knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )

The interpretation of the Price Sensitivity Meter analysis in its original publication^[Van Westendorp, P (1976) "NSS-Price Sensitivity Meter (PSM) -- A new approach to study consumer perception of price" *Proceedings of the ESOMAR Congress*, 139--167. Online available at the ESOMAR website.] focuses on a graphical inspection of the results. All relevant price points are defined by the intersection of two distribution functions, which are identified by visualizing the results. Even if this package provides a summary of the main results, most users will probably also be interested in visualizing them.

This vignette provides a short overview of efficient ways to provide visualizations of the results, both using R's base graphics as well as the ggplot2 package. The plots are designed to show all main outcomes and can serve as a reference visualization, which may be reused for the purpose of a particular project.

I will use the same set of (random) example data throughout this vignette.

library(pricesensitivitymeter) set.seed(20) toocheap <- round(rnorm(n = 250, mean = 8, sd = 1.5), digits = 2) cheap <- round(rnorm(n = 250, mean = 12, sd = 2), digits = 2) expensive <- round(rnorm(n = 250, mean = 13, sd = 1), digits = 2) tooexpensive <- round(rnorm(n = 250, mean = 15, sd = 1), digits = 2) # optional: purchase intent for "cheap" and "expensive price" (Newton Miller Smith Extension) pi_cheap <- sample(x = c(1:5), size = length(cheap), replace = TRUE, prob = c(0.1, 0.1, 0.2, 0.3, 0.3)) pi_expensive <- sample(x = c(1:5), size = length(expensive), replace = TRUE, prob = c(0.3, 0.3, 0.2, 0.1, 0.1)) data.psm <- data.frame(toocheap, cheap, expensive, tooexpensive, pi_cheap, pi_expensive) output.psm <- psm_analysis(toocheap = "toocheap", cheap = "cheap", expensive = "expensive", tooexpensive = "tooexpensive", data = data.psm, pi_cheap = "pi_cheap", pi_expensive = "pi_expensive", validate = TRUE)

The resulting *output.psm* object contains the results which will be used for plotting. In particular, the slot *data_vanwestendorp* includes a dataframe with all prices that are included in the input data and the corresponding values of the cumulative distribution functions for the different price levels. Let's have a quick look at one row in this dataset (row 260):

```
output.psm$data_vanwestendorp[260, ]
```

This means the following: For the price of `r output.psm$data_vanwestendorp[260, "price"]`

, ...

`r round(output.psm$data_vanwestendorp[260, "ecdf_toocheap"], digits = 2) * 100`

% of respondents think that this price is not too cheap (*not*too cheap because the function for the "too cheap" answers is reversed in every Price Sensitivity Meter analysis as explained in the original article)`r round(output.psm$data_vanwestendorp[260, "ecdf_cheap"], digits = 2) * 100`

% of respondents think that this price is cheap`r round(output.psm$data_vanwestendorp[260, "ecdf_expensive"], digits = 2) * 100`

% of respondents think that this price is expensive`r round(output.psm$data_vanwestendorp[260, "ecdf_tooexpensive"], digits = 2) * 100`

% of respondents think that this price (or more) is too expensive`r round(output.psm$data_vanwestendorp[260, "ecdf_not_cheap"], digits = 2) * 100`

% of respondents think that this price is*not*cheap - this is the inverse of edfc_cheap (so both values always add up to 100%)`r round(output.psm$data_vanwestendorp[260, "ecdf_not_expensive"], digits = 2) * 100`

% of respondents think that this price is*not*expensive - this is the inverse of edfc_expensive (so both values always add up to 100%)

We can now visualize this information for all the different price points.

Below is some example code that uses the information stored in the results object to create a graph that includes

- all possible price points
- a semi-transparent area to show the range of acceptable prices (between the Point of Marginal Cheapness and the Point of Marginal Expensiveness)
- the empirical cumulative distribution function for the "too cheap", "not cheap", "not expensive" and "too expensive" price perceptions
- highlights for the Indifference Price Point and the Optimal Price Point

library(ggplot2) # all plot elements without any labels psmplot <- ggplot(data = output.psm$data_vanwestendorp, aes(x = price)) + annotate(geom = "rect", # shaded background area for range of acceptable prices xmin = output.psm$pricerange_lower, xmax = output.psm$pricerange_upper, ymin = 0, ymax = Inf, fill="grey50", alpha = 0.3) + geom_line(aes(y = ecdf_toocheap, # line: too cheap colour = "too cheap", linetype = "too cheap"), size= 1) + geom_line(aes(y = ecdf_tooexpensive, # line: too expensive colour = "too expensive", linetype = "too expensive"), size = 1) + geom_line(aes(y = ecdf_not_cheap, # line: not cheap colour = "not cheap", linetype = "not cheap"), size = 1) + geom_line(aes(y = ecdf_not_expensive, # line: not expensive colour = "not expensive", linetype = "not expensive"), size = 1) + annotate(geom = "point", # Indifference Price Point (intersection of "cheap" and "expensive") x = output.psm$idp, y = output.psm$data_vanwestendorp$ecdf_not_cheap[output.psm$data_vanwestendorp$price == output.psm$idp], size = 5, shape = 18, colour = "#009E73") + annotate(geom = "point", # Optimal Price Point (intersection of "too cheap" and "too expensive") x = output.psm$opp, y = output.psm$data_vanwestendorp$ecdf_toocheap[output.psm$data_vanwestendorp$price == output.psm$opp], size = 3, shape = 17, colour = "#009E73") # Labels and Colours psmplot + labs(x = "Price", y = "Share of Respondents (0-1)", title = "Price Sensitivity Meter Plot", caption = "Shaded area: range of acceptable prices\nData: Randomly generated") + scale_colour_manual(name = "Legend", values = c("too cheap" = "#009E73", "not cheap" = "#009E73", "not expensive" = "#D55E00", "too expensive" = "#D55E00")) + scale_linetype_manual(name="Legend", values = c("too cheap" = "dotted", "not cheap" = "solid", "not expensive" = "solid", "too expensive" = "dotted")) + annotate(geom = "text", # Label of Indifference Price Point x = output.psm$idp + 1.5, y = output.psm$data_vanwestendorp$ecdf_not_cheap[output.psm$data_vanwestendorp$price == output.psm$idp], label = paste("IDP: ", output.psm$idp)) + annotate(geom = "text", # Label of Optimal Price Point x = output.psm$opp + 1.5, y = output.psm$data_vanwestendorp$ecdf_toocheap[output.psm$data_vanwestendorp$price == output.psm$opp], label = paste("OPP: ", output.psm$opp)) + theme_minimal()

Below is an additional example of the same plot, using R's built-in plotting function. Note that there are a few slight differences, such as the lack of transparency in the shaded range of acceptable prices.

par(cex.sub = 0.66) # reducing the font size of the subtitle # Setting up the plot: empty canvas plot(x = output.psm$data_vanwestendorp$price, y = output.psm$data_vanwestendorp$ecdf_toocheap, type = "n", xlab = "", ylab = "") grid() # adding gridlines # Shaded Area: Range of Accetable Prices rect(xleft = output.psm$pricerange_lower, ybottom = 0, xright = output.psm$pricerange_upper, ytop = 1, col = "grey85", border = 0) # adding line: too cheap lines(x = output.psm$data_vanwestendorp$price, y = output.psm$data_vanwestendorp$ecdf_toocheap, lty = "dotted", col = "#009E73") # adding line: not cheap lines(x = output.psm$data_vanwestendorp$price, y = output.psm$data_vanwestendorp$ecdf_not_cheap, lty = "solid", col = "#009E73") # adding line: not expensive lines(x = output.psm$data_vanwestendorp$price, y = output.psm$data_vanwestendorp$ecdf_not_expensive, lty = "solid", col = "#D55E00") # adding line: too expensive lines(x = output.psm$data_vanwestendorp$price, y = output.psm$data_vanwestendorp$ecdf_tooexpensive, lty = "dotted", col = "#D55E00") # adding the axes descriptions title(main = "Price Sensitivity Meter Plot", xlab = "Price", ylab = "Share of Respondents (0-1)") # Right-align of subtitle adj.old <- par()$adj # saving old default value (= centered) par(adj = 1) # setting the new default to right-align title(sub = "Shaded area: Range of acceptable prices.\nData: Randomly generated") # writing the actual subtitle par(adj = adj.old) # restore standard (so that next main title is again centered) # adding the Indifference Price Point (intersection of "cheap" and "expensive") points(x = output.psm$idp, y = output.psm$data_vanwestendorp$ecdf_not_cheap[which(output.psm$data_vanwestendorp$price == output.psm$idp)], cex = 2, pch = 18, col = "#009E73") text(x = output.psm$idp, y = output.psm$data_vanwestendorp$ecdf_cheap[which(output.psm$data_vanwestendorp$price == output.psm$idp)], labels = paste("IDP:", output.psm$idp), pos = 4) # adding the Optimal Price Point (intersection of "too cheap" and "too expensive") points(x = output.psm$opp, y = output.psm$data_vanwestendorp$ecdf_toocheap[which(output.psm$data_vanwestendorp$price == output.psm$opp)], cex = 2, pch = 17, col = "#009E73") text(x = output.psm$opp, y = output.psm$data_vanwestendorp$ecdf_toocheap[which(output.psm$data_vanwestendorp$price == output.psm$opp)], labels = paste("OPP:", output.psm$opp), pos = 4) # adding legend legend("bottomleft", title = "Legend", c("too cheap", "not cheap", "not expensive", "too expensive"), lty = c("dotted", "solid", "solid", "dotted"), col = c("#009E73", "#009E73", "#D55E00", "#D55E00"), cex = 0.66)

When we have information of the purchase intent at the "cheap" and at the "expensive" price point, we can also model the optimal price to maximize trial or revenue based on the Newton Miller Smith Extension

library(ggplot2) # Plot for Optimal Trial ggplot(data = output.psm$data_nms, aes(x = price)) + geom_line(aes(y = trial)) + # trial curve geom_vline(xintercept = output.psm$price_optimal_trial, linetype = "dotted") + # highlighting the optimal price geom_text(data = subset(output.psm$data_nms, trial == max(trial)), aes(x = price + 0.5, y = trial), label = paste("Optimal Price:", output.psm$price_optimal_trial), hjust = 0) + # labelling the optimal price labs(x = "Price", y = "Likelihood to Buy (Trial)", title = "Price Sensitivity Meter: Price for Optimal Trial", caption = "Data: Randomly generated") + theme_minimal() # Plot for Optimal Revenue ggplot(data = output.psm$data_nms, aes(x = price)) + geom_line(aes(y = revenue)) + # revenue curve geom_vline(xintercept = output.psm$price_optimal_revenue, linetype = "dotted") + # highlighting the optimal price geom_text(data = subset(output.psm$data_nms, revenue == max(revenue)), aes(x = price + 0.5, y = revenue), label = paste("Optimal Price:", output.psm$price_optimal_revenue), hjust = 0) + # labelling the optimal price labs(x = "Price", y = "Revenue", title = "Price Sensitivity Meter: Price for Optimal Revenue", caption = paste("Combined revenue for a sample of n =", output.psm$total_sample - output.psm$invalid_cases, "respondents\nData: Randomly generated")) + theme_minimal()

Below are versions of the same plots with R's base plot function instead of ggplot. For those simpler plots, the differences between ggplot and base plot are less pronounced.

par(cex.sub = 0.66) # reducing the font size of the subtitle # a) Plot for Optimal Trial # Main plot: Line, axis descriptions plot(x = output.psm$data_nms$price, y = output.psm$data_nms$trial, type = "l", main = "Price Sensitivity Meter: Price for Optimal Trial", xlab = "Price", ylab = "Likelihood to Buy (Trial)") grid() # adding gridlines # Right-align of subtitle adj.old <- par()$adj # saving old default value (= centered) par(adj = 1) # setting the new default to right-align title(sub = "Data: Randomly generated") # writing the actual subtitle par(adj = adj.old) # restore standard (so that next main title is again centered) # drawing additional line to highlight optimal price abline(v = output.psm$data_nms$price[which.max(output.psm$data_nms$trial)], lty = "dotted") # annotating the optimal price text(x = output.psm$data_nms$price[which.max(output.psm$data_nms$trial)], y = max(output.psm$data_nms$trial), labels = paste("Optimal Price:", output.psm$price_optimal_trial), pos = 4) # b) Plot for Optimal Revenue # Main plot: Line, axis descriptions plot(x = output.psm$data_nms$price, y = output.psm$data_nms$revenue, type = "l", main = "Price Sensitivity Meter: Price for Optimal Revenue", xlab = "Price", ylab = "Revenue") grid() # adding gridlines # Right-align of subtitle adj.old <- par()$adj # saving old default value (= centered) par(adj = 1) # setting the new default to right-align title(sub = paste("Combined revenue for a sample of n =", output.psm$total_sample - output.psm$invalid_cases, "respondents\nData: Randomly generated")) # writing the actual subtitle par(adj = adj.old) # restore standard (so that next main title is again centered) # drawing additional line to highlight optimal price abline(v = output.psm$data_nms$price[which.max(output.psm$data_nms$revenue)], lty = "dotted") # annotating the optimal price text(x = output.psm$data_nms$price[which.max(output.psm$data_nms$revenue)], y = max(output.psm$data_nms$revenue), labels = paste("Optimal Price:", output.psm$price_optimal_revenue), pos = 4)

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