By Marshall Chasin, AuD
The following post was first published in Hearing Review in July 2014 and is republished here by permission.
You should always make time for family, and one thing I have learned as a father is to always do things with your children. Several years ago, I approached my daughter CJ and said, “What would you like to do today? It’s a beautiful sunny day. We could go for a walk, throw around a Frisbee, or even calculate the probability of error if a word list was terminated before the 25 words were all given.”
What daughter wouldn’t jump at a chance to calculate using the Poisson statistical distribution? We first had to clarify what we were going to calculate and decided on the following: If a person had gotten all of the words correct so far, what would be the probability of error, if the audiologist had stopped after 10 words, after 15 words, and after 20 words, in a 25-word list? More specifically, we calculated the probability of error for getting more than three words wrong (after having gotten them all correct) for a shortened word list. Anything below 88% correct in a word recognition task would be viewed as “abnormal.”
Now this is not as bizarre as it sounds- well, actually it is, but my daughter was a graduate student and in her undergraduate years she had concentrated on statistics, calculus, physics, (as well as art history and the civilizations of early man)- I think the word “eclectic” comes to mind. We actually published the answer to this question in the September 2007 Hearing Review (HR).
Of the many statistical probability density distributions available, we selected the Poisson Distribution. This is a discrete distribution that is used for rare events. An example used by one of my own statistics professors was “If one person is killed crossing the street en route to class every year (because they were not paying attention to where they were going), what is the probability that two would be killed in a year? What about 6? This distribution will give exact probabilities for questions like this.
What about the question, “if a person has gotten the first 10 words in a list correctly, what is the probability that in the next 15 words (not given), they would get 1 error, 2 errors, or 3 errors? These word errors are relatively rare events, especially if the first 10 words (or 15 words, or 20 words, are correct). Clinicians do take short cuts from time to time, and if not completing a word list, it would be nice to know the degree of trouble you may be in if the full 25-word list would have resulted in a percentage correct of less than 88%- the clinical limit of normal for many speech-based tests.
For those who have forgotten the important calculations, the distribution is given as:
P(x) = e-avg(avg)x/x!
And for those who have not forgotten the formula, unfortunately it is still the same thing.
In this formula, “avg” refers to the average number of errors (such as the number of people crossing the street that don’t quite make it, in one year). The variable x allows us to ask the question “What if x number of people are killed? What is the probability of that happening?” Clearly if x happens to be close to the average, then the probability is relatively high, and if x is far from the average, the probability is much lower.
Without further delay, figure 1 is from our article in HR. It shows that if only 5 words are given (and there are no errors), then there is a 22.15% chance that there will be more than 3 errors if a full 25-word list is given. However, if the first 15 words are given without errors, then there is only a 3.4% chance that it would be incorrect to say that the patient’s word recognition score is normal. If 20 words are given without error, we can probably stop the word list since there would be only 0.34% of a chance that we were wrong.
It should be pointed out that these calculations are independent of the word list being used, and independent of hearing status. Implicit in the selection of the 88% to 100% range being accepted as “normal” is the nature of the word lists, the hearing status, and the slope of the performance-intensity function. It also becomes an entirely different statistical problem once errors are made, as this has been shown to be a binomial distribution1.
None of this should be taken as an endorsement of routinely cutting short your word recognition testing, but at least we now know the odds of an error if we do so. So go ahead and spend time with your children. Who knows? You may find out about something important.
1. Raffin MJM, Schafer D. Application of a probability model based on the binomial distribution to speech-discrimination scores. J Speech Hear Res. 1980; 23: 570-575.