# Non-categorical choice: multinomial logit or something different?

My case is quite simple.

Customers are faced with series of choices of unique alternatives. For example:

• Customer 1 chooses one of 1,2,3,4,5,6
• Customer 1 chooses one of 7,8,9
• Customer 2 chooses one of 10,11,12,13
• Customer 2 chooses one of 14,15,16 etc

All the alternatives 1,2,... are unique and can't be naturally categorized. Every alternative has the same set of attributes. I'm interested in effects of alternatives' attributes on customer's choice.

For example, a person travels through a country and chooses a hotel to stay. Every day she is in a new city and the hotels to choose are completely new. The hotels differ by price and rating. One can expect that price has negative and rating has positive effect on customer's choice.

It seems that multinomial logit is not applicable here as it needs a categorical dependent variable.

My understanding is that there should be a simple statistical method for the problem because mathematics behind it seems to be quite clear. Indeed, assume the alternatives have 2 continuous attributes.

Let $U= \alpha X+\beta Y$ be the utility function of the attributes $X$ and $Y$. Every observation can be represented by several points on the $(X,Y)$-plane. Assume there are four alternatives $A, B, C, D$ and $A$ is chosen. For every vector with tail at $A$ calculate the fraction of points in the left half-plane given by this vecor. Thus one gets a circle partitioned into the union of arcs and corresponding numbers so that the sum of two numbers corresponding to the opposite arcs equals 1. After aggregation a similar partitioned circle appears. Then the tangents of the vectors from the arc with maximal number are the estimators of $\alpha/\beta$.

• I think you mean categorical instead of categorised. And from your description, it sounds like A1 - A4 are categorical, which means that a multinomial model should work. – mkt - Reinstate Monica Jul 7 '17 at 20:57
• Is the customer's choice binary (e.g. buy vs. not buy)? If so you don't need multinomial logistic regression. – Will Jul 7 '17 at 20:58
• @Will, thanks. The customer chooses one of the alternatives. – 8k14 Jul 7 '17 at 21:07
• I'm getting a 404 on the picture file, too. The business about lines and circular arcs makes no sense to me: I do not see how these are an adequate metaphor for any statistical model. – whuber Jul 7 '17 at 21:39
• @whuber Thanks. We can assume that the customer knows the option set. I understand that this data may be not sufficient to draw conclusions about relative impact of atributes. For example in some cases one can get a well-fit estimator which belongs to a large interval so that even the sign of $alpha/beta$ can not be defined. – 8k14 Jul 13 '17 at 5:33

I extract a simple case : choosing between three hotels $H_1,H_1,H_2$ depending on their prices $(x_1,x_2,x_3)$.

This sounds like multinomial logistic regression but is not (as far as I can see). Like, in multinomial regression, the feature vector is $X=(x_1,x_2,x_3)$ and there are three "fixed" categories (even if they are not the same hostels, it does not matter). But in multinomial logistic regression the linear predictor associated to each category depends on the whole $X$. In terms of utility function, it's like the utility function of each hostel would depend on all $(x_1,x_2,x_3)$. Actually, in your model, the utility function of $H_1$ depends only on $x_1$ : $\beta x_1$. More than this the utility function of $H_2$ depends on $x_2$ with the same coefficient (no reason why the "second" hostel should be treated differently from the first one) : $\beta x_2$

One solution is to get inspired by multinomial logistic regression and build your own model. I use the same notations as https://en.wikipedia.org/wiki/Multinomial_logistic_regression. The intercept can be dropped because it plays a redundant role (with $Z$). Call $h$ the choice variable (outcome) with values in $\{1,2,3\}$.

The model can be:

$P(h=k)=\frac{1}{Z}e^{\beta x_k}$ with $Z$ being the re-normalization coefficient equal to $Z=\displaystyle\sum_{k=1}^3e^{\beta x_k}$

Now when the number of hostels varies, no problem, call $n$ the number of hostels:

$P(h=k|n\text{ choices})=\frac{1}{\displaystyle\sum_{k=1}^n e^{\beta x_k}}e^{\beta x_k}$

Then you can find $\beta$ with MLE.

NOTE : this question may be strongly related to your problem : Alternatives to the multinomial logit model. What I described seens to be exactly conditional logit

• Thanks a lot. Your idea seems to be one of the ways to resolve my problem – 8k14 Jul 19 '17 at 16:03