I. IntroductionII. Bayes Theorem and Conditional ProbabilityTroughPeakDurationDurationIntervalRatioOct. 28, 2008 LEC #10 ECON 240A-1 L. PhillipsProbability ModelsI. IntroductionSo far we have used probability as a foundation for the binomial and normal distributions. Then we have used these distributions as the models underlying our statistics, such as the z variable, normal with mean zero and variance one. Interval estimation and hypothesis testing were based on such models. Student’s t-distribution wasbased on a more elaborate model for a random variable that was the ratio of a z variable to a Chi-Square variable.But probability can be used directly for analysis. We will begin by studying an application of Bayes Theorem, using conditional probability, joint probability, and marginal probability to calculate the likelihood a patient has a disease given that his/her test for the disease was positive.II. Bayes Theorem and Conditional ProbabilityThere are two underlying events in this example of epidemiology and statistics. First, an individual is sick (has the disease), event S, or does not, event S, and second, the individual tests positive for the disease, event P, or does not, event P. A two by two tableau is depicted in Table 1, showing the four joint probabilities, such as the individual testing positive and having the disease, p(SP). The marginal probabilities are shown along the right side and the bottom of this tableau.The underlying facts are that (1) the incidence of the disease in the population is one in a thousand, p(S) = 0.001, (2) the probability of testing positive for the disease given that you have it is ninety nine out of one hundred, p(P/S) = 0.99, and (3) the probability of a false positive, i.e. testing positive even if you do not have the disease, isOct. 28, 2008 LEC #10 ECON 240A-2 L. PhillipsProbability Modelsonly two in a hundred, i.e.)/( SPp = 0.02. Note that the test seems reasonably accurate or reliable.The issue for the patient and the physician is what is the probability the patient has the disease if the test comes back positive, i.e. what is p(S/P)?--------------------------------------------------------------------------------Table 1: Testing for a Disease, Joint and Marginal ProbabilitiesS: DiseasedS: HealthyP: Test Positive p(SP) p(SP) p(P)P: Test negative p(SP) p(PS ) p(P)p(S) p(S) 1------------------------------------------------------------------------------------------We use the facts to fill out the tableau, as indicated in Table 2.--------------------------------------------------------------------------------Table 2: Testing for a Disease, Joint and Marginal Probabilities, Some NumbersS: DiseasedS: HealthyP: Test Positive p(SP) p(SP) p(P)P: Test negative p(SP) p(PS ) p(P)p(S) = 0.001 p(S) = 0.999 1------------------------------------------------------------------------------------------We can use conditional probability and our facts to extend our knowledge:p(P/S) p(S) = p(PS) , (1)0.02 * 0.999 = 0.01998 (2)andp(P/S) p(S) = p(PS) (3)0.99*0.001 = 0.00099 (4)Oct. 28, 2008 LEC #10 ECON 240A-3 L. PhillipsProbability ModelsThe calculation of these two joint probabilities can be used to fill in the tableau even more:----------------------------------------------------------------------------------Table 3: Testing for a Disease, Joint and Marginal Probabilities, Sufficient InfoS: DiseasedS: HealthyP: Test Positive p(SP) = 0.00099 p(SP) = 0.01998 p(P)P: Test negative p(SP) p(PS ) p(P)p(S) = 0.001 p(S) = 0.999 1------------------------------------------------------------------------------------------There is now sufficient information to fill in all of the remaining blanks.We are interested in the conditional probability, p(S/P), the probability we have the disease given that we tested positive:p(S/P) p(P) = p(SP) (5)p(S/P) 0.02097 = 0.00099 (6)p(S/P) = 0.0472. (7)So, despite the apparent accuracy of the test, with a conditional probability of testing positive if you have the disease of 0.99, and a conditional probability of testing positive ifyou are healthy of 0.02, the calculated conditional probability the patient has the disease given the test comes back positive is less than 5%. So the test is not all that informative. This is called the false positive paradox and is another good reason to have a good doctor,and to be involved in your own health care. Both the patient and the doctor need to conduct further tests and further diagnoses before settling on a course of treatment.Oct. 28, 2008 LEC #10 ECON 240A-4 L. PhillipsProbability ModelsThe tableau provided a cognitive and organized methodology for using the facts to derive their implications for decision making. We could take a short cut by combining Eq.’s (3) and (5):p(P/S) p(S) = p(PS) = p(S/P) p(P) (8)p(S/P) = [p(P/S) p(S)]/p(P) = 0.99*0.001/p(P) (9)and we still need p(P) to get the conditional probability that answers our question. Looking at Table 1, we seep(P) = p(SP) + p(SP) (10)= p(P/S) p(S) + p(P/S ) p(S) (11)So, by combining Eq.’s (9) and (11) we obtain:p(S/P) = [p(P/S) p(S)]/[ p(P/S) p(S) + p(P/S ) p(S)] (12)This is called Bayes Theorem.III. Duration Models, Failure Times, and the Exponential DistributionDuration studies or failure time analysis involve issues such as (1) the length of time an individual is unemployed before obtaining a job, (2) the length of time an individual is onwelfare before leaving the rolls, (3) the duration of economic expansions or booms, and (4) the length of life for a patient after receiving a therapy, such as a heart bypass operation. From the last example, you can see that this type of statistics has a history in medicine as well as economics, and terms such as “number at risk” carry over.For illustration, we will look at the length of expansions since World War II, the number of months between the trough of the business cycle and its peak. These turning points are established by the National Bureau of Economic Research, and the informationOct. 28, 2008 LEC #10 ECON 240A-5 L. PhillipsProbability Modelsis published on occasion in Survey of Current Business, published monthly by the U.S. Department of Commerce.The timing of the post-war business cycles is reproduced in Table 4.---------------------------------------------------------------------------------Table 4: Duration of Post-War Economic Expansions in MonthsTrough Peak DurationOctober 1945 November 1948 37October 1949 July
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