Biological variation and analytical performance specifications

Steady state.

The calculation of biological variation data assumes that the individuals assessed are in a “steady state,” that is, the homeostatic set points do not change during the duration of the study. If the population displays a trend over the sampling period, data should be transformed to a “steady state,” for example, by correcting for trends using regression analysis or using other methods such as multiple of medians (MoM) and its natural logarithm.

Data transformation/normal distribution.

If one wishes to estimate the CI for the components of biological variation, most methods assume that data is normally distributed. As most biological data are naturally logarithmically distributed, the examination and calculations must be performed on the logarithms of the observations. This both helps in extracting the CVs and ensuring that the data distribution is closer to normal. It is important to specify that the normality relates to the model effects; that is, both the analytical variation around the true sample value and the individuals’ variation around their homeostatic set point are normally distributed. It does not relate to the total pooled data. Pooling the standardized residuals for each level, namely, residuals from replicates (difference between replicates and mean of replicates from each sample), residuals from samples (difference between mean of samples and mean for the individual), and residuals from subjects (differences between individual means and total mean), can be used to assess normality.

Example: Assume observations of 2.25, 2.50, and 2.75 are from individual A and 3.50, 4.00, and 4.50 from individual B. Standardized residuals are generated by first dividing by the mean for each individual. Individual A will have an average of 2.50, so standardized observations will be 0.90, 1.00, and 1.10, and corresponding standardized residuals −0.10, 0.00, and 0.10. Individual B will have an average of 4.00, standardized observations 0.875, 1.00, and 1.125, and corresponding residuals −0.125, 0.00, and 0.125. The pooled standardized residuals will be −0.10, 0.00, 0.10, −0.125, 0.00, and 0.125. The standardized residuals can then be examined using Kolmogorov-Smirnov or Anderson-Darling or other techniques for the assessment of normality.

If a log-transformation is applied, it is important to transform the estimated SDs back to CVs afterward. An alternative approach is using the CV-ANOVA as described by Røraas and colleagues. Using this approach, data are transformed by dividing each subject’s measurement values by that subject’s mean value, so a distribution of values around 1 is obtained, and the CV _A and CV _I can be derived directly, as all subjects have a mean of 1. However, this approach does not lend itself to estimation of CV _G .

Outliers.

The assessment of outliers is important, because such aberrant values will lead to erroneous estimates of the components of biological variation if applying the method of Fraser and Harris, or a similar approach. It is important that this assessment is done using the same measure of variability that is estimated; that is, if CVs are estimated, which is usually the case, all the calculations should be performed using CVs. This can be achieved by normalizing data, for example, through log-transformation as described above. After any transformation, outlier assessments are performed at three levels: (1) between duplicates or replicates, (2) between samples within an individual, and (3) between individuals. For levels 1 and 2, typically Cochrane’s test is used on the CV , but applied to SD if we work on log-transformed data. Failure to remove outliers in the replicates can result in a falsely high CV _A and an erroneous CV _I estimate, while failure to remove outliers from results from each of the individuals can result in a falsely increased CV _I . Finally, outliers among the mean values of the individuals (level 3) are assessed. A simple strategy to perform this process is to use Reed’s criterion, where the difference between any mean value and the next highest or lowest value in the series should be less than one-third of the absolute range of all values. Another useful approach for the assessment of outliers between individuals is a simple graphical approach in which the mean values and the range of all these values are plotted for each individual (on the y -axis) against concentration or activity (on the x -axis). An example is provided in Fig. 8.2 and discussed later in this chapter. Failure to exclude outliers of the mean values of the different individuals will result in a falsely large CV _G , and because the overall mean value will be different; this may also affect CV _A and CV _I depending on the transformation chosen. The number of outliers and concentrations that have been found and the number of datapoints used to derive the components of biological variation should always be reported.

Homogeneity.

Applications of within-subject biological variation data, particularly for estimation of reference change values (RCVs) (which will be dealt with later), depend on the within-subject variation data, for example, the variances being homogeneously distributed. If the data are derived from a study cohort with heterogeneously distributed within-subject estimates, the results may not be representative of the population, except for an “average” individual. To remedy this, stratified analysis by subgroups should be applied if possible, or an alternative approach for deriving biological variation estimates, such as Bayesian statistics, should be used. Thus although estimates of CV _I and RCVs can be calculated, these are not generalizable to the overall population. It is therefore always necessary to check that the variances in specimens drawn from a population are “homogeneous” by definition, and consequently, that the ranked cumulative distributions of these variances are distributed around the true variance of the population according to χ ² /df ( χ ² ) distribution for degrees of freedom (df) according to the individual sample sizes. In contrast, when a series of different variances has a dispersion around the pooled variance different from a χ ² /df distribution, they are considered to be heterogeneous. Ideally if we have normality, a balanced design and a low CV _A compared to CV _I , then homogeneity can be illustrated by plotting the cumulated ranked fractions of within-subject variations as a function of the within-subject variation estimates on a Rankit scale ( Fig. 8.3 ). If homogeneous, this curve will fit to the theoretical of the square root of the pooled variance multiplied by χ ² /df. Variance homogeneity can also be tested further by Bartlett’s test. Although Cochrane’s test of the variances of the mean values is primarily an outlier test, it also can be used as a test for homogeneity.

After testing for homogeneity and outliers, it is important to indicate how many individuals and results have to be removed to obtain homogeneity of the data used to estimate the CV _I . This again provides an indication of the representative nature of the data and underscores its suitability for wide application. It is also recommended to check if excluded individuals share common traits that can explain their heterogeneity, which can be valuable information to the applicability of the estimated CV _I .

Calculation of analytical, within-subject, and between-subject variation.

There are several estimators available: ANOVA, maximum likelihood (ML), restricted maximum likelihood (REML), minimum variance quadratic unbiased estimator (MIVQUE), and weighted analysis of means (WAM). For balanced designs, the choice of estimator is not important; however, for unbalanced designs, the estimators can yield different results and should be chosen carefully. These estimators are available in most statistical software packages using general linear models or generalized linear models. In historical publications, many have not used formal ANOVA but instead have simply used subtracted variances. The assumption is that, because preanalytical sources of variation have been minimized and can be considered negligible, the total CV (CV _T ) of a set of results from each cohort of individuals includes CV _A , CV _I , and CV _G . Then, because CV _T = [(CV _A ) ² + (CV _I ) ² + (CV _G ) ² ] ^½ , the components can be calculated by simple subtraction. However, this approach will require calculations including degrees of freedom and therefore a formal ANOVA is a simpler approach for correct calculations and can take into account unbalanced designs and further give the ability to calculate CIs. Additionally, in many of these studies, CV _A has been estimated based on quality control materials, the possibility of outliers has not been assessed, and a normal distribution and homogeneity of data are assumed; as a consequence, estimates of the components of biological variation are likely to be less precise.