Editing 1725: Linear Regression

{{comic
| number    = 1725
| date      = August 26, 2016
| title     = Linear Regression
| image     = linear_regression.png
| titletext = The 95% confidence interval suggests Rexthor's dog could also be a cat, or possibly a teapot.
}}

==Explanation==
{{w|Linear regression}} is a method for modeling the relationship between multiple variables. In the simplest case, it can be used for two variables wherein the model determines a "{{w|least squares|best-fit}}" line through a {{w|scatter plot}} of the datasets, together with a {{w|coefficient of determination}}, usually denoted ''r''<sup>2</sup> or ''R''<sup>2</sup>. When only two variables are included in the regression, ''R''<sup>2</sup> is merely the square of the correlation between the two variables. ''R''<sup>2</sup> is a number between 0 and 1 that indicates how well one variable can be used to predict the value of another. A value of 1 means perfect correlation, while a value close to 0 indicates a weak relationship between the variables.

A constellation is pattern created by linking the apparent positions of stars as seen in the sky from Earth. (Astronomers, in technical contexts, usually refer to these as {{w|Asterism_(astronomy)|asterisms}}, reserving "{{w|Constellation_(astronomy)|constellations}}" for the 88 regions into which the sky is divided, each named for the most prominent asterism it contains, although "constellation" is used informally in place of "asterism" by even seasoned astronomers.) Different civilizations have recognized different constellations, and one could create their own constellations by connecting assorted points, the way Randall connected points in his plot to make "Rexthor."

In this comic, a set of data has had linear regression and some form of statistical analysis applied to it, indicating that there is low correlation between the two. The data points are so widely scattered that (as noted in the comic) it is easier to connect the data points in a constellation-like pattern than it is to determine whether the correlation is negative or positive (without looking at the trendline, of course). Because of this, [[Randall]] suggests we should be suspicious of any conclusions drawn from this data.

The comic is somewhat misleading, since the data in the graph actually has an ''R''<sup>2</sup> of 0.02, only a third of what Randall claims.  An example of published research with an ''R''<sup>2</sup> of 0.06 where the association in the graph is noticeable (if not strong) can be found [http://www.i-jmr.org/2012/1/e1/ here] (figure 2 has ''r'' = 0.25 which corresponds to ''R''<sup>2</sup> = 0.06). In addition, it is hard to see the association in the comic's graph because relatively few points are plotted. In a data set with 1000 observations and ''R''<sup>2</sup> = 0.06, any association between the two variables would be quite clear.

The lines connecting the stars in this "constellation" create a crude illustration of a person with an outstretched arm holding up a dog, which could be a reference to the film {{w|Life is Beautiful}} where a waiter carries a dog on his tray without realizing. The name "Rexthor the Dog Bearer" spoofs the fact that numerous Greek-derived constellation names have both a proper name and an epithet (for example, "Orion, the Hunter"). The fact that "Rex" is an archetypal dog name (but also meaning {{w|Rex (title)|king}} as in king of the dinosaurs <i>Tyrannosaurus rex</i>), adds to the humor.

The 95% {{w|confidence interval}} in statistics is such a range of an estimate, that it is expected to contain the real value (the estimated population parameter) 95% of the time. The confidence interval is a standard method to provide evaluation of the estimation error in statistics. On the right panel the resulting estimate seems to be a drawing, so the 95% confidence interval would be a set of drawings, expected to contain the correct drawing in 95% of samples where it is calculated. According to the title text, the interval in this particular sample also includes a cat and a teapot, so we can only make extremely vague statements in order to maintain 95% confidence.

The teapot may be a reference to {{w|Russell's_teapot|Russell's teapot}}, or possibly to the {{w|Sagittarius_(constellation)#Visualizations|"teapot" asterism in the constellation Sagittarius.}} Alternatively it is just because the "dog" actually looks more like a teapot than a dog, and Randall noticed this and added it in the title text. In the latter case, the two first suggestions are just another example on how humans see patterns also where there are none to find, like those of {{w|pareidolia}} mentioned in [[1551: Pluto]].

==Transcript==
:[Two square panels show identical sets of scattered black dots, with only the red additions being different.]

:[The left panel shows a slightly rising red line drawn through the middle of the panel, passing near a few dots but not obviously related to most of them. A red text is below the dots:]
:<span style="color:red">R<sup>2</sup>=0.06</span>

:[The right panel shows many of the dots connected by red lines to form a stick figure of a man resembling the constellation Orion, with the hand on the reader's right raised and holding an object. A red text is below the dots:]
:<span style="color:red">Rexthor, the Dog-Bearer</span>

:[A caption is below and spanning both panels:] 
:I don't trust linear regressions when it's harder to guess the direction of the correlation from the scatter plot than to find new constellations on it.

{{comic discussion}}

[[Category:Comics with color]]
[[Category:Scatter plots]]
[[Category:Math]]
[[Category:Statistics]]
[[Category:Astronomy]]
[[Category:Animals]]
[[Category:Dogs]]