brownelearning

I just watched a good TED speech by Peter Donnelly, a statistician who explains that most people have just enough knowledge of statistics to be dangerous. Among other examples, he walks through claims of test accuracy and how deceiving they can be. He leads us through the following hypothetical example with HIV:

In a population of 1,000,000, 100 are HIV-positive; 999,900 are HIV-negative.

A test for HIV is 99% accurate.

What is the chance that a person who tests positive is actually negative?

About 99%.

See, out of the 100 people who are actually positive, 99 will test positive and 1 will test negative – that’s 99% accuracy. Of the 999,900 people who are negative, 989,901 will test negative while 9,999 (1 out of every 100) will test positive even though they are actually negative.

So then we have 10,098 people who tested positive, but 9,999 of them are actually negative. 9,999 / 10,098 = .9902. About 99% of the positive results are false even though the test has 99% accuracy.

Weird, eh? The BBC draws on this principle in an editorial about cameras that supposedly identify terrorists. And I heard that a running magazine completed a similar analysis of tests for performance enhancing drugs: If there are 100 dopers out of 10,000 runners, and the test is 99% accurate, half of those caught will be innocent.

But don’t make the mistake of assuming that accuracy statistics are meaningless; on the contrary, they can be very useful if we know the degree to which the target condition exists in the tested sample. Donnelly’s hypothetical HIV test produced so many false positives because HIV was actually very rare in his hypothetical sample.

Suppose we develop a test for a condition that is more common than HIV – say, a DNA test for left-handedness – and this test is 99% accurate. About 10% of the population is left-handed, but 8.3% (1 out of 12) of those our test identifies as left-handed are actually right-handed. This still may not be what you expect from a 99% accurate test, but it’s a whole lot better than our HIV test that also claimed 99% accuracy. In other words, even if two tests have equal accuracy, the one that tests for the more common condition will have a lower ratio of false positives to true positives.

With a 99% accurate DNA test for an even-more-common condition – femaleness – we will get results very close to the claimed accuracy because being female is more common than having HIV or being left-handed. In fact, if we assume (wrongly, I know) that each sex occurs equally in the world, 1% of those identified as males will actually be females, and 1% of those identified as females will be males. That’s pretty good.

So, we just need to make sure to only test for common conditions and we won’t have this problem, right? Well, no. See, the tradeoff is that, as the condition becomes more common, your rate of false negatives increases. This figure shows how 99% accurate tests produce different percentages of false positives and false negatives depending on how common or rare the condition is in the tested sample.

I know it can be hard to teach full-time – or substitute regularly – and take graduate courses. Many of my advisees take at least two courses each summer so they can carry fewer credits during the fall and spring semesters. I advise you to do this, with a few caveats…

1. There are two five-year limits:
   1. If you have received your initial teaching certification, you have five years from the date of your initial certification to complete your professional certification (master’s degree). For example, if you graduated with a teaching certificate in June of 2007, you must complete your professional certification by June of 2012. This deadline may be extended, but you must apply for an extension with the state. Do not wait until the clock runs out before applying for your extension.
   2. The college only gives you five years to complete a graduate degree. If you began your graduate program in the fall of 2009, you have until the end of summer ’14 to earn your degree. Again, this may be extended through the college, but don’t wait until the last semester to apply for the extension.
2. To be considered enrolled at half-time, and therefore eligible for federal loans, you must be enrolled in at least 6 credits per semester. Your summer enrollment will not count towards this; so don’t plan on taking six credits over the summer and then just three in the fall. Also, you must be enrolled in those six credits when the financial aid office distributes the federal loans. In other words, don’t wait until the add-drop deadline to register.
3. If you wish to take advantage of the New York State Tuition Assistance Program (TAP), you must enroll in 12 credits each semester. In my opinion, that is impractical for a full-time teacher.
4. If financial aid is not a concern to you (e.g. your district is reimbursing your tuition), you may take as few as one course each year and remain in good standing with the college. If you go two semesters without completing a course, you must apply for a leave of absence from the program. Again, do not wait to do this. You must also check with your district for their policies as well.

The bottom line: Plan to take at least two courses (6 credits) each fall and spring semester. If you need to enroll in TAP, you must take four courses (12 credits) each semester. Otherwise, make sure you complete the program within five years of your initial certification, within five years of beginning your graduate work, and take at least one course per year. If you miss any one of those, fill out the necessary extension/leave of absence applications as soon as possible.

Many texts explain basic operant conditioning and behaviorism in Skinnerian terms. They usually can’t avoid mentioning B. F. Skinner in the first paragraph or two, but then they explain the difference between reinforcement and punishment like this:

Basically, the distinction between punishment and reinforcement is this: Something that decreases behavior is punishment, while reinforcement increases behavior. They can be positive – if the stimuli is administered – or negative – if the stimulus is removed.

There are three problems with this model: First, as Blackman (1974) put it, this definition is “rather different from that prompted by common sense.” This model’s definition of punishment is wildly different from the vernacular definition that dates to the 12th century. (According to the Random House Dictionary, poena, the root of punish, is related to penalty and pain.)

Second, despite claims to the contrary, there is no functional distinction between “positive punishment” and “negative reinforcement.” For example, two psychologists tried to explain:

You have an electronic fence around your yard. Your dog wears a collar that gives him a small electric shock every time he tries to cross the wire buried in the yard. The aversive consequence is the shock, the behavior that is reduced or eliminated is walking or running across the barrier. After several exposures to the shock the dog learns by punishment learning not to run or walk across the barrier.

…

Negative reinforcement is a different way of learning where a behavior is made more likely to occur because some unpleasant consequence is removed or avoided. Where punishment decreases or eliminates a behavior, negative reinforcement has the opposite effect of increasing behavior.

So when the dog stays in the yard (an increased behavior) because he is not getting shocked (an unpleasant consequence avoided) that’s negative reinforcement. But when the dog doesn’t leave the yard (decreased behavior) because he gets shocked when he does (unpleasant consequence administered), it’s punishment. Got that?

Third, and most importantly, this model of positive/negative/reinforcement/punishment does not approach Skinner’s theories or his experimental results. I do not mean that the model is wrong (though my second point reveals a fundamental weakness), but it’s problematic that many texts present Skinner as the father of operant conditioning, and then present ideas he rejected without noting the incongruence.

Here is how Skinner himself described positive reinforcement in Walden Two (pp. 259-260; emphasis added):

[If] it’s in our power to create any of the situations which a person likes or to remove any situation he doesn’t like, we can control his behavior. When he behaves as we want him to behave, we simply create a situation he likes, or remove one he doesn’t like. As a result, the probability that he will behave that way again goes up, which is what we want. Technically it’s called ‘positive reinforcement’.

While this explanation may fit into the table’s classification of positive reinforcement, it encroaches on “negative reinforcement” because he explicitly states that removing an unpleasant stimulus is also positive reinforcement. For Skinner, the positive/negative dichotomy referred to whether the consequence of the behavior was appetitive (pleasant) or aversive (unpleasant).

Skinner also explained that it is dangerous to place punishment, “removing a situation a person likes or setting up one he doesn’t like,” in symmetry with positive reinforcement. Punishment does not lead to lasting behavioral modification and it presents a slew of socially and individually damaging side effects. In fact, I believe the reason Skinner preferred the term punishment over negative reinforcement – even though he saw them as synonymous – was that he wanted to emphasize that they are not on the same plane.

So, where did this commonly-accepted model originate if not from the father of behaviorism? As Holth pointed out in a 2005 article in Behaviorism Analyst Today, Azrin and Holz first proposed this newer, less effectual model in 1966 (in a chapter on punishment in a volume edited by Honig). For them, giving a cookie to a child when he doesn’t wet his bed is actually punishment because its purpose is to reduce a behavior.

As ridiculous as that sounds, it is more preposterous that this view would be ascribed to Skinner.