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Statistics - Classification Tests - True Positive Rate

Posted Aug 24 2010 12:00am

Classification tests are tests used to separate people into separate groups.  In other words, they categorize.  This is different than, say, an IQ test, which is primarily designed to provide information regarding one's various cognities abilities.  yes, IQ tests can be used to classify, but that is not their primary purpose, nor is that the way their data is presented. 

A pregnancy test is more of a classification test - women are separated into two groups, eith "pregnant" or "not pregnant."  With a classification test, we want to be able to differentiate between two groups of individuals as much as we can, without making errors in our classifications.  For example, a pregnancy test with a fifty percent error rate would be pretty useless - you can get those kinds of results simply by flipping a coin.  In addition, we'd have a lot of very upset people who had been wrongly classified. 

This post will briefly define "True Positive Rate."  Simply, the True Positive Rate (TPR) is the proportion of the group of interest who generate a positive score on the classification test being used.  In keeping with the example above, let's say our group of interest is women who are pregnant.  We want to develop the best pregnancy test possible, to include an excellent TPR.  So, let's say that a positive score on our pregnancy test indicates that the test-taker has scored positively - in this case, the test is saying the person is pregnant.  The TPR is the proportion of pregnant women who obtain a positive score when they take the test.  In other words, if the TPR is .8, then out of 100 pregnant women who take this test, 80 will test positive. 

Is this good?  It depends upon multiple factors, including the seriousness of the issue, a comparison to other techniques, etc.  It also depends upon the False Positive Rate (FPR), or how often an individual who does not have the condition also scores positive.  In the current example, if 100 non-pregnant women are administered the test, and 10 test positive (that is, the test says they are pregnant when they are not), then the FPR is .1. 

I doubt these ratios would be acceptable in the case of pregnancy testing, but these types of decisions are made in all sorts of situations, including in psychology.  We can also adjust the cut-off scores to increase the TPR or FPR, with an understanding that manipulating the cut-off score to enhance one rate will also impact the other.  This can get tricky, and it also depends on various factors.    here is one more example.

Let's say there is a new cancer screen, based on blood work.  The screen predicts the development of cancer in the next five years.  It is cheap ($10), easy to administer, and if positive, one can have a more thorough test (%100 accrate) as a follow up for $1000, as well as a series of procedures to then reduce the likelihood of onset by 50%.  In this case, the screen will be set to maximize TPR; we would want to capture as many people who might develop cancer as possible, in order to then administer a readily available follow-up test.  Yes, there will be more false positives, but the temporary anxiety over the possibility of having cancer would be overshadowed by being able to capture all of the people who do have the marker, along with the treatment intervention.  In this case, we would want as many true positives as possible, and tolerate false negatives it would save lives.

Conversely, let's say someone develops a test that categorizes people as "not college material."  Here, we would want the exact opposite, in that we would seek to minimize the incorrect classification of someone not being able to complete college, when they actually would be able to.  The risk of false positives in this case carries the more significant impact; it would be better for more individuals to go to college and ultimately not succeed (but having been given the opportunity to try), than to prevent individuals would would benefit from attending college (and successfully graduating) from ever having the chance.

These are just random hypotheticals, made up in order to demonstrate the importance of TPR and FPR.  In different situations, each can be the primary ratio.  Statistically, the important thing to note is that both measure someone scoring positively for the category being measured, either positive (correctly - TPR) or positive incorrectly - FPR).  The terms associated with scoring negative on classification tests will be (hopefully) addressed in another post.  For a more technical discussion, start here  and here .

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