Logistic regression

The easiest (and most basic) way to start analyzing the conjoint data is with logistic regression. Note that I’m not endorsing this use of logistic regression in conjoint analysis, because nowadays it has become a standard to compensate for mixed effects (see package lme4). However, for the purposes of this post, I’m going to carry on with the simple glm which is sufficiently good for our illustration. In any case, my experience is that the models yield similar results in most cases.

glm_set <- class_action_conjoint %>% 
  mutate(score_selection = score_selection/100) %>% 
  mutate(compensation_push_pull = factor(compensation_push_pull,
                                         levels = c("pull", "push"),
                                         ordered = F),
         compensation_type = factor(compensation_type,
                                    levels = 
                                      c("another_product",
                                        "refund_next_purchase",
                                        "coupon",
                                        "bank_cheque",
                                        "credit_cart"),
                                    ordered = F)) %>% 
  select(-`Set Number`, -`Card Number`, -`Response ID`) %>% 
  mutate(compensation_amount_ILS = factor(compensation_amount_ILS, levels = c(25, 50, 75, 100)))

conjoint_glm_model <- glm(data = glm_set %>% 
      select(score_selection, compensation_push_pull, compensation_amount_ILS, compensation_type),
    formula = score_selection ~ .,
    family = binomial())
summary(conjoint_glm_model)
## 
## Call:
## glm(formula = score_selection ~ ., family = binomial(), data = glm_set %>% 
##     select(score_selection, compensation_push_pull, compensation_amount_ILS, 
##         compensation_type))
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -1.8635  -0.8111  -0.5139   0.8796   2.6557  
## 
## Coefficients:
##                                       Estimate Std. Error z value Pr(>|z|)
## (Intercept)                           -3.49667    0.11341 -30.832   <2e-16
## compensation_push_pullpush             0.74140    0.05807  12.767   <2e-16
## compensation_amount_ILS50              1.01476    0.09385  10.813   <2e-16
## compensation_amount_ILS75              1.73623    0.09224  18.823   <2e-16
## compensation_amount_ILS100             2.40149    0.09281  25.876   <2e-16
## compensation_typerefund_next_purchase  0.08431    0.10399   0.811    0.418
## compensation_typecoupon                1.10396    0.09588  11.514   <2e-16
## compensation_typebank_cheque           1.53888    0.09473  16.245   <2e-16
## compensation_typecredit_cart           1.89640    0.09586  19.782   <2e-16
##                                          
## (Intercept)                           ***
## compensation_push_pullpush            ***
## compensation_amount_ILS50             ***
## compensation_amount_ILS75             ***
## compensation_amount_ILS100            ***
## compensation_typerefund_next_purchase    
## compensation_typecoupon               ***
## compensation_typebank_cheque          ***
## compensation_typecredit_cart          ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 8936.7  on 7019  degrees of freedom
## Residual deviance: 7295.6  on 7011  degrees of freedom
## AIC: 7313.6
## 
## Number of Fisher Scoring iterations: 4

Note how most variables (actually all but compensation_typerefund_next_purchase) are significant and with a positive estimate (i.e., odds ratio > 1). This means means that when a certain variable increases, the probability of choosing the package increases, i.e.:

• Passively getting the compensation (“Push”) is better than a required act to get the compensation (“pull”) .
• Any sum of money (50, 75, 100) is better than 25, in an increasing odds ratio.
• Most compensation types (credit card payback, bank cheque, coupon) are significantly better than a complimentary product.

Now comes the interesting part: for example, compare the following three packages. Try to guess which one is more attractive:

It is not that easy to determine between the three. In this situation there is no single strategy which is superior to the others, we can however plot these three packages with the logistic regression response and standard errors. First let’s put them all in a tibble (I also added the best and worst packages).

package_comparison <- tribble(
  ~package_name, ~compensation_push_pull, ~compensation_amount_ILS, ~compensation_type,
  "pkg1", "pull", 25, "credit_cart",
  "pkg2", "pull", 75, "refund_next_purchase",
  "pkg3", "push", 25, "coupon",
  "worst", "pull", 25, "another_product",
  "best", "push", 100, "credit_cart"
) %>% 
  mutate(compensation_amount_ILS = factor(compensation_amount_ILS)) # need to convert to factor - which is how it is modeled in the glm.

predicted_responses <- predict(conjoint_glm_model, newdata = package_comparison, type = "response", se.fit = T)

# lets join these together

package_responses <- package_comparison %>% 
  mutate(fit = predicted_responses$fit,
         se.fit = predicted_responses$se.fit)

package_responses
## # A tibble: 5 x 6
##   package_name compensation_pu~ compensation_am~ compensation_ty~    fit
##   <chr>        <chr>            <fct>            <chr>             <dbl>
## 1 pkg1         pull             25               credit_cart      0.168 
## 2 pkg2         pull             75               refund_next_pur~ 0.158 
## 3 pkg3         push             25               coupon           0.161 
## 4 worst        pull             25               another_product  0.0294
## 5 best         push             100              credit_cart      0.824 
## # ... with 1 more variable: se.fit <dbl>
# P.S. - excuse the "credit_cart" typo (I build the model that way, only then noticed...)

ggplot(package_responses %>% slice(1:3) , aes(x = package_name, y = fit)) + 
  geom_point() +
  geom_errorbar(aes(ymin = fit - se.fit, ymax = fit + se.fit)) + 
  ggtitle("Package comparison (packages 1-3)", subtitle = "Error bars represent the SE") + 
  ylab("Predicted response (glm logit)") + 
  xlab("Package name") +
  scale_y_continuous(labels = scales::percent_format(accuracy = 1))

ggplot(package_responses, aes(x = package_name, y = fit)) + 
  geom_point() +
  geom_errorbar(aes(ymin = fit - se.fit, ymax = fit + se.fit)) + 
  ggtitle("Package comparison (including best and worst packages)", subtitle = "Error bars represent the SE") + 
  ylab("Predicted response (glm logit)") +
  xlab("Package name") +
  scale_y_continuous(labels = scales::percent_format(accuracy = 1))

We see that the three packages (pkg1, pkg2, and pkg3) are relatively similar, within one standard error from one another. When compared to the worst package they are roughly \(\sim8\) times better (via odds ratio), but the best package is \(\sim5\) times better than packages 1-3.

One can use these concepts to illustrate the benefits of each parameter on the different packages, and let the user experience how different features make the packages more or less “attractive”.

As an experiment, I prepared a nice little shiny app which lets the user experiment with the different features: build two packages and then compare them. You can checkout the code at the github repo, or check out the live app here.

Conclusions

Surveys are a popular tool used in class actions (at least in Israel). They can be used to estimate the tradeoffs between various types of compensation or settlement, for example with the use of conjoint analysis.

With a glm model one can tell the differences of various packages, and the odds ratio is a way to illustrate to decision makers a comparison of various options (and how much “more attractive” is one package over another).

A shiny app can be a nice way to illustrate the results of a conjoint analysis, and to let the user experiment with how different features make a specific option better or worse than another option.