## Paired combinatorial logit model

Koppelman and Wen (2000) proposed the paired combinatorial logit model, which is a nested logit model with nests composed by every combination of two alternatives. This model is obtained by using the following \(G\) function:

\[ G(y_1, y_2, ..., y_n)=\sum_{k=1}^{J-1}\sum_{l=k+1}^J\left(y_k^{1/\lambda_{kl}}+y_l^{1/\lambda_{kl}} \right)^{\lambda_{kl}} \]

The pcl model is consistent with random utility maximisation if \(0<\lambda_{kl}\leq 1\) and the multinomial logit results if \(\lambda_{kl}=1 \;\forall (k,l)\). The resulting probabilities are:

\[ P_l = \frac{\sum_{k\neq l}e^{V_l/\lambda_{lk}}\left(e^{V_k/\lambda_{lk}} + e^{V_l/\lambda_{lk}}\right)^{\lambda_{lk}-1}} {\sum_{k=1}^{J-1}\sum_{l=k+1}^{J}\left(e^{V_k/\lambda_{lk}} + e^{V_l/\lambda_{lk}}\right)^{\lambda_{lk}}} \]

which can be expressed as a sum of \(J-1\) product of a conditional probability of choosing the alternative and the marginal probability of choosing the nest:

\[ P_l=\sum_{k\neq l}P_{l\mid lk} P_{lk} \]

with:

\[ P_{l \mid lk} = \frac{e^{V_l/\lambda_{lk}}}{e^{V_k/\lambda_{lk}} + e^{V_l/\lambda_{lk}}} \]

\[ P_{lk}= \frac{\left(e^{V_k/\lambda_{lk}} + e^{V_l/\lambda_{lk}}\right)^{\lambda_{lk}}}{\sum_{k=1}^{J-1}\sum_{l=k+1}^{J}\left(e^{V_k/\lambda_{lk}} + e^{V_l/\lambda_{lk}}\right)^{\lambda_{lk}}} \]

We reproduce the example used by Koppelman and Wen (2000) on the same subset of the `ModeCanada` than the one used by Bhat (1995). Three modes are considered and there are therefore three nests. The elasticity of the train-air nest is set to one. To estimate this model, one has to set the `nests` argument to `"pcl"`. All the nests of two alternatives are then automatically created. The restriction on the nest elasticity for the train-air nest is performed by using the `constPar` argument.

    library(mlogit)
    busUsers <- with(ModeCanada, case[choice == 1 & alt == 'bus'])
    Bhat <- subset(ModeCanada, ! case %in% busUsers &
                               alt != 'bus' & noalt == 4)
    Bhat$alt <- Bhat$alt[drop = TRUE]
    Bhat <- dfidx(Bhat, idx = c("case", "alt"), choice = "choice",
                  idnames = c("chid", "alt"))
    pcl <- mlogit(choice ~ freq + cost + ivt + ovt, Bhat, reflevel = 'car',
                  nests = 'pcl', constPar=c('iv:train.air'))
    gaze(pcl)

                        Estimate Std. Error z-value Pr(>|z|)
    (Intercept):train  1.3043932  0.1654423   7.884 3.11e-15
    (Intercept):air    1.9901292  0.3557061   5.595 2.21e-08
    freq               0.0653783  0.0043569  15.006  < 2e-16
    cost              -0.0244856  0.0031657  -7.735 1.04e-14
    ivt               -0.0076154  0.0006737 -11.303  < 2e-16
    ovt               -0.0322399  0.0023710 -13.598  < 2e-16
    iv:car.train       0.4212904  0.0861343   4.891 1.00e-06
    iv:car.air         0.2712332  0.0906132   2.993  0.00276

## The rank-ordered logit model

Sometimes, in stated-preference surveys, the respondents are asked to give the full rank of their preference for all the alternative, and not only the prefered alternative. The relevant model for this kind of data is the rank-ordered logit model, which can be estimated as a standard multinomial logit model if the data is reshaped correctly.1

The ranking can be decomposed in a series of choices of the best alternative within a decreasing set of available alternatives. For example, with 4 alternatives, the probability that the ranking would be 3-1-4-2 can be writen as follow:

- alternative 3 is in the first position, the probability is then \(\frac{e^{\beta^{\top}x_3}}{e^{\beta^{\top}x_1}+e^{\beta^{\top}x_2}+e^{\beta^{\top}x_3}+e^{\beta^{\top}x_4}}\),
- alternative 1 is in second position, the relevant probability is the logit probability that 1 is the chosen alternative in the set of alternatives (1-2-4): \(\frac{e^{\beta^{\top}x_1}}{e^{\beta^{\top}x_1}+e^{\beta^{\top}x_2}+e^{\beta^{\top}x_4}}\),
- alternative 4 is in third position, the relevant probability is the logit probability that 4 is the chosen alternative in the set of alternatives (2-4): \(\frac{e^{\beta^{\top}x_4}}{e^{\beta^{\top}x_2}+e^{\beta^{\top}x_4}}\),
- the probability of the full ranking is then simply the product of these 3 probabilities.

This model can therefore simply be fitted as a multinomial logit model ; the ranking for one individual amoung J alternatives is writen as \(J-1\) choices among \(J, J-1, ..., 2\) alternatives.

The estimation of the rank-ordered logit model is illustrated using the `Game` data set (Fok, Paap, and Van Dijk 2012). Respondents are asked to rank 6 gaming platforms. The covariates are a dummy `own` which indicates whether a specific platform is curently owned, the age of the respondent (`age`) and the number of hours spent on gaming per week (`hours`). The data set is available in wide (`game`) and long (`game2`) format. In wide format, the consists on \(J\) columns which indicate the ranking of each alternative.

    data("Game", package = "mlogit")
    data("Game2", package = "mlogit")
    head(Game,2)
    ##   ch.Xbox ch.PlayStation ch.PSPortable ch.GameCube ch.GameBoy ch.PC
    ## 1       2              1             3           5          6     4
    ## 2       4              2             3           5          6     1
    ##   own.Xbox own.PlayStation own.PSPortable own.GameCube own.GameBoy
    ## 1        0               1              0            0           0
    ## 2        0               1              0            0           0
    ##   own.PC age hours
    ## 1      1  33  2.00
    ## 2      1  19  3.25
    head(Game2, 7)
    ##   age hours    platform ch own chid
    ## 1  33  2.00     GameBoy  6   0    1
    ## 2  33  2.00    GameCube  5   0    1
    ## 3  33  2.00          PC  4   1    1
    ## 4  33  2.00 PlayStation  1   1    1
    ## 5  33  2.00  PSPortable  3   0    1
    ## 6  33  2.00        Xbox  2   0    1
    ## 7  19  3.25     GameBoy  6   0    2
    nrow(Game)
    ## [1] 91
    nrow(Game2)
    ## [1] 546

Note that `Game` contains 91 rows (there are 91 individuals) and that `Game2` contains 546 rows (\(91\) individuals \(times\) 6 alternatives)

To use `dfidx`, the `ranked` argument should `TRUE`:

    G <- dfidx(Game, varying = 1:12, choice = "ch", ranked = TRUE,
               idnames = c("chid", "alt"))
    G <- dfidx(Game2, choice = "ch", ranked = TRUE, idx = c("chid", "platform"),
               idnames = c("chid", "alt"))
    print(G, n = 3)

    ~~~~~~~
     first 3 observations out of 1820
    ~~~~~~~
      age hours    ch own    idx
    1  33     2 FALSE   0 1:eBoy
    2  33     2 FALSE   0 1:Cube
    3  33     2 FALSE   1   1:PC

    ~~~ indexes ~~~~
      idx1 chid      alt
    1    1    1  GameBoy
    2    1    1 GameCube
    3    1    1       PC
    indexes:  1, 1, 2

Note that the choice variable is now a logical variable and that the number of row is now 1820 (91 individuals \(\times (6+5+4+3+2)\) alternatives).

Using `PC` as the reference level, we can then reproduce the results of the original reference:

    mlogit(ch ~ own | hours + age, G, reflevel = "PC") |> gaze()

                            Estimate Std. Error z-value Pr(>|z|)
    (Intercept):GameBoy      1.57038    1.60025   0.981 0.326429
    (Intercept):GameCube     1.40410    1.60348   0.876 0.381218
    (Intercept):PSPortable   2.58356    1.62078   1.594 0.110930
    (Intercept):PlayStation  2.27851    1.60699   1.418 0.156227
    (Intercept):Xbox         2.73377    1.53610   1.780 0.075127
    own                      0.96337    0.19040   5.060 4.20e-07
    hours:GameBoy           -0.23561    0.05213  -4.520 6.19e-06
    hours:GameCube          -0.18707    0.05102  -3.667 0.000246
    hours:PSPortable        -0.23369    0.04941  -4.729 2.25e-06
    hours:PlayStation       -0.12920    0.04468  -2.891 0.003834
    hours:Xbox              -0.17301    0.04570  -3.786 0.000153
    age:GameBoy             -0.07359    0.07863  -0.936 0.349344
    age:GameCube            -0.06757    0.07763  -0.870 0.384055
    age:PSPortable          -0.08867    0.07942  -1.116 0.264230
    age:PlayStation         -0.06701    0.07936  -0.844 0.398515
    age:Xbox                -0.06666    0.07520  -0.886 0.375423

## Bibliography

Beggs, S., S. Cardell, and J. Hausman. 1981. “Assessing the Potential Demand for Electric Cars.” Journal of Econometrics 16: 1–19.
Bhat, Chandra R. 1995. “A Heteroscedastic Extreme Value Model of Intercity Travel Mode Choice.” Transportation Research Part B: Methodological 29 (6): 471–83. https://doi.org/10.1016/0191-2615(95)00015-6.
Chapman, R., and R. Staelin. 1982. “Exploiting Rank Ordered Choice Set Data Within the Stochastic Utility Model.” Journal of Marketing Research 14: 288–301.
Fok, Dennis, Richard Paap, and Bram Van Dijk. 2012. “A Rank-Ordered Logit Model with Unobserved Heterogeneity in Ranking Capatibilities.” Journal of Applied Econometrics 27 (5): 831–46. https://doi.org/10.1002/jae.1223.
Hausman, J., and P. Ruud. 1987. “Specifying and Testing Econometric Models for Rank-Ordered Data.” Journal of Econometrics 34: 83–103.
Koppelman, Frank S., and Chieh-Hua Wen. 2000. “The Paired Combinatorial Logit Model: Properties, Estimation and Application.” Transportation Research Part B: Methodological 34 (2): 75–89. https://doi.org/10.1016/S0191-2615(99)00012-0.

1. See for example Beggs, Cardell, and Hausman (1981), Chapman and Staelin (1982) and Hausman and Ruud (1987).↩︎