Are We Estimating the Mean and Variance Correctly in the Presence of Observations Outside of Measurable Range?

Laboratory measurements used for safety assessments in clinical trials are subject to the limits of the used laboratory equipment. These limits determine the range of values which the equipment can accurately measure. When observations fall outside the measurable range, this creates a problem in estimating parameters of the normal distribution. It may be tempting to use methods of estimation that are easy to implement, however selecting an incorrect method may lead to biased estimates (under- or overestimation) and change the research outcomes, for example, incorrect result of two-sample test about means when comparing two populations or biased estimation of regression line. In this article, we consider the use of four methods: ignoring unmeasured observations, replacing unmeasured observations with a multiple of the limit, using a truncated normal distribution, and using a normal distribution with censored observations. To compare these methods we designed a simulation study and measured their accuracy in several different situations using relative error <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow> <mml:mfrac> <mml:mrow><mml:mover><mml:mi>μ</mml:mi> <mml:mo>̂</mml:mo></mml:mover> <mml:mo>-</mml:mo> <mml:mi>μ</mml:mi></mml:mrow> <mml:mi>μ</mml:mi></mml:mfrac> </mml:mrow> <mml:annotation>$$ \frac{\hat{\mu}-\mu }{\mu } $$</mml:annotation></mml:semantics> </mml:math> , ratio <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow> <mml:mfrac><mml:mover><mml:mi>σ</mml:mi> <mml:mo>̂</mml:mo></mml:mover> <mml:mi>σ</mml:mi></mml:mfrac> </mml:mrow> <mml:annotation>$$ \frac{\hat{\sigma}}{\sigma } $$</mml:annotation></mml:semantics> </mml:math> , and mean square errors of both parameters. Based on the results of this simulation study, if the amount of observations outside of measurable range is below 40%, we recommend using a normal distribution with censored observations in practice. These recommendations should be incorporated into guidelines for good statistical practice. If the amount of observations outside of measurable range exceeds 40%, we advise not to use the data for any statistical analysis. To illustrate how the choice of method can affect the estimates, we applied the methods to real-life laboratory data.