Some say the cure for this is to switch to something called “Bayesian analysis.” Alas, while Bayesian procedures help some, they are far from a panacea, because they rely on the same foundational error those traditional procedures do.
Let me explain, but I warn you it won’t be easy going.
When data is collected in science to aide in answering a question, something like the following often happens.
First, “parameterized” probability (or statistical) models are created. These quantify the uncertainty of some measurable thing, and usually show how that uncertainty changes with different assumptions. The “parameters” can be thought of as knobs or dials that when turned change the probabilities.
Let’s use weight loss for an example, but keep in mind that what I have to say goes for any use of probability models.
We want to quantify in an experiment the uncertainty we have in weight loss among groups of people. Let’s suppose half the people in our experiment took some new drug which is touted to lead to greater weight loss.
The first mistake, and one that is ubiquitous, is to assume that probability models can prove the drug caused any observed weight loss. The drug may or may not work, but the model used to analyze it cannot be used as proof of this. These models, after all, begin with the assumption the drug is important or causal. To conclude because the model fits the data well that cause has been shown is to argue in a circle.
The opposite error is also common. The drug could be a cause of weight loss, but the model is poor at representing it, or the data, which is part of the model, is insufficient or flawed.
Incidentally (and you can skip this), it is curious that while the data is an integral unremovable part of the model, data is often thought of as something independent of the model. This is true mathematically, but only in the formal academic creation of model types. It is never true philosophically or in practice. Failure to recognize this exacerbates the misconception that probability models can identify cause.
Here is where it becomes interesting, and a lot more difficult.
There are two main streams of statistical thinking on models, both of which mistakenly take probability as a real thing. It is not obvious at first, but this is the fundamental error. This mistaken idea that probability is a real property of the world is why it is thought probability models can identify cause. Instead, probability is only a way to quantify our uncertainty. It is part of our thoughts, not part of the world.
Those two modeling streams are frequentism and Bayesianism, and a lot of fuss is made about their differences. But it turns out they’re much the same.
Frequentism leads to “null hypothesis significance testing”, and the hunt for wee p-values based on statements about the parameters of models. Bayesianism does testing in a slightly different way, but it, too, keeps its focus on the parameters of models. Again, these parameters are the guts inside models, which both frequentism and Bayes treats as real things. In our example, weight loss is real, but thoughts about weight loss are not real and not part of the world.
The idea behind hypothesis testing or Bayesian analysis is the same. Both suppose that if the model fits well, the model parameter associated with weight loss (or whatever we’re interested in) in reality takes some real value (hypothesis testing) or that the uncertainty in its value can be quantified (Bayes). If parameters are a part of the world, like rocks or electrons or whatever are, then it makes sense that if parameters take the “wrong” or the “right” value, they say something about the world. Just like weight says something real about the world. Yet parameters are like thoughts of weight loss, not weight loss itself.
Proving that parameters are purely matters of thought, and are not real, takes some doing (I have a book on the subject, but beware, it is not light reading). However, I offer a simple intuitive argument here.
Models are like maps of reality: and maps are not the territory. They are only abstractions of it. The figures and symbols on maps are not real, but they help give an idea of reality. Confusing the symbols on the map as reality would be the same mistake that is made in assuming the name of thing is the thing itself. And do not forget that there are many different maps that can be made of the same location, some better or less useful than others. Those symbols on the maps can’t all be real. Same thing with models and their parameters.
What needs to happen in statistical modeling, then, is to remove focus from inside models, stop obsessing over parameters, and put the focus back on observable reality. Have models make testable (probability) predictions about reality, and stop all indirect unverifiable statements about parameters. Make predictions of real things, actual measurable entities in the world.
If models are good, they will make good predictions. If they are bad, they won’t. It’s really as simple as that.
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