This material can be read in conjunction with pages 50 to 51 of Cleveland’s book.
Some batches of data may show a systematic change in spread vs. location; a good example is stereogram fusion time data in Cleveland’s book where an increase in location translates into an increase in spread. Such dependency is often undesirable (e.g. in an ANOVA for instance) and preferably removed in an analysis. A plot well suited for visualizing this dependency is the spread-level plot, s-l (or spread-location plot as Cleveland calls it).
The plot compares a measure of the spread’s residual to the location (usually the median) for each batch of data. The spread is usually distilled down to its residual (what remains after subtracting each batch value by the batch median) then transformed by taking the square root of its absolute value. The following block of code recreates the s-l plot in figure 2.25 from Cleveland’s book.
library(dplyr) library(ggplot2) df <- read.csv("http://mgimond.github.io/ES218/Data/fusion.csv") # Create two new columns: group median and group residual df1 <- df %>% group_by(nv.vv) %>% mutate( Median = median(time), Residuals = sqrt( abs(time - Median))) # Generate the s-l plot ggplot(df1, aes(x = Median, y = Residuals)) + geom_jitter(alpha = 0.4, width = 0.2) + stat_summary(fun.y = median, geom = "line", col = "red") + ylab(expression(sqrt(abs(" Residuals "))))
The red line in the plot helps identify the type of relationship between spread and location (defined by the group median in this plot). If the line slopes upward, there is an increasing spread as a function of increasing location; if it slopes downward, there is a decreasing spread as a function of increasing location; and if the slope is flat, there is no change in spread as a function of location.
In the data fusion example, there is an obvious increase in spread as a function of location.
On page 60, Cleveland compares spread vs location for three different batches of values: two, three and mixed dimensional ecosystems. The values represent the mean chain length of a web (defined by the number of hierarchical prey-predator links in the web) for the three different ecosystem types.
df <- read.csv("http://mgimond.github.io/ES218/Data/Food_web.csv") # Create two new columns: group median and group residuals df1 <- df %>% group_by(dimension) %>% mutate( Median = median(mean.length), Residuals = sqrt( abs( mean.length - Median))) # Generate the s-l plot ggplot(df1, aes(x = Median, y = Residuals)) + geom_jitter(alpha = 0.4, width = 0.05, height = 0) + stat_summary(fun.y = median, geom = "line", col = "red") + ylab(expression(sqrt( abs(" Residuals ")))) + geom_text(aes(x = Median, y = 1.75, label = dimension))
A monotonic spread is apparent in this dataset too, i.e. as the median chain length increases, so does the spread.
In this next example, we will compare water temperature spreads with median temperatures across 12 months using Gulf of Maine buoy data. Since we have many categories to work with, we will do without the jitter and stack the points within each month group.
df <- read.table("http://mgimond.github.io/ES218/Data/buoy_44005_2012.dat", header=T) # Create two new columns: group median and group residual df1 <- df %>% group_by(MM) %>% mutate(Median = median(WTMP), Residuals = sqrt(abs(WTMP - Median))) # Generate the s-l plot ggplot(df1, aes(x = Median, y = Residuals)) + geom_point(alpha = 0.1) + stat_summary(fun.y = median, geom = "line", col = "red") + ylab(expression(sqrt(abs( " Residuals ")))) + geom_text(aes(x = Median, y = 2.5, label = MM), cex = 3)
Note that the discrete appearance of the values reflects the level of precision used to record water temperature data (a tenth of a degree Celsius). Unlike our previous two examples, a systematic spread as a function of increasing water temperature is not immediately apparent.
Another version of the S-L plot (and one that seems to be the most popular) pits the log of the inter-quartile spread vs the log of the median. This approach only works for positive values (this may require that values be adjusted so that the minimum value be no less than or equal to 0).
This approach is appealing in that the slope of the best fit line can be used to come up with a power transformation (a topic covered in next week’s lecture) via power = 1 - slope.
This variant of the s-l plot can be computed in R as follows (we will use the buoy data as an example).
sl <- df %>% group_by(MM) %>% summarise (level = log(median(WTMP)), IQR = IQR(WTMP), # Computes the interquartile range spread = log(IQR)) ggplot(sl, aes(x = level, y = spread)) + geom_point() + stat_smooth(method = "lm", se = FALSE) + xlab("Median (log)") + ylab("Spread (log)") + geom_text(aes(x = level, y = spread, label = MM), vjust = 1.21, cex = 2.8)
Note how this plot differs from Cleveland’s plot in that we are only displaying each month’s median spread value and we are fitting a straight line to the points.
The slope suggests that there may be a monotonic increase in spread vs location, but a close examination of the regression model indicates that the slope may not be significantly different from 0 (the p-value of 0.12 suggests that there is a 12% chance that the computed slope is consistent with a slope we would expect to get if there was no linear relationship between spread and location values).
Call: lm(formula = spread ~ level, data = sl) Residuals: Min 1Q Median 3Q Max -1.6491 -0.2979 -0.1910 0.6701 1.4585 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -2.2585 1.4327 -1.576 0.146 level 0.9896 0.5907 1.675 0.125 Residual standard error: 0.8912 on 10 degrees of freedom Multiple R-squared: 0.2192, Adjusted R-squared: 0.1411 F-statistic: 2.807 on 1 and 10 DF, p-value: 0.1248