Bland-Altman Plot for Censored Variables

Abstract

The comparison of two measurement methods turns out to be a statistical challenge if some of the observations are below the limit of quantification or detection. Here we show how the Bland-Altman plot can be modified for censored variables. The reference lines (bias and limits of agreement) in the Bland-Altman plot have to be estimated for censored variables. In a simulation study, we compared three different estimation methods: Restricting the data set to fully quantifiable pairs of observations (complete case analysis), naïvely substituting missing values with half of the limit of quantification, and a multiple imputation procedure based on a maximum likelihood approach for bivariate lognormally distributed variables with censoring. The results show that simple ad-hoc solutions may lead to bias in the results when comparing two measurement methods with censored observations, whereas the presented multiple imputation approach of the Bland-Altman method allows adequate consideration of censored variables. The method works similarly for other distribution assumptions.

Keywords: difference plot; limited variable; nondetects.

FIGURE 1

Bland–Altman plots of a bivariate lognormally distributed data set $N=100,{\mu }_x={\mu }_y=0,{\sigma }_x={\sigma }_y=1,\rho =0.9$. (A) Shows the classical Bland–Altman plot according to Bland and Altman with the complete data. In (B–D) $x=10\%$ and $y=30\%,$ of the observations are left‐censored and different approaches are presented: (B) complete case analysis, (C) naïve imputation with half of the censoring cut‐off, and (D) multiple imputation approach.

FIGURE 2

Bias and mean square error (MSE) of three different estimation methods (complete case analysis, naïve imputation with half of the censoring cut‐off, multiple imputation approach) for the reference lines of the Bland–Altman plot from a simulation study with bivariate lognormally distributed data (${\mu }_x={\mu }_y=0,{\sigma }_x={\sigma }_y=1,\rho =0.9$) with 2000 iterations for each scenario. The 15 scenarios differed by samples size ($N=30,50,$ or $100$) and the proportion of left‐censored observations per variable $x=10\%$ or $30\%$ and $y=10\%,20\%,30\%,$ or $40\%$.

FIGURE 3

Bland–Altman plots for censored variables of two allergens (Can f 1 and Fel d 1) comparing storage of extracts from electrostatic dust collectors at −20°C and −80°C. Three different approaches are presented: Complete case analysis, naïve imputation with half of the censoring cut‐off, and a multiple imputation approach based on maximum likelihood estimation. For the multiple imputation approach, an underlying bivariate lognormal distribution of the data is assumed for each allergen. Censored observations are multiple imputed according to the estimated distributions and symbolized by 20 imputed values each. Estimated reference lines (and their 95% confidence intervals) are displayed as magenta lines (and grey dashed lines).

FIGURE 4

Bland–Altman plots for censored variables of two samplers measuring welding fume. Three different approaches are presented: Complete case analysis, naïve imputation with half of the censoring cut‐off, and a multiple imputation approach based on maximum likelihood estimation. For the multiple imputation approach, an underlying bivariate lognormal distribution of the data is assumed. Censored observations are multiple imputed according to the estimated distribution and symbolized by 15 imputed values each. Estimated reference lines (and their 95% confidence intervals) are displayed as magenta lines (and grey dashed lines).