Statistical issues with applying VAM

There’s a wonderful statistical discussion of Michael Winerip’s NYT article critiquing the use of value-added modeling in evaluating teachers, which I referenced in a previous post. I wanted to highlight some of the key statistical errors in that discussion, since I think these are important and understandable concepts for the general public to consider.

• Margin of error: Ms. Isaacson’s 7th percentile score actually ranged from 0 to 52, yet the state is disregarding that uncertainty in making its employment recommendations. This is why I dislike the article’s headline, or more generally the saying, “Numbers don’t lie.” No, they don’t lie, but they do approximate, and can thus mislead, if those approximations aren’t adequately conveyed and recognized.
• Reversion to the mean: (You may be more familiar with this concept as “regression to the mean,” but since it applies more broadly than linear regression, “reversion” is a more suitable term.) A single measurement can be influenced by many randomly varying factors, so one extreme value could reflect an unusual cluster of chance events. Measuring it again is likely to yield a value closer to the mean, simply because those chance events are unlikely to coincide again to produce another extreme value. Ms. Isaacson’s students could have been lucky in their high scores the previous year, causing their scores in the subsequent year to look low compared to predictions.
• Using only 4 discrete categories (or ranks) for grades:
  • The first problem with this is the imprecision that results. The model exaggerates the impact of between-grade transitions (e.g., improving from a 3 to a 4) but ignores within-grade changes (e.g., improving from a low 3 to a high 3).
  • The second problem is that this exacerbates the nonlinearity of the assessment (discussed next). When changes that produce grade transitions are more likely than changes that don’t produce grade transitions, having so few possible grade transitions further inflates their impact.
    Another instantiation of this problem is that the imprecision also exaggerates the ceiling effects mentioned below, in that benefits to students already earning the maximum score become invisible (as noted in a comment by journalist Steve Sailer: 

    Maybe this high IQ 7th grade teacher is doing a lot of good for students who were already 4s, the maximum score. A lot of her students later qualify for admission to Stuyvesant, the most exclusive public high school in New York.
    But, if she is, the formula can’t measure it because 4 is the highest score you can get.

• Nonlinearity: Not all grade transitions are equally likely, but the model treats them as such. Here are two major reasons why some transitions are more likely than others.
  • Measurement ceiling effects: Improving at the top range is more difficult and unlikely than improving in the middle range, as discussed in this comment:
    

    Going from 3.6 to 3.7 is much more difficult than going from 2.0 to 2.1, simply due to the upper-bound scoring of 4.

    However, the commenter then gives an example of a natural ceiling rather than a measurement ceiling. Natural ceilings (e.g., decreasing changes in weight loss, long jump, reaction time, etc. as the values become more extreme) do translate into nonlinearity, but due to physiological limitations rather than measurement ceilings. That said, the above quote still holds true because of the measurement ceiling, which masks the upper-bound variability among students who could have scored higher but inflates the relative lower-bound variability due to missing a question (whether from carelessness, a bad day, or bad luck in the question selection for the test). These students have more opportunities to be hurt by bad luck than helped by good luck because the test imposes a ceiling (doesn’t ask all the harder questions which they perhaps could have answered).

  • Unequal responses to feedback: The students and teachers all know that some grade transitions are more important than others. Just as students invest extra effort to turn an F into a D, so do teachers invest extra resources in moving students from below-basic to basic scores.
    More generally, a fundamental tenet of assessment is to inform the students in advance of the grading expectations. That means that there will always be nonlinearity, since now the students (and teachers) are “boundary-conscious” and behaving in ways to deliberately try to cross (or not cross) certain boundaries.
• Definition of “value”: The value-added model described compares students’ current scores against predictions based on their prior-year scores. That implies that earning a 3 in 4th grade has no more value than earning a 3 in 3rd grade. As noted in this comment:
  

  There appears to be a failure to acknowledge that students must make academic progress just to maintain a high score from one year to the next, assuming all of the tests are grade level appropriate.

  Perhaps students can earn the same (high or moderate) score year after year on badly designed tests simply through good test-taking strategies, but presumably the tests being used in these models are believed to measure actual learning. A teacher who helps “proficient” students earn “proficient” scores the next year is still teaching them something worthwhile, even if there’s room for more improvement.

These criticisms can be addressed by several recommendations:

1. Margin of error. Don’t base high-stakes decisions on highly uncertain metrics.
2. Reversion to the mean. Use multiple measures. These could be estimates across multiple years (as in multiyear smoothing, as another commenter suggested), or values from multiple different assessments.
3. Few grading categories. At the very least, use more scoring categories. Better yet, use the raw scores.
4. Ceiling effect. Use tests with a higher ceiling. This could be an interesting application for using a form of dynamic assessment for measuring learning potential, although that might be tricky from a psychometric or educational measurement perspective.
5. Nonlinearity of feedback. Draw from a broader pool of assessments that measure learning in a variety of ways, to discourage “gaming the system” on just one test (being overly sensitive to one set of arbitrary scoring boundaries).
6. Definition of “value.” Change the baseline expectation (either in the model itself or in the interpretation of its results) to reflect the reality that earning the same score on a harder test actually does demonstrate learning.

Those are just the statistical issues. Don’t forget all the other problems we’ve mentioned, especially: the flaws in applying aggregate inferences to the individual; the imperfect link between student performance and teacher effectiveness; the lack of usable information provided to teachers; and the importance of attracting, training, and retaining good teachers.

Some limitations of value-added modeling

Following this discussion on teacher evaluation led me to a fascinating analysis by Jim Manzi.

We’ve already discussed some concerns with using standardized test scores as the outcome measures in value-added modeling; Manzi points out other problems with the model and the inputs to the model.

1. Teaching is complex.
2. It’s difficult to make good predictions about achievement across different domains.
3. It’s unrealistic to attribute success or failure only to a single teacher.
4. The effects of teaching extend beyond one school year, and therefore measurements capture influences that go back beyond one year and one teacher.

I’m not particularly fond of the above list—while I agree with all the claims, they’re not explained very clearly and they don’t capture the below key issues, which he discusses in more depth.

1. Inferences about the aggregate are not inferences about an individual.

More deeply, the model is valid at the aggregate level, “but any one data point cannot be validated.” This is a fundamental problem, true of stereotypes, of generalizations, and of averages. While they may enable you to make broad claims about a population of people, you can’t apply those claims to policies about a particular individual with enough confidence to justify high-stakes outcomes such as firing decisions. As Manzi summarizes it, an evaluation system works to help an organization achieve an outcome, not to be fair to the individuals within that organization.

This is also related to problems with data mining—by throwing a bunch of data into a model and turning the crank, you can end up with all kinds of difficult-to-interpret correlations which are excellent predictors but which don’t make a whole lot of sense from a theoretical standpoint.

2. Basing decisions on single instead of multiple measures is flawed.

From a statistical modeling perspective, it’s easier to work with a single precise, quantitative measure than with multiple measures. But this inflates the influence of that one measure, which is often limited in time and scale. Figuring out how to combine multiple measures into a single metric requires subjective judgment (and thus organizational agreement), and, in Manzi’s words, “is very unlikely to work” with value-added modeling. (I do wish he’d expanded on this point further, though.)

3. All assessments are proxies.

If the proxy is given more value than the underlying phenomenon it’s supposed to measure, this can incentivize “teaching to the test”. With much at stake, some people will try to game the system. This may motivate those who construct and rely on the model to periodically change the metrics, but that introduces more instability in interpreting and calibrating the results across implementations.

In highlighting these weaknesses of value-added modeling, Manzi concludes by arguing that improving teacher evaluation requires a lot more careful interpretation of its results, within the context of better teacher management. I would very much welcome hearing more dialogue about what that management and leadership should look like, instead of so much hype about impressive but complex statistical tools expected to solve the whole problem on their own.