice probabilities of two brands, say, k and l:

Fig. 16.3 Two cases requiring a nonlinear regression.

If we introduce another brand, this ratio will not change. In other words, the model implies that the introduction of a new alternative will diminish the probability of choosing the preexisting alternative proportionally. However, an often cited counterexample involves two soft-drinks, say a cola and a lemonade. Assume that the choice probabilities are the same, 50% and 50%. If we add a sugar-free cola, it is not quite sensible to assume that the two probabilities will change by the same amount; sales of cola are likely to be much more influenced by the introduction of a similar product. Alternative choice models have been proposed in order to overcome this limitation.