1. Probit response models
The logistic response function suits well to situations where we have a zero-one valued response variable. However, any function that restricts the model predictions to be in the range between zero and one would do the same job.
The following lesson you learn how to use the probit response function for explaining customer purchase decisions.
2. Probit response function
Although the logistic and the probit response functions are extremely similar, we can see that the logistic has only slightly flatter tails. In marketing research, there is a deeper theoretical model underpinning the usage of the probit response function:
We can think of the purchase probability as the underlying latent propensity that a customer purchases a certain product. If that latent propensity passes a threshold - typically 0 - then the purchase event will occur.
3. A probit model for beer demand
Much like for the logistic response model, for the probit response model, the latent propensity to purchase a certain product is influenced by the marketing activities - But we theoretically assume an underlying regression for a continuous unobservable response variable.
In our example, we assume that the latent propensity depends on the price-dot-ratio predictor. The corresponding probit response model can again be estimated by using the function glm(). We only have to add the statement 'link equals probit' to the 'binomial' 'family' argument. The result is stored in an object named 'probit-dot-model'.
As usual, the model coefficients can be obtained by applying the coefficient function on the probit-dot-model object. In the probit response model, the price-dot-ratio coefficient is taken to be the z-value of a normal distribution. Therefore, lower values for the price-dot-ratio coefficient mean that the event is less likely to happen.
4. Logistic vs. probit
We compare the coefficients of the logistic and the probit response models by using the function cbind(). We see that the coefficients for both models differ because they are estimated on a different scale. In practice, the coefficients differ by a factor of about 1-point-6 because of the different distributional assumptions underlying the models.
5. Average marginal effects
Compared to the logistic response model, there is a big difference in terms of interpreting the output for the probit response model. The coefficient of the logistic response model is directly interpretable as a log-odds - while the results obtained for the probit response model are essentially uninterpretable z-values. Therefore we have to be careful about the interpretation of the effect estimates. But we can still interpret the change in the latent propensity for each individual and average this effect over all individuals.
We calculate the average marginal effects for the logistic and the probit-dot-model by using the function margins() from the margins package.
If you look at marginal effects, the logistic and probit.model will make essentially the same predictions.
As both models give the same result, but the coefficients of the logistic model are easier to interpret in practice, the logistic model is often preferred.
6. Let's practice!
Let’s compare the probit to the logistic model in practice.