Fishing in the Bay

Few statistics are more oft-quoted by empirical researchers than r-squared. While applauding the value of an intuitive interpretation in principle, it is pretty clear that the interpretation is wrong. Apart from honesty, the main reason I care about this is that it gets me into trouble with (the more discerning) students.

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A common lament of the naughties is that we are drowning in data. Wouldn’t it be nice if we could swim instead of drown? I have recently become aware of a new data visualization tool called Pivot, developed by Microsoft Live Labs.

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The Monty Hall problem is a terrific little example of how intuition, especially the intuition of those trained in statistics, can go horribly wrong. It appears from recent calculations of the economist, M. Keith Chen that some of the most famous experiments in psychology may have fallen victim to the same logical fallacy.

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Regression output almost universally lists estimates and standard errors as the main descriptors of the involvement of each predictor in the model. Yet in many fields, people are very confortable with correlations. Statisticians should also like correlations as they are unit free and allow a direct comparison of the strength of the association of each predictor. Bearing in mind that “correlation and regression” are the “meat and veg” of statistics (or the Laurel and Hardy?), why are correlations not reported in standard regression output?

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In class, I often explain standard error as the typical error or unreliability of an estimator. Standard deviation is typical deviation of the data around the mean. So I make it a point to interpret the word “standard” in the commonly understood sense. The problem is reconciling this with P-values. If an estimate differs from the null by one standard error then the one-sided P-value is 16% which is fairly unlikey in an informal sense, though not “significant” according to traditional choice of significance levels.

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I’m interested in how people think about statistical ideas—that’s ‘statistical cognition’. How do ‘read’ a CI? How do you interpret it, how would you explain its message to someone? We know the 95% refers to the % of intervals in a particular infinite sequence of intervals that include mu. But we’d like to think that our interval, calculated from our sample, is somehow pretty typical of the whole set. That’s probably the basis for our interpretation of our interval. If we’ve been unlucky and our interval is in the 5% that miss mu, we hope it’s a near miss.

Today’s CI comment and question is about this notion of typicality: is our obtained single CI a good exemplar of the set of all such CIs? Consider the simulation pictures below. Each shows 50 independent samples from a normal population, with the 95% CIs shown, those being based on t and the sample SD. Sample sizes of 30, 10, and 3. Horizontal scales differ, as signalled by tick marks on top axis.

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As a software magpie (anything shiny attracts my attention) I often find myself trawling through obscure reaches of the web, looking for interesting things. I’m especially interested in different ways of organizing and communicating statistical thinking. With this in mind, I’ve been experimenting with some nice software for mapping arguments called Reason!Able. My goal was to see if I could represent all the necessary inferential steps for a simple statistical exercise - a one-way analysis of variance for a four-level factor. Such a representation might be a useful teaching tool, or help communicate statistical results to non-statisticians, and possibly even reveal some of the more obscure corners of how we statisticians think. It turned out to be a little more complicated than I had suspected …
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