## Slope Graphs

The most used graph for visualizing the relationship between two numeric variables is the scatter plot.

But there is one alternative that can be useful and is increasingly popular: the slope chart or slope graph.

### Tufte’s Slope Graph

Two articles on slope graphs with examples:

Tufte showed this example in The Visual Display of Quantitative Information:

Some features of the data that are easy to see:

• order of the countries within each year;

• how each country’s values changed;

• how the rates of change compare;

• the country (Britain) that does not fit the general pattern.

The chart uses no non-data ink.

The chart in this form is well suited for small data sets or summaries with modest numbers of categories.

Scalability in this full form is limited, but better if labels and values are dropped.

The idea can be extended to multiple periods, though two periods or levels is most common when labeling is used. Without labeling this becomes a parallel coordinates plot.

### Barley Mean Yields

Averaging yield over the different varieties for each site and year produces

gbsy <- group_by(barley, site, year)
absy <- summarize(gbsy, avg_yield = mean(yield))
## # A tibble: 6 x 3
## # Groups:   site [3]
##   site            year  avg_yield
##   <fct>           <fct>     <dbl>
## 1 Grand Rapids    1932       20.8
## 2 Grand Rapids    1931       29.1
## 3 Duluth          1932       25.7
## 4 Duluth          1931       30.3
## 5 University Farm 1932       29.5
## 6 University Farm 1931       35.8

The year variable in the summary is a factor with the levels in the wrong order, so we need to fix that:

levels(absy$year) ## [1] "1932" "1931" absy <- mutate(absy, year = fct_rev(year)) levels(absy$year)
## [1] "1931" "1932"

The core of a slope graph for these means is

p <- ggplot(absy, aes(x = year, y = avg_yield, group = site)) + geom_line()
p

Adding the labels can be done as

p + geom_text(aes(label = paste0(site, ", ", round(avg_yield, 1))),
hjust = "outward")

The labal positions could use further adjusting; using geom_text_repel from the ggrepel package handles this well:

library(ggrepel)
p <- p + geom_text_repel(aes(label = paste0(site, ", ", round(avg_yield, 1))),
hjust = "outward", direction = "y")
basic_barley_slopes <- p
p

Finally, clear out non-data ink and move axis labels to the top:

p + theme(panel.background = element_blank(),
panel.grid=element_blank(),
axis.ticks=element_blank(),
axis.text.y=element_blank(),
axis.title=element_blank(),
panel.border=element_blank()) +
scale_x_discrete(position = "top")

### Father-Son Heights

The father.son data set has 1078 observations, which is too large for the labeled slope graph, but the basic representation is useful.

To make creating the graph easier we can convert the data frame into one with a height variable, a variable indicating whether the height is for father or son, and a variable identifying the pair:

fs <- mutate(father.son, id = seq_len(nrow(father.son)))
##    fheight  sheight id
## 1 65.04851 59.77827  1
## 2 63.25094 63.21404  2
## 3 64.95532 63.34242  3
## 4 65.75250 62.79238  4
## 5 61.13723 64.28113  5
## 6 63.02254 64.24221  6
fs <- gather(fs, who, height, 1:2)
##   id     who   height
## 1  1 fheight 65.04851
## 2  2 fheight 63.25094
## 3  3 fheight 64.95532
## 4  4 fheight 65.75250
## 5  5 fheight 61.13723
## 6  6 fheight 63.02254

The basic plot is quite simple:

ggplot(fs, aes(x = who, y = height)) + geom_line(aes(group = id))

With an axis adjustment and using a reduced alpha level:

ggplot(fs, aes(x = who, y = height)) +
geom_line(aes(group = id), alpha = 0.1) +
scale_x_discrete(expand = c(.1, 0))

This very clearly shows the famous regression to the mean effect:

• taller parents tend to be taller than their children;
• shorter parents tend to be shorter than their children.

Conversely,

• taller children tend to be taller than their parents;
• shorter children tend to be shorter than their parents.

## Scatter Plots

A scatter plot of two variables plots maps the values on one variable to the vertical axis and the other to the horisontal axis of a cartesian coordinate system and places a marker for each observation at the resulting point.

Conventions:

• Plot A versus/against B means A is mapped to the vertical, or $$y$$, axis, and B to the horizontal, or $$x$$ axis.

• If we can think of variation A as being partly explained byt B then we usually plot A against B.

• If we can thing of B as helping to precict A, then we usually plot A against B.

### Barley Mean Yields

For a scatter plot of mean yield in 1932 against mean yield in 1931 for the different sites it is useful to have a data frame containing variables for each year.

This requires the inverse of a gather operation, called a spread:

sabsy <- spread(absy, year, avg_yield, sep="")
## # A tibble: 6 x 3
## # Groups:   site [6]
##   site            year1931 year1932
##   <fct>              <dbl>    <dbl>
## 1 Grand Rapids        29.1     20.8
## 2 Duluth              30.3     25.7
## 3 University Farm     35.8     29.5
## 4 Morris              29.3     41.5
## 5 Crookston           43.7     31.2
## 6 Waseca              54.3     41.9

The basic scatter plot of y = year1932 against x = year1931:

p <- ggplot(sabsy, aes(x = year1931, y = year1932)) + geom_point()
p

Adding labels using geom_text_repel identifies the Morris site:

p <- p + geom_text_repel(aes(label = site), vjust = "top")
p

To recognize the reversal we can add the 45 degree line:

p + geom_abline(aes(intercept = 0, slope = 1), linetype = 2)

### Father and Son Heights

The basic scatter plot:

p0 <-  ggplot(father.son, aes(x = fheight, y = sheight))
p1 <- p0 + geom_point()
p1

Adding a line with slope one helps identify the regression to the mean phenomenon:

p2 <- p1 + geom_abline(aes(intercept = mean(sheight) - mean(fheight),
slope = 1),
linetype = 2, color = "red")
p2

Adding a regression line helps further:

p2 + geom_smooth(method = "lm")

But for showing the regression effect it is hard to beat the scatter plot of sheight - fheight against fheight:

ggplot(father.son) +
geom_point(aes(x = fheight, y = sheight - fheight)) +
geom_hline(aes(yintercept = 0), linetype = 2, color = "red")

### Old Faithful Eruptions

A scatter plot of the waiting times until the next eruption against the duration of the current eruption for the faithful data set shows the two clusters corresponding to the short and long eruptions:

ggplot(faithful) + geom_point(aes(x = eruptions, y = waiting))

For the geyser data set from the MASS package a plot of the two variables shows a different pattern:

ggplot(geyser) + geom_point(aes(x = duration, y = waiting))

The reason for the difference is that in the geyser data set the waiting time reflects the time since the previous eruption, not the time until the next one.

For this time order it would be more natural to plot duration against time since the last eruption:

ggplot(geyser) + geom_point(aes(x = waiting, y = duration))

We can adjust these data to pair durations with waiting times until the next eruption using the lag function from dplyr. This produces the same basic pattern as for the faithful data set:

ggplot(geyser) + geom_point(aes(x = lag(duration), y = waiting))
## Warning: Removed 1 rows containing missing values (geom_point).