This document discusses why it is a good idea to plot your data before you run descriptive or predictive statistical models. It also discusses degrees of freedom, and why the idea is important.
library(ggplot2)
library(datasauRus)
library(dplyr)
Attaching package: 'dplyr'The following objects are masked from 'package:stats':
filter, lagThe following objects are masked from 'package:base':
intersect, setdiff, setequal, unionlibrary(stats)
library(graphics)Here are a bunch of scatter plots.
ggplot(datasaurus_dozen, aes(x = x, y = y, colour = dataset))+
geom_point() +
theme_void() +
theme(legend.position = "none")+
facet_wrap(~dataset, ncol = 3)
They look pretty different, don’t they?
Except that these all have roughly the same summary statistics.
datasaurus_dozen %>%
group_by(dataset) %>%
summarize(
mean_x = mean(x),
mean_y = mean(y),
std_dev_x = sd(x),
std_dev_y = sd(y),
corr_x_y = cor(x, y),
nxy = length(x)
)# A tibble: 13 × 7
dataset mean_x mean_y std_dev_x std_dev_y corr_x_y nxy
<chr> <dbl> <dbl> <dbl> <dbl> <dbl> <int>
1 away 54.3 47.8 16.8 26.9 -0.0641 142
2 bullseye 54.3 47.8 16.8 26.9 -0.0686 142
3 circle 54.3 47.8 16.8 26.9 -0.0683 142
4 dino 54.3 47.8 16.8 26.9 -0.0645 142
5 dots 54.3 47.8 16.8 26.9 -0.0603 142
6 h_lines 54.3 47.8 16.8 26.9 -0.0617 142
7 high_lines 54.3 47.8 16.8 26.9 -0.0685 142
8 slant_down 54.3 47.8 16.8 26.9 -0.0690 142
9 slant_up 54.3 47.8 16.8 26.9 -0.0686 142
10 star 54.3 47.8 16.8 26.9 -0.0630 142
11 v_lines 54.3 47.8 16.8 26.9 -0.0694 142
12 wide_lines 54.3 47.8 16.8 26.9 -0.0666 142
13 x_shape 54.3 47.8 16.8 26.9 -0.0656 142Think about that a minute. If we didn’t plot the data and only had the means, standard deviations, and correlation values available from some table in a paper, we’d think these sets of variables were essentially the same, wouldn’t we?
These plots come from the datasauRus (Davies et al., 2022) package inspired by a paper by (Matejka & Fitzmaurice, 2017). The (Matejka & Fitzmaurice, 2017) is based on Anscombe’s quartet (Anscombe, 1973), which is also available as the anscombe package. Here are the originals from (Anscombe, 1973).
Below, I use the code that accompanies the anscombe package verbatim, slightly reordered to plot first.
ff <- y ~ x
mods <- setNames(as.list(1:4), paste0("lm", 1:4))
for(i in 1:4) {
ff[2:3] <- lapply(paste0(c("y","x"), i), as.name)
## or ff[[2]] <- as.name(paste0("y", i))
## ff[[3]] <- as.name(paste0("x", i))
mods[[i]] <- lmi <- lm(ff, data = anscombe)
#print(anova(lmi))
}ff <- y ~ x
mods <- setNames(as.list(1:4), paste0("lm", 1:4))
for(i in 1:4) {
ff[2:3] <- lapply(paste0(c("y","x"), i), as.name)
## or ff[[2]] <- as.name(paste0("y", i))
## ff[[3]] <- as.name(paste0("x", i))
mods[[i]] <- lmi <- lm(ff, data = anscombe)
# print(anova(lmi))
}op <- par(mfrow = c(2, 2), mar = 0.1+c(4,4,1,1), oma = c(0, 0, 2, 0))
for(i in 1:4) {
ff[2:3] <- lapply(paste0(c("y","x"), i), as.name)
plot(ff, data = anscombe, col = "red", pch = 21, bg = "orange", cex = 1.2,
xlim = c(3, 19), ylim = c(3, 13))
abline(mods[[i]], col = "blue")
}
mtext("Anscombe's 4 Regression data sets", outer = TRUE, cex = 1.5)
Here are some summary statistics for these pairs.
summary(anscombe) x1 x2 x3 x4 y1
Min. : 4.0 Min. : 4.0 Min. : 4.0 Min. : 8 Min. : 4.260
1st Qu.: 6.5 1st Qu.: 6.5 1st Qu.: 6.5 1st Qu.: 8 1st Qu.: 6.315
Median : 9.0 Median : 9.0 Median : 9.0 Median : 8 Median : 7.580
Mean : 9.0 Mean : 9.0 Mean : 9.0 Mean : 9 Mean : 7.501
3rd Qu.:11.5 3rd Qu.:11.5 3rd Qu.:11.5 3rd Qu.: 8 3rd Qu.: 8.570
Max. :14.0 Max. :14.0 Max. :14.0 Max. :19 Max. :10.840
y2 y3 y4
Min. :3.100 Min. : 5.39 Min. : 5.250
1st Qu.:6.695 1st Qu.: 6.25 1st Qu.: 6.170
Median :8.140 Median : 7.11 Median : 7.040
Mean :7.501 Mean : 7.50 Mean : 7.501
3rd Qu.:8.950 3rd Qu.: 7.98 3rd Qu.: 8.190
Max. :9.260 Max. :12.74 Max. :12.500 I don’t remember my stats professor showing us Anscombe’s datasets. The datasauRus came along much later. But I’m pretty sure this was what he had in mind when he exhorted the class to “plot your data!” It’s excellent advice, and I try to follow it as an author and as a reviewer.
One way to think about what’s going on here is that we analysts want an efficient way to summarize complex sets of data. Statistics like the mean or median, standard deviation, and correlation often do that. If a big set of data is well described by one of these statistics, then we’ve saved ourselves a lot of time and headache. If we think about a summary of some dataset as having \(m\) degrees of freedom, what we mean is that we only need \(m\) numbers for the summary. So, if we summarize a big dataset with the mean, then our summary (or model) has only one degree of freedom. If we include the standard deviation, then our summary has two degrees of freedom. If we use a linear regression to predict y ~ x then we usually have a slope estimate and an intercept estimate, or two degrees of freedom.
If we’re summarizing a raw dataset with \(n=142\) points in X and \(n=142\) datapoints in Y like the datasauRus sets above, being able to summarize with just a few numbers (degrees of freedom) is powerful and efficient. And in general, we analysts favor more parsimonious models of data–those with fewer degrees of freedom.
The problem arises when our simple models with few degrees of freedom don’t really capture important information about data sets, or when they persuade us to think (wrongly) that data sets with the same summary statistics are necessarily comparable. They might be, but they might not be. We have to look at the data–plot it–and confirm that we haven’t lost information in simplifying a dataset with a small number of statistics. The datasauRus and anscombe plots show us perverse situations when the small degree of freedom summaries are deeply misleading.
The converse is also true. We can describe a dataset increasingly precisely with a large number of degrees of freedom. For example, imagine that our model of one of the datasauRus sets has \(2*n\) degrees of freedom, where \(n=142\). If those 284 numbers are equal to the data points themselves, our summary has captured the data exactly. But we haven’t really gained anything, have we? We’ve just redescribed the data without reducing it to a summary with fewer degrees of freedom than existed in the first place. So, the best summaries (models) of data are small (in the number of degrees of freedom), but not too small.
When Simmons, Nelson, & Simonsohn (2011) discuss ‘researcher degrees of freedom’, they extend these ideas in a new direction. How many steps (degrees of freedom) did the analysts take to reach their conclusions? Ideally, as few as possible, and certainly far less than the number of data points.
So, how many researcher degrees of freedom are too many? That’s a judgment call. But whatever you determine, justify your argument. And maybe see whether the authors plotted their data in a way that helps make a case one way or the other.