18  Seeing Relationships

Modules 1 to 6 were about one variable at a time: how is yield distributed, what is the mean, what is the spread. This module is about pairs of variables: does yield change with fertilizer, does price move with quality, do two crops rise and fall together.

Almost every interesting question in data analysis is a question about a relationship. The rest of the course is, in one way or another, about formalizing these questions.

Learning Objectives

By the end of this chapter you should be able to:

  1. Read a scatter plot before computing anything from it.
  2. Say whether a relationship looks linear, curved, or absent.
  3. Spot outliers and explain why one point can move a fitted line.
  4. Recognize spread that grows across the plot, and hidden subgroups.

18.1 Look First

You met the scatter plot in Chapter 10: one variable on each axis, a point for every observation. Here we use it for a different purpose. There it was a way to show a relationship to a reader. Now it is a way to find out what the relationship is, before you put a number on it.

The order matters. Every summary in this module – the correlation, the fitted line – is a single number standing in for a whole cloud of points. Each one assumes something about the shape of that cloud, and each one will hand you an answer whether or not the assumption holds. The plot is how you check.

Load the trial data we will use for the rest of the module:

library(tidyverse)

trial <- read_csv("canola_trial.csv")

It has 120 canola fields, each with a nitrogen rate (fertilizer_kg_ha), growing-season rainfall (rainfall_mm), the variety grown, and the yield_bu_acre harvested.

ggplot(trial, aes(x = fertilizer_kg_ha, y = yield_bu_acre)) +
  geom_point()

18.2 Four Things to Look For

Is it a line, or a curve?

The methods in this module fit straight lines. If the cloud bends, a straight line will still be fitted, and it will still report a slope, and the slope will be misleading.

Curvature is common in agronomy. Fertilizer response is the standard case: the first 50 kg/ha of nitrogen buys a lot of yield, the next 50 buys less, and past some point more nitrogen does nothing. That is a curve, and forcing a line through it understates the effect at low rates and overstates it at high ones.

Is one point running the show?

A single observation far from the rest can pull a fitted line a long way, particularly if it sits at the far left or right of the plot. A point out at the edge has leverage: it is a long way from the centre of the data, so tilting the line toward it costs little elsewhere.

When you see one, find out what it is before you decide what to do. A field that yielded 9 bu/ac because it was hailed out is a real observation, but it is not telling you about the fertilizer response, and it does not belong in a regression that is meant to. A yield of 340 bu/ac is a data-entry error. Neither should be dropped silently – say what you removed and why.

Does the spread stay the same?

Look at how tightly the points hug the trend at the left of the plot, and compare it with the right. If the scatter fans out – tight at low values, wide at high ones – the relationship is less predictable at one end than the other.

The fitted line can still be a reasonable summary. What breaks is the uncertainty around it, which is calculated assuming the scatter is even. We will not do anything about that in this module; Modules 11 to 13 are where uncertainty gets treated properly. For now, notice it and say so.

Is the cloud actually two clouds?

Sometimes what looks like one relationship is two groups sitting in different places. Two varieties with different yield potential, two soil zones, two years with different weather. Colour the points by a third variable and the structure appears:

ggplot(trial, aes(x = fertilizer_kg_ha, y = yield_bu_acre, colour = variety)) +
  geom_point()

This matters more than it looks. A relationship computed across mixed groups can be quite different from the relationship within each one – occasionally it even points the other way. That is a real phenomenon with a name, Simpson’s paradox, and the multiple regression in Section 20.6 is one way of dealing with it.

18.3 Anscombe’s Quartet

The argument for plotting first has a famous demonstration. Frank Anscombe built four small datasets that share almost every summary statistic: the same means, the same standard deviations, the same correlation, the same fitted line. Plotted, they look nothing alike – one is a tidy linear relationship, one is a clean curve, one is a perfect line with a single outlier dragging it, and one is a vertical stack with one point off to the side.

It ships with R, so you can see it yourself:

anscombe

Four numbers agreed on four datasets that have nothing in common. Any of the summaries in this module can be computed on data it has no business describing. Plot first.

18.4 Test Bank Sample

  1. (Reading.) You plot yield against seeding rate and the points bend downward at high rates. What does fitting a straight line get wrong?
  2. (Outliers.) A single field sits far to the right of every other point. Why does its position on the x axis matter, not just how unusual its yield is?
  3. (Spread.) What does it mean if the scatter fans out from left to right, and what does it affect?
  4. (Subgroups.) Give an agricultural example where one relationship across a whole dataset could differ from the relationship within each group.

18.5 Practice Exercises

  1. Plot yield_bu_acre against fertilizer_kg_ha for the trial data. Describe the shape in one sentence before computing anything.
  2. Colour the same plot by variety. Does the picture change?
  3. Plot yield against rainfall_mm. Which of the two predictors looks more strongly related to yield?
  4. Load anscombe and plot all four pairs. Compute the correlation of each. Explain what the exercise is for.