Multinomial Logit Model

Multinomial Logit Model in Discrete Choice Experiments

Health preference studies usually report a table of coefficients, and in a large share of them those coefficients come from a multinomial logit model fitted to the choices respondents made. The model links each observed choice to the attributes of the options on offer through random utility theory, and the resulting weights become the trade-offs and willingness-to-pay values used in benefit-risk work and cost-benefit analysis. Soekhai and colleagues reported in 2019 that, among health discrete choice experiments published between 2013 and 2017, 39 per cent reported mixed logit models and 39 per cent multinomial logit models. Knowing what the multinomial logit model assumes, and which of its outputs survive those assumptions, is therefore part of reading most stated preference studies in health.

Random utility and the logit choice probability

The model starts from the idea that a respondent picks the alternative that gives the highest utility, as in utility maximisation, but the analyst sees only part of that utility. The observed part depends on attributes such as effectiveness, risk and cost, and the rest is treated as random. McFadden derived the conditional logit model from this setup, and Train notes that Luce had earlier derived the same formula from assumptions about choice probabilities, with McFadden completing the analysis by showing that the logit formula implies extreme value errors.

$$U_{njt} = \beta' x_{njt} + \varepsilon_{njt}$$

where $U_{njt}$ is the utility respondent $n$ obtains from alternative $j$ in choice task $t$, $x_{njt}$ is the vector of attribute levels describing that alternative, $\beta$ is the vector of preference weights to be estimated and $\varepsilon_{njt}$ is the unobserved part of utility.

If each $\varepsilon_{njt}$ is independently and identically distributed with a type I extreme value distribution, the probability that alternative $i$ is chosen takes a closed form.

$$P_{nit} = \frac{\exp(\beta' x_{nit})}{\sum_{j \in C_{nt}} \exp(\beta' x_{njt})}$$

where $P_{nit}$ is the probability that respondent $n$ chooses alternative $i$ in task $t$, $C_{nt}$ is the choice set shown in that task, $j$ indexes every alternative in the set including $i$, and $\exp$ is the exponential function. The probabilities lie between zero and one and sum to one across the set. Only differences in utility affect them, so adding the same constant to every alternative changes nothing, and an alternative-specific constant can be estimated for all but one alternative, such as an opt-out.

Scale and what the coefficients identify

Utility has no natural unit, so the model fixes one. Train explains that setting the variance of the error to $\pi^2/6$, about 1.6, is the same as normalising the scale of utility. If the true error variance is $\sigma^2$ times that value, the estimated coefficients are the true weights divided by $\sigma$.

$$\hat{\beta} = \frac{\beta^{*}}{\sigma}$$

where $\hat{\beta}$ is the coefficient vector the multinomial logit model returns, $\beta^{*}$ is the vector of underlying preference weights and $\sigma$ is a scale factor proportional to the standard deviation of the unobserved utility.

A coefficient therefore measures preference relative to the noise in choices. A group whose choices are more consistent has a smaller $\sigma$ and larger coefficients, even when its trade-offs are identical to another group's. Swait and Louviere showed that identical utility specifications estimated on data with unequal error variances will produce coefficients of different magnitude, and they proposed a relative scaling test for comparing multinomial logit parameters across data sets. A ratio of two coefficients from the same model cancels $\sigma$, which is why trade-offs carry more meaning than raw coefficients.

Independence of irrelevant alternatives and its consequences

Because the errors are independent and share one variance, the odds of choosing one alternative over another depend only on those two alternatives. Train notes that this ratio is independent of the other alternatives in the set, which is the origin of the name independence of irrelevant alternatives (IIA).

$$\frac{P_{nit}}{P_{nkt}} = \exp\left(\beta' x_{nit}-\beta' x_{nkt}\right)$$

where $P_{nit}$ and $P_{nkt}$ are the probabilities of choosing alternatives $i$ and $k$ in the same task, and the right-hand side contains only the attributes of those two alternatives.

The practical result is proportional substitution. When an alternative is added or withdrawn, the model takes share from, or returns share to, all remaining alternatives in proportion to their existing probabilities. Train's red bus and blue bus example shows the failure case, in which logit overestimates the probability of taking either bus and underestimates the probability of taking a car once a second, near-identical bus is offered. In health, the same problem arises when two similar treatments are compared alongside a no-treatment option, since respondents who leave one of them are more likely to move to the other.

Train sets out the limits of logit under three headings. The model can represent taste variation that is linked to observed respondent characteristics but not random taste variation, it implies proportional substitution across alternatives, and it cannot handle repeated choices in which unobserved factors are correlated over time for the same person. Each heading corresponds to a feature of most health DCEs, where preferences differ in ways the survey does not record and every respondent completes a series of tasks.

Conditional logit and multinomial logit as names

Two specifications share the logit formula but differ in what varies. Hoffman and Duncan set out the distinction for demographers. In the multinomial logit model in the strict sense, explanatory variables describe the individual and are the same for every alternative, so they influence choice only through coefficients that differ by alternative. In McFadden's conditional logit model, the explanatory variables describe the alternatives and differ across them, which is the structure of a DCE.

$$V_{njt} = \beta' x_{njt} + \gamma_j' z_n$$

where $V_{njt}$ is the observed utility, $x_{njt}$ holds the alternative-specific attributes with a common coefficient vector $\beta$, $z_n$ holds characteristics of respondent $n$ such as age or disease severity, and $\gamma_j$ is a coefficient vector specific to alternative $j$, with $\gamma_j$ set to zero for one reference alternative.

A model with only the first term is a conditional logit model, a model with only the second is a multinomial logit model in the strict sense, and both are special cases of the same likelihood. Both labels appear in health DCE reporting, so the specification matters more than the name. The ISPOR good research practices report on statistical methods by Hauber and colleagues presents conditional logit as one of the most common approaches to analysing DCE data. Respondent characteristics enter a conditional logit through interactions with attributes or with alternative-specific constants, because a variable that is the same for every alternative cancels from the utility differences.

Estimation by maximum likelihood with repeated choices

The coefficients are found by maximum likelihood estimation. The log-likelihood adds the log probability of the alternative actually chosen across every task and respondent.

$$LL(\beta) = \sum_{n=1}^{N} \sum_{t=1}^{T} \sum_{j \in C_{nt}} y_{njt} \ln P_{njt}$$

where $LL(\beta)$ is the log-likelihood, $N$ is the number of respondents, $T$ is the number of tasks each completes, $y_{njt}$ equals 1 if respondent $n$ chose alternative $j$ in task $t$ and 0 otherwise, and $P_{njt}$ is the logit probability.

Train credits McFadden with showing that this log-likelihood is globally concave in the parameters for utility that is linear in parameters, so standard numerical routines find a single maximum. Fit is often summarised by the likelihood ratio index, one minus the ratio of the maximised log-likelihood to the log-likelihood with all parameters at zero, which Train describes as lying between zero, when the estimates do no better than zero parameters, and one, when they predict every choice perfectly. The index is not an $R^2$ in the regression sense and should not be judged against the same benchmarks.

The standard model treats every task as an independent observation. In a DCE where each respondent answers several tasks, that assumption understates uncertainty, and analyses commonly cluster standard errors by respondent or move to a panel model. Comparisons between nested specifications, such as a model with and without interaction terms, use the likelihood ratio test.

Trade-offs and willingness to pay from coefficient ratios

The quantities that inform decisions come from ratios of coefficients, because the scale factor cancels in a ratio. The marginal rate of substitution between two attributes is the ratio of their coefficients with the sign reversed, and when one attribute is a cost the ratio becomes marginal willingness to pay.

$$WTP_a = -\frac{\beta_a}{\beta_c}$$

where $WTP_a$ is the money a respondent would give up for a one-unit increase in attribute $a$, $\beta_a$ is the coefficient on that attribute and $\beta_c$ is the coefficient on the cost attribute, which is expected to be negative.

A ratio of two estimates has a skewed sampling distribution, so its confidence interval needs care. Hole compared the delta, Fieller, Krinsky Robb and bootstrap methods on simulated data and found that in most scenarios all four were reasonably accurate and gave similar results. The delta method was the most accurate when the data were well conditioned, and the bootstrap was more robust to noisy data and misspecification.

Illustrative worked example with a three-option treatment choice

All numbers in this example are illustrative. A DCE offers patients two unlabelled treatments and a no-treatment option. Attributes are the chance of meaningful symptom improvement, coded in units of 10 percentage points, the risk of a serious side effect in percentage points, and a monthly out-of-pocket cost in GBP. A conditional logit fit gives a treatment constant of 0.50, a coefficient of 0.30 per 10 points of improvement, minus 0.25 per percentage point of risk and minus 0.02 per GBP. The no-treatment option has utility fixed at zero.

Step 1 (observed utility). Treatment A offers 60 per cent improvement, 4 per cent risk and GBP 30 per month, so $V_A = 0.50 + 0.30 \times 6-0.25 \times 4-0.02 \times 30 = 0.70$. Treatment B offers 40 per cent improvement, 1 per cent risk and GBP 10, so $V_B = 0.50 + 0.30 \times 4-0.25 \times 1-0.02 \times 10 = 1.25$. No treatment has $V_C = 0$.

Step 2 (exponentials). To four decimal places, $\exp(0.70) \approx 2.0138$, $\exp(1.25) \approx 3.4903$ and $\exp(0) = 1$, so the denominator is $D = 2.0138 + 3.4903 + 1 = 6.5041$.

Step 3 (choice probabilities). $P_A = 2.0138 / 6.5041 = 0.3096$, $P_B = 3.4903 / 6.5041 = 0.5366$ and $P_C = 1 / 6.5041 = 0.1537$. If a respondent chose B, that task adds $\ln 0.5366 \approx -0.6224$ to the log-likelihood.

Step 4 (IIA in action). Removing the no-treatment option gives $P_A = 2.0138 / 5.5041 = 0.3659$ and $P_B = 3.4903 / 5.5041 = 0.6341$. The odds of B over A stay at $P_B / P_A = 3.4903 / 2.0138 = 1.7332$, and both treatments gain by the same factor, $f = 1 / (1-0.1537) \approx 1.18$. If B and no treatment shared unobserved features, such as avoiding a clinic visit, people leaving no treatment would move to B more than this and the logit prediction would be wrong.

Step 5 (trade-offs). Willingness to pay for 10 more percentage points of improvement is $WTP_e = -0.30 / (-0.02) = 15$, or GBP 15 per month. For a 1 percentage point increase in risk it is $WTP_r = -(-0.25) / (-0.02) = -12.5$, so a 1 point reduction in risk is worth GBP 12.50 per month. The marginal rate of substitution is $MRS = 0.25 / 0.30 \approx 0.8333$, so about 8.3 percentage points of extra improvement offset 1 point of extra risk.

Step 6 (scale). If the same people answered with half the noise, every coefficient would double. Utilities become 1.40 and 2.50, and the probabilities move to about 0.2353 for A, 0.7067 for B and 0.0580 for no treatment. The willingness to pay for improvement is unchanged at $WTP_e = -0.60 / (-0.04) = 15$.

The probabilities in step 3 are predicted shares for this hypothetical choice and depend on which alternatives are offered, while the ratios in step 5 hold across scale. A comparison of raw coefficients between two samples would confuse the two, as step 6 shows.

Models that relax the multinomial logit assumptions

Each extension relaxes one of the restrictions above, at the cost of more parameters and more demanding estimation. The ISPOR report by Hauber and colleagues describes random-parameters logit, hierarchical Bayes and latent-class finite-mixture models as alternatives to conditional logit for health DCE data.

• Mixed logit, also called the random parameter model, lets coefficients vary across respondents according to a chosen distribution and can link a respondent's tasks. McFadden and Train showed that, under mild regularity conditions, any discrete choice model derived from random utility maximisation can be approximated as closely as desired by a mixed logit model.
• The latent class model assumes a finite number of preference groups, each with its own conditional logit coefficients, and estimates the probability that each respondent belongs to each group.
• Heteroskedastic and scale-adjusted logit models let the scale factor differ between respondent groups, tasks or data sources, which is the setting Swait and Louviere addressed. Fiebig and colleagues proposed the generalised multinomial logit model to account for scale heterogeneity and coefficient heterogeneity together.
• Nested logit groups similar alternatives so that substitution within a nest is stronger than substitution across nests, which addresses the red bus and blue bus problem directly.

A richer model is not automatically better. Scale heterogeneity and preference heterogeneity can be hard to separate, and willingness-to-pay distributions from random coefficient models can have long tails when the cost coefficient is allowed near zero.

Boundaries with related models and methods

Binary logistic regression is the special case with two outcomes. A paired choice task with no opt-out can be estimated as a logistic regression, without an intercept, on the differences in attribute levels between the two alternatives. Conjoint analysis is the wider family of attribute-based methods, and rating or ranking versions do not need a choice model at all. Best-worst scaling records more than one pick per set, so a logit analysis of those data has to state how the best and worst choices relate. Efficient designs for a DCE, discussed under choice experiment design, are computed for an assumed choice model, and the multinomial logit is a common assumption in practice, so the design and the analysis model are linked before any data are collected.

Limitations and common misreadings

The multinomial logit model describes stated choices from a particular choice set and sample. Its restrictions are easy to lose sight of once coefficients and ratios are copied into a model or report.

1. A larger coefficient in one subgroup means that subgroup cares more about the attribute. The difference may reflect a smaller error variance, so subgroup comparisons should use ratios or a model that estimates relative scale.
2. Independence of irrelevant alternatives only matters when forecasting uptake. If the substitution pattern that IIA imposes is wrong, the model is misspecified, so its coefficients and willingness-to-pay estimates can be biased as well as its forecasts. The forecasting error is largest for a new option that resembles an existing one.
3. Clustered standard errors fix the panel problem. They correct the uncertainty but leave the coefficients assuming no correlation in tastes across a respondent's tasks.
4. Willingness-to-pay intervals can be built from separate coefficient intervals. Dividing the ends of two separate intervals does not give a valid interval for a ratio, and a cost coefficient close to zero makes the ratio unstable whatever method is used.

Sources

• McFadden D. Conditional logit analysis of qualitative choice behavior. In: Zarembka P, ed. Frontiers in Econometrics. New York: Academic Press; 1974:105-142.
• Train KE. Discrete Choice Methods with Simulation. 2nd ed. Cambridge: Cambridge University Press; 2009. Chapters 2 and 3. https://eml.berkeley.edu/choice2/ch3.pdf
• Hoffman SD, Duncan GJ. Multinomial and conditional logit discrete-choice models in demography. Demography. 1988;25(3):415-427. https://doi.org/10.2307/2061541
• Hauber AB, González JM, Groothuis-Oudshoorn CGM, et al. Statistical methods for the analysis of discrete choice experiments: a report of the ISPOR Conjoint Analysis Good Research Practices Task Force. Value in Health. 2016;19(4):300-315. https://doi.org/10.1016/j.jval.2016.04.004
• Swait J, Louviere J. The role of the scale parameter in the estimation and comparison of multinomial logit models. Journal of Marketing Research. 1993;30(3):305-314. https://doi.org/10.1177/002224379303000303
• McFadden D, Train K. Mixed MNL models for discrete response. Journal of Applied Econometrics. 2000;15(5):447-470. https://ideas.repec.org/a/jae/japmet/v15y2000i5p447-470.html
• Fiebig DG, Keane MP, Louviere J, Wasi N. The generalized multinomial logit model: accounting for scale and coefficient heterogeneity. Marketing Science. 2010;29(3):393-421. https://doi.org/10.1287/mksc.1090.0508
• Hole AR. A comparison of approaches to estimating confidence intervals for willingness to pay measures. Health Economics. 2007;16(8):827-840. https://doi.org/10.1002/hec.1197
• Soekhai V, de Bekker-Grob EW, Ellis AR, Vass CM. Discrete choice experiments in health economics: past, present and future. PharmacoEconomics. 2019;37(2):201-226. https://doi.org/10.1007/s40273-018-0734-2