SUMMARY This study examined the effects of different sample sizes and different levels of bias (systematic error) between replicated measurements on the accuracy of estimates of random error calculated using two common formulae: Dahlberg’s and the ‘method of moments’ estimator (MME). Computer-based numerical simulations were used to generate clinically realistic measurements involving random errors with a known distribution. For each simulation, two sets of ‘measured values’ were generated to provide the replicated data necessary for the estimation of the random error. Dahlberg’s and the MME formula were applied to these paired data sets and the resulting estimates of error compared with the ‘true’ error. Nine different sample sizes (n = 2, 5, 10, 15, 20, 25, 30, 50, and 100) and two different types of bias (additive and multiplicative) were examined for their effect on the estimated error. In each case, the estimates of the random error were based on the distribution of 5000 separate simulations. The results indicate that with a sample of less than 25–30 replicated measurements, the resulting estimates of error are potentially unreliable and may under or overestimate the true error, irrespective of the formula used in the calculation. Where, however, a bias exists between the replicate measurements, Dahlberg’s formula can be expected to overestimate the true value of the random error even where the biases are small and difficult to detect by standard statistical tests. No such distorting effect was found for the MME formula, which provided estimates of error that were not meaningfully different from the true value even where relatively large biases existed between the replicates. These results suggest the following: 1. A sample of at least 25–30 cases should be replicated to provide an estimate of the random error. 2. Where the original study contains fewer than 20 cases, the estimate of error will be unreliable. In these circumstances, it would be helpful if a confidence interval for the true error was also quoted. 3. Unless one can be absolutely sure that no bias exists between the replicate measurements, Dahlberg’s formula should be avoided and the MME formula used instead.

The effect of sample size and bias on the reliability of estimates of error: a comparative study of Dahlberg's formula

In a large class of statistical inverse problems it is necessary to suppose that the transformation that is inverted is known. Although, in many applications, it is unrealistic to make this assumption, the problem is often insoluble without it. However, if additional data are available, then it is possible to estimate consistently the unknown error density. Data are seldom available directly on the transformation, but repeated, or replicated, measurements increasingly are becoming available. Such data consist of ``intrinsic'' values that are measured several times, with errors that are generally independent. Working in this setting we treat the nonparametric deconvolution problems of density estimation with observation errors, and regression with errors in variables. We show that, even if the number of repeated measurements is quite small, it is possible for modified kernel estimators to achieve the same level of performance they would if the error distribution were known. Indeed, density and regression estimators can be constructed from replicated data so that they have the same first-order properties as conventional estimators in the known-error case, without any replication, but with sample size equal to the sum of the numbers of replicates. Practical methods for constructing estimators with these properties are suggested, involving empirical rules for smoothing-parameter choice.

On deconvolution with repeated measurements

Comparison of optical, needle probe and ultrasonic techniques for the measurement of articular cartilage thickness

Kinetic theory of granular flow was verified experimentally for flow of fluid catalytic cracking particles in a vertical pipe. Measurements of particle pressure using a differential transducer and granular temperature with a digital camera as a function of bulk density, determined using an X-ray densitometer, showed that a relation exists among pressure, temperature, and density, analogous to the ideal gas law. In the limit of zero solids volume fraction: (Solid Pressure)/[(Granular Temperature) × (Bulk Density)] = 1.0. Measurements of radial distribution functions using the digital camera showed that their peak values occur at particle contact and lie between the predictions from the Bagnold equation and Carnahan–Starling equation. The hard sphere model was corrected for a cohesive pressure using the minimum in the measured radial distribution function. The new model agrees with the pressure measurements in the dense regime.

Equation of state and radial distribution functions of FCC particles in a CFB

In many contexts involving multivariate linear regression models, some or all of the independent (predictor) variables are measured with error. It is argued that if the goal is to assess the relationship of the dependent variables to the true predictor variables, then it is important to determine the reliability matrix Λ of the measurement X of the vector of true predictors. If Λ is singular, then the slope matrix B is not identifiable. If Λ is nearly singular, then B cannot be accurately estimated. A two-step estimation procedure is proposed in which Λ is estimated from data on the measured predictors X and on prior information from reliability studies on these predictors and the parameters of the (latent) linear relationship are estimated by classical linear regression methods. This approach not only allows the use of available software but also lends itself to traditional diagnostic methods. Large-sample properties of the estimators resulting from this approach are derived. It is shown how a c...

John L. Jaech

Papers

Statistical Analysis of Measurement Errors