Simple computation of a bivariate normal integral arising from a problem of misclassification with applications to the diagnosis of hypertension

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Abstract
The joint distribution of the true and observed values of a variable that is subject to measurement error is bivariate normal.An important special case occurs when we want the joint probability of the true value being below a cutoff point and the observed value above it.In that case the required integral can be simply evaluated using a Gaussian quadrature formula, which can easily be evaluated using a calculator.This formula is used to estimate the probabilities of misclassification of participants in screening programs for hypertension.It shows that basing a diagnosis on a single visit, at which a single measurement was made leads to a very high risk of misclassification.The probability of a subject having a blood pressure below the cutoff point, given that the observed pressure is above it, would be 0.45.Increasing the number of visits to three, and measuring the blood pressure twice at each visit, as advocated by Rosner and Polk (1979), would bring the probability down to 0.29.