Econometrics cares not only about statistical significance but also usefulness/economic usefulness.
Causal inference builds on base statistics and ML, but its strength lies in how it uses design and assumptions to isolate causality. Tools like sensitivity analysis, robustness checks, and falsification tests help assess whether the causal story holds up. My one beef is that these tools still lean heavily on the assumption that the underlying theoretical model is correctly specified. In other words, causal inference helps stress-test assumptions, but it doesn’t always provide a clear way to judge whether one theoretical framework is more valid than another!
> Using Effect Size—or Why the P Value Is Not Enough
> Statistical significance is the least interesting thing about the results. You should describe the results in terms of measures of magnitude –not just, does a treatment affect people, but how much does it affect them.
– Gene V. Glass
And if we increase N enough we will be able to find these 'good measurements' and 'statistically significant differences' everywhere.
Worse still if we did not agree in advance what hypotheses we were testing, and go looking back through historical data to find 'statistically significant' correlations.
One interesting thing to keep in mind is that Ronald Fisher did most of his work before the publication of Kolmogorov's probability axioms (1933). There's a real sense in which the statistics used in social sciences diverged from mathematics before the rise of modern statistics.
So there's a lot of tradition going back to the 19th century that's misguided, wrong, or maybe just not best practice.
N being big means that small real effects can plausibly be detected as being statistically significant.
It doesn't mean that a larger proportion of measurements are falsely identified as being statistically significant. That will still occur at a 5% frequency or whatever your alpha value is, unless your null is misspecified.
But even though you know the measurement can't be exactly 0.000 (with infinitely many decimal places) a priori, you don't know if your measurement is any good a priori or whether you're measuring the right thing.
In a finite or countable number of trials you won't see a measure zero event.
> they're estimating the probability of rejecting the null if the null was true.
Right, but the null hypothesis is usually false and so it's a weird thing to measure. It's a proxy for the real thing you want, which is the probability of your hypothesis being true given the data. These are just some of the reasons why many statisticians consider the tradition of null hypothesis testing to be a mistake.
Significance does not tell you this. The p-value can be arbitrarily close to 0 while the probability of the null hypothesis being true is simultaneously arbitrarily close to one
When wielded correctly, statistical significance is a useful guide both to what's a real signal worth further investigation, and it filters out meaningless effect sizes.
A bigger problem even when statistical significance is used right is publication bias. If, out of 100 experiments, we only get to see the 7 that were significant, we already have a false:true ratio of 5:2 in the results we see – even though all are presented as true.
(effect size) / (noise / sqrt(n))
Note that bigger test statistic means smaller p-value.
So very low p-values usually come from bigger effects or from very large sample sizes (n). That's why you can technically get p<0.001 with a microscopic effect, but only if you have astronomical sample sizes. In most empirical studies, though, p<0.001 does suggest the effect is going to be large because there are practical limits on the sample size.
I'm regularly working with datasets in the hundreds of thousands to millions, and that's small fry compared with what's out there.
The use of regression, for me at least, is not getting that p-gotcha for a paper, but as a posh pivot table that accounts for all the variables at once.
For example, I’ve encountered the belief that just by recording something at ultra high temporal resolution gives you “millions of datapoints”. This then has all sorts of effects on the breakdown of statistics and hypothesis testing (seemingly).
In reality, the replicability of the entire setup, the day it was performed, the person doing it, etc. means the n for the day is probably closer to 1. So to ensure replicability you’d have to at least do it on separate days, with separately prepared samples. Otherwise, how can you eliminate the chance that your ultra finicky sample just happened to vibe with that day’s temperature and humidity?
But they don’t teach you in statistics what exactly “n” means, probably because a hundred years ago it was much more literal in nature. 100 samples is because I counted 100 mice, 100 peas, or 100 surveys.
There’s a lot of folks out there though who learned the mechanics of linear regression in a bootcamp or something without gaining an appreciation for the underlying theories, and those folks are looking for low p-value and as long as they get it it’s good enough.
I saw this link yesterday and could barely believe it, but I guess these folks really live among us.
https://stats.stackexchange.com/questions/185507/what-happen...
For example, if your phenomenon is observable at 50 Hz, maybe even 10 Hz, then any higher temporal resolution does not give you new information, because any two adjacent datapoints in the time-series are extremely correlated. Going the other way, at a very low sampling frequency you'd just get the mean, which might not reveal anything of interest.
If you bin 100 Hz data at 50 Hz, are they the same? Is the Fourier spectrum the same? If you have samples of different resolution you must choose the lowest common denominator for a fair statistical comparison. Otherwise, a recording between a potato and an advanced instrument would always be "statistically different", which doesn't make sense.
If you don't find "anything", the old adage goes "the absence of evidence is not the evidence of absence", so statistics don't really fail here. You can only conclude that your method is not sensitive enough.
As an example, read just about any health or nutrition research article referenced in popular media and there's very often a pretty weak effect size even though they've achieved "statistical significance." People then end up making big changes to their lifestyles and habits based on research that really does not justify those changes.
People interpret "statistically significant" to mean "notable"/"meaningful". I detected a difference, and statistics say that it matters. That's the wrong way to think about things.
Significance testing only tells you the probability that the measured difference is a "good measurement". With a certain degree of confidence, you can say "the difference exists as measured".
Whether the measured difference is significant in the sense of "meaningful" is a value judgement that we / stakeholders should impose on top of that, usually based on the magnitude of the measured difference, not the statistical significance.
It sounds obvious, but this is one of the most common fallacies I observe in industry and a lot of science.
For example: "This intervention causes an uplift in [metric] with p<0.001. High statistical significance! The uplift: 0.000001%." Meaningful? Probably not.