The theory of reference values. part 6. presentation of observed values related to reference values

Clinica Chimica Acta, 127 (1983) 441F-448F Elsevier Biomedical Press

INTERNATIONAL SCIENTIFIC

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FEDERATION COMMITTEE,

OF CLINICAL CHEMISTRY

CLINICAL

EXPERT PANEL ON THEORY OF REFERENCE

SECTION VALUES (EPTRV)

IFCC Document (1982) Stage 2, Draft 2 (1982-07-27) with a proposal for an IFCC Recommendation THE THEORY OF REFERENCE VALUES. PART 6. PRESENTATION OBSERVED VALUES RELATED TO REFERENCE VALUES

OF

Prepared by: R. Dybkaer Comments on these proposals should be sent before H.E. Solberg, MD Department of Clinical Chemistry Rikshospitalet Oslo 1 Norway Comments

from the viewpoint

of languages

1983-05-31 to

other than English

are encouraged.

Preface This paper forms a part of a series of Recommendations on the Theory of Reference Values. Other parts comprise: Part 1. The concept of reference values (1978). Clin Chem 1979; 25: 1506-1508; Clin Chim Acta 1978; 87: 459F-465F; J Clin Chem Clin Biochem 1979; 17: 337-339. Part 2. Selection of healthy individuals for the production of reference values. Part 3. Preparation of individuals and specimen collection for the production of reference values. EP Members: C. PetitClerc (CDN) 1977-1983, H.E. Solberg, Chairman (N) 1977-1983, D. Stamm (GFR) 1979-1982, P. Wilding (USA) 197661984 Former Members: R. G&beck (SF) 1970-1978, G. Siest (F) 1973-1979, T. Whitehead (GB) 1970-1976, G.Z. Williams (USA) 1970-1976. List of abbreviations and symbols: AM, arithmetic mean; B, blood; EPTRV, Expert Panel on Theory of Reference Values; IFCC, International Federation of Clinical Chemistry; lg, logarithm of; N, number; nonp., non-parametric; OV, observed value; P, cumulative number fraction (as ordinate value of a fractile); param., parametric; ref. int., reference interval: RV, reference value; SD, standard deviation. The decimal sign is a comma.

Part 4. Control of analytical variation values. Part 5. Treatment of collected reference

in the production

and use of reference

values. Determination

of reference

limits.

The first document in this series, ‘The Concept of Reference Values’, describes the subject of reference values and defines various terms. It should be consulted prior to the reading of the present document for a thorough understanding. Stage 1, draft 6 was extensively circulated to IFCC National Representatives and IFCC Corporate Affiliate Representatives, to Associate Members and Members of the Expert Panel on Theory of Reference Values, and to IFCC Officers and Members of the Congress and Education Committees. Apart from general endorsements, a number of suggestions for changes were received. The largest part of the proposals has been incorporated in the present Stage 2, Draft 2 Document. Contents Preface ................................................ 1. Introduction ............................................ 2. Presentation of reference values ............................... 3. Transformation of observed value to show relation to reference 4. Conclusions ............................................. 5. References ..............................................

values

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1. Introduction I. 1. Why reference values An observed value of a quantity provides meaningful information only when compared to relevant reference values. In clinical chemistry, the basis of comparison is either other values of the same type of quantity obtained from the same person or from comparable reference individuals. The present document deals with the latter situation [l-3]. I. 2. Statistical problem Usually, the procedure is not a comparison in the strict statistical sense, because the population underlying the observed value is not the same as that underlying the reference values, which generally are obtained at earlier dates. Still, comparing has proven clinically useful. 1.3. Matching test individual to reference individual However, a comparison may be misleading if the person under investigation does not match the reference individuals adequately as regards, e.g. sex, age, fasting, exercise, and body posture or if the specimen collection procedure is different. (Such factors are discussed in Parts 2 and 3 of this series of documents.) The clinician is ultimately responsible for selecting the appropriate set of reference values.

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1.4. Information

about reference

values

Therefore, the clinician should always be supplied with information about the available sets of reference values. The minimum information comprises definition of reference individual (and subclassification), method of selection, preparation of individual, procedure for specimen collection, analytical method, number of reference individuals (in each subclass). 1.5. Observed

value and reference

values

For convenience, each set of reference values usually is not presented in detail, but after some form of data reduction. This first level of reporting, stating observed value and reference values or reference interval, is discussed in Section 2. 1.6. Observed value and relation to reference

values

In order to facilitate the process of comparison, the clinician may wish a summary of the relationship between the observed value and each set of relevant reference values. Such second level of reporting, consisting in giving the observed value and its transformation, is presented in Section 3. 2. Presentation of reference values 2.0. Format of presentation The set of reference values or characteristic values derived from it can be made available to the clinicians in the form of tables, graphs or a few figures, either on result reports (laboratory sheets) or by separate publication. A printed set of reference data on reports is recommended only if one can ensure that it is relevant to the observed value to be reported. When necessary, different sets should be given according to age, sex, activity and posture (ambulant, supine), etc. 2.1. All values given When few reference values are available, say 15, it is safe and informative to list them all in ascending order of magnitude. Even large numbers of values may be listed in tables and graphs when sufficient space is available. 2.2. Reference interval A widely used practice is to state the limits of an interval * that depends on location and dispersion of the reference values [4]. In clinical chemistry it is customary to calculate a closed interval comprising a central number fraction ** of 0,95 (or 95%) of the reference values. Other number fractions or an asymmetrical

* The term ‘interval’ is preferred to ‘range’ which should be restricted to the difference between upper and lower limit of an interval (or class). ** The kind of quantity ‘number fraction’ is defined as the number of elements in a given subset (or individual class) divided by the number of elements in the set (or number distribution). The ambiguous term ‘frequency’ is often used for this kind of quantity.

REFERENCE INTERVAL8 FRACTILE

Q95-rd ;int.(poram.l

B-Hemoglobin(Fe),substance

concentration

mmolll

Fig. 1. The cumulative number fraction distribution of 50 reference values for the substance concentration of hemoglobin in blood of female adult persons. Observed values are indicated by points, a fitted Gaussian curve as an unbroken S-curve. The non-parametric and parametric central 0,95interfractile intervals are shown as unbroken bars; the estimations of non-parametric and parametric fractiles. corresponding to a reference value of 9,0 mmol/l, are shown by broken arrows.

position of the reference interval may be more appropriate in particular cases; the actual choice should always be explicitly stated. Of the several possible statistical types of reference intervals, two merit special attention and will be described as central 0,95-intervals. 2.2.1. Non-parametric central 0,9.5-interfractile interval. The reference values are listed in ascending order of magnitude and a number fraction of 0,025 of the values are cut off at each end. The lowest and the highest values of the remaining number fraction of 0,95 are the limits (the 0,025- and 0,975-fractiles ***). In Fig. 1, where the number of reference values was 50, one value has been cut off at each end to yield (the nearest approximation to) the 0,95-ref. int. (nonp.) of 7,0 to 10,3. This procedure is non-parametric, i.e. independent of a hypothesis regarding the type of distribution underlying the observed reference distribution. Still, the interval may be misleading at low numbers of reference values, even with 50. 2.2.2. Parametric central 0,95-interfractile interval. A mathematically known distribution is fitted to the reference distribution. Usually, a Gaussian hypothesis is applied directly or after some transformation of the reference values (e.g. lg RV, *** A fractile is a variate value (e.g. a reference value) equal to and below which lies a stated fraction the cumulative number fraction distribution.

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m). Then, the central interfractile interval is taken from the model distribution. For the Gaussian distribution, a central 0,95-interval is obtained by calculating the limits arithmetic mean - 1,96 . standard deviation and arithmetic mean + 1,96 . standard deviation. In Fig. 1, the 0,95-ref. int. (param.) is somewhat narrower than the non-parametric version due to random effects at only 50 reference values. With one decimal, the interval becomes 7,0 to 10,l. Seriously misleading limits may occur if the hypothetical distribution is wrong, as may well occur with a low number of reference values, even with 50. 2.2.3. Other types. Other types of intervals are being advocated, especially the tolerance interval with confidence probability [4-61 and the prediction interval [4,6,7]. With a moderate number of reference values, they may lead to erroneous results due to the choice of a wrong model or because the assumption of random sampling is not fulfilled. With a large number, all the intervals become virtually identical. (For the estimation of reference intervals and the confidence interval around each reference limit, see Part 5 of this series of documents.) 3. Transformation

of observed value to show relation to reference values

3.0. Principles of transformation Many relations between observed value and reference values have been advocated with non-parametric or parametric procedure, linear or non-linear transformation and based on the location or dispersion of the reference distribution. A few different types may be discussed. 3.1. Classification of observed value into one of three classes On the basis of the two reference limits of a reference interval (Section 2.2) it is possible to classify an observed value as ‘unusually low’ when situated below the lower reference limit, ‘usual’ if between or equal to either of the reference limits or ‘ unusually high’ when above the upper reference limit. (Terms such as ‘abnormal’, ‘pathological’ or ‘normal’ are not recommended, both because of possible confusion with mathematical and physiological meanings of ‘normal’ and because an ‘abnormal value’ may be a purely statistical, non-pathological occurrence.) A value falling into one of the three intervals may also be said to have the transformed value of - 1 arbitrary unit, 0 arbitrary unit or 1 arbitrary unit, respectively (or the classes may be labelled 1, 2, and 3). Another possibility is to flag an unusually low or high observed value with a convenient symbol. In any case, the three-class characterization, which constitutes a linear transformation dependent on dispersion and location of the reference distribution, is crude and a waste of the information contained in the set of reference values. It should also be remembered that an observed value within the (interindividual) reference interval may be a rare value for a particular individual according to his intraindividual reference distribution and, therefore, a significant sign of disease for that individual.

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3.2. Observed value minus arithmetic mean, divided by standard deviation

This parametric statistic, (OV - AM)/SD, is a linear transformation of the observed value and depends on the two parameters of a Gaussian distribution, i.e. on both location and dispersion. The function has been used frequently [8-l 11, sometimes with ambiguous names such as ‘SD unit’, ‘SDU’ or ‘normal equivalent deviate’. The parametric central 0,95-reference interval has the transformed limits - 1,96 and + 1,96. The results may be misleading if the reference distribution is definitely non-Gaussian; therefore, this transformation cannot be recommended, in general. It is useful, however, when the number of reference values is high, preferably around 500, and the distribution approximates well to the Gaussian type. 3.3. Fractile This statistic [12,13] is the observed value itself with an indication of the corresponding ordinate value on the appropriate cumulative number fraction reference distribution. The distribution curve may be observed or fitted according to a hypothesis, i.e. by non-parametric or parametric procedure, respectively. In essence, the fractile provides the coordinates of a point on the cumulative distribution curve and is a non-linear, curve-fitting transformation. For example (Fig. l), the substance concentration of hemoglobin(Fe) in blood (of a certain female patient at a stated time) may be reported as 9,0 mmol/l (P = 0,76; nonp.), indicating that a number fraction of 0,76 of the reference values were equal to or less than 9,0 mmol/l. If the Gaussian hypothesis had been applied, the fractile would be reported as (P = 0,71; param). The reference limits of the central 0,95-interval will have (P = 0,025) and (P = 0,975), respectively. If the observed value is outside the interval of registered reference values, a fractile cannot be stated with the non-parametric procedure. Probably, this is not a drawback, since any statistical measure is of doubtful credibility in these regions of extrapolation. It is safer to indicate (lowest RV = xyz . unit) or (highest RV = pqr . unit) after the observed value, when it is below the lowest or above the highest reference value, respectively. 3.4. Other transformations A number of other transformations reference.

found

in the literature

are given

here for

3.4.1. Two, four or more classes. Instead of three classes (Section 3.1) the interval of possible observed values may have two, four or more, e.g. [ 14,151. Two classes, i.e. a binary classification, may be relevant where observed values are either below or above a stated limit and can be symbolized by 0 and 1, respectively. (This convention is preferred to ‘minus’ and ‘plus’, respectively.) Four or more classes permit a refinement of the three-way classification mentioned in 3.1. The observed value may be related to 3.4.2. Observed value divided by a fractile. a specified fractile of the observed cumulative reference distribution or the fitted (Gaussian-based) cumulative reference distribution [16], e.g. by dividing the observed value by the observed median (0,5-fractile) or arithmetic mean (0,5-fractile), respectively, or by the upper reference limit (e.g. the 0,975-fractile).

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The statistic given in Section 3.2 has been 3.4.3. Analogues of (OV - AM)/SD. modified in several ways, retaining the principle, but changing the magnitude of the transformed values [ 17-2 11. Parametric techniques based on 3.4.4. Bayes’ estimative or prospective approach. Bayes’ theorem permit calculation of either the probability that the patient ‘belongs to’ the reference population, given the observed value [22-241, or the probability that the observed value does not ‘belong to’ the reference values (‘index of atypicality’) [25-271. 4. Conclusions 4.1. The observed value should always be stated The original observed value of a given quantity must be reported - irrespective of the way in which it is related to reference values - to allow, e.g. comparisons with other types of quantities and metabolic calculations. 4.2. Reference values or reference interval A minimum of additional data is a presentation of relevant reference values or a specified reference interval derived from them, together with accessible information on how the values were obtained. The non-parametric central 0,95_interfractile interval (Section 2.2.1) is recommended. 4.3. Three classes, distance or fractile Important possibilities for transforming the observed value are a classification into one of three classes (Section 3.1), a statistical measure of distance from arithmetic mean of the reference values (Section 3.2), and the non-parametric fractile (Section 3.3). The last one is informative and free of assumptions. 4.4. Influences on choice The selection of a suitable way of presenting reference values and the relation the observed value to them depends on the clinical situation, e.g. health screening monitoring of therapy, and on available computing facilities.

of or

5. References 1 Dybkaer R, Grlsbeck R. Theory of reference values (Editorial), Stand J Clin Lab Invest 1973; 32: l-7. 2 Dybkaer R. Content and lay-out of clinical chemical results. In: Siest G. ed. Organisation des laboratoires et interpretation des rtsultats. Biologic Prospective, Troisitme Colloque International, Pont-&Mousson 1975. Paris; L’Expansion Scientifique Francaise, 1975: 795-800. 3 G&beck R, Dybkaer R, Winkel P. Relating observed values to reference values. Ann Biol Clin 1978; 36: 193-194. 4 Winkel P, Statland BE. Reference values. In: Henry JB, ed. Clinical diagnosis and management by laboratory methods, 16th ed. Philadelphia, PA: Saunders, 1979: 29-52. 5 Scientific Tables. Documenta Geigy, 6th ed. Basle: J.R. Geigy, 1962.

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Part I. Tables, examples and applications. J 6 Hahn GJ. Statistical intervals for a normal population. Qua1 Technol 1970; 2: 115-125. 7 Hahn GJ, Nelson W. A survey of prediction intervals and their applications. J Qua1 Technol 1973; 5: 178-188. 8 Grasbeck R, Fellman J. Normal values and statistics (Editorial). Stand J Clin Lab Invest 1968; 21: 193-195. 9 Gullick HD, Schauble MK. SD unit system for standardized reporting and interpretation of laboratory data. Am J Clin Path01 1972; 57: 517-525. 10 Rushmer RF. Accentuate the positive. A display system for clinical laboratory data. J Am Med Ass 1968; 206: 836-838. 11 Bold AM. Clinical chemistry reporting. Problems and proposals. Lancet 1976; i: 951-955. 12 Hald A. Statistical theory with engineering applications, 5th printing. New York, NY: John Wiley and Sons, Inc., 1962. 13 Rossing RG, Hatcher III WE. A computer program for estimation of reference percentile values in laboratory data. Comput Progr Biomed 1979; 9: 69-74. 14 Casey AE, Terhune SR, Franklin R, Hale K, Yilmaz M, Gross J, Rennecker C. North Alabama Whites and Indians compared anthropometrically with ancient and modern people using 36 blood pigment and skeletal factors (Abstract). Proc South Med Ass (St. Louis, Miss) 1960; (Quoted in 5.15). 15 Casey AE, Downey E. Further use of statens in the recording, reporting, analysis, and retrieval of automated computerized laboratory and clinical data. Am J Clin Path01 1970; 53: 748-754. 16 Lederer WH, Gerstbrein HL. Expressing results of enzyme assays. Clin Chem 1974; 20: 916-917. 17 Bliss CI. The method of probits, Science, New Series 1934; 79: 38839; 409-410 (A correction). 18 Gabrieli ER. Enhancing the meaning of clinical laboratory data. Crit Rev Clin Lab Sci 1970; 1: 65-85. 19 Hoffman RG. Statistics in the practice of medicine. J Am Med Assoc 1963; 185: 864-873. 20 Lo JS, Kellen JA. A proposal for a more uniform output in laboratory data. Clin Chim Acta 1972; 41: 239-245. 21 Delwaide PA, Buret J, Albert A. Le concept de ‘valeur normale’ en chimie clinique. Rev MM Liege 1972; 27: 694-709. 22 Anderson JA, Boyle JA. Computer diagnosis: statistical aspects. Br Med Bull 1968; 24: 230-235. 23 Jacquez JA. Algorithmic diagnosis: a review with emphasis on Bayesian methods. In: Jacquez JA, ed. Computer diagnosis and diagnostic methods. Springfield, IL: Charles C. Thomas, 1972: 374-393. 24 Starmer CF, Lee KL. A mathematical approach to medical decisions: application of Bayes’ rule to a mixture of continuous and discrete clinical variables. Comput Biomed Res 1976; 9: 531-541. IR. Statistical prediction analysis. Cambridge, UK: Cambridge University 25 Aitchison J, Dunsmore Press, 1975: 212-237. 26 Aitchison J, Habbema JDF, Kay JW. A critical comparison of two methods of statistical discrimination. Appl Statist 1977; 26: 15-25. indices as reference values for laboratory data. Am J Clin Path01 1981; 76: 27 Albert A. Atypicality 421-425.