## Miscellaneous Data

Outliers Within any set of observations it is not unusual to obtain occasional data points that vary substantially from others found in the same group of subjects. These outlier data may be the result of unknown errors or may represent genuine information, and indeed, whether such data should be included or rejected is the subject of much debate. Methods for detecting outlier data and the use of robust methods of statistical analysis to accommodate such data are complex issues, beyond the scope of this chapter; a thorough treatment of the subject is presented by Barnett and Lewis -40] . to which the reader is referred.

BLQ Data Statisticians sometimes warn against left-censored distributions, a term evocative of political intrigue and vaguely racy data sets that actually refers to the more bland reality of omitting results that are below the levels of quantitation of the assay (BLQ). BLQ results are fairly common in clinical and preclinical biomarker analyses and can cause serious problems with interpretation of the data if they are not taken into account, since unreported values are unusable in any statistical analysis of the data. Data imputation, a process to replace unreported BLQ values with valid estimates, can also affect the data interpretation, since all values in a data set affect the measures of central tendency, error, and statistical power. It seems clear that estimating a theoretical value for BLQ data points would allow for more accurate conclusions to be drawn from the data than simply removing data through left-censoring, but what value should indeed be assigned to BLQ data points?

Some researchers report BLQ values as zero, perhaps due to the influence of the sometimes black-and-white perspective of assay validation-style cutoffs. Assigning a zero value regardless of the sensitivity of a given test could be misleading, however, since zero is an absolute, whereas BLQ represents a scale from zero up to an upper limit that differs from one test to another in an assay - specific manner. Furthermore, assigning zero values creates other mathematical difficulties, such as not being able to compute fold-differences of another value with respect to the zero, or not being able to work with log transforms of data.

One common approach to address these issues in the data imputation process has been to assign arbitrary nonzero values to BLQ data, most commonly either the limit of detection (LOD) divided by 2 or divided by the square root of 2, depending on the skewness of the data set in question [41]. Both appreciable bias and loss of power have been demonstrated with this type of approach, however [42,43]. Succop et al. compared the bias of alternative methods for imputing BLQ values and concluded that by imputing the values based on either median percentiles below the detection limit or based on predicting the BLQ value from an equation model fit to the noncensored data set, both produced a good correlation between predicted and reported low values. In their example data set, bias between the predicted and true values was only 2.9% this way, compared to 348% overestimation bias when LOD/2 was used to impute the BLQ values [44].

Standard curves of analytical assays typically provide an equation model fit that can be used to calculate values down to the zero level, but assay validation typically limits the data reported to levels that have met a cutoff level that is deemed to have accuracy and precision within proscribed acceptance limits (i.e., the lower limit of quantitation). The study above suggests that although equation-predicted values below the levels of quantitation of an assay may not meet acceptance criteria per se, they still represent more accurate estimations for BLQ data than either arbitrary imputation methods or left- censoring of data sets. Succop et al. recommended that analytical labs should provide a numerical result for all samples analyzed, with a flag of those values that are below the detection limit [44] .

## Project Management Made Easy

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