Modeling observations with a detection limit using a truncated normal distribution with censoring

Abstract

Background: When data are collected subject to a detection limit, observations below the detection limit may be considered censored. In addition, the domain of such observations may be restricted; for example, values may be required to be non-negative.

Methods: We propose a method for estimating population mean and variance from censored observations that accounts for known domain restriction. The method finds maximum likelihood estimates assuming an underlying truncated normal distribution.

Results: We show that our method, tcensReg, has lower bias, Type I error rates, and mean squared error than other methods commonly used for data with detection limits such as Tobit regression and single imputation under a range of simulation settings from mild to heavy censoring and truncation. We further demonstrate the consistency of the maximum likelihood estimators. We apply our method to analyze vision quality data collected from ophthalmology clinical trials comparing different types of intraocular lenses implanted during cataract surgery. All of the methods yield similar conclusions regarding non-inferiority, but estimates from the tcensReg method suggest that there may be greater mean differences and overall variability.

Conclusions: In the presence of detection limits, our new method tcensReg provides a way to incorporate known domain restrictions in dependent variables that substantially improves inferences.

Trial registration: ClinicalTrials.gov NCT01510717 NCT01424189.

Keywords: Contrast sensitivity; Limited domain; Visual acuity; limited dependent variables.

Conflict of interest statement

Hyung-Woo Kim is a former employee of Alcon Laboratories, Inc.

Figures

Fig. 1

CSV-1000E1 Contrast Sensitivity Chart for 12.0 CPD. 1This testing chart is distributed by Vector Vision and was accessed from http://www.vectorvision.com/csv1000-contrast-sensitivity/ on 29NOV2018

Fig. 2

Marginal Histograms for Monofocal and Multifocal Lens at 12 CPD under Dim Lighting. Log contrast sensitivity scores are converted from contrast sensitivity threshold scores via Table 1. Detection limit for 12 CPD occurs at ν=0.61. Histogram is shown in the background with Gaussian kernel density estimate in the foreground with bandwidth set to 0.2

Fig. 3

Truncated Normal Distribution with Censoring. Potential density for a left truncated normal distribution with left censoring. The density above was created with μ=0.8,σ=0.5,a=0, and ν=0.61. In our application, a=0 and ν=0.61

Fig. 4

Performance Metrics for μ from Six Different Estimation Methods in Single Mean Model. GS = Gold Standard, i.e., uncensored observations with truncation adjustment; Uncens NT = Uncensored data with no truncation adjustment; DL = detection limit; Tobit = Tobit censored regression with no truncation adjustment; tcensReg = Censored regression with truncation adjustment

Fig. 5

Performance Metrics for σ from Six Different Estimation Methods in Single Mean Model. GS = Gold Standard, i.e., uncensored observations with truncation adjustment; Uncens NT = Uncensored data with no truncation adjustment; DL = detection limit; Tobit = Tobit censored regression with no truncation adjustment; tcensReg = Censored regression with truncation adjustment

Fig. 6

Average Bias for δ from Six Different Estimation Methods in Two Population Model. The vertical dashed black line corresponds to the case when δ=0, i.e., μ1=μ2. GS = Gold Standard, i.e., uncensored observations with truncation adjustment; Uncens NT = Uncensored data with no truncation adjustment; DL = detection limit; Tobit = Tobit censored regression with no truncation adjustment; tcensReg = Censored regression with truncation adjustment

Fig. 7

Performance of Maximum Likelihood Estimate for μ as Function of Sample Size. The vertical dashed black line on the left figure corresponds to zero bias. GS = Gold Standard, i.e., uncensored observations with truncation adjustment; Uncens NT = Uncensored data with no truncation adjustment; DL = detection limit; Tobit = Tobit censored regression with no truncation adjustment; tcensReg = Censored regression with truncation adjustment

Fig. 8

95% Confidence Intervals for Separate Standard Deviation for Monofocal vs Multifocal Lens at 12 CPD

Fig. 9

90% Confidence Intervals for Difference in Monofocal vs Multifocal Lens at 12 CPD. The horizontal dashed line at δ=−0.15 indicates the non-inferiority margin. DL = detection limit; Tobit = Tobit censored regression with no truncation adjustment; tcensReg = Censored regression with truncation adjustment

Fig. 10

Estimate of Common Standard Deviation in Monofocal vs Multifocal Lens at 12 CPD. DL = detection limit; Tobit = Tobit censored regression with no truncation adjustment; tcensReg = Censored regression with truncation adjustment

Fig. 11

Average Log Mean Squared Error for δ from Six Different Estimation Methods in Two Population Model. The vertical dashed black line corresponds to the case when δ=0, i.e., μ1=μ2. GS = Gold Standard, i.e., uncensored observations with truncation adjustment; Uncens NT = Uncensored data with no truncation adjustment; DL = detection limit; Tobit = Tobit censored regression with no truncation adjustment; tcensReg = Censored regression with truncation adjustment

Fig. 12

Performance Metrics for Common σ from Six Different Estimation Methods in Two Population Model. The vertical dashed black line corresponds to the case when δ=0, i.e., μ1=μ2. GS = Gold Standard, i.e., uncensored observations with truncation adjustment; Uncens NT = Uncensored data with no truncation adjustment; DL = detection limit; Tobit = Tobit censored regression with no truncation adjustment; tcensReg = Censored regression with truncation adjustment

Fig. 13

Performance of Maximum Likelihood Estimate for σ as Function of Sample Size. The vertical dashed black line on the left figure corresponds to zero bias. GS = Gold Standard, i.e., uncensored observations with truncation adjustment; Uncens NT = Uncensored data with no truncation adjustment; DL = detection limit; Tobit = Tobit censored regression with no truncation adjustment; tcensReg = Censored regression with truncation adjustment