Data visualization toolbox

# 6 Multiway Data

- Multway dot plots
- Additive fits
- Color scatterplots

Multiway data consist of one quantative variable, the response, and two or more categorical variables as factors. There is a value of the response for each combination of levels of the categorical variables. The general task with this type of data is to determine how the response depends on the factors.

## Data sets

This chapter includes examples for two data sets:

Livestock
  Counts from a census of 5 types of farm animal in 26 countries. The goal of the analysis is to visualize the variations in livestock populations by country.
Barley
  Yields of 10 varieties of barley grown at 6 sites during two years. The goal of the analysis is to determine the relative importance of the factors site, variety and year. This is a famous data set for design of experiments and analysis of variance.

## Multiway Dot Plots `(multidotplot.m)`

The multiple panel approach is also helpful with dot plots. For the multiway plot in Figure 6.1, the counts have been log transformed to cope with their large range. The countries are arranged in decreasing order of median number of livestock. The panels are arranged so that the livestock medians are in order.

| Figure 6.1 Multiway dot plot of the livestock data. (book 6.1)

This organization of the graphical data makes it easy to examine the data for consistency and interesting values.

## Additive Fit

The simplest function which can be fit to multiway data is the additive function. Performing an additive fit to the livestock data leads to the result in Figure 6.2.

| Figure 6.2 Dot plot of the bisquare estimates of the country and livestock main effects. (book 6.4)

The livestock main effects account for more variation than the country main effects.

Details of Additive Fit `(addfit.m)`
The additive function to fit a response y to two factors a and b is of the form where mu is the mean of the response and The additive function is fit by least squares, which produces parameter estimates given by Iterative bisquare fitting can also be used to suppress the effects of outliers.

The fitted function accounts for much of the variation in the data. (Figure 6.3)

| Figure 6.3 R-F spread plot of the additive fit and its residuals for the livestock data.

## Color Scatterplot

The barley data have been analyzed many times over the years, for the most part taking the data at face value. Recent display of the data using multiway dot plots strongly suggests that the Morris data has been interchanged between the two years. Figure 6.4 shows some of the evidence in a very compact presentation.

| Figure 6.4 Annual changes in barley yield.

The same issue is revealed in a mean difference plot. (Figure 6.5) Both figures also raise a question about one point at Grand Rapids and at University Farm.

| Figure 6.5 Mean difference plot of the barley data. The open cirles are positive versions of the points with a yield decrease.

Despite years of analysis, the anomalous data were overlooked. Earlier visualization could have resolved the data quality problem while the experimenters were still alive and improved the value of subsequent analyses.

1 Introduction | 4 Trivariate Data
2 Univariate Data | 5 Hypervariate Data
3 Bivariate Data | 6 Multiway Data

Send feedback to author@datatool.com | Go to Data visualization home