Assessing a Binary Measurement System with Varying Misclassification Rates Using a Latent Class Random Effects Model

Abstract

When no gold standard measurement system is available, we can assess a binary measurement system by making repeated measurements on a random sample of parts and then using a latent class model for the analysis. However, there is widespread criticism of the model assumptions that, given the true state of the part, the repeated measurements are independent and have the same misclassification probability. We propose a latent class random effects model that relaxes these assumptions by modeling the distribution of the two misclassification rates with Beta distributions. We start by finding the likelihood, the maximum likelihood estimates (MLEs) and their approximate standard deviations with the standard assessment plan that selects parts at random from the process. However, to estimate the model parameters with reasonable precision, the standard plan requires extremely large sample sizes in the common industrial situation where the proportion of conforming parts is high and the misclassification probabilities are small. More realistic sample sizes are possible when we instead sample randomly from the population of previously failed parts and supplement the likelihood with baseline information on the overall pass rate. We show using simulation that, for feasible designs, the asymptotic standard deviation based on the expected information provides a reasonably close approximation to the simulated standard deviation. We then use these approximations to investigate how the properties of the MLEs for the unknown parameters depend on the baseline size, the number of parts in the sample, and the number of repeated measurements per part.

• Beta Binomial
• Binary Measurement Systems
• Latent Class Model
• Misclassification Rates
• Random Effects

Additional information

Notes on contributors

Oana Danila

Dr. Danila is a recent Ph.D. graduate from the Department of Statistics and Actuarial Science at the University of Waterloo. Her email address is [email protected].

Stefan H. Steiner

Dr. Steiner is a Professor in the Department of Statistics and Actuarial Science at the University of Waterloo and Director of the Business and Industrial Statistics Research Group. He is a senior member of ASQ. His email address is [email protected].

R. Jock MacKay

Dr. MacKay is an Adjunct Professor in the Department of Statistics and Actuarial Science at the University of Waterloo. He is a member of ASQ. His email address is [email protected].