1. Statistics & graphics for the laboratory 25 Biological variation Introduction Estimation (ANOVA application) Index-of-individuality Comparison of a result with a reference interval ("Grey-zone") Reference change value (RCV) Biological variation

2. Statistics & graphics for the laboratory 26 Reference interval and biological variation Reference interval encompasses: pre-analytical imprecision analytical imprecision within-subject biological variation between-subject biological variation Usefulness of reference intervals Reference intervals are of most use when the between-subject variation is smaller than the within-subject variation: CV B-S 1 If not, individuals may have values which lie within reference limits but are highly unusual for them:  See also later: Index of individuality Uses of biological variation Setting desirable analytical goals Assessing significance of changes in serial results Assessing utility of population based reference values Deciding number of specimens required to assess homeostatic set point of an individual Determining optimum mode for result reporting Comparing available tests Assessing clinical utility of tests Abbreviations for biological variation CV W-S = Within-subject CV B-S = Between-subject CV G = "Group" = SQRT(CV 2 W-S + CV 2 B-S ) = ¼ of the reference interval Note: sometimes CV G is used to designate CV B-S (e.g. in the Westgard database on analytical specifications) Biological variation

3. Statistics & graphics for the laboratory 27 Estimation of biological variation (ANOVA-application) Approaches One overall analysis using nested ANOVA based on dedicated software (SAS, SPSS, BMDP etc.). Stepwise approach with computation of variances at each level. General requirement: Repeated measurements at each level, i.e. at least duplicates, in order to resolve the variance components. a time period. Inspection of data on biological variation Generally, ANOVA is robust towards moderate deviations from normality, but it is sensitive towards outliers. Occurrence of outliers Within subgroups or one subgroup versus the bulk of the rest of subgroups Within-subject variation Homogeneity/heterogeneity Within- & between subject biological variation Example Creatinine Biological variation

4. Statistics & graphics for the laboratory 28 Analytical & biological variation Estimation of biological variation Remark Consider the importance of analytical imprecision. Inter (  B-S )/intra (  W-S ) -individual biological variation Analytical variation (  A ) Total dispersion of a single measurement of individuals*:  T 2 =  B-S 2 + (  W-S 2 +  A 2 ) =  B-S 2 +  T, W-S 2 Index of individuality (I i ) (see also later):  T,W-S /  B-S ~  W-S /  B-S *Ignoring pre-analytical variation Components of variation-ANOVA Between (B-S)- and within-individual(W-S) biological variation* Inclusive analytical variation (σ T,W-S ) The within-biological variation is here obtained as an average value for all the studied individuals, which generally is preferable. Biological variation

5. Statistics & graphics for the laboratory 29 Analytical & within biological variation Analytical variation can be minimized by collecting and running the samples within one run. Shortcut computational principles Duplicate sets of measurements SD = [Σd 2 /2n] 0.5 for n duplicate measurements, where d is the difference between pairs of measurements. Examples Measurements of duplicate samples to derive the SD A Duplicate samples from each individual to derive SD T,W-S Practical approach Obtain repeated samples over a suitable time period, e.g. 5 samples, from a collection of individuals, e.g. 10-20 subjects, and measure each sample in duplicate. Inspect/test data for outliers and variance homogeneity. Carry out a simple ANOVA on the means of duplicates and derive SD B-S and SD T,W-S Derive SD A from duplicates Derive SD W-S from the relation: SD T,W-S 2 = 0.5 SD A 2 + SD W-S 2 (the factor 0.5 enters because the means of duplicate measurements entered the analysis) Biological variation Cochran&Bartlett; ANOVA

6. Statistics & graphics for the laboratory 30 Index-of-individuality ( II ) CV W-S /CV B-S is called "Index-of-individuality" = Ratio between within- and between-subject variation Often, the analytical variation is included, to give II = (CV W-S 2 + CV A 2 ) ½ /CV B-S Harris EK Clin Chem 1974;20:1535-42 Examples Calculation of CV G Range = 27 Mean = 18 ¼ of the RI in % of the mean = 36% Biological variation

7. Statistics & graphics for the laboratory 31 Index-of-individuality ( II ) II = CV W-S /CV B-S If II < 0.6 high degree of individuality reference ranges of limited utility (better use RCV!) If II > 1.4 low degree of individuality reference ranges are more useful Most analytes have II < 1.4 !! Examples Biological variation

8. Statistics & graphics for the laboratory 32 Comparison of a result with a reference-interval The "grey-zone" Example 2 measurement results for serum glucose: (1) 88 mg/100 ml (2) 109 mg/100 ml with a reference-interval: 60 - 100 mg/100 ml. Question: Is result (1) actually inside, result (2) outside the reference-interval? Data :CV a = 2 %; CV i = 6.1 % CV tot = SQRT(CV a 2 +CV i 2 ) = SQRT(2 2 + 6.1 6.1) = 6.4% Transform CV tot into s tot : 6.4 mg/100 ml (at 100 mg/100 ml) Calculate the “grey-zone” around the limit of 100 mg/100 ml with 95 (90) % probability (Note: one-sided): ±1.65 (1.29)  s tot or 1.65 (1.29)  6.4 mg/100 ml = ±10.6 (8.3) mg/100 ml. Grey zone at 95 % = 89.4 - 110.6 mg/100 ml Grey zone at 90 % = 91.7 - 108.3 mg/100 ml. Thus, result (1) lies inside the reference-interval in both cases. Result (2) lies outside the reference-interval only in the case of 90% probability. Conclusion: If one wants to miss few pathological cases, one would calculate the grey zone below the limit with 95 % probability, the one above the limit with 90 %. Glucose – "Grey-zone" Alternative approach With the statistical concept of Power: How big is the chance that we decide "healthy", when the patient is sick, in fact (ß-error; false negatives). Biological variation The power concept will be explained later. It is important for: Sample size method comparison Limit of detection (LOD) IQC

9. Statistics & graphics for the laboratory 33 P = 95% Biology, only 1.65 CV W-S "Grey zone" for results at reference limits P = 95% Total 1.65 [U A 2 +CV W-S 2 ] U A Analytical uncertainty (see "Goals") Biological variation

10. Statistics & graphics for the laboratory 34 Reference Change Value (RCV), or Medically significant difference (  med ) 95 % interval for difference between two samplings and measurements  med = 1.96 [2 (CV 2 W-S + CV 2 Anal )] ½ The RCV is particular important for analytes with high CV B-S.  med for the “creatinine clearance” C: ml plasma cleared per min per standard body surface U cr : concentration of creatinine in urine (mg/ml) P cr : concentration of creatinine in plasma (mg/ml) V: volume urine flow in ml per min A: body surface in m 2 For U cr : 1 mg/ml P cr : 0.01mg/ml V: 1 ml/min A: 1.88 m 2 (man of 70 kg and 1.75 m tall) is C = 92 ml/min/1.73 m 2 Calculation of s a : When CV a = 2% for U cr =1 mg/ml, then s a = 0.02 mg/ml. When CV a = 2% for P cr = 0.01 mg/ml, then s a = 0.0002 mg/ml. CV a for V is not considered. Note: in our example, s y = s a ; y = C s a = 92 0.0283 = 2.6 ml/min/1.73m 2 Biological variation

11. Statistics & graphics for the laboratory 35  med for the “creatinine clearance” Calculation: CV i = 15 % or at C = 92 ml/min/1.73 m 2, s i = 13.8 ml/min/1.73 m 2 (*) s tot = SQRT[2.6 2 + 13.8 2 ] = 14 ml/min/1.73m 2 Thus:  med = 1.96 SQRT[2] 14 = 38.9 ml/min/1.73 m 2 in other words a decrease in creatinine clearance of 42% is to consider medically significant. (*) Tietz Textbook of Clinical Chemistry 2nd Ed, Burtis CA, Ashwood ER, eds. WB Saunders Company, Philadelphia, 1994: p 1536. Note: If we also take the CV a for V into account, f.e. 5%, then : This example thus makes it obvious that in particular the biological variation contributes to the medically signifcant value. It is for that reason that Tietz recommends: “...Sequential determinations of creatinine clearance and averaging of values are required to reduce this variation appreciably”. Biological variation = 5.3 ml/min/1.73 m 2 = 14.7 ml/min/1.73 m 2 = 40.7 ml/min/1.73 m 2

12. Statistics & graphics for the laboratory 36 Notes