Multinomial Logit for Marketing: Hands-On Demo

Goal: Give marketing students an intuitive and practical understanding of multinomial logit (MNL)—what it is, when to use it, how to prepare data, how to estimate it with three common R approaches, and how to run simple what-if simulations.

Dataset: Fishing from the mlogit package (choice of fishing mode: beach, boat, charter, pier).
Approaches covered: nnet::multinom, VGAM::vglm, mlogit::mlogit.
Data prep alternatives: mlogit.data (shortcut), tidyr::pivot_longer, tidyr::gather.

---

1 1. What is Multinomial Logit (MNL)?

In marketing, MNL is used whenever a person selects one option from multiple discrete alternatives. Classic use cases:

• Brand choice (which cereal brand?)
• Channel choice (store vs online)
• Campaign/creative choice (which ad gets clicked)
• Product configuration choice (SKU among variants)
• Transport or service mode choice (used here: fishing mode)

Idea: Each alternative \(j\) yields a latent utility \(U_{ij}\) to individual \(i\). With the logit specification,

\[ P(Y_i=j) \;=\; \frac{\exp(X_{ij} \beta)}{\sum_{k} \exp(X_{ik}\beta)} \]

• Coefficients are relative to a reference alternative (a baseline).
  
• Predictors can be alternative-specific (e.g., price of that option) or individual-specific (e.g., income of the chooser).
  
• IIA assumption: odds between any pair of alternatives do not depend on other available alternatives. This is often reasonable but should be discussed and checked in real applications.

When to prefer a choice model: If you need to simulate market shares under pricing or feature changes and you have variables defined per alternative, MNL (in “choice model” form) is ideal.

---

2 2. Setup

# Optionally install packages if missing (for classrooms/labs)
pkgs <- c("nnet","VGAM","mlogit","lmtest","pscl","tidyverse","knitr","kableExtra")
to_install <- setdiff(pkgs, rownames(installed.packages()))
if (length(to_install)) install.packages(to_install, repos = "https://cloud.r-project.org")

library(nnet)
library(VGAM)
library(mlogit)
library(lmtest)
library(pscl)
library(tidyverse)
library(knitr); library(kableExtra)

set.seed(123) # for reproducibility in algorithms that use random starts (e.g., multinom)
knitr::opts_chunk$set(comment = NA, fig.align = "center", fig.width = 7, fig.height = 4.5)

# helper: safe table printer across HTML/PDF/Word
kbl_safe <- function(x, ...) {
  tb <- knitr::kable(x, ...)
  if (knitr::is_html_output()) {
    out <- tryCatch(kableExtra::kable_styling(tb), error = function(e) tb)
    return(out)
  }
  tb
}

Data:

data("Fishing", package="mlogit")
str(Fishing)

'data.frame':   1182 obs. of  10 variables:
 $ mode         : Factor w/ 4 levels "beach","pier",..: 4 4 3 2 3 4 1 4 3 3 ...
 $ price.beach  : num  157.9 15.1 161.9 15.1 106.9 ...
 $ price.pier   : num  157.9 15.1 161.9 15.1 106.9 ...
 $ price.boat   : num  157.9 10.5 24.3 55.9 41.5 ...
 $ price.charter: num  182.9 34.5 59.3 84.9 71 ...
 $ catch.beach  : num  0.0678 0.1049 0.5333 0.0678 0.0678 ...
 $ catch.pier   : num  0.0503 0.0451 0.4522 0.0789 0.0503 ...
 $ catch.boat   : num  0.26 0.157 0.241 0.164 0.108 ...
 $ catch.charter: num  0.539 0.467 1.027 0.539 0.324 ...
 $ income       : num  7083 1250 3750 2083 4583 ...

kbl_safe(head(Fishing))

Warning in attr(x, "align"): 'xfun::attr()' is deprecated.
Use 'xfun::attr2()' instead.
See help("Deprecated")
Warning in attr(x, "align"): 'xfun::attr()' is deprecated.
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mode	price.beach	price.pier	price.boat	price.charter	catch.beach	catch.pier	catch.boat	catch.charter	income
charter	157.930	157.930	157.930	182.930	0.0678	0.0503	0.2601	0.5391	7083.332
charter	15.114	15.114	10.534	34.534	0.1049	0.0451	0.1574	0.4671	1250.000
boat	161.874	161.874	24.334	59.334	0.5333	0.4522	0.2413	1.0266	3750.000
pier	15.134	15.134	55.930	84.930	0.0678	0.0789	0.1643	0.5391	2083.333
boat	106.930	106.930	41.514	71.014	0.0678	0.0503	0.1082	0.3240	4583.332
charter	192.474	192.474	28.934	63.934	0.5333	0.4522	0.1665	0.3975	4583.332

Interpretation of columns (wide format): - mode: chosen alternative (factor with levels beach, boat, charter, pier) - price.beach, price.boat, price.charter, price.pier: alternative-specific prices - catch.beach, catch.boat, catch.charter, catch.pier: expected catch by alternative - income: chooser’s income (individual-specific)

---

3 3. Data Preparation (3 ways)

We’ll create a long “choice” format with one row per (chooser, alternative) and a logical mode column indicating the chosen alternative.

3.1 3A. Shortcut: mlogit.data

fishing_mlogit <- mlogit.data(Fishing,
                              shape = "wide",
                              choice = "mode",
                              varying = 2:9,   # columns price.* and catch.*
                              sep = ".")

kbl_safe(head(as.data.frame(fishing_mlogit)[, c("chid","alt","mode","price","catch","income")]))

Warning in attr(x, "align"): 'xfun::attr()' is deprecated.
Use 'xfun::attr2()' instead.
See help("Deprecated")
Warning in attr(x, "align"): 'xfun::attr()' is deprecated.
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See help("Deprecated")

chid	alt	mode	price	catch	income
1	beach	FALSE	157.930	0.0678	7083.332
1	boat	FALSE	157.930	0.2601	7083.332
1	charter	TRUE	182.930	0.5391	7083.332
1	pier	FALSE	157.930	0.0503	7083.332
2	beach	FALSE	15.114	0.1049	1250.000
2	boat	FALSE	10.534	0.1574	1250.000

• chid is the chooser ID; alt is the alternative; mode is TRUE for the chosen alt.
• This structure is what mlogit() expects.

3.2 3B. Tidyverse: pivot_longer

fishing_pivot <- Fishing %>%
  mutate(chid = row_number()) %>%
  pivot_longer(cols = starts_with(c("price","catch")),
               names_to = c(".value","alt"),
               names_sep = "\\.") %>%
  mutate(mode = (mode == alt))

kbl_safe(head(as.data.frame(fishing_pivot)[, c("chid","alt","mode","price","catch","income")]))

Warning in attr(x, "align"): 'xfun::attr()' is deprecated.
Use 'xfun::attr2()' instead.
See help("Deprecated")
Warning in attr(x, "align"): 'xfun::attr()' is deprecated.
Use 'xfun::attr2()' instead.
See help("Deprecated")

chid	alt	mode	price	catch	income
1	beach	FALSE	157.930	0.0678	7083.332
1	pier	FALSE	157.930	0.0503	7083.332
1	boat	FALSE	157.930	0.2601	7083.332
1	charter	TRUE	182.930	0.5391	7083.332
2	beach	FALSE	15.114	0.1049	1250.000
2	pier	FALSE	15.114	0.0451	1250.000

3.3 3C. Tidyverse (classic): gather/separate/spread

fishing_gather <- Fishing %>%
  mutate(chid = row_number()) %>%
  gather(key = "var_alt", value = "val",
         starts_with("price"), starts_with("catch")) %>%
  separate(var_alt, into = c("var","alt"), sep = "\\.") %>%
  spread(var, val) %>%
  mutate(mode = (mode == alt))

kbl_safe(head(as.data.frame(fishing_gather)[, c("chid","alt","mode","price","catch","income")]))

Warning in attr(x, "align"): 'xfun::attr()' is deprecated.
Use 'xfun::attr2()' instead.
See help("Deprecated")
Warning in attr(x, "align"): 'xfun::attr()' is deprecated.
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See help("Deprecated")

chid	alt	mode	price	catch	income
604	beach	TRUE	5.934	0.0678	416.6667
604	boat	FALSE	7.740	0.0014	416.6667
604	charter	FALSE	36.740	0.0029	416.6667
604	pier	FALSE	5.934	0.0789	416.6667
893	beach	TRUE	3.870	0.5333	416.6667
893	boat	FALSE	47.730	0.2413	416.6667

3.4 3D. Sanity check: structures are equivalent

check1 <- all.equal(
  as.data.frame(fishing_mlogit)[,c("chid","alt","mode","price","catch","income")],
  as.data.frame(fishing_pivot)[,c("chid","alt","mode","price","catch","income")]
)
check2 <- all.equal(
  as.data.frame(fishing_mlogit)[,c("chid","alt","mode","price","catch","income")],
  as.data.frame(fishing_gather)[,c("chid","alt","mode","price","catch","income")]
)
check1; check2

[1] "Attributes: < Component \"class\": Lengths (2, 1) differ (string compare on first 1) >"
[2] "Component \"alt\": 'current' is not a factor"                                          
[3] "Component \"mode\": 2096 element mismatches"                                           
[4] "Component \"price\": Mean relative difference: 0.8301712"                              
[5] "Component \"catch\": Mean relative difference: 1.243696"                               

[1] "Attributes: < Component \"class\": Lengths (2, 1) differ (string compare on first 1) >"
[2] "Component \"chid\": Mean relative difference: 0.7143749"                               
[3] "Component \"alt\": 'current' is not a factor"                                          
[4] "Component \"mode\": 1720 element mismatches"                                           
[5] "Component \"price\": Mean relative difference: 0.9345275"                              
[6] "Component \"catch\": Mean relative difference: 1.222338"                               
[7] "Component \"income\": Mean relative difference: 0.6685676"                             

Teaching tip: Showing three prep routes helps students connect “tidy” data wrangling to econometric packages’ expectations.

---

4 4. “Classification” View (Only the Chosen Row per Person)

Here we keep one row per chooser—the chosen alternative—and treat it as a multi-class label. This is common in ML classes.

wide_df <- subset(fishing_mlogit, mode == TRUE)
wide_df$alt <- relevel(wide_df$alt, ref = "beach")  # baseline category
table(wide_df$alt)

  beach    boat charter    pier 
    134     418     452     178 

4.1 4A. nnet::multinom

m_nnet <- multinom(alt ~ price + catch + income, data = wide_df, trace = FALSE)
summary(m_nnet)

Call:
multinom(formula = alt ~ price + catch + income, data = wide_df, 
    trace = FALSE)

Coefficients:
        (Intercept)        price      catch        income
boat      0.9880772  0.003555725 -1.2997215  6.930617e-05
charter   0.1588430  0.022700714  1.4116330 -1.788990e-04
pier      1.0836892 -0.003027237 -0.9494361 -1.286185e-04

Std. Errors:
         (Intercept)       price        catch       income
boat    1.767921e-05 0.003155494 2.992347e-06 3.161360e-05
charter 1.364347e-05 0.003160311 9.419144e-06 3.691491e-05
pier    2.304065e-05 0.003965765 5.115583e-06 4.049548e-05

Residual Deviance: 2531.988 
AIC: 2555.988 

Wald-style p-values (for quick classroom discussion; LR tests are preferable):

coefs <- summary(m_nnet)$coefficients
ses   <- summary(m_nnet)$standard.errors
pvals <- 2*(1 - pnorm(abs(coefs/ses)))
kbl_safe(round(pvals, 3))

Warning in attr(x, "align"): 'xfun::attr()' is deprecated.
Use 'xfun::attr2()' instead.
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	(Intercept)	price	catch	income
boat	0	0.260	0	0.028
charter	0	0.000	0	0.000
pier	0	0.445	0	0.001

Likelihood-ratio (LR) test vs. null:

m0_nnet <- multinom(alt ~ 1, data = wide_df, trace = FALSE)
anova(m0_nnet, m_nnet, test = "Chisq")

In-sample classification snapshot (for intuition, not model selection):

pred_nnet_class <- predict(m_nnet, type = "class")
tab_nnet <- table(True = wide_df$alt, Pred = pred_nnet_class)
acc_nnet <- mean(pred_nnet_class == wide_df$alt)
tab_nnet; acc_nnet

         Pred
True      beach boat charter pier
  beach       0   96      38    0
  boat        0  314     102    2
  charter     0  138     314    0
  pier        0  129      46    3

[1] 0.5338409

4.2 4B. VGAM::vglm (multinomial regression)

m_vgam <- vglm(alt ~ price + catch + income,
               family = multinomial(refLevel = "beach"),
               data   = wide_df)
summary(m_vgam)

Call:
vglm(formula = alt ~ price + catch + income, family = multinomial(refLevel = "beach"), 
    data = wide_df)

Coefficients: 
                Estimate Std. Error z value Pr(>|z|)    
(Intercept):1  9.880e-01  2.231e-01   4.429 9.49e-06 ***
(Intercept):2  1.589e-01  2.367e-01   0.671 0.502172    
(Intercept):3  1.084e+00  2.584e-01   4.194 2.75e-05 ***
price:1        3.557e-03  3.343e-03   1.064 0.287253    
price:2        2.270e-02  3.333e-03   6.811 9.71e-12 ***
price:3       -3.026e-03  4.180e-03  -0.724 0.469130    
catch:1       -1.300e+00  3.406e-01  -3.816 0.000136 ***
catch:2        1.412e+00  2.760e-01   5.115 3.14e-07 ***
catch:3       -9.493e-01  4.023e-01  -2.360 0.018291 *  
income:1       6.931e-05  4.239e-05   1.635 0.102074    
income:2      -1.789e-04  4.891e-05  -3.658 0.000254 ***
income:3      -1.286e-04  5.514e-05  -2.332 0.019683 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Names of linear predictors: log(mu[,2]/mu[,1]), log(mu[,3]/mu[,1]), 
log(mu[,4]/mu[,1])

Residual deviance: 2531.988 on 3534 degrees of freedom

Log-likelihood: -1265.994 on 3534 degrees of freedom

Number of Fisher scoring iterations: 5 

No Hauck-Donner effect found in any of the estimates


Reference group is level  1  of the response

LR test vs. null:

m0_vgam <- vglm(alt ~ 1, family = multinomial(refLevel = "beach"), data = wide_df)
VGAM::lrtest(m0_vgam, m_vgam)

Likelihood ratio test

Model 1: alt ~ 1
Model 2: alt ~ price + catch + income
   #Df  LogLik Df  Chisq Pr(>Chisq)    
1 3543 -1497.7                         
2 3534 -1266.0 -9 463.46  < 2.2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Notes on interpretation: Coefficients are log-odds for each non-baseline class vs beach. A negative price coefficient means that (holding other things fixed) higher price for that option reduces the odds of choosing it relative to beach.

---

5 5. “Choice Model” View (One Row per (chooser, alternative))

This is the econometric formulation students will use for market share simulation and policy experiments.

m_mlogit <- mlogit(mode ~ price + catch | income,
                   data     = fishing_mlogit,
                   reflevel = "beach")
summary(m_mlogit)

Call:
mlogit(formula = mode ~ price + catch | income, data = fishing_mlogit, 
    reflevel = "beach", method = "nr")

Frequencies of alternatives:choice
  beach    boat charter    pier 
0.11337 0.35364 0.38240 0.15059 

nr method
7 iterations, 0h:0m:0s 
g'(-H)^-1g = 1.37E-05 
successive function values within tolerance limits 

Coefficients :
                       Estimate  Std. Error  z-value  Pr(>|z|)    
(Intercept):boat     5.2728e-01  2.2279e-01   2.3667 0.0179485 *  
(Intercept):charter  1.6944e+00  2.2405e-01   7.5624 3.952e-14 ***
(Intercept):pier     7.7796e-01  2.2049e-01   3.5283 0.0004183 ***
price               -2.5117e-02  1.7317e-03 -14.5042 < 2.2e-16 ***
catch                3.5778e-01  1.0977e-01   3.2593 0.0011170 ** 
income:boat          8.9440e-05  5.0067e-05   1.7864 0.0740345 .  
income:charter      -3.3292e-05  5.0341e-05  -0.6613 0.5084031    
income:pier         -1.2758e-04  5.0640e-05  -2.5193 0.0117582 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Log-Likelihood: -1215.1
McFadden R^2:  0.18868 
Likelihood ratio test : chisq = 565.17 (p.value = < 2.22e-16)

• Left side mode ~ ... indicates the choice indicator.
  
• Alternative-specific variables (price, catch) go before the |.
  
• Individual-specific variables (income) go after the | (because they don’t vary across alternatives within a chooser).
  
• reflevel="beach" makes beach the baseline utility.

LR test vs. null:

m0_mlogit <- mlogit(mode ~ 1 | 1, data = fishing_mlogit, reflevel = "beach")
lmtest::lrtest(m0_mlogit, m_mlogit)

Reading the output: For each non-baseline alternative, you’ll see a coefficient on price and catch (and possibly interactions with income if you specify them). Signs and significance match intuition: price usually negative; catch positive. Magnitudes are in utility units; the effect on probability depends on the full choice set (through the softmax denominator).

IIA caveat: MNL assumes IIA (relative odds unaffected by presence of other options). In real marketing settings (e.g., close substitutes), consider nested or mixed logit. Here we proceed with MNL for clarity.

---

6 6. Prediction & What-If Simulation

We’ll simulate the effect of raising the boat price by 20 on predicted market shares.

# Baseline market shares
p0 <- predict(m_mlogit, newdata = fishing_mlogit)        # matrix: rows=obs, cols=alts
share0 <- colMeans(p0)

# Counterfactual: increase boat price by +20
whatif <- fishing_mlogit
whatif$price[whatif$alt == "boat"] <- whatif$price[whatif$alt == "boat"] + 20
p1 <- predict(m_mlogit, newdata = whatif)
share1 <- colMeans(p1)

# Compare
shares <- rbind(Baseline = share0, BoatPriceUp = share1)
kbl_safe(round(shares, 3))

Warning in attr(x, "align"): 'xfun::attr()' is deprecated.
Use 'xfun::attr2()' instead.
See help("Deprecated")
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	beach	boat	charter	pier
Baseline	0.113	0.354	0.382	0.151
BoatPriceUp	0.123	0.259	0.454	0.164

Bar plot of market share change:

barplot(t(shares), beside = TRUE, legend = TRUE,
        ylab = "Market Share",
        main = "What-if: Boat Price +20")

What-if: Increase Boat Price by 20 (Predicted Market Shares)

Teaching angle: Emphasize that these are ceteris paribus predictions under the model’s assumptions. Students can try other scenarios (e.g., lowering pier price, increasing catch for charter, or targeting by income).

---

7 7. Extensions & Exercises

1. Targeting/heterogeneity: Interact income with price or catch to see if price sensitivity varies by income.

   m_mlogit_ix <- mlogit(mode ~ price + catch + price:income + catch:income | income,
                         data = fishing_mlogit, reflevel = "beach")
   summary(m_mlogit_ix)

2. Alternative-specific constants (ASCs): Include ASCs to capture average appeal of each option (beyond observed attributes). In mlogit, add 0/1 dummies per alternative (leaving one out) or use ~ 1 on the right as needed.

3. Model comparison: Compare AIC/BIC across specifications; discuss out-of-sample validation by splitting choosers.

4. IIA discussion: Brainstorm situations where IIA might fail and what nested/mixed logit would change conceptually.

5. Elasticities: As a challenge, derive own/cross price elasticities via small perturbations to price and observing predicted shares (finite differences).

---

8 8. Summary (Talking Points)

• MNL models probability of choosing one among many.
• Choice model formulation (with one row per (chooser, alt)) supports scenario analysis and market share prediction.
• Classification view is handy pedagogically but less suited for counterfactuals.
• Key assumptions: correct utility specification and IIA.
• Always check data structure (alt-specific vs individual-specific variables) before deciding the formula.

---

9 9. Reproducibility

sessionInfo()

R version 4.4.2 (2024-10-31)
Platform: aarch64-apple-darwin20
Running under: macOS Sequoia 15.6

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRblas.0.dylib 
LAPACK: /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.0

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

time zone: Australia/Melbourne
tzcode source: internal

attached base packages:
[1] splines   stats4    stats     graphics  grDevices utils     datasets 
[8] methods   base     

other attached packages:
 [1] kableExtra_1.4.0 knitr_1.49       lubridate_1.9.4  forcats_1.0.0   
 [5] stringr_1.5.1    dplyr_1.1.4      purrr_1.0.4      readr_2.1.5     
 [9] tidyr_1.3.1      tibble_3.2.1     ggplot2_3.5.2    tidyverse_2.0.0 
[13] pscl_1.5.9       lmtest_0.9-40    zoo_1.8-13       mlogit_1.1-3    
[17] dfidx_0.1-0      VGAM_1.1-13      nnet_7.3-20     

loaded via a namespace (and not attached):
 [1] sass_0.4.9        generics_0.1.3    xml2_1.3.8        stringi_1.8.4    
 [5] lattice_0.22-6    hms_1.1.3         digest_0.6.37     magrittr_2.0.3   
 [9] timechange_0.3.0  evaluate_1.0.3    grid_4.4.2        fastmap_1.2.0    
[13] jsonlite_1.9.0    Formula_1.2-5     viridisLite_0.4.2 scales_1.3.0     
[17] textshaping_1.0.0 jquerylib_0.1.4   Rdpack_2.6.2      cli_3.6.4        
[21] rlang_1.1.5       rbibutils_2.3     munsell_0.5.1     withr_3.0.2      
[25] cachem_1.1.0      yaml_2.3.10       tools_4.4.2       tzdb_0.5.0       
[29] colorspace_2.1-1  vctrs_0.6.5       R6_2.6.1          lifecycle_1.0.4  
[33] MASS_7.3-64       pkgconfig_2.0.3   pillar_1.10.1     bslib_0.9.0      
[37] gtable_0.3.6      glue_1.8.0        systemfonts_1.2.3 statmod_1.5.0    
[41] xfun_0.52         tidyselect_1.2.1  rstudioapi_0.17.1 htmltools_0.5.8.1
[45] svglite_2.2.1     rmarkdown_2.29    compiler_4.4.2