Editing 2533: Slope Hypothesis Testing

{{comic
| number    = 2533
| date      = October 25, 2021
| title     = Slope Hypothesis Testing
| image     = slope_hypothesis_testing.png
| titletext = "What? I can't hear--" "What? I said, are you sure--" "CAN YOU PLEASE SPEAK--"
}}

==Explanation==
"Slope hypothesis testing" is a method of testing the significance of a hypothesis involving a scatter plot.

In this comic, [[Cueball]] and [[Megan]] are performing a study comparing student exam grades to the volume of their screams. Student A has the worst grade and softest scream, but Student B has the ''best'' grades and Student C the ''loudest'' scream. A trendline has been plotted, indicating a positive correlation between grades and volume...but the p-value is extremely high, indicating little statistical significance to the trend. P-value is based on both how well the data fits the trendline and how many data points have been taken; the more data points and the better they fit, the lower the p-value and more significant the data.

Megan complains about the insignificance of their results, so Cueball suggests having each student scream into the microphone a few more times. (The three students are still there as they can be seen behind them. The three students look like schoolkids; one of them is [[Jill]].) 

Having the students scream again will not help though, because it only provides more data on the screaming without providing more data on its relation to exam scores and is a joke around poor statistical calculations likely made in the field today. The p-value is incorrectly recalculated based on the increased number of measurements without accounting for the fact that observations are nested within students. Each student has exactly the same test scores (probably referencing the same datum as before) and have vocal volume ranges that don't drift far either (each seems to have a range of scream that is fairly consistent and far from overlapping). Megan is pleased by these results, but Cueball belatedly realizes this technique may not be scientifically valid. Cueball is correct (presuming that they are using simple linear regression). A more appropriate technique would account for the non-independence of the data (that multiple data points come from each person). Examples of such techniques are multilevel modeling and Huber-White robust standard errors. 

Measuring data multiple times can be a way to increase its accuracy but does not increase the number of data points with regard to another metric, and the horizontally clustered points on the chart make this visually clear. A more effective and scientifically correct way of gathering data test would be to test other students and add their figures to the existing data, rather than repeatedly testing the same three students.

Common statistical formulae assume the data points are statistically independent, that is, that the test score and volume measurement from one point don't reveal anything about those of the other points. By measuring each individual's scream multiple times, Cueball and Megan violate the independence assumption (a person's scream volume is unlikely to be independent from one scream to the next) and invalidate their significance calculation. This is an example of pseudoreplication. Furthermore, Megan and Cueball fail to obtain new test scores for each student, which would further limit their statistical options.

Another strange aspect of their experiment is that the p-values obtained during a typical linear regression assume there is uncertainty in the y-values, but the x-values are fully known, whereas in this experiment, they are reducing uncertainty in the x-values of their data, while doing nothing to improve knowledge of the y-values.

Moreover, even if the new data were statistically independent, this still appears to be a classic example of "p-hacking", where new data is added until a statistically significant p-value is obtained.

In current AI, there's a push toward "few-shot learning", where only a few data items are used to form conclusions, rather than the usual millions of them.  This comic displays danger associated with using such approaches without understanding them in depth.

Additionally, a common theme in some research is the discovery of correlations that do not survive independent reproduction.  This is because randomness with too few samples produces apparent correlations, and Randall has repeatedly made comics about this hopeful error (see [[111]], [[925]] and [[882]] among others).

In the title text, Megan and Cueball are trying to yell over each other, asking each other to speak up so they can be heard; they are presumably suffering tinnitus or other hearing problems after listening to so much shouting.

==Transcript==
:[Three points are labeled "Student A", "Student B" and "Student C" from left to right in a scatter plot with axes labeled "Stats exam grade" (60-100) and "Scream loudness (decibels)" (86-94) with a trend line. Student B has the highest exam grade, followed by Student C and then Student A.]
:[A line goes from the trend line to a box containing the following:]
:β=1.94 
:p=0.586

:[In a frameless panel, Megan reads a piece of paper while facing Cueball while three students look at them from the background.]
:Megan: Darn, not significant.
:Cueball: We need more data. Have them each try yelling in to the mic a few more times.

:[The same scatter plot as in the first panel except with more points for each of the students with slightly different decibel values, and the text in the text box changed to:]
:β=1.94
:p=0.037*  
:<nowiki>*</nowiki>Significant!

:[Similar panel to the second one]
:Megan: Perfect!
:Cueball: Are you ''sure'' we're doing slope hypothesis testing right?

{{comic discussion}}

[[Category:Comics featuring Megan]]
[[Category:Comics featuring Cueball]]
[[Category:Comics featuring Jill]] <!-- The other two kids are also, well, kids, and thus not Hairy or Megan -->
[[Category:Statistics]]
[[Category:Charts]]
[[Category:Scientific research]]
[[Category:Kids]]