Statistics

Quantity, Quality, Social Science

September 17th, 2009  |  Published in Social Science, Statistics

Henry Farrell expresses the duality of social scientific thought by invoking a passage from one of my favorite books, Calvino’s Invisible Cities. The comments spin out the eternal quantitative vs. qualitative research debate, in both more and less interesting permutations.

Historically and philosophically, the whole qual-quant divide is an important object of social science, since it is itself a consequence of the same process of modernity and capitalist development that produces social science itself. It is only when society and its institutions appear at a scale too large for the human mind to grasp all at once that we require abstractions–particularly statistical and mathematic ones–to simplifiy and describe our social world to ourselves.

Within academia, however, there is a seemingly inescapable sense that qualitative and quantitative epistemologies are locked in some kind of zero-sum competition. These days people like to talk about “mixed methods”, but I agree with some commenters in the above thread that this too often amounts to doing a quantitative study and then using qualitative examples (from interviews or ethnography or whatever) as examples or window dressing.

It seems to me that a lot of this is driven by a misapprehension about what either approach is really good for.  The problem is that we expect quantitative and qualitative approaches to do the same kind of thing; that is, to collect data and use them to test well-defined hypotheses. I find that quantitative approaches are generally quite useful for taking well-defined concepts, and reasonably precise operationalizations of those concepts, and testing the interrelations between them. If your question is “do high tax rates inhibit economic growth”, and you have acceptable definitions and data for the subject and object of that hypothesis, then you can make useful–though never definitive–inferences using quantitative methods.

Qualitative methods are less often (though sometimes) suited to this kind of thing, because they are by nature rooted in the idiosyncracies of specific cases and hence are difficult to generalize. What qualitative work is really good for, I think, is in generating concepts. Quantitative analysis presupposes a huge conceptual apparatus: from the way ideas are operationalized, to the way survey questions are written, to the way variables are defined, to the way models are parameterized. Some of these presuppositions can be adjusted in the course of an analysis, but others are deeply encoded in the information we use. If  you want to know whether the categories of a “race” variable are appropriate, the best strategy is probably a qualitative one, which will examine how racial categories are experienced by people, and how they operate in everyday life. Likewise, new hypotheses can arise from “thick description” which would not be apparent from consulting large tables of numbers.

This, however, brings up an issue that will probably be uncomfortable for a lot of qualitative social scientists, particular those who are concerned with defending the “scientific” credentials of their work. Namely, can we draw a clear boundary between qualitative social science, journalism, and even fiction, with regards to their utility for driving the concept-formation process? Social science typically differentiates itself from mere journalism by its greater rigour; yet in my reading, the kind of rigour which is most important to qualitative work will be its interpretive rigour, rather than its precision in research design and data-gathering. Whether one is starting with ethnographic field notes or with The Wire, the point is to draw out and develop concepts and hypotheses in a sufficiently precise way that they can be tested with larger-scale (which is to say, generally quantitative) empirical data.

To put things this way seems to slide into a kind of cultural studies, except that the latter tends to set itself up as oppositional, rather than complementary, to quantitative empirical work. We would do far better, I think, to recognize that data analysis without qualitative conceptual interpretation is sterile and stagnant, while qualitative analysis without large-scale empiricism will tend to be speculative and inconclusive.

The ontology of statistics

September 4th, 2009  |  Published in Social Science, Statistics

Bayesian statistics is sometimes differentiated from its frequentist alternative with the claim that frequentists have a kind of platonist ontology, which treats the parameters they seek to estimate as being fixed by nature; Bayesians, in contrast, are said to hold a stochastic ontology in which there is variability “all the way down”, as it were. This distinction implies that frequentist measurements of uncertainty refer solely to epistemological uncertainty:  if we estimate that a certain variable has a mean of 50 and a standard error of two, we are saying only that we do not have enough information to specify the mean more precisely. In contrast, the Bayesian perspective (according to the view just elucidated) would hold that a measure of uncertainty includes not only epistemological but also ontological uncertainty: even with a sample size approaching infinity, the mean of the variable in question is the realization of some probability distribution and not a fixed quantity, and therefore can never be specified without uncertainty.

As regards the frequentist-Bayesian distinction, the above distinction is misleading and unhelpful.  Andrew Gelman is, by any sensible account, one of the leading exponents and practitioners of Bayesian statistics, and yet he says here that “I’m a Bayesian and I think parameters are fixed by nature. But I don’t know them, so I model them using random variables.” Compare this to the comment of another Bayesian, Bill Jefferys: “I’ve always regarded the main difference between Bayesian and classical statistics to be the fact that Bayesians treat the state of nature (e.g., the value of a parameter) as a random variable, whereas the classical way of looking at it is that it’s a fixed but unknown number, and that putting a probability distribution on it doesn’t make sense.”

For Gelman, the choice of Bayesian methods is not primarily motivated by ontological commitments, but is rather a kind of pragmatism: he adopts techniques such as shrinkage estimators, prior distributions, etc. because they give good predictions about the state of the world in cases where frequentist methods fail or cannot be applied. This, I suspect, corresponds to the inclinations and motivations of many applied researchers, who as often as not will be uninterested in the ontology implied by their methods, so long as the techniques give reasonable answers.

Moreover, if it is possible to be a Bayesian with a Platonist ontology, it is equally possible to wander into a stochastic view of the world without reaching beyond the well-accepted “classical” methods. Consider, for example, the logistic regression, which is by now a part of routine introductory statistical instruction in every field of social science.  A logistic regression model does not directly predict a binary outcome y, which can be 0 or 1. Rather, it predicts the probability of such an outcome, conditional on the predictor variables.  There are two ways to think about such models. One of them, the so-called “latent variable” interpretation, posits that there is some unobservable continuous variable Z, and that the outcome y is 0 if this Z variable is below a certain threshold, and 1 otherwise. If one holds to this interpretation, it is perhaps possible to hold to a Platonist ontology, by stipulating that the value of Z is “fixed by nature”. However, this fixed parameter is at the same time unobservable, leading to the unsatisfying conclusion that a the propensity of event y occuring for a given subject is at once fixed and unknowable.

In the latent variable interpretation,  the predicted probabilities generated by a logistic regression are simply emanations from the “true” quantity of interest, the unobserved value of Z. An alternative interpretation is that the predicted probabilities are themselves the quantities of interest. Ontologically, this means that rather than having an exact value for Z, each case is associated with a certain probability that for that case, y=1. Of course, in the actual world we observe, each case in our dataset is either 1 or 0. But this second interpretation of the model implies that if we “ran the tape of history over again”, to paraphrase Stephen Jay Gould, the values of y for each individual case might be different; only the overall distribution of probabilities is assumed to be constant.

Thus the distinction between the Platonist and stochastic ontologies in statistics turns out to be quite orthogonal to the distinction between frequentist and Bayesian. And it is an important distinction to be aware of, because it has real practical implications for applied researchers.  It will affect, for example, the way in which we assess how well a model fits the data.

In the case of logistic regression, the Platonist view would imply that the best model possible would predict every case correctly: that is, it would yield a predicted probability of more than 0.5 when y=1, and less than 0.5 when y=0. On the stochastic view, however, that degree of predictive accuracy is a priori held to be impossible, and achieving it indicates overfitting of the model. The best one can really aim for, on this view, is a model which gets the probabilities right–so that for 10 cases with predicted probabilities of 0.1, there should should be one case where y=1 and nine where y=0.

This conundrum arises even for Ordinary Least Squares regression, even though in that case the outcome variable is continuous and the model predicts it directly. It has long been traditional to assess OLS model fit using R-squared, the proportion of variance explained by the model. Many people unthinkingly assume that because the theoretical upper bound of the R-squared statistic is 1, the maximum possible value in any particular empirical situation is also 1. But this assumption 0nce again rests on an implicit Platonist ontology. It assumes that sigma, the residual standard error of a regression, reflects only omitted variables rather than inherent variability in the outcome in question. But as Gary King observed a long time ago, if some portion of sigma is due to intrinsic, ontological variability, then the maximum value of R-squared is some unknown value less than 1.* In this case, once again, high values of R-squared may be indicators of overfitting rather than signs of a well-constructed model.

Statistics, even in its grubbiest, most applied forms, is philosophical; we ignore that aspect of quantitative practice at our peril. I am put in mind of Keynes’ remark about economic common sense and theoretical doctrine, which I will not repeat here as it is already ubiquitous.

*In practice, the residual variability may truly ontological in the sense that it is rooted in the probabilistic behavior of the physical world at the level of quantum mechanics, or it may be that all variation can be accounted for in principle, but that residual variation is irreducible in practice, because of the exteremely large number of very minor causes that contribute to the outcome. In either case, the consequence for the applied researcher is the same.

http://www.stat.columbia.edu/~cook/movabletype/archives/2007/12/intractable_is.html#more