A new method for estimating a standardized prevalence of child malnutrition from anthropometric indicators.

Research!Recherche A new method for estimating a standardized prevalence of child malnutrition from anthropometric indicators J.O. Mora1 Although anthropometric indicators are widely used for assessing the nutritional status of children, lack of consensus on the cut-off points for prevalence estimates has precluded the use of standard analytical methods in population surveys. A simple method for estimating a standardized prevalence of child malnu- trition from anthropometric indicators is presented. The method is based on comparing the distribution of the indicator with that of the normalized NCHS reference population, the underlying assumption being that both distributions are nearly normal. Standardized prevalence is defined as the proportion of cases in the observed population that is outside the normal distribution of the reference values, which can be estimated from the mean and stan- dard deviation of the standardized Z-scores of the population, by using a formula based on the mathemati- cal properties of the normal probability curve. A reference table is included which provides computer-estimated prevalence rates for different mean Z-scores and standard deviations of normally distributed anthropometric indicators. Introduction Nutritional anthropometry remains the most practi- cal and useful means for the assessment of the nutri- tional status of the population, particularly among infants and young children (1, 2). In cross-sectional surveys, the use of appropriate anthropometric indi- cators allows the identification of the nature and extent of energy-protein malnutrition in the com- munity. Repeated assessments are also useful for follow-up of populations, comparisons between groups, evaluation of programmes, and statistical comparisons in epidemiological research (3, 4). Despite the popularity and recognized useful- ness of nutritional anthropometry in assessments of health and nutrition, there have been many dis- cussions and conflicting recommendations about the cut-off points to be used for estimating the preva- lence of anthropometric abnormality which is con- ventionally taken to indicate "undernutrition" (1-10). 1 Senior Medical Nutritionist, International Nutrition Unit, Logical Technical Services Corp., 7222 47th Street, Chevy Chase, MD 20815, USA. Different cut-off points and classification systems have been proposed and used for estimating the prevalence of malnutrition in population surveys; thus the reported rates are often not comparable and sometimes questionable. This confusion and the con- sequent lack of standard analytical methods have apparently legitimized an unfortunate tendency to leave every country (or group) open to set up its own criteria, depending on the local circumstances (political or other), for the sake of practicality. However, as stated elsewhere (11), "practical con- siderations are extraneous to the concept of bio- metric or functional abnormality" on which true prevalence estimates should be based. This paper proposes a simple method for esti- mating a standardized prevalence of malnutrition in cross-sectional population studies. The proposed method makes any further discussion on cut-off points for prevalence estimates irrelevant. While the method could be applied regardless of the source of "'reference values" (a topic which is not discussed in the paper), it does require the availability of a healthy, well-nourished reference population whose distribution of anthropometric values is normalized, Bulletin of the World Health Organization, 67 (2): 133-142 (1989) c World Health Organization 1989 133 J.o. Mora such as the NCHS/CDC (U.S. National Center for Health Statistics/Centers for Disease Control) growth reference. It also requires expressing the anthropometric indicator as the difference between the observed value and the age/sex reference value, in units of standard deviations (Z-scores) of the refer- ence population. Cut-offpoints for prevelence estimates One of the first known attempts to use anthropo- metric measurements for estimating the prevalence of child malnutrition was the popular Gomez classi- fication (5), which uses 90 per cent weight-for-age as the cut-off point for identifying children with nutri- tion and health problems. This classification method was originally designed as a guide to the prognosis of hospitalized malnourished children, and not as a yardstick for prevalence estimates, but over the past decades it has been used increasingly in developing countries for the assessment of malnutrition in the community. With widespread use, questions have been raised about this classification system (12-13), mostly because of its relatively high cut-off point, which is equivalent to about one standard deviation below the reference mean. This appears to grossly overesti- mate the prevalence of child malnutrition by includ- ing a sizeable proportion (about 15.9 per cent) of "false positives" (i.e., those whose weights are within the normal range of the reference population distribution). What has been particularly dubious is the supposed abnormality of those children labeled as having first-degree malnutrition (75 to 90 per cent weight-for-age). This is often circumvented by reporting prevalence rates for cases below 75 per cent weight-for-age, that is, only second and third degree (14-16). Recognizing these problems, as well as those related to the use of percentages to express anthro- pometric indicators, the World Health Organization (WHO) advocated expressing the deviation from the anthropometric measurement of the reference median in terms of standard deviations or Z scores, and strongly urged the adoption of the NCHS refer- ence population data (17) as normative values for international use. To use the Z-scores method, the NCHS/CDC growth reference curves had to be transformed into a Z-score representation with approximately normal distribution (18, 19). Normal- ized growth curves developed at the Centers for Disease Control are being used worldwide since 1978 to assess the nutritional status of populations. WHO also proposed that the normal range for any popu- lation should be between plus and minus two stan- dard deviation (± 2 S.D.) units of the median (2, 8-10), a range that includes 95.4 per cent of the refer- ence population, and would yield only about 2.3 per cent false positives on each side. Besides its statistical justification, the WHO cut-off point, below which the values are seen as potentially abnormal, has been further supported by studies of "functional outcomes" showing a signifi- cant increase in the risk of death (20-24), as well as a decreased immune response (25) when anthropo- metric indicators drop below such a point. Thus, the WHO recommendation has been generally adopted (10, 26) and the cut-off point at two standard devi- ation units below the reference median (or the third percentile) has been widely used lately for estimating the prevalence of malnutrition in national surveys. As expected, however, the shift of the cut-off point from one to two standard deviations resulted in dramatically lower prevalence rates of malnutri- tion in developing countries, thus suggesting that, in contrast with the former cut-off point, the latter might tend to underestimate the magnitude of the problem. WHO then suggested (27), as part of its methodology for measuring change in nutritional status, using either one or two standard deviations as the dividing line between normality and abnor- mality, but adjusting the resulting prevalence by sub- tracting the proportion of cases expected below such a cut-off point in the normal distribution (either 15.9 per cent or 2.3 per cent). This adjustment for "false positives" does not really solve the problem, since applying either of the two cut-off points to a given population yields quite contrasting prevalence estimates. As an example, the following disparate figures result from using the two cut-off points and their corresponding adjustments with the weight-for-age data of the 1977-80 Colom- bian National Health Survey (28): Measured Cut-off prevalence points (%) -1 S.D. -2 S.D. 49.2 16.9 Adjusted Adjust- prevalence .ment (%) (%) -15.9 -2.3 33.3 14.6 Clearly, in spite of the WHO-suggested adjust- ment, arbitrary cut-off points selected for prevalence studies have profound implications, and might be misused for non-scientific purposes to comply with political and other interests. A floating cut-off point could indeed be moved up or down depending on whether the interest is to dramatize the seriousness of the problem or to show that it is of much less magnitude. Keller has recently contended that "if reasonable simple statistical methods were available, 134 Estimating child malnutritlon it would be more desirable to compare distributions rather than prevalences, which to some degree distort biological realities" (29). Nutritional anthropometry as a diagnostic test Nutritional anthropometry may be conceived as a diagnostic test to identify and count the malnour- ished by classifying persons as malnourished or well- nourished in relation to a specific level of the diagnostic indicators (4). As such, it would be com- posed of both an indicator and a cut-off point for that indicator which, as in other diagnostic tests, results in some misclassification of subjects. Thus, while some well-nourished individuals may be wrongly classified as malnourished (false positives), some malnourished persons are classified as well- nourished (false negatives). This is because of the usual overlapping of the distribution of biological indicators among the diseased and the non-diseased individuals. The ideal cut-off point for a given indi- cator would be one which results in a complete separation of normals from abnormals within the population (3). A diagnostic test is supposed to reflect a true underlying reality, usually a disease entity, which is accurately diagnosed or directly measured by other means. However, in the case of nutritional anthro- pometry, valid external criteria and instruments for a direct measure (diagnosis) of the true reality (e.g., nutritional status/body composition) are not readily available. In the absence of objective criteria for diagnosis, the following three basic approaches are used in clinical epidemiology to set up criteria for abnormality, which may be defined either as being sick, being treatable, or being unusual (30): (a) The identification of a significant association (and eventually break-off or threshold points) between the test indicator(s) and changes in function- al outcomes, such as the risk of health impairment, disability or death. This approach has been attempt- ed in evaluating anthropometric indicators using either concurrent (immune response) or long-term (mortality) functional outcomes (20-25). Unfor- tunately, these validation criteria are not totally appropriate because such functional outcomes are known to be affected by factors other than nutri- tional status, which is the true underlying reality of concern. (b) The demonstration of selective response to a treatment intended to modify a disease determinant in subjects chosen on the basis of the results of the diagnostic test, as in iron-deficiency anaemia (31).' ' Freire, W. Use of hemoglobin levels to determine iron defi- ciency in high-prevalence areas of iron-deficiency anemias. Ph.D. thesis. Ithaca, New York, Cornell University, 1981. This approach can rarely be used in population work, and it is clearly of no use with weight indica- tors because the weight response to increased food intake may well exceed desirable healthy limits. (c) The so called "normative reality" (3, 30), using probability estimations based on the statistical properties of the normal distribution of values from a supposedly healthy reference population. This implies comparing the observed values of an indica- tor with those derived from a known normal popu- lation (e.g., the NCHS reference values obtained from a well-nourished, healthy population growing under optimal environmental conditions), and con- sidering those values outside the normal population as abnormal. The normative reference values are seen as rep- resenting the underlying reality, and their distribu- tion is supposed to meet the statistical properties of the normal distribution, provided that all the abnor- mals and only the abnormals were excluded. The ref- erence population distribution of anthropometric values is thought to reflect only the individual varia- bility of the genetic potential for growth, which is expected to be fully realized under presumably optimal environmental conditions. This normative reality is the implicit basis of anthropometry refer- ence values and provides the best known criteria for identifying the malnourished, and is therefore a useful yardstick for prevalence estimates in cross- sectional assessments. Method The prevalence of child malnutrition, as defined by anthropometric abnormality, could be estimated on the basis of the normative reality by using the fol- lowing well-known simple formula: SP = MP - FP + FN where SP = standardized prevalence, defined as the proportion of individuals in the observed population who are outside the normal distribution of the refer- ence population; MP = measured prevalence, calcu- lated from the observed population as the proportion of children under a given cut-off point of the reference population; FP = false positives, esti- mated as the proportion of values expected under the cut-off point in the reference population; FN = false negatives, estimated as the excess pro- portion of subjects above the cut-off point in the observed population as compared to the reference population distribution. In most populations, the three common anthro- pometric indicators (weight-for-age, length or height- for-age, and weight-for-height or length) have bell- 135 J.O. Mora shaped, more or less symmetrical distributions (2, 8, 10, 27-29). In developing countries, the distri- bution of anthropometric indicators is approxi- mately Gaussian but shifted to the left of the normal distribution of the reference population, with only a slight skewness and a variable degree of overlapping depending on the distance between the two distribu- tions. As an example, Fig. 1 shows a hypothetical Gaussian distribution of a population indicator whose mean is one standard deviation to the left of the reference population distribution (i.e., its mean standardized Z-score is -1.0) and its standard devi- ation is also 1.0. There is some overlapping between the two curves, and their intersection occurs at half the distance between their means; this is always the case when the standard deviation of both distribu- tions is of the same size. Adjusting only for false positives, as suggested by WHO, would tend to underestimate the preva- lence of malnutrition. By adjusting for both false positives and false negatives, a standardized and more accurate estimation of the prevalence would be obtained. Calculating the proportions of false posi- tives at a given cut-off point is a straightforward pro- cedure; they are defined as the proportion of cases found below that point in the normal distribution of the reference population, e.g., 15.9 per cent below one standard deviation and 2.3 per cent below two standard deviations of the reference mean. As seen in Fig. 1, the proportion of false positives (i.e., area (c) for two standard deviations below the mean, and areas (c) plus (d) for one standard deviation) exclu- sively depends on the cut-off point used, regardless of the distance between the curves; their calculation can be mathematically expressed using the cumula- tive distribution function (c.d.f. = I) of the stan- dardized normal curve (32), as FP = 1 (- K), where -K = cut-off point. Unlike the false positives, the proportion of false negatives would depend not only on the cut-off point but also, to a great extent, on the degree of overlap- ping, i.e., on the closeness of the two curves. False negatives may be defined as the excess proportion of cases found in the observed population above the cut-off point, as compared to the reference popu- lation. Thus, in Fig. 1, area (b) would represent the proportion of false negatives for a cut-off point of one standard deviation, and areas (a) plus (b) for two standard deviations. It should be noted that false negatives are located in the interval between the cut-off and the intersection of the two curves. The estimation of the proportion of false nega- tives can also be expressed mathematically using the c.d.f. of the normal curve. Both (a) and (b) or any other excess fraction of the observed population dis- tribution can be estimated (from the table of areas under the normal curve (32)) as the difference in the proportion of cases between the corresponding equivalent intervals of the reference and the observed population, provided that the two distributions are Gaussian and the distance between the two curves is expressed in units of standard deviation (Z-scores) of the reference population, i.e., as the mean (or median) standardized Z-score of the observed popu- lation indicator. In Fig. 1, the proportion of false negatives for a cut-off point (K) of - 2.OZ can be estimated as the difference between the proportion of the population expected from -1.OZ to +0.5Z as seen in the observed curve at the left (areas a + b + d + e), and that from -2.OZ to -0.5Z as seen in the reference curve at the right (areas d + e). These proportions can be calculated from the Table of areas under the normal probability curve (32), as follows: - l.OZ to 0 = 0.8413 - 0.5000 = 0.3413 (a + b) +0.5Z to 0 = 0.6915 - 0.5000 = 0.1915 (b + e) - 1.OZ to +0.5Z = 0.3413 + 0.1915 = 0.5328 (a + b + d +t) -2.OZ to -0.5Z = 0.9773 - 0.6915 = 0.2858 (d + e) (a) + (b) = 0.5328 - 0.2858 = 0.2470 = 24.7 per cent Similarly, false negatives for K =-1.0 (area b) would be: +0.5Z to 0 = 0.6915 - 0.5000 = 0.1915 (b + e) -1.OZ to -0.5Z = 0.8413 - 0.6915 = 0.1498 (e) (b) = 0.1915 - 0.1498 = 0.0417 = 4.2 per cent Standardized prevalence rates for different cut-off points and distances between similar Gauss- ian curves with standard deviations of one Z have been calculated in Table 1, after adjusting the mea- sured prevalence for the proportion of both false positives and false negatives. Theoretical calculations have been made for distances between the two curves (differences between means or medians) equivalent to 1.0, 1.5 and 2.0 standard deviations of the reference 136 Estimating child malnutrition Table 1: Estimated percentage prevalence of malnutrition at selected cut-off points for different distances (in Z-scores) between the observed and the reference population distributions of an anthropometric Indicator (when the observed standard deviation Is equal to one) Measured False False Standardized Mean Z-score Cut-off point prevalence positives negatives prevalence' Difference (-Z) (-K) (MP) (FP) (FN) (SP) (MP - SP) 0.5 1.00 30.9 15.9 4.7 19.7 11.2 1.28 (p10)b 21.7 10.0 8.0 19.7 2.0 1.50 15.9 6.7 10.5 19.7 -3.8 1.88 (P3)b 8.3 3.0 14.4 19.7 -11.4 2.00 6.7 2.3 15.3 19.7 -13.0 1.0 1.00 50.0 15.9 4.2 38.3 11.7 1.28 (P10) 39.0 10.0 9.3 38.3 0.7 1.50 30.9 6.7 14.1 38.3 -7.4 1.88 (P3) 18.9 3.0 22.4 38.3 -19.4 2.00 15.9 2.3 24.7 38.3 -22.4 1.5 1.00 69.2 15.9 1.4 54.7 14.5 1.28 (P10) 58.7 10.0 6.0 54.7 4.0 1.50 50.0 6.7 11.4 54.7 -4.7 1.88 (P3) 35.2 3.0 22.5 54.7 -19.5 2.00 30.9 2.3 26.1 54.7 -23.8 2.0 1.00 84.2 15.9 0 68.3 15.9 1.28 (P10) 76.4 10.0 1.9 68.3 8.1 1.50 69.2 6.7 5.8 68.3 0.9 1.88 (P3) 54.8 3.0 16.5 68.3 -13.5 2.00 50.0 2.3 20.6 68.3 -18.3 a SP = MP - FP + FN b P10 and P3 refer to the 10th and 3rd percentile, respectively. values, and for commonly used cut-off points at 1.28 (10th percentile) and 1.88 (3rd percentile) standard deviations below the reference median. For a given distance between curves, the same standardized prevalence rate is found irrespective of the cut-off points (indeed, there is only one cut-off curve). As expected, the prevalence is a function of the distance between the observed and the reference curve and not of the cut-off points; in fact, these become irrelevant when the prevalence is estimated by adjusting the measured prevalence for the propor- tions of both false positives and false negatives, as calculated by the method proposed here. It is also observed in Fig. 1 that, when the stan- dard deviation of the observed distribution is also one, a cut-off point located at half the distance between the two Gaussian curves would yield no false negatives, thus making it possible to obtain standardized prevalence estimates from the mea- sured prevalence adjusted (by subtraction) for false positives only, as proposed by WHO. In this case, SP = MP - FP. Therefore, this adjustment is appro- priate only for that particular cut-off point when the observed distribution is nearly normal and its stan- dard deviation is one. When the above definitions of false positives and false negatives are applied, it then becomes clear that the estimated standardized prevalence is rep- resented by the shaded area in Fig. 1, i.e., by that portion of the observed distribution that does not overlap (it is indeed outside) the reference population distribution. Thus a statistical method could be developed to compare the two distributions as sug- gested by Keller (29), using a cut-off curve rather than a cut-off point. The method would be based on the presumption that under optimal environmental conditions everybody will grow within the bound- Fig. 1. Overlapping of the Gaussian distributions of an anthropometric Indicator In the observed and In the ref- erence population. The shaded area represents the stan- dardized prevalence of abnormality In the observed population. -3 -2 -1 0 .1 *2 +3 Obser,,ed Standadrdedvations in standardized Z scores of the reference popultion distribution 137 J.O. Mora Table 2: Estimated prevalence of abnormality for different mean Z-scores and standard deviations of a normally distributed anthropometric Indicator Standard deviation Mean Z-acore 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00 0.00 0.0 1.2 2.3 3.4 4.4 5.4 6.3 7.2 8.1 8.9 9.7 10.4 11.2 11.9 12.5 13.2 13.8 14.4 15.0 15.6 16.1 0.05 2.0 2.6 3.6 4.6 5.6 6.5 7.4 8.3 9.1 9.9 10.7 11.4 12.1 12.8 13.4 14.1 14.7 15.3 15.9 16.4 16.9 0.10 4.0 4.3 5.1 6.0 6.8 7.7 8.6 9.4 10.2 11.0 11.7 12.4 13.1 13.7 14.4 15.0 15.6 16.2 16.7 17.2 17.8 0.15 6.0 6.1 6.7 7.4 8.2 9.0 9.8 10.6 11.3 12.1 12.8 13.4 14.1 14.7 15.3 15.9 16.5 17.0 17.6 18.1 18.6 0.20 8.0 8.0 8.4 9.0 9.7 10.4 11.1 11.8 12.5 13.2 13.9 14.5 15.1 15.7 16.3 16.9 17.4 18.0 18.5 19.0 19.4 0.25 9.9 9.9 10.2 10.6 11.2 11.8 12.5 13.1 13.8 14.4 15.0 15.6 16.2 16.8 17.3 17.9 18.4 18.9 19.4 19.8 20.3 0.30 11.9 11.8 11.9 12.3 12.7 13.3 13.8 14.4 15.0 15.6 16.2 16.7 17.3 17.8 18.3 18.9 19.3 19.8 20.3 20.7 21.2 0.35 13.9 13.7 13.7 14.0 14.3 14.8 15.3 15.8 16.3 16.8 17.4 17.9 18.4 18.9 19.4 19.9 20.3 20.8 21.2 21.7 22.1 0.40 15.9 15.6 15.6 15.7 16.0 16.3 16.7 17.2 17.6 18.1 18.6 19.1 19.5 20.0 20.5 20.9 21.3 21.8 22.2 22.6 23.0 0.45 17.8 17.5 17.4 17.4 17.6 17.9 18.2 18.6 19.0 19.4 19.8 20.3 20.7 21.1 21.5 22.0 22.4 22.8 23.1 23.5 23.9 0.50 19.7 19.4 19.2 19.2 19.3 19.4 19.7 20.0 20.4 20.7 21.1 21.5 21.9 22.3 22.6 23.0 23.4 23.8 24.1 24.5 24.8 0.55 21.7 21.2 21.0 20.9 20.9 21.0 21.2 21.5 21.8 22.1 22.4 22.7 23.1 23.4 23.8 24.1 24.4 24.8 25.1 25.4 25.8 0.60 23.6 23.1 22.8 22.6 22.6 22.6 22.7 22.9 23.1 23.4 23.7 24.0 24.3 24.6 24.9 25.2 25.5 25.8 26.1 26.4 26.7 0.65 25.5 25.0 24.6 24.4 24.2 24.2 24.3 24.4 24.6 24.8 25.0 25.2 25.5 25.7 26.0 26.3 26.6 26.8 27.1 27.4 27.7 0.70 27.4 26.8 26.4 26.1 25.9 25.8 25.8 25.9 26.0 26.1 26.3 26.5 26.7 26.9 27.2 27.4 27.6 27.9 28.1 28.4 28.6 0.75 29.2 28.6 28.1 27.8 27.6 27.4 27.3 27.3 27.4 27.5 27.6 27.8 27.9 28.1 28.3 28.5 28.7 28.9 29.2 29.4 29.6 0.80 31.1 30.4 29.9 29.5 29.2 29.0 28.9 28.8 28.8 28.9 28.9 29.0 29.2 29.3 29.5 29.6 29.8 30.0 30.2 30.4 30.6 0.85 32.9 32.2 31.6 31.2 30.8 30.6 30.4 30.3 30.2 30.2 30.3 30.3 30.4 30.5 30.6 30.8 30.9 31.1 31.2 31.4 31.5 0.90 34.7 34.0 33.4 32.9 32.5 32.2 31.9 31.8 31.7 31.6 31.6 31.6 31.7 31.7 31.8 31.9 32.0 32.1 32.3 32.4 32.5 0.95 36.5 35.7 35.1 34.5 34.1 33.7 33.5 33.2 33.1 33.0 32.9 32.9 32.9 32.9 33.0 33.0 33.1 33.2 33.3 33.4 33.5 1.00 38.3 37.5 36.8 36.2 35.7 35.3 35.0 34.7 34.5 34.4 34.3 34.2 34.1 34.1 34.1 34.2 34.2 34.3 34.3 34.4 34.5 1.05 40.0 39.2 38.5 37.8 37.3 36.8 36.5 36.2 35.9 35.7 35.6 35.5 35.4 35.3 35.3 35.3 35.3 35.4 35.4 35.4 35.5 1.10 41.8 40.9 40.1 39.4 38.9 38.4 38.0 37.6 37.3 37.1 36.9 36.7 36.6 36.6 36.5 36.5 36.4 36.4 36.4 36.5 36.5 1.15 43.5 42.6 41.7 41.0 40.4 39.9 39.4 39.0 38.7 38.4 38.2 38.0 37.9 37.8 37.7 37.6 37.5 37.5 37.5 37.5 37.5 1.20 45.1 44.2 43.4 42.6 42.0 41.4 40.9 40.5 40.1 39.8 39.5 39.3 39.1 39.0 38.8 38.7 38.7 38.6 38.5 38.5 38.5 1.25 46.8 45.8 45.0 44.2 43.5 42.9 42.4 41.9 41.5 41.1 40.8 40.6 40.3 40.2 40.0 39.9 39.8 39.7 39.6 39.5 39.5 1.30 48.4 47.4 46.5 45.7 45.0 44.4 43.8 43.3 42.8 42.5 42.1 41.8 41.6 41.4 41.2 41.0 40.9 40.7 40.6 40.6 40.5 1.35 50.0 49.0 48.1 47.2 46.5 45.8 45.2 44.7 44.2 43.8 43.4 43.1 42.8 42.5 42.3 42.1 42.0 41.8 41.7 41.6 41.5 1.40 51.6 50.6 49.6 48.7 48.0 47.3 46.6 46.1 45.5 45.1 44.7 44.3 44.0 43.7 43.5 43.2 43.1 42.9 42.7 42.6 42.5 1.45 53.2 52.1 51.1 50.2 49.4 48.7 48.0 47.4 46.9 46.4 46.0 45.6 45.2 44.9 44.6 44.4 44.1 43.9 43.8 43.6 43.5 1.50 54.7 53.6 52.6 51.7 50.8 50.1 49.4 48.8 48.2 47.7 47.2 46.8 46.4 46.1 45.8 45.5 45.2 45.0 44.8 44.6 44.5 1.55 56.2 55.1 54.0 53.1 52.3 51.5 50.7 50.1 49.5 48.9 48.4 48.0 47.6 47.2 46.9 46.6 46.3 46.1 45.8 45.6 45.4 1.60 57.6 56.5 55.5 54.5 53.6 52.8 52.1 51.4 50.8 50.2 49.7 49.2 48.8 48.4 48.0 47.7 47.4 47.1 46.9 46.6 46.4 1.65 59.1 57.9 56.9 55.9 55.0 54.2 53.4 52.7 52.0 51.4 50.9 50.4 49.9 49.5 49.1 48.8 48.4 48.1 47.9 47.6 47.4 1.70 60.5 59.3 58.3 57.3 56.3 55.5 54.7 54.0 53.3 52.7 52.1 51.6 51.1 50.6 50.2 49.8 49.5 49.2 48.9 48.6 48.4 1.75 61.8 60.7 59.6 58.6 57.7 56.8 56.0 55.2 54.5 53.9 53.3 52.7 52.2 51.8 51.3 50.9 50.6 50.2 49.9 49.6 49.3 1.80 63.2 62.0 60.9 59.9 59.0 58.1 57.2 56.5 55.7 55.1 54.5 53.9 53.4 52.9 52.4 52.0 51.6 51.2 50.9 50.6 50.3 1.85 64.5 63.3 62.2 61.2 60.2 59.3 58.5 57.7 57.0 56.3 55.6 55.0 54.5 54.0 53.5 53.0 52.6 52.2 51.9 51.5 51.2 1.90 65.8 64.6 63.5 62.5 61.5 60.6 59.7 58.9 58.1 57.4 56.8 56.2 55.6 55.0 54.5 54.1 53.6 53.2 52.9 52.5 52.2 1.95 67.0 65.9 64.8 63.7 62.7 61.8 60.9 60.1 59.3 58.6 57.9 57.3 56.7 56.1 55.6 55.1 54.7 54.2 53.8 53.5 53.1 2.00 68.3 67.1 66.0 64.9 63.9 63.0 62.1 61.2 60.4 59.7 59.0 58.4 57.7 57.2 56.6 56.1 55.7 55.2 54.8 54.4 54.1 2.05 69.5 68.3 67.2 66.1 65.1 64.1 63.2 62.4 61.6 60.8 60.1 59.4 58.8 58.2 57.7 57.1 56.7 56.2 55.8 55.4 55.0 2.10 70.6 69.4 68.3 67.3 66.2 65.3 64.4 63.5 62.7 61.9 61.2 60.5 59.9 59.3 58.7 58.1 57.6 57.2 56.7 56.3 55.9 2.15 71.8 70.6 69.5 68.4 67.4 66.4 65.5 64.6 63.8 63.0 62.3 61.6 60.9 60.3 59.7 59.1 58.6 58.1 57.7 57.2 56.8 2.20 72.9 71.7 70.6 69.5 68.5 67.5 66.6 65.7 64.8 64.0 63.3 62.6 61.9 61.3 60.7 60.1 59.6 59.1 58.6 58.1 57.7 2.25 73.9 72.8 71.6 70.6 69.5 68.6 67.6 66.7 65.9 65.1 64.3 63.6 62.9 62.3 61.7 61.1 60.5 60.0 59.5 59.0 58.6 2.30 75.0 73.8 72.7 71.6 70.6 69.6 68.7 67.8 66.9 66.1 65.3 64.6 63.9 63.2 62.6 62.0 61.5 60.9 60.4 59.9 59.5 2.35 76.0 74.8 73.7 72.7 71.6 70.6 69.7 68.8 67.9 67.1 66.3 65.6 64.9 64.2 63.6 63.0 62.4 61.8 61.3 60.8 60.3 2.40 77.0 75.8 74.7 73.7 72.6 71.6 70.7 69.8 68.9 68.1 67.3 66.6 65.8 65.2 64.5 63.9 63.3 62.7 62.2 61.7 61.2 2.45 77.9 76.8 75.7 74.6 73.6 72.6 71.7 70.8 69.9 69.1 68.3 67.5 66.8 66.1 65.4 64.8 64.2 63.1 63.6 62.6 62.1 2.50 78.9 77.7 76.7 75.6 74.6 73.6 72.6 71.7 70.8 70.0 69.2 68.4 67.7 67.0 66.3 65.7 65.1 64.5 63.9 63.4 62.9 aries of the reference population distribution, so that allow straightforward calculations based on the all individuals growing outside of this distribution observed mean Z-score and standard deviation. This do so because of environmental constraints. formula makes use of the cumulative distribution A simple mathematical formulation has been function (c.d.f.) of the normal probability curve; thus developed for estimating the standardized prevalence the underlying assumption is that the observed dis- of malnutrition in population studies, which would tribution is nearly normal. The estimated stan- 138 Estimating child malnutrition dardized prevalence (SP) can be obtained by using the following formula: SP = (Z_ aZ2+2c2Ina-2lncr) +4(OZa Z2+2Ina2na) ( >1) where D = c.d.f. of the standardized normal distribu- tion; Z = mean standardized Z-score of the observed population; a = standard deviation of the stan- dardized Z-scores of the observed population. For the special case of a = 1, SP = 200 2 - 1 The calculation of Z-scores of anthropometric indicators usually requires computer facilities and software (e.g., the CDC software package for the analysis of anthropometric data (33)). To facilitate a rapid assessment of prevalence whenever the observed distribution of Z-scores is approximately normal, Table 2 shows the mathematically estimated prevalence rates for increasing negative mean Z- scores (from zero to 2.50) and standard deviations (from 1.00 to 2.00) of the observed distribution. These ranges would cover most, if not all, probable values to be found in anthropometric studies of prevalence. As an example, for a population whose mean Z score and standard deviation for an anthro- pometric indicator are 1.50 and 1.20, respectively, the estimated prevalence of malnutrition would be 50.9 per cent, as indicated by the intersection of the two values in Table 2. Estimates were obtained using a special computer program.b The accuracy of these estimates to reflect the true standardized prevalence rates is obviously con- tingent upon the extent to which both the reference values and the observed Z-scores are normally dis- tributed. In most populations the distribution of height-for-age is approximately normal, whereas those of weight-for-age and weight-for-height are somewhat skewed (2, 8, 10, 27-29). The original NCHS/CDC reference distributions of weight-for- age and weight-for-height were slightly skewed; thus in constructing the normalized NCHS reference tables (18) the population was divided into two halves at the median, and standard deviations were calculated for each half. The frequent skewness in the distributions observed in developing countries may introduce some underestimation in the calculations; however, b Readers interested in this computer program may write to the author. Fig. 2. Differences In prevalence estimations between the method proposed here (Mora) and the Gomez and WHO methods for Increasing dlstance between distribu- tions (Ul - U2) and a standard deviation of 1.0Z. 100 _ 80- ~60- ~40 -GOMEZ MORA 20 ---- WHO 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Ul - U2 exact estimations applying our method to actual data from nutrition surveys in developing countries showed that the magnitude of the error is negligible (under 10% of the total prevalence) compared with the one resulting from the presence of false positives and false negatives when using conventional cut-ofl points. Therefore, distributions of anthropometric indicators can be compared to the normalized NCHS/CDC reference, and standard analytical tests based on the assumption of a normal distribution can be applied to the Z-values so derived (34). Although adjusting for skewness in prevalence esti- mates is theoretically feasible, for practical purposes this would be an unnecessary sophistication. The differences in the prevalence estimations between the method proposed here and the Gomez and WHO methods are graphically shown in Fig. 2, using estimates for increasing negative mean Z- scores with standard deviation of 1.OZ. The largest differences occur between 0.5Z and 2.OZ, which is the range covering most common situations in developing countries, and become negligible above 3.OZ, which is likely to be found only in extreme famine conditions. Discussion The method here proposed for estimating the stan- dardized prevalence of child malnutrition in popu- lation studies provides a useful tool for standardizing 139 J.O. Mora the analysis of anthropometric data from cross- sectional surveys for the assessment of the nutri- tional status in the community. The method yields useful estimates of the population prevalence by comparing the observed and the reference popu- lation distributions of the anthropometric indicator, as suggested by Keller (29), using a cut-off curve rather than a cut-off point. The same method can be used for estimating a standardized prevalence of overweight. For comparison purposes, standard sta- tistical tests can be applied to the mean Z-scores and standard deviations of the observed distributions. Standardized prevalence rates can be obtained from Table 2 for any of the anthropometric indica- tors, provided that it is expressed in terms of Z- scores of the normalized NCHS/CDC reference population and that its distribution is nearly normal and not grossly skewed. Standardized prevalence is defined as the proportion of cases in that portion of the observed distribution which does not overlap the normal reference population distribution, and its estimation is based on the mathematical properties of the normal probability curve, even when the stan- dard deviation of the observed distribution is differ- ent from one (indeed, it is usually greater than one). Our concept of standardized prevalence chal- lenges the traditional epidemiological dogma calling for case definition and counting of individual cases to estimate disease prevalence. Indeed, the case defi- nition approach is impractical when there are no feasible means to individually identify "false posi- tives" and "false negatives" so as to count only the true diseased, as is the case in nutritional anthro- pometry. A different approach for estimating a stan- dardized population prevalence is used which does not look at individual cases but at the whole dis- tribution of groups of individuals. The principles of this "mixed distribution analysis" method have been used for estimating the prevalence of anaemia as the proportion of individuals whose haemoglobin values are shifted downwards relative to a distribution of haemoglobin values of non-anaemic individuals (35, 36). By using this method, cut-off points become irrelevant for prevalence estimates, thus making any further controversy on the topic actually unneces- sary. Cut-off points would remain important for screening purposes, when the aim is setting targets for action rather than estimating prevalence rates; in this case, the best level for screening would be the one that yields just the proportion of individuals for which the resources suffice (3). Cut-off points may also be useful for educational purposes, such as in the charts commonly used in growth monitoring. The proposed method is obviously not applic- able for the individual assessment of nutritional status. As it is totally based on the mathematical properties of the normal probability distribution, its outcome is a standardized population estimate and not an individual diagnosis. In fact, for a given indi- vidual value of an indicator within the area of over- lapping, it will be practically impossible with that information alone to ascertain whether or not it belongs to the reference population; this is also true with the adjustment for false positives suggested by WHO (27). The individual assessment of nutritional status should be based on longitudinal observations of growth (incremental growth, growth curves), com- plemented by clinical and other evaluations. Acknowledgements This work was supported by the Latinamerican Research Company, Latinreco, Nestec Ltd (Switzerland), and Logical Technical Services Corporation (USA). The author gratefully acknowledges the key contri- bution of J. Ricardo Mora in developing the mathematical formulations and obtaining the computer estimations of this paper. The valuable suggestions made by Dr W. Keller (including the calculations for Fig. 2) and Dr J.P. Habicht are also acknowledged. Resume Nouvelle methode pour evaluer la prevalence normallsee de la malnutrition Infantile a partlr des Indicateurs anthropometriques Les mesures et indicateurs anthropometriques servent couramment a evaluer l'etat nutritionnel des individus et des populations. Dans les pays en developpement, on les utilise dans des enqudtes transversales pour evaluer la prevalence de la malnutrition chez les enfants. Les estimations de la prevalence sont g6neralement fond6es sur des seuils conventionnels pour les valeurs de ref6- rence obtenues dans des populations censees etre bien nourries. Toutefois, les divergences d'opinion quant au choix de ces seuils empechent d'utiliser des methodes analytiques normalisees pour cal- culer les taux de prevalence. La correction recom- mand6e par l'OMS pour tenir compte des faux positifs produit des resultats non concordants lorsque la valeur seuil utilisee est differente. Des seuils "flottants" pourraient Otre utilisee a des fins non scientifiques. Une methode normalisee a ete mise au point pour estimer la prevalence de la malnutrition (caracterisee par des indicateurs anthropome- triques inferieurs a la norme) a l'aide de la 140 Estimating child malnutrition formule suivante: SP = MP - FP + FN avec: SP = Prevalence normalisee, definie par la proportion des individus de la population observ6e qui sont en dehors de la distribution normale de la population de reference; MP = Prevalence mesuree, calculee comme etant la proportion d'individus au-dessous d'un seuil donne de la population de reference; FP = Faux positifs, soit la proportion estim6e d'individus que l'on s'attend a trouvbr au-dessous du seuil de la population de ref6rence. FN = Faux negatifs, soit la proportion excedentaire de sujets qui se trouvent au-dessus du seuil dans la population observee par compa- raison avec la distribution de la population de reference. La prevalence estimee est essentiellement fonction de la distance entre la distribution obser- vee et la distribution de reference. Une methode statistique a ete mise au point pour comparer les deux distributions et estimer un taux de preva- lence normalisee, selon la definition ci-dessus. La formulation mathematique fait appel a la fonction de distribution cumulative de la courbe normale de probabilite; elle est fondee sur les ecarts moyens (Z-scores) observes et sur l'ecart-type. L'hypothese fondamentale est que les distributions de la population observee et de la population de reference sont a peu pres normales. Pour faciliter l'evaluation rapide de la preva- lence, on a etabli a I'aide d'un ordinateur une table qui indique les taux de prevalence estimes mathematiquement pour des valeurs negatives croissantes de l'ecart moyen et de l'ecart-type, et qui couvrent les valeurs les plus susceptibles d'etre rencontrees dans les etudes anthro- pometriques de prevalence. Les valeurs estimees sont plus faibles que celles obtenues par la me- thode de la classification de Gomez et plus ele- vees que celles qui resultent de I'application de la methode recommandee par l'OMS, methode qui consiste a fixer le seuil a deux ecarts-types au- dessous de la moyenne de la population de refe- rence. Notre conception de la prevalence normalisee remet en question le dogme epidemiologique tra- ditionnel qui repose sur une d6finition des cas et sur le comptage des cas individuels pour estimer la prevalence d'une maladie. Notre methode ne consiste pas a examiner les cas individuels, mais a appliquer les principes de I'analyse des distribu- tions mixtes et c comparer les distributions en considerant le seuil comme une ligne plutot qu'un point. 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