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84 CHAPTER 3 SOME BASIC PROBABILITY CONCEPTS
confirmed that 43 of the 71 cases were bucket-handle tears. The cases may be cross-classified by
“bow-tie sign” status and surgical results as follows:
Tear Surgically Tear Surgically Confirmed As
Confirmed (D) Not Present ( D) Total
Positive Test 38 10 48
(absent bow-tie sign) 1T2
Negative Test 5 18 23
(bow-tie sign present) 1T2
Total 43 28 71
Source: Theodore A. Dorsay and Clyde A. Helms, “Bucket-Handle Meniscal Tears of the Knee: Sensitivity
and Specificity of MRI Signs,” Skeletal Radiology, 32 (2003), 266–272.
(a) What is the sensitivity of testing to see if the absent bow-tie sign indicates a meniscal tear
(b) What is the specificity of testing to see if the absent bow-tie sign indicates a meniscal tear
(c) What additional information would you need to determine the predictive value of the test
3.5.3 Oexle et al. (A-7) calculated the negative predictive value of a test for carriers of X-linked ornithine
transcarbamylase deficiency (OTCD—a disorder of the urea cycle). A test known as the “allopurinol
test” is often used as a screening device of potential carriers whose relatives are OTCD patients.
They cited a study by Brusilow and Horwich (A-8) that estimated the sensitivity of the allopurinol
test as .927. Oexle et al. themselves estimated the specificity of the allopurinol test as .997.
Also they estimated the prevalence in the population of individuals with OTCD as 132000. Use
this information and Bayes’s theorem to calculate the predictive value negative of the allopurinol
screening test.
3.6 SUMMARY
In this chapter some of the basic ideas and concepts of probability were presented. The
objective has been to provide enough of a “feel” for the subject so that the probabilistic
aspects of statistical inference can be more readily understood and appreciated when
this topic is presented later.
We defined probability as a number between 0 and 1 that measures the likelihood
of the occurrence of some event. We distinguished between subjective probability and
objective probability. Objective probability can be categorized further as classical or relative
frequency probability. After stating the three properties of probability, we defined
and illustrated the calculation of the following kinds of probabilities: marginal, joint, and
conditional. We also learned how to apply the addition and multiplication rules to
find certain probabilities. We learned the meaning of independent, mutually exclusive,
and complementary events. We learned the meaning of specificity, sensitivity, predictive
value positive, and predictive value negative as applied to a screening test or disease
symptom. Finally, we learned how to use Bayes’s theorem to calculate the probability
that a subject has a disease, given that the subject has a positive screening test result
(or has the symptom of interest).