In the multinomial logit model, each case in the data has the same set of possible unordered outcomes. If the outcome is a choice, one can think about the multinomial logit model as one in which each individual has the same set of alternatives, and chooses one among them.
A conditional logit model provides a different way of fitting a multinomial logit model. The multinomial logit approach we have already covered is simpler, and there is no reason to use conditional logit for an ordinary multinomial logit model.
But, the conditional logit model is more flexible. One way it is more flexible is that the conditional logit model provides a way of adding information that varies over the possible outcomes.
Here are some examples that illustrate what we mean by alternative-specific information:
When a conditional logit model is used to fit the equivalent of a multinomial logit model with alternative-specific variables, this is sometimes called a McFadden’s choice model.
The example we will use involves a survey conducted in 1989 among people who had engaged in a fishing trip in Southern California. For each trip, there were four possible options:
For each trip, each of these options has an associated price (in dollars) and an associated quality (in terms of the expected number of target fish caught). These are alternative-specific variables. In addition, the data include a measure of monthly income for the respondent (measured in thousands of dollars), which is a case-specific variable.
When we have alternative-specific data, we want the data arranged so that each alternative is on a different row, with some id variable identifying which rows belong to the same case. A binary variable should indicate which of the alternatives is the selected outcome.
For the fishing data, there are four alternatives, so each case will have four rows. We will list the first few cases here.
library(tidyverse)
library(haven)
library(marginaleffects)
data <- read_dta("../dta/fishing.dta") %>%
mutate(alternative = as_factor(alternative, levels="label"),
alternative = relevel(alternative, ref = "pier"),
choice = as_factor(choice, levels="label")) %>%
rename(alt = alternative) %>%
arrange(id, alt, choice, chosen)# A tibble: 12 × 6
id alt chosen price quality income
<dbl> <fct> <dbl> <dbl> <dbl> <dbl>
1 1 pier 0 158. 0.0503 7.08
2 1 beach 0 158. 0.0678 7.08
3 1 private boat 0 158. 0.260 7.08
4 1 charter boat 1 183. 0.539 7.08
5 2 pier 0 15.1 0.0451 1.25
6 2 beach 0 15.1 0.105 1.25
7 2 private boat 0 10.5 0.157 1.25
8 2 charter boat 1 34.5 0.467 1.25
9 3 pier 0 162. 0.452 3.75
10 3 beach 0 162. 0.533 3.75
11 3 private boat 1 24.3 0.241 3.75
12 3 charter boat 0 59.3 1.03 3.75. use ../dta/fishing.dta, clear (Data from fishing survey by Thomson and Crooke (1991)) . list id alternative chosen price quality income in 1/12, sep(4) +-----------------------------------------------------------+ | id alternative chosen price quality income | |-----------------------------------------------------------| 1. | 1 beach 0 157.93 .0678 7.083332 | 2. | 1 charter boat 1 182.93 .5391 7.083332 | 3. | 1 pier 0 157.93 .0503 7.083332 | 4. | 1 private boat 0 157.93 .2601 7.083332 | |-----------------------------------------------------------| 5. | 2 beach 0 15.114 .1049 1.25 | 6. | 2 charter boat 1 34.534 .4671 1.25 | 7. | 2 pier 0 15.114 .0451 1.25 | 8. | 2 private boat 0 10.534 .1574 1.25 | |-----------------------------------------------------------| 9. | 3 beach 0 161.874 .5333 3.75 | 10. | 3 charter boat 0 59.334 1.0266 3.75 | 11. | 3 pier 0 161.874 .4522 3.75 | 12. | 3 private boat 1 24.334 .2413 3.75 | +-----------------------------------------------------------+
Above, the variable \(\texttt{id}\) identifies the individual cases. The variable \(\texttt{alternative}\) indicates the alternative corresponding to each row. \(\texttt{chosen}\) indicates which alternative was used; for example, for the first observation, the trip was by charter boat.
\(\texttt{price}\) and \(\texttt{quality}\) are alternative-specific variables and vary across alternatives. The variable \(\texttt{income}\) is case-specific and we can see that it is the same within each case.
Stata: rearranging data with alternative-specific values. You may have data with alternative-specific variables in which the information is all in a single row, with different variables containing the different alternative-specific values for a given measure. The \(\texttt{reshape long}\) command is what you use to re-arrange this data so that each row represents a different alternative.
The conditional logit model can be written as:
\[ \begin{equation} u_{ij}=\mathbf{x}_{ij}^{A}\mathbf{\beta }^{A}+\mathbf{x}_{i}^{C}% \mathbf{\beta }_{j \mathrm{\ vs\ } base}^{C}+\varepsilon _{ij} \end{equation} \]
where:
The predicted probability of category \(m\) being the chosen category is then calculated as:
\[ \begin{equation} \Pr \left( y=m\mid \mathbf{x}\right) =\frac{\exp \left( \mathbf{x}_{m}^{A}% \mathbf{\beta }^{A}+\mathbf{x}^{C}\mathbf{\beta }_{m \mathrm{\ vs\ } base}^{C}\right) }{\sum_{j=1}^{k}\exp \left( \mathbf{x}_{j}^{A}\mathbf{\beta }^{A}+% \mathbf{x}^{C}\mathbf{\beta }_{j \mathrm{\ vs\ } base}^{C}\right) } \end{equation} \]
This is like with multinomial logit, where for each observation there is a separate \(\mathbf{x\beta}\) that can be computed for each alternative. We can then sum the \(\exp(\mathbf{x\beta})\) over each alternative and this is the denominator of our predicted probability, where the \(\exp(\mathbf{x\beta})\) for alternative \(m\) is the numerator for calculating \(\Pr(y=m)\).
In R, we will fit the model using the clogit() function from the survival package.
chosen: the binary variable that is 1 for the option that was selectedprice and quality are included in the model in the familiar wayincome is included as an interaction with the variable that identifies each of the alternatives (alt)id) is included in the model as strata(id).Call:
coxph(formula = Surv(rep(1, 4728L), chosen) ~ price + quality +
(alt * income) + strata(id), data = data, method = "exact")
n= 4728, number of events= 1182
coef exp(coef) se(coef) z Pr(>|z|)
price -0.025117 0.975196 0.001732 -14.504 < 2e-16 ***
quality 0.357782 1.430154 0.109773 3.259 0.001117 **
altbeach -0.777959 0.459342 0.220494 -3.528 0.000418 ***
altprivate boat -0.250681 0.778271 0.203940 -1.229 0.219000
altcharter boat 0.916406 2.500289 0.207265 4.421 9.81e-06 ***
income NA NA 0.000000 NA NA
altbeach:income 0.127577 1.136073 0.050640 2.519 0.011758 *
altprivate boat:income 0.217017 1.242365 0.050058 4.335 1.46e-05 ***
altcharter boat:income 0.094285 1.098873 0.050060 1.883 0.059640 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
exp(coef) exp(-coef) lower .95 upper .95
price 0.9752 1.0254 0.9719 0.9785
quality 1.4302 0.6992 1.1533 1.7735
altbeach 0.4593 2.1770 0.2982 0.7077
altprivate boat 0.7783 1.2849 0.5218 1.1607
altcharter boat 2.5003 0.4000 1.6656 3.7533
income NA NA NA NA
altbeach:income 1.1361 0.8802 1.0287 1.2546
altprivate boat:income 1.2424 0.8049 1.1263 1.3704
altcharter boat:income 1.0989 0.9100 0.9962 1.2122
Concordance= 0.757 (se = 0.011 )
Likelihood ratio test= 846.9 on 8 df, p=<2e-16
Wald test = 322.7 on 8 df, p=<2e-16
Score (logrank) test = 596 on 8 df, p=<2e-16In the output above, you’ll notice the coefficient for income is indicated as “NA.” However, there are different coefficients for income for each of the three alternatives (vs. the base category), just like in multinomial logit.
In the conditional logit model, coefficients are only estimated for terms that vary within a case (in this case, within the same value of id). Income does not vary within a person. However, when we include the interaction terms, we implicitly refer to a term that does vary within a person: income \(\times\) beach, for example, equals the person’s income for their beach row and is 0 for all other rows.
We will reformat the output to be easier to read:
library(modelsummary)
coef_rename <- c(
"price" = "Price of option",
"quality" = "Expected catch",
"altbeach:income" = "Income: Beach",
"altprivate boat:income" = "Income: Private boat",
"altcharter boat:income" = "Income: Charter boat",
"altbeach" = "Intercept: Beach",
"altprivate boat" = "Intercept: Private boat",
"altcharter boat" = "Intercept: Charter boat")
# Create the formatted table with renamed coefficients
modelsummary(model,
stars = TRUE,
estimate = "{estimate} ({std.error}){stars}",
statistic = NULL,
coef_map = coef_rename, # Apply the coefficient renaming
include.LogLike = TRUE)Model matrix is rank deficient. Parameters `income` were not estimable.| (1) | |
|---|---|
| Price of option | -0.025 (0.002)*** |
| Expected catch | 0.358 (0.110)** |
| Income: Beach | 0.128 (0.051)* |
| Income: Private boat | 0.217 (0.050)*** |
| Income: Charter boat | 0.094 (0.050)+ |
| Intercept: Beach | -0.778 (0.220)*** |
| Intercept: Private boat | -0.251 (0.204) |
| Intercept: Charter boat | 0.916 (0.207)*** |
| Num.Obs. | 4728 |
| AIC | 2446.3 |
| BIC | 2498.0 |
| RMSE | 0.39 |
In Stata, fitting a conditional logit model for simple choice data is done in two parts. First, the \(\texttt{cmcset}\) command is used to specify two things: (1) what variable identifies each observation and (2) what variable identifies each alternative within each observation. In our example these variables are conveniently named \(\mathtt{id}\) and \(\mathtt{alternative}\).
. cmset id alternative Case ID variable: id Alternatives variable: alternative
Then, we estimate the model using the command cmclogit, using “pier” as our base category:
. cmclogit chosen price quality, casevar(income) base(2) Iteration 0: log likelihood = -1270.0164 Iteration 1: log likelihood = -1217.7258 Iteration 2: log likelihood = -1215.1499 Iteration 3: log likelihood = -1215.1376 Iteration 4: log likelihood = -1215.1376 Conditional logit choice model Number of obs = 4,728 Case ID variable: id Number of cases = 1182 Alternatives variable: alternative Alts per case: min = 4 avg = 4.0 max = 4 Wald chi2(5) = 252.98 Log likelihood = -1215.1376 Prob > chi2 = 0.0000 ------------------------------------------------------------------------------ chosen | Coefficient Std. err. z P>|z| [95% conf. interval] -------------+---------------------------------------------------------------- alternative | price | -.0251166 .0017317 -14.50 0.000 -.0285106 -.0217225 quality | .357782 .1097733 3.26 0.001 .1426302 .5729337 -------------+---------------------------------------------------------------- beach | income | .1275771 .0506395 2.52 0.012 .0283255 .2268288 _cons | -.7779594 .2204939 -3.53 0.000 -1.21012 -.3457992 -------------+---------------------------------------------------------------- pier | (base alternative) -------------+---------------------------------------------------------------- private_boat | income | .217017 .0500582 4.34 0.000 .1189047 .3151293 _cons | -.2506806 .2039395 -1.23 0.219 -.6503948 .1490336 -------------+---------------------------------------------------------------- charter_boat | income | .0942854 .05006 1.88 0.060 -.0038303 .1924012 _cons | .9164063 .2072648 4.42 0.000 .5101748 1.322638 ------------------------------------------------------------------------------
The signs of these results mean that the price of an option is negatively associated with it being chosen, while an option’s quality is positively associated with being chosen. Unsurprisingly, customers are attracted to lower prices and higher quality.
Income is positively associated with choosing any option versus fishing off a pier, but most strongly with fishing by private boat.
The signs of the intercepts mean that, net of everything else, fishing by charter boat is the most popular choice, while fishing off the beach is least popular.
As a logit model, we can exponentiate conditional logit coefficients and interpret those.
library(modelsummary)
coef_rename <- c(
"price" = "Price of option",
"quality" = "Expected catch",
"altbeach:income" = "Income: Beach",
"altprivate boat:income" = "Income: Private boat",
"altcharter boat:income" = "Income: Charter boat",
"altbeach" = "Intercept: Beach",
"altprivate boat" = "Intercept: Private boat",
"altcharter boat" = "Intercept: Charter boat")
# Create the formatted table with renamed coefficients
modelsummary(model,
exponentiate=TRUE,
stars = TRUE,
estimate = "{estimate} ({std.error}){stars}",
statistic = NULL,
coef_map = coef_rename, # Apply the coefficient renaming
include.LogLike = TRUE)Model matrix is rank deficient. Parameters `income` were not estimable.| (1) | |
|---|---|
| Price of option | 0.975 (0.002)*** |
| Expected catch | 1.430 (0.157)** |
| Income: Beach | 1.136 (0.058)* |
| Income: Private boat | 1.242 (0.062)*** |
| Income: Charter boat | 1.099 (0.055)+ |
| Intercept: Beach | 0.459 (0.101)*** |
| Intercept: Private boat | 0.778 (0.159) |
| Intercept: Charter boat | 2.500 (0.518)*** |
| Num.Obs. | 4728 |
| AIC | 2446.3 |
| BIC | 2498.0 |
| RMSE | 0.39 |
These can be obtained in Stata using the \(\texttt{or}\) option.
. cmclogit chosen price quality, casevar(income) or base(2) Iteration 0: log likelihood = -1270.0164 Iteration 1: log likelihood = -1217.7258 Iteration 2: log likelihood = -1215.1499 Iteration 3: log likelihood = -1215.1376 Iteration 4: log likelihood = -1215.1376 Conditional logit choice model Number of obs = 4,728 Case ID variable: id Number of cases = 1182 Alternatives variable: alternative Alts per case: min = 4 avg = 4.0 max = 4 Wald chi2(5) = 252.98 Log likelihood = -1215.1376 Prob > chi2 = 0.0000 ------------------------------------------------------------------------------ chosen | Odds ratio Std. err. z P>|z| [95% conf. interval] -------------+---------------------------------------------------------------- alternative | price | .9751962 .0016887 -14.50 0.000 .971892 .9785117 quality | 1.430154 .1569927 3.26 0.001 1.153303 1.773462 -------------+---------------------------------------------------------------- beach | income | 1.136073 .0575302 2.52 0.012 1.02873 1.254615 _cons | .4593424 .1012822 -3.53 0.000 .2981616 .7076546 -------------+---------------------------------------------------------------- pier | (base alternative) -------------+---------------------------------------------------------------- private_boat | income | 1.242365 .0621906 4.34 0.000 1.126263 1.370436 _cons | .7782709 .1587202 -1.23 0.219 .5218397 1.160712 -------------+---------------------------------------------------------------- charter_boat | income | 1.098873 .0550096 1.88 0.060 .996177 1.212157 _cons | 2.500289 .518222 4.42 0.000 1.665582 3.753309 ------------------------------------------------------------------------------ Note: Exponentiated coefficients represent odds ratios for alternative-specific variables (first equation) and relative-risk ratios for case-specific variables. Note: _cons estimates baseline relative risk for each outcome.
As the note at the bottom of the Stata output indicates, the results are an unusual hybrid of odds ratios (for the alternative-specific variables) and relative risk ratios (for the case-specific variables).
The coefficients for the alternative-specific variables can be interpreted as:
The coefficients for the case-specific variables can be interpreted as:
Consistent with its terminology with multinomial logit, Stata refers to the exponentiated coefficients for case-specific variables as relative risk ratios. As before, we do not do this because it invites confusion with the more usual use of “relative risk.”
Presently, I have only implemented this for Stata, and have not figured out how to do it in R.
For the case-specific variables, the average marginal change can be obtained in Stata using \(\texttt{margins}\) with the \(\texttt{dydx()}\) option:
. margins, dydx(income) Average marginal effects Number of obs = 4,728 Model VCE: OIM Expression: Pr(alternative|1 selected), predict() dy/dx wrt: income ------------------------------------------------------------------------------- | Delta-method | dy/dx std. err. z P>|z| [95% conf. interval] --------------+---------------------------------------------------------------- income | _outcome | beach | .0034878 .003541 0.98 0.325 -.0034524 .010428 pier | -.0144069 .004369 -3.30 0.001 -.0229701 -.0058437 private boat | .0266822 .00515 5.18 0.000 .0165885 .0367759 charter boat | -.0157631 .00559 -2.82 0.005 -.0267193 -.0048069 -------------------------------------------------------------------------------
From these results, we can see that, on average, a marginal increase in income increases the likelihood of choosing to fish by private boat by .027, and that this increase comes as the result of a decrease in the probability of fishing via charter boat of .016 and of fishing from a pier of .014.
For the alternative-specific variables, the Stata output is more complex to interpret. Here are the average marginal effects for price:
. margins, dydx(price) Average marginal effects Number of obs = 4,728 Model VCE: OIM Expression: Pr(alternative|1 selected), predict() dy/dx wrt: price -------------------------------------------------------------------------------------------- | Delta-method | dy/dx std. err. z P>|z| [95% conf. interval] ---------------------------+---------------------------------------------------------------- price | _outcome#alternative | beach#beach | -.0021102 .0001727 -12.22 0.000 -.0024487 -.0017717 beach#pier | .0009075 .0000997 9.10 0.000 .0007121 .0011029 beach#private boat | .0005558 .0000458 12.14 0.000 .000466 .0006455 beach#charter boat | .0006469 .0000563 11.50 0.000 .0005366 .0007572 pier#beach | .0009075 .0000997 9.10 0.000 .0007121 .0011029 pier#pier | -.0025593 .000177 -14.46 0.000 -.0029062 -.0022123 pier#private boat | .0007361 .0000543 13.55 0.000 .0006296 .0008425 pier#charter boat | .0009157 .000067 13.67 0.000 .0007844 .001047 private boat#beach | .0005558 .0000458 12.14 0.000 .000466 .0006455 private boat#pier | .0007361 .0000543 13.55 0.000 .0006296 .0008425 private boat#private boat | -.0050457 .0003323 -15.18 0.000 -.005697 -.0043944 private boat#charter boat | .0037539 .0002957 12.70 0.000 .0031743 .0043334 charter boat#beach | .0006469 .0000563 11.50 0.000 .0005366 .0007572 charter boat#pier | .0009157 .000067 13.67 0.000 .0007844 .001047 charter boat#private boat | .0037539 .0002957 12.70 0.000 .0031743 .0043334 charter boat#charter boat | -.0053165 .0003461 -15.36 0.000 -.0059948 -.0046381 --------------------------------------------------------------------------------------------
In this output, a row is labeled by a pair of alternatives separated by \(\texttt{\#}\). The alternative before the \(\texttt{\#}\) is the alternative for which the price is being increased by the marginal amount. The alternative after the hash is the alternative whose change in predicted probability is being evaluated.
The results in the highlighted rows at the bottom evaluate the changes in the probability for an increase in the price of using a charter boat.
The predicted probability of using a charter boat decreases by .0053, or a little more than half a percentage point. Most of that decrease is offset by an increase in the probability of using a private boat instead, which increases the probability by an average of .0038. The probabilities of using the beach or the pier increase by smaller amounts.