Interpolation in Small Samples

2018-10-21

Sparse price information can become a problem when running Price Sensitivity Meter Analyses, particularly in small samples. In order to deal with this issue, the psm_analysis package has an option interpolate to approximate values on the price curves for all values where the actual information from respondents is missing.

This vignette illustrates the underlying problem and explains how the interpolate argument provides a solution.

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)
#> Registered S3 methods overwritten by 'ggplot2':
#>   method         from
#>   [.quosures     rlang
#>   c.quosures     rlang
#>   print.quosures rlang

# Plot with price steps
psmplot_steps <-  ggplot(data = psm_steps$data_vanwestendorp, aes(x = price)) + geom_line(aes(y = ecdf_cheap, # line: cheap colour = "cheap"), size = 1) + geom_line(aes(y = ecdf_expensive, # line: expensive colour = "expensive"), size = 1) + geom_rug() + annotate(geom = "point", # Indifference Price Point (intersection of "cheap" and "expensive") x = psm_steps$idp,
y = psm_steps$data_vanwestendorp$ecdf_cheap[psm_steps$data_vanwestendorp$price == psm_steps$idp], size = 5, shape = 18, colour = "#009E73") psmplot_steps + labs(x = "Price", y = "Share of Respondents (0-1)", title = "Price Sensitivity Meter Plot (without interpolation)") + scale_colour_manual(name = "Legend", values = c("cheap" = "#009E73", "expensive" = "#D55E00")) + annotate(geom = "text", # Label of Indifference Price Point x = psm_steps$idp + 1,
y = psm_steps$data_vanwestendorp$ecdf_cheap[psm_steps$data_vanwestendorp$price == psm_steps$idp], label = paste("IDP: ", psm_steps$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 “cheap” and the “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 “expensive” curve is higher than the “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.

Using the interpolate argument

The psm_analysis function includes an optional argument interpolate. If this argument is set to TRUE, the function uses linear interpolation to estimate the price curves between the price points that are provided by the respondents. Consider an easy example: If the “cheap” function is at 20% for a price of 5.99 and at 21% for a price of 6.01, the function would estimate that it is at 20.5% for a price of 6.00. The function uses fixed price steps of 0.01 units between the smallest and the largest price for which the respondents provide information.1

Below are the results for the same data when the interpolate function is set to TRUE.

# Running the analysis with interpolation
psm_interpolated <- psm_analysis(toocheap = toocheap,
cheap = cheap_alt,
expensive = expensive_alt,
tooexpensive = tooexpensive,
validate = TRUE,
interpolate = TRUE)

# Plot with interpolated price steps

psmplot_interpolated <-  ggplot(data = psm_interpolated$data_vanwestendorp, aes(x = price)) + geom_line(aes(y = ecdf_cheap, # line: cheap colour = "cheap"), size = 1) + geom_line(aes(y = ecdf_expensive, # line: expensive colour = "expensive"), size = 1) + geom_rug() + annotate(geom = "point", # Indifference Price Point (intersection of "cheap" and "expensive") x = psm_interpolated$idp,
y = psm_interpolated$data_vanwestendorp$ecdf_cheap[psm_interpolated$data_vanwestendorp$price == psm_interpolated$idp], size = 5, shape = 18, colour = "#009E73") psmplot_interpolated + labs(x = "Price", y = "Share of Respondents (0-1)", title = "Price Sensitivity Meter Plot (with interpolation)") + scale_colour_manual(name = "Legend", values = c("cheap" = "#009E73", "expensive" = "#D55E00")) + annotate(geom = "text", # Label of Indifference Price Point x = psm_interpolated$idp + 1,
y = psm_interpolated$data_vanwestendorp$ecdf_cheap[psm_interpolated$data_vanwestendorp$price == psm_interpolated$idp], label = paste("IDP: ", psm_interpolated$idp)) +
coord_cartesian(xlim = c(10, 20)) +
theme_minimal()

The rug plot demonstrates now that the data includes information for more pricing steps due to the interpolation.

The following plot demonstrates the close relationship between the original data and the interpolated data: The thicker semi-transparent line shows the interpolated values of the “cheap” and “expensive” curves. The thinner solid lines are the original data.

1. This might not always be practical, for instance in currencies where such a subdivision does not exist (such as the Chilean peso or the Tanzanian shilling).