Illustrating the Problem
Consider an example with the following price information.
set.seed(1976)
# create random variables
toocheap <- sort(round(rnorm(n = 50, mean = 8, sd = 1.5), digits = 2))
tooexpensive <- sort(round(rnorm(n = 50, mean = 20, sd = 1), digits = 2))
cheap <- sort(round(rnorm(n = 50, mean = 15, sd = 3), digits = 2))
expensive <- sort(round(rnorm(n = 50, mean = 16, sd = 3), digits = 2))
# setting up alternative "cheap"/"expensive" variables with a price step
pricestep_cheap <- which(cheap > 14.25 & cheap < 17.5)
cheap_alt <- cheap
cheap_alt[pricestep_cheap] <- round(mean(cheap_alt[pricestep_cheap]), digits = 2)
pricestep_expensive <- which(expensive > 14 & expensive < 16)
expensive_alt <- expensive
expensive_alt[pricestep_expensive] <- round(mean(expensive_alt[pricestep_expensive]), digits = 2)When we run a Price Sensitivity Meter Analysis on the data and visualize it, we get the following result. (The plot is slightly simplified: It does not show the “too cheap” and “too expensive” curves, as they are not relevant for the example. Moreover, it is slightly zoomed in.)
library(pricesensitivitymeter)
# Running the analysis with price steps and without interpolation
psm_steps <- psm_analysis(toocheap = toocheap,
cheap = cheap_alt,
expensive = expensive_alt,
tooexpensive = tooexpensive,
validate = TRUE,
interpolate = FALSE)
library(ggplot2)
# Plot with price steps
psmplot_steps <- ggplot(data = psm_steps$data_vanwestendorp, aes(x = price)) +
geom_line(aes(y = ecdf_not_cheap, # line: not cheap
colour = "not cheap"),
size = 1) +
geom_line(aes(y = ecdf_not_expensive, # line: not expensive
colour = "not expensive"),
size = 1) +
geom_rug() +
annotate(geom = "point", # Indifference Price Point (intersection of "not cheap" and "not expensive")
x = psm_steps$idp,
y = psm_steps$data_vanwestendorp$ecdf_not_expensive[psm_steps$data_vanwestendorp$price == psm_steps$idp],
size = 5,
shape = 18,
colour = "#D55E00")
psmplot_steps +
labs(x = "Price",
y = "Share of Respondents (0-1)",
title = "Price Sensitivity Meter Plot (without interpolation)") +
scale_colour_manual(name = "Legend",
values = c("not cheap" = "#009E73",
"not expensive" = "#D55E00")) +
annotate(geom = "text", # Label of Indifference Price Point
x = psm_steps$idp + 1,
y = psm_steps$data_vanwestendorp$ecdf_not_expensive[psm_steps$data_vanwestendorp$price == psm_steps$idp] - 0.05,
label = paste("IDP: ", psm_steps$idp)) +
coord_cartesian(xlim = c(10, 20)) +
theme_minimal()
The Indifference Price Point (IDP) in this plot is not at the actual intersection of the “not cheap” and the “not expensive” curve. This is due to the data structure: We have sparse information for prices between 14 and 15. This can be seen in the rug plot close to the x-axis: Each vertical line indicates a price for which we have information in our data - however, there is a considerable gap around 14-15 (caused by the initial data manipulation that added price steps). The combination of price steps (as can be seen in the steep slope of the curves in this area) and sparse data points creates this problem. If the interpolate argument is set to FALSE, the psm_analysis() function uses the first price for which the “not expensive” curve is higher than the “not cheap” curve, which is the case for a price of 15.13 in our example.
A larger sample size is usually sufficient to overcome this issue. However, this might not always be possible. Therefore, the psm_analysis() function has an optional argument which helps to interpolate the pricing curve in cases of sparse information.
