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I am not an expert in regression, but I have a problem that I believe should be solved by logistic regression. The problem is rather specific, so I try to describe it using a more tangible example.

Say there are five airline carriers, A,..., E, that offer flights to several locations. Let's also assume the following quality of service for the carriers: Q(A) > Q(B) > ... > Q(E), that is, carrier A offers the "best" service during the flight. However, this comes at the cost of a higher price, i.e., P(A) > P(B) > ... > P(E). Therefore, there is a clear trade-off between quality of service and price.

Also, to get the general preference of a user, we ask her/him several questions in the form of the following example: "Which carrier would you select to get a flight to New York given the following prices: p(A) = \$1000, P(B) = \$900,..., p(E) = \$650?". Before asking the questions, we let the user know about the quality of service provided by each carrier, so that she/he considers it in their decision making.

Once we have enough sample points (say 30), we use them to build a model that can predict user's choice for scenarios that are not in the sample points. To put it concretely, I am looking for a model that can take as input the list of prices, and provide as output the user's preferred carrier.

Following are few points that I have figured about the model (which may not be correct):

  • it seems to be a multinomial logistic regression model, where the outcome is one of the five carriers.
  • explanatory variables need not include characteristics of the user, instead they should include attributes of the alternatives (i.e., the price of each carrier).
  • the decision is made by comparing all the alternatives in a scenario, that is, an undesirable alternative in one scenario might be considered desirable in another scenario.

I would really appreciate any suggestions.

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  • $\begingroup$ Lots of discrete choice models use multinomial logistic regression, as you suggest: see en.wikipedia.org/wiki/Discrete_choice, for example. $\endgroup$ – Adrian Apr 12 '15 at 14:05
  • $\begingroup$ You're going to need a lot more than 30 people. The choice will depend not only on a carrier's price, but the relative prices of all other carriers simultaneously. That means you will need to include interactions. $\endgroup$ – gung - Reinstate Monica Apr 12 '15 at 14:09
  • $\begingroup$ Thanks for your replies. Apart from the scale of the user-study, do you think multinomial logistic regression is suitable for this problem? Can you please give me an example of how the model should look like? $\endgroup$ – arima Apr 12 '15 at 14:16
  • $\begingroup$ The example here may be useful $\endgroup$ – user20650 Apr 12 '15 at 14:43
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    $\begingroup$ Thanks @user2065, I am reading the document and it seems to be relevant. $\endgroup$ – arima Apr 12 '15 at 15:17
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The choice of model depends on what you are trying to accomplish. There are many estimation strategies available, including multinomial logit (MNL), multinomial probit (MNP), nested logit (NL, which is a specific instance of Generalized Extreme Value models, GEV), and mixed "flavors" of the aforementioned models. I think it is worth mentioning that despite advances in discrete choice models (a lot of this work was done in the later parts of the 20th century), MNL model remains a popular choice for modeling discrete outcomes for the reason that it is computationally simple. However, one key assumption of MNL, the independence of irrelevant alternatives (IIA), is often violated. In layman's terms, IIA assumes "equal competition" between outcome choices. Essentially, if A is preferable to B, then adding a third choice, C should not make B preferable to A (adding alternatives should not affect preference). This is often illustrated with the "red bus/blue bus" problem:

Suppose commuters have a choice between taking a car or a red bus to work. If these were the only two choices, then the probability of choosing mode of transport is equal, that is: $P(car)=P(red bus)=0.5$, giving us an odds ratio of 1

Suppose that a third option--a blue bus--is now available. In real world situations, it is safe to assume that commuters probably don't care what color bus they take if they prefer to take buses. In this scenario, $P(car)=0.5$, and $P(red bus)=P(blue bus)=0.25$. The addition of the blue bus no longer makes the odds ratio of choosing car over red bus equal to 1, now that the probability of choosing the red bus has changed (probability now smaller with the introduction of the blue bus, which can be taken as a perfect substitute of the red bus).

By contrast, IIA assumes that the addition of the blue bus will not affect selection probabilities, that is, under IIA, it is expected that $P(car)=P(red bus)=P(blue bus)$, which in this case would equal to 0.33. This is not realistic, since people are likely not going to change their preferences (be less likely to take the car) because of the color of the bus.

The good news for you is, you can formally test whether the IIA assumption was met using the Hausman-McFadden test, and your statistical package likely has that test built in (See summary in Train 2009 [and Hausman and McFadden (1984) if you're interested in the gory details]). The other discrete choice models I described above (MNP, mixed MNL, NL) relax the IIA assumption, so those are viable alternatives for you. I also read a working paper a couple of years ago that compares the MNL with MNP and mixed MNL models, which concluded that MNL can still be used even if IIA was severely violated (see Kropko 2010).

I mentioned at the beginning of this response that choice depends on what you are trying to accomplish. If this analysis is for a class project, then you can probably get away with MNL and testing the IIA assumption as described above. If you are trying to publish this in a peer-reviewed journal, then you need to know your audience: the more purist the audience in their methods (statisticians, econometricians, etc.), the less likely they will be satisfied with MNL if the IIA assumption was violated. More applied journals are probably less concerned with the technical details (although I advocate doing it the "most correct" way, even if you are not going to be grilled too much about your methods).

You mentioned that you have price and quality data for each alternative. You can most certainly model those with MNL (if you choose MNL over other methods) by entering the price of the alternative as a separate covariate in the model (e.g. $price_A$, $price_B$, etc.) if your data are in wide format (one line in your dataset for each decision maker). OR, you can model using alternative-specific conditional logit (ASCL) if your data are in long format (each decision maker has the same number of lines as alternatives). MNL and ASCL basically give you the same results (Cameron and Trivedi 2009).

If you are using Stata, use -mlogit- for multinomial logit and -asclogit- for the alternative-specific conditional logit model.

References

  1. Cameron, C and Trivedi P (2009). Microeconometrics using Stata. College Station, TX: Stata Press.

  2. Hausman, J and McFadden D. (1984). Specification tests for the multinomial logit model. Econometrica, 52, 1219-1240

  3. Kropko, J. (2010). A comparison of three discrete choice estimators. Working paper.

  4. Train, K. E. (2009). Discrete choice methods with simulation (2nd ed.). New York, New York: Cambridge University Press. The complete book is available on Train's website (which @gung linked to above). Here's the direct link to the book chapters: http://eml.berkeley.edu/books/choice2.html

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  • $\begingroup$ Thanks a lot for your detailed reply! I still have some doubts about violation of IIA in this problem. For example, if the prices are P(A) = 1000, P(B) = 999, ..., P(E) = 995, then the user will probably select carrier A, knowing that it has a better quality. But if the prices are P(A) = 1000, P(B) = 800, ..., P(E) = 550, then most likely the user will not select A. However, I am not sure this can be explained by IIA, that is, in the first scenario, the user would still prefer A over B irrespective of other choices. $\endgroup$ – arima Apr 12 '15 at 20:35
  • $\begingroup$ If I understand correctly, MNL calculates the utility for each alternative "individually", i.e., without considering the impacts of other alternatives. For this problem, however, I am not sure if this is enough! Instead, I'd like to know if it would be better to consider the interactions between variables of all alternatives in order to calculate the utility for each. $\endgroup$ – arima Apr 12 '15 at 20:35
  • $\begingroup$ Can you elaborate on what other variables you are controlling for? For example, how is "quality" measured? When you enter it in the model, for example as a separate variable for each alternative, then you can get the own- AND cross- effects of that variable on the probability of selection for each alternative. If you do not control for quality though quality is related to price, then you may see some strange results (e.g., higher priced alternatives being preferred over lower priced alternatives) $\endgroup$ – Marquis de Carabas Apr 12 '15 at 20:55
  • $\begingroup$ Utility is a latent variable (we cannot measure it directly). Discrete choice models assume that people will act in order to maximize utility, so the alternative selected is one that gives the highest utility to the decision-maker. I strongly recommend reading Chapters 2 of the Train book for discussion on utility and Chapter 3 for logit models $\endgroup$ – Marquis de Carabas Apr 12 '15 at 21:00
  • $\begingroup$ Wouldn't it be indifferent to consider a separate variable for quality knowing that it is fixed in all situations? I assumed (but may be I am wrong!) that it's effect on the utility will be reflected on the intercept of each alternative. By cross-effects do you mean interaction between variables? If yes, then don't you think I should consider the cross-effects of the price variable on the utility of each alternative? $\endgroup$ – arima Apr 12 '15 at 21:23

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