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.

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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.
  2. 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.

  3. Basing decisions on single instead of multiple measures is flawed.
  4. 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.)

  5. All assessments are proxies.
  6. 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.

The value-added wave is a tsunami

Edweek ran an article earlier this week in which economist Douglas N. Harris attempts to encourage economists and educators to get along.

He unfortunately lost me in the 3rd paragraph.

Drawing on student-level achievement data across years, linked to individual teachers, statistical techniques can be used to estimate how much each teacher contributed to student scores—the value-added measure of teacher performance. These measures in turn can be given to teachers and school leaders to inform professional development and curriculum decisions, or to make arguably higher-stakes decisions about performance pay, tenure, and dismissal.

Emphasis mine.

Economists and their education reform allies frequently make this claim, but it is not true, at least not yet. Value-added measures are based on standardized-test scores and neither currently provide information an educator can actually use to make professional development or curriculum decisions. When the scores are released, administrators and teachers receive a composite score and a handful of subscores for each student. In math, these subscores might be for topics like “Number and Operation Sense” and “Geometry”.

It does not do an educator any good to know last year’s students struggled with a topic as broad as “Number and Operation Sense”. Which numbers? Integers? Decimals? Did the students have problems with basic place value? Which operations? The non-commutative ones? Or did they have specific problems with regrouping and carrying? In what way are the students struggling? What errors are they making? What misconceptions might these errors point to? None of this information is contained in a score report. So, as an educator faced with test scores low in “Number and Operation Sense” (and which might be low in other areas as well), where do you start? Do you throw out the entire curriculum? If not, how do you know which parts of it need to be re-examined?

People trained in education recognize a difference between formative assessment—information collected for the purpose of improving instruction and student learning, and summative assessment—information collected to determine whether a student or other entity has reached a desired endpoint. Standardized tests are summative assessments—bad scores on them are like knowing that your football team keeps losing its games. This information is not sufficient for helping the team improve.

Why do economists see the issue so differently?

An economist myself, let me try to explain. Economists tend to think like well-meaning business people. They focus more on bottom-line results than processes and pedagogy, care more about preparing students for the workplace than the ballot box or art museum, and worry more about U.S. economic competitiveness. Economists also focus on the role financial incentives play in organizations, more so than the other myriad factors affecting human behavior. From this perspective, if we can get rid of ineffective teachers and provide financial incentives for the remainder to improve, then students will have higher test scores, yielding more productive workers and a more competitive U.S. economy.

This logic makes educators and education scholars cringe: Do economists not see that drill-and-kill has replaced rich, inquiry-based learning? Do they really think test preparation is the solution to the nation’s economic prosperity? Economists do partly recognize these concerns, as the quotations from the recent reports suggest. But they also see the motivation and goals of human behavior somewhat differently from the way most educators do.

This false dichotomy makes me cringe. As a trained education research scientist who is no stranger to statistical models, value-added is not ready for prime time because its primary input—standardized test scores—is deeply flawed. In science and statistics, if you put garbage data into your model, you will get garbage conclusions out. It has nothing to do with valuing art over economic competitiveness, and everything to do with the integrity of the science.

The divide between economists and others might be more productive if any of the reports provided specific recommendations. For example, creating better student assessments and combining value-added with classroom assessments are musts.

Thank you. Here where I start agreeing—if only that had been the central point of the article. I don’t dismiss value-added modeling as a technique, but I do not believe we have anything resembling good measures of teaching and learning.

We also have to avoid letting the tail wag the dog: Some states and districts are trying to expand testing to nontested grades and subjects, and to change test instruments so the scores more clearly reflect student growth for value-added calculations. This thinking is exactly backwards.

I agree completely, but that won’t stop states and districts from desperately trying to game the system. Since economists focus so much on financial incentives, this should be easy for them to understand: when the penalty for having low standardized test scores (or low value-added scores) is losing your funding, you will do whatever will get those scores up fastest. In most cases, that is changing the rules by which the scores are computed. Welcome to Campbell’s law.

The (other) dark side of standardized testing

Dan DiMaggio, professional standardized test-scorer, writes harrowingly in Monthly Review:

No matter at what pace scorers work, however, tests are not always scored with the utmost attentiveness. The work is mind numbing, so scorers have to invent ways to entertain themselves. The most common method seems to be staring blankly at the wall or into space for minutes at a time. But at work this year, I discovered that no one would notice if I just read news articles while scoring tests. So every night, while scoring from home, I would surf the Internet and cut and paste loads of articles—reports on Indian Maoists, scientific speculation on whether animals can be gay, critiques of standardized testing—into what typically came to be an eighty-page, single-spaced Word document. Then I would print it out and read it the next day while I was working at the scoring center. This was the only way to avoid going insane. I still managed to score at the average rate for the room and perform according to “quality” standards. While scoring from home, I routinely carry on three or four intense conversations on Gchat. This is the reality of test scoring.

The central assumption behind the push to link student test performance to teacher merit pay is that test scores accurately reflect student learning. We apparently need a system like this because we don’t trust teachers to fairly score their own students’ work. But we trust these guys.

Supplementary reading by Todd Farley:

One of the tests I scored had students read a passage about bicycle safety. They were then instructed to draw a poster that illustrated a rule that was indicated in the text. We would award one point for a poster that included a correct rule and zero for a drawing that did not.

The first poster I saw was a drawing of a young cyclist, a helmet tightly attached to his head, flying his bike over a canal filled with flaming oil, his two arms waving wildly in the air. I stared at the response for minutes. Was this a picture of a helmet-wearing child who understood the basic rules of bike safety? Or was it meant to portray a youngster killing himself on two wheels?

And the followup…

Since 1994, when I first got hired as a lowly temp for measly wages to spend mere seconds glancing at and scoring standardized tests, until the release of my non‐bestselling book last fall, I had steadfastly believed that large‐scale assessment was a lame measure of student learning that really only benefitted the multi‐national corporations paid millions upon millions upon millions of dollars to write and score the tests. I began to see the error of my ways last Thanksgiving, however, just as soon as my huge son popped from his mother’s womb, keening and wailing, demanding massive amounts of food, a closet full of clothing, and the assistance of various costly household staff (baby‐sitter, music teacher, test‐prep tutor, etc.). Only then, as my little boy first began his mantra of “more, more, more,” did I finally see standardized testing for what it really is: a growth industry.

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