1. Topics Reference interval studies The importance of seeing Parametric, Non-parametric and other Estimating the error of a reference interval study Sample size for a reference interval study Outlier exclusion Partitioning for age, sex or other Data mining techniques Requirements for reference interval sharing

2. The importance of seeing This is a workshop on statistical techniques The human brain is a very powerful mathematical engine The best inputs are graphical not numerical ALWAYS graph your data ALWAYS think about your data

3. The importance of numbers Which distribution is Gaussian? Both! N=50 N=2000

4. Parametric statistics Can be used on parametric distributions Parametric distributions are those which can be described by parameters Gaussian Distribution defined by 2 parameters: Mean (average) – indication of the center Standard deviation – indication of scatter –Symmetrical distribution (not skewed) –68.3% within +/- 1SD –95.4% within +/- 2SD –99.7% within +/- 3SD

5. Non-parametric statistics No assumptions about distribution Percentiles determined by ranking Measure of centre is median (50 th percentile) Measure of scatter is percentiles (eg 2.5 th and 97.5 th )

6. Non-parametric statistics xth percentile is X × (n+1) th lowest sample Example: 75 th centile, n=138 75 th = 0.75 x 139 th lowest sample = 104 th lowest = 9

7. Non-Parametric - numbers To determine two percentiles P% apart Need at least (100/P)-1 observations Examples –95 th Centile (separate from 90 th ) –Need (100/5)-1 = 19 observations –97.5 th Centile (separate from 95 th ) –Need (100/2.5)-1 = 39 observations –99 th Centile (separate from 98 th ) –Need (100/1)-1 = 99 observations

8. Robust Techniques Methods giving more weight to the more common (central) values than to the peripheral results Described by Amadeo Pesce –Estimating reference intervals with n=20! –Horne PS, Pesce AJ, Copeland BE. Clin Chem 1998;44:622-631. Techniques not readily available *** Data-mining techniques may be considered “robust”

9. Confidence Intervals Reference interval studies are experiments There is “Experimental error” This is revealed when more than one reference interval study is performed. Even if every other factor is the same, a different sampling of a population will produce a different result The confidence interval of the Upper and Lower reference intervals describe this error

10. Statistical Imprecision of RI study Estimates of reference limits has limitations Expressed as the confidence interval of the Reference Limits, eg 90% CI of the upper and lower reference limits Confidence intervals decrease as the number of people in the study increases. Large n Small n

11. CI - Parametric Mean +{z 1 s +/- z 2 *SQRT[s 2 /n + (z 1 2 *s 2 )/2n]} s = SD n=sample size z 1 = probit value related to percentile (=1.96 for 97.5th percentile) z 2 = covering factor for confidence level (= 1.64 for 90%)

12. CI - Parametric +/-2SD +/- 1.64*SQRT[s 2 /n + (1.96 2 *s 2 )/2n]} Mean = 20, SD = 10

13. EXAMPLE 2.5 th Centile, n=250 97.5 th = 0.025 x (n-1) = 0.025 x 249 = 6 th lowest sample 90% confidence interval is 3rd to 12 th lowest samples

14. Outlier exclusion “some observation whose discordancy from the majority of the sample is excessive in relation to the assumed distribution model for the sample, thereby leading to the suspicion that it is not generated by this model.” A vital part of a reference interval study using parametric or non-parametric statistics Particularly difficult with “logarithmic” data –(BNP data)

15. Outlier exclusion Dixons criteria If D (distance of outlier from next sample) is > 1/3 x R (range of entire data set): exclude For groups of outliers treat each individually –NCCLS, Horn and Pesce Other: remove any data outside +/- 4SD “Reliable statistical detection of outliers in reference interval data remains a challenge” –Solberg and Lahti, Clin Chem 2005;51:2326-2332

16. Distributions Commonly “assumed” distributions –Gaussian –Square root –Logarithmic –More skewed

17. Box - Cox Transformations A family of transformations y = (x λ - 1)/λ y = ln(x+c) λ=0 Covers many forms of transformation λ = 1 linear transformation (unchanged) λ = 0.5 square root transformation λ = 0.2 skewed right (less skewed than log) λ = 0 (or close to zero) logarithmic transformation λ = -0.2 Heavily skewed right (more than log) λ < 0 “Over-log” transformation Normalises data more skewed than log distribution

18. Transformations of ALT NHANES III: ALT, male, age 20 to 80, n=6423 Raw data Lambda=1 Logarithmic Lambda=0 “Over-Log” Lambda=-0.5

19. 11 19 27 33 41 49 57 NORIP STUDY Female ALT (n=1220) Female URL: 45.6 (90% CI 42.5 – 49.3, n=1220) Male URL: 68 (90% CI 63.4 – 73.6, n=1080) ALT (U/L)

20. Partitioning Provision of separate reference intervals for subgroups Sex and age (paediatric & geriatric) most common Others may include race, menopausal status, stage of gestation or menstrual cycle. Historically Harris and Boyd has been recommended. New theories –Lahti A et al. Clin Chem 2002;48:338-352

21. Lahti et al Criteria depends on asterisk rate of subgroups when common intervals are applied. <3.2% asterisk rate of either subgroup: NO >4.1% asterisk rate of either subgroup: YES In-between: consider other factors Note: non-parametric approach also described –Very complex –Clin. Chem., May 2004; 50: 891 - 900.

22. Data Mining Bhattacharya, LG. Journal of the Biometric Society. 1967;23:115-135. Example data: Frequency Distribution of the forkal length of the Porgy caught by pair-trawl fishery in the East China Sea.

23. Bhattacharya Assumptions –Gaussian or Log Gaussian distributions –Most results unaffected by reason for testing blood –Ideal for “profiles” –No systematic effect of source on results. Eg Inpatients with low sodium and albumin Outpatients with delayed separation Beware –No confidence limits for results –User-influence on results

24. GJ - Excel Bhattacharya