On Tuesday I commented here on the ASA (in their words) “Position on p-values: context, process, and purpose.” A number of readers replied, some of them positive, some mistakenly thinking I don’t think statistical inferences are needed, and some claiming I overinterpreted the ASA’s statement. I’ll respond in the current post, and will devote most of it to what I believe are the proper alternatives.
First, though, in order to address the question, “What did the ASA really mean?”, I think it may be helpful discuss why the ASA suddenly came out with a statement. What we know is that the ASA statement itself opens with George Cobb’s wonderfully succinct complaint about the vicious cycle we are caught in: “We teach [significance testing] because it’s what we do; we do it because it’s what we teach.” The ASA then cites deep concerns in the literature, with quotes such as “[Significance testing] is science’s dirtiest secret” with “numerous deep flaws.”
As the ASA also points out, none of this is new. However, there is renewed attention, in a more urgent tone than in the past. The ASA notes that part of this is due to the “replicability crisis,” sparked by the Ionnidis paper., which among other things led the ASA to set up a last-minute (one might even say “emergency”) session at JSM 2014. Another impetus was a ban on p-values by a psychology journal (though I’d add that the journal’s statement led some to wonder whether their editorial staff was entirely clear on the issue).
But I also speculate that the ASA’s sudden action came in part because of a deep concern that the world is passing Statistics by, with our field increasingly being seen as irrelevant. I’ve commented on this as being sadly true, and of course many of you will recall then-ASA president Marie Davidian’s plaintive column title, “Aren’t WE Data Science?” I suspect that one of the major motivations for the ASA’s taking a position on p-values was to dispel the notion that statistics is out of data and irrelevant.
In that light, I stand by the title of my blog post on the matter. Granted, I might have used language like “ASA Says [Mostly] No to P-values,” but I believe it was basically a No. Like any statement that comes out of a committee, the ASA phrasing adopts the least common (read most status quo) denominator and takes pains not to sound too extreme, but to me the main message is clear. For example, I believe my MovieLens data example captures the essence of what the ASA said.
What is especially telling is that the ASA gave no examples in which p-values might be profitably used. Their Principle 1 is no more than a mathematical definition, not a recommendation. This is tied to the issue I brought up, that the null hypothesis is nearly always false a priori, which I will return to later in this blog.
Well, then, what should be done instead? Contrary to the impression I seem to have given some readers, I certainly do not advocate using visualization techniques and the like instead of statistical inference. I agree with the classical (though sadly underemphasized) alternative to testing, which is to form confidence intervals.
Let’s go back to my MovieLens example:
Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 3.4725821 0.0482655 71.947 < 2e-16 *** age 0.0033891 0.0011860 2.858 0.00436 ** gender 0.0002862 0.0318670 0.009 0.99284 ...
The age factor is found to have a “highly significant ” impact on movie rating, but in fact the point estimate, 0.0033891, shows that the impact is negligible; a 10-year difference in age corresponds to about 0.03 point in mean rating, minuscule in view of the fact that ratings range from 1 to 5.
A confidence interval for the true beta coefficient, in this case, (0.0011,0.0057), shows that. Tragically, a big mistake made by many who teach statistics is to check whether such an interval contains 0 — which entirely defeats the purpose of the CI. The proper use of this interval is to note that the interval is near 0, in fact even at its upper bound.
So slavish use of p-values would have led to an inappropriate conclusion here (the ASA’s point), and moreover, once we have that CI, the p-value is useless (my point).
My point was also that we knew a priori that H0: β1 = 0 is false. The true coefficient is not 0.0000000000… ad infinitum. In other words, the fundamental problem — “statistics’ dirtiest secret” — is that the hypothesis test is asking the wrong question. (I know that some of you have your favorite settings in which you think the null hypothesis really can be true, but I would claim that closer inspection would reveal that that is not the case.) So, not only is the test providing no additional value, once we have a point estimate and standard error, it is often worse than useless, i.e. harmful.
Problems like the above can occur with large samples (though there were only 949 users in the version of the movie data that I used). The opposite can occur with small samples. Let’s use the current U.S. election season. Say a staffer for Candidate X commissions a small poll, and the result is that a CI for p, the population proportion planning to vote for X, is (0.48,0.65). Yes, this contains 0.50, but I submit that the staffer would be remiss in telling X, “There is no significant difference in support between you and your opponent.” The interval is far more informative, in this case more optimistic, than that,
One of the most dramatic examples of how harmful testing can be is a Wharton study in which the authors took real data and added noise variables. In the ensuing regression analysis, lo and behold, the fake predictors were found to be “significant.”
Those of us who teach statistics (which I do, in a computer science context, as well as a member of a stat department long ago) have a responsibility to NOT let our students and our consulting clients follow their natural desire for simple, easy, pat answers, in this case p-values. Interpreting a confidence interval takes some thought, unlike p-values, which automatically make our decisions for us.
All this is fine for straightforward situations such as estimation of a single mean or a single regression coefficent. Unfortunately, though, developing alternatives to testing in more advanced settings can be challenging, even in R. I always praise R as being “Statistically Correct, written by statisticians for statisticians,” but regrettably, those statisticians are the ones George Cobb complained about, and R is far too testing-oriented.
Consider for instance assessing univariate goodness of fit for some parametric model. Note that I use the word assessing rather than testing, especially important in that the basic R function for the Kolmogorov-Smirnov procedure, ks.test(), is just what its name implies, a test. The R Core Team could easily remedy that, by including an option to return a plottable confidence band for the true CDF. And once again, the proper use of such a band would NOT be to check whether the fitted parametric CDF falls entirely within the band; the model might be quite adequate even if the fitted CDF strays outside the band somewhat in some regions.
Another example is that of log-linear models. This methodology is so fundamentally test-oriented that it may not be clear to some what might be done instead. But really, it’s the same principle: Estimate the parameters and obtain their standard errors. If for example you think a model with two-way interactions might suffice, estimate the three-way interactions; if they are small relative to the lower-order ones, you might stop at two. (Putting aside here the issue of after-the-fact inference, a problem in any case.)
But even that is not quite straightforward in R (I’ve never used SAS, etc. but they are likely the same). The loglin() function, for instance, doesn’t even report the point estimates unless one pro-actively requests them — and even if requested, no standard errors are available. If one wants the latter, one must use glm() with the “Poisson trick.”
In the log-linear situation, one might just informally look at standard errors, but if one wants formal CIs on many parameters, one must use simultaneous inference techniques, which brings me to my next topic.
The vast majority of those techniques are, once again, test-oriented. One of the few, the classic Scheffe’ method, is presented in unnecessarily restrictive form in textbooks (linear model, normality of Y, homoscedasticity, F-statistic and so on). But with suitable centering and scaling, quadratic forms of asymptotically normally distributed vectors have an asymptotic chi-squared distribution, which can be used to get approximate simultaneous confidence intervals. R should add a function to do this on vcov() output.
In short, there is a lot that could be done, in our teaching, practice and software. Maybe the ASA statement will inspire some in that direction.