### Don't Believe the "Defenders" of Teachers: Teachers Do Matter

Family and income are surely important, but the "10% of variance" argument is wrong for at least two reasons:

First, in statistical terms, saying that teachers account for 10% of the variance in student test scores does NOT mean that teachers are unimportant. Wrong, wrong, wrong. (At the end of the blog post, I say more about what explaining variance means.)

The eminent Harvard professors Rosenthal and Rubin explained this in a 1982 article, "A Simple, General Purpose Display of Magnitude of Experimental Effect," Journal of Educational Psychology 74 no. 2: 166-69 (that article isn't available online, but is described here).

As luck would have it, Rosenthal and Rubin address the precise example of a case wherein 10% of the variance was explained:

We found experienced behavioral researchers and experienced statisticians quite surprised when we showed them that the Pearson r of .32 associated with a coefficient of determination (r^{2}) of only .10 was the correlational equivalent of increasing a success rate from 34% to 66% by means of an experimental treatment procedure; for example, these values could mean that a death rate under the control condition is 66% but is only 34% under the experimental condition. We believe . . . that there may be a widespread tendency to underestimate the importance of the effects of behavioral (and biomedical) interventions . . . simply because they are often associated with what are thought to be low values of r^{2}.

By analogy, saying that teacher quality explains 10% of the variance would be equivalent to saying that teachers can raise the passing rate from 34% to 66%. That's nothing to sneeze at, and it certainly isn't a reason for teachers to throw up their hands in dismay at the hopelessness of their task.

Second, the fact that teachers account for 10% of variance NOW, given a particular set of data points, tells us little or nothing about the true causal importance of teachers. As Richard Berk explains in his book

*Regression Analysis: A Constructive Critique,*"Contributions to explained variance for different predictors do not represent the causal importance of a variable." 10% isn't a Platonic ceiling on what teachers can accomplish, and the proportion of variance explained tells us very little about how much impact teachers

*really do*have.

A simple hypothetical example makes this clear: Imagine that all teachers in a school were of equal quality. Given equal teachers, any variation in student test scores would automatically have to arise from

*something other than*differing quality of teaching. So a regression equation in that context might tell us that demographics explain a huge amount of the variation in test scores, while teaching quality explains nothing. But it would be

*completely wrong*to conclude that demographics are inherently more important than teaching quality, or even that teaching quality doesn't matter.

*The exact opposite might be the case*, for all that such a regression could tell us.

Moreover, if all teachers became twice as effective as they are now, there would still be variance among teachers and variance among student test scores, and teachers collectively might still "account" for a "small" amount of variance, but student performance might be much higher. The fact that teachers account for 10% of variance today (as large as that actually is) simply does not give us any sort of limit on how much student achievement could rise if the mean teacher effectiveness shifted sharply to the right.

So the would-be defenders of teachers can breathe a sigh of relief: value-added modeling might still be a shaky idea for several other reasons, but there's no need to denigrate the potential of teachers.

* * *

A more detailed statistical explanation:

The proportion of variance explained means is that if you take the Pearson product-moment correlation, and square it, you end up (after some algebra) with the following:

What does this mean? The denominator is calculated by taking all the individual Y's (in the education context, all of the student test scores that you're trying to explain), subtracting the average Y value, squaring all of the differences, and adding up all of the squared values. In the context of the following graph, the denominator gives us a measure of the total squared distance (in the vertical direction) that all of the red dots deviate from the average Y value.

The numerator tells us how far the regression line deviates from the predicted Y values. The regression line predicts that the Y values will be along the line itself, which obviously isn't exactly true. So the

*predicted*Y values (that's what the little ^ sign over the Y means) have the average Y value subtracted, the difference is squared, and then all the squared differences are added up.

All in all, the "proportion of variance explained" figure is just a way to represent

*how close a regression line based on X will come to the actual red dots in the graph, compared to how close a line based on just the average red dot will come*.

For the same reason that correlation is not causation, accounting for variance does not provide an upper limit for the true causal importance of a variable. As noted above, the level of variance "explained" is a bad way to determine how important X actually is. See D'Andrade and Hart, for example. UPDATE: See Cosma Shalizi's post on explaining variance.

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