# COMPUTATION OF CENTILES AND Z-SCORES FOR HEIGHT-FOR-AGE, WEIGHT-FOR-AGE AND BMI-FOR-AGE

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1 COMPUTATION OF CENTILES AND Z-SCORES FOR HEIGHT-FOR-AGE, WEIGHT-FOR-AGE AND BMI-FOR-AGE The mehod used o consruc he 2007 WHO references relied on GAMLSS wih he Box-Cox power exponenial disribuion (Rigby and Sasinopoulos, However, he final seleced models simplied o he LMS model (Cole and Green, 992 since none of he references required adjusmen for kurosis. As a resul, he compuaion of perceniles and -scores for all hree icaors uses formulae based on he LMS mehod. However, a resricion was imposed on all icaors o enable he derivaion of perceniles only wihin he inerval corresponding o - scores beween -3 and 3. The underlying reasoning is ha perceniles beyond ±3 SD are invarian o changes in equivalen -scores. The loss accruing o his resricion is small since he inclusion range corresponds o he 0.35h o h perceniles. For all icaors, he abulaed fied values of Box-Cox power, median and coefficien of variaion corresponding o age (or heigh are denoed by, M( and, respecively. Ceniles and -scores for heigh-for-age For his icaor, is equal o, simplying he Box-Cox normal disribuion used in he LMS mehod o he normal disribuion. Therefore, dferences beween adjacen sandard deviaions (e.g. beween 2 SD and 3 SD were consan for a specic age bu varied a dferen ages. In his case, he ceniles a age can be esimaed from: C 00 [ + Z ] M ( [ + Z ] ( M ( M ( + SDev( Z, 3 Z where Z is he normal equivalen deviae for ail area, C00( is he 00-h cenile, and SDev( is he sandard deviaion a age (derived from muliplying by M(. The ividual -score for a measuremen y a age was compued as: 3 y M ( y M ( SDev( Page of 5

2 Ceniles and -scores for weigh-for-age and BMI-for-age The weigh-based icaors presened righ-skewed disribuions. When modelled correcly, righ skewness in daa has he effec of making disances beween posiive -scores increase progressively he farher away hey are from he median, while disances beween negaive - scores decrease progressively. The LMS mehod fis skewed daa adequaely by using a Box-Cox normal disribuion, which follows he empirical daa closely. The drawback, however, is ha he ouer ails of he disribuion are highly affeced by exreme daa poins even only very few (e.g. less han %. Following he same mehodology applied o he WHO Child Growh Sandards, a resriced applicaion of he LMS mehod was hus used for he 2007 WHO weigh-based icaors, limiing he Box-Cox normal disribuion o he inerval corresponding o -scores where empirical daa were available (i.e. beween -3 SD and 3 SD. Beyond hese limis, he sandard deviaion a each age was fixed o he disance beween ±2 SD and ±3 SD, respecively. This approach avoids making assumpions abou he disribuion of daa beyond he limis of he observed values (WHO Mulicenre Growh Reference Sudy Group, As a resul of his adjusmen, he -score disribuion can depar slighly from normaliy a he exreme ails (beyond ±3 SD, alhough he expeced pracical impac is minimal. The ceniles were calculaed as follows: [ + Z ] L (, 3 Z 3 C ( ( 00 M The following procedure is recommended o calculae a -score for an ividual child wih measuremen y a age :. Calculae 2. Compue he final -score ( y M ( of he child for ha icaor as: y SD3pos 3 + SD23pos y SD3neg 3 + SD23neg 3 > 3 < 3 Page 2 of 5

3 where SD3 pos is he cu-off 3 SD calculaed a by he LMS mehod: [ (3 ] SD 3pos M ( + ; SD3 neg is he cu-off -3 SD calculaed a by he LMS mehod: [ + ( 3 ] SD3neg M ( ; SD23 pos is he dference beween he cu-offs 3 SD and 2 SD calculaed a by he LMS mehod: [ + (3] M ( [ (2] SD 23pos M ( + ; and SD23neg mehod: is he dference beween he cu-offs -2 SD and -3 SD calculaed a by he LMS [ + ( 2 ] M ( [ + ( 3 ] SD23neg M ( Page 3 of 5

4 To illusrae he procedure, an example wih BMI-for-age for boys is provided below and displayed in Figure x Child 2 Body Mass Index (kg/m² x Child x Child Age (years Figure Examples of children/adolescens ranked according o he 2007 WHO BMI-forage reference Child : year-old boy wih BMI30. L-.7862; M6.9392; S0.070; ( [ 30.0 ] (.7862 >3 [ + ( (3] ( [ + ( (2] ( SD 3 pos SD 2 pos SD 23 pos Child 2: 6 year-old boy wih BMI4. L-.3529; M20.495; S0.2579; Page 4 of 5

5 ( [ 4.0 ] < (.3529 [ + ( ( 2 ] ( [ + ( ( 3 ] ( SD2 neg SD 3 neg SD 23 neg Child 3: 9 year-old boy wih BMI9 L-.638; M6.0490; S0.0038; ( [ 9 ] and 3 (LMS -score (.638 Bibliography Cole TJ, Green PJ (992. Smoohing reference cenile curves: he LMS mehod and penalied likelihood. Saisics in Medicine, : Rigby RA, Sasinopoulos DM (2004. Smooh cenile curves for skew and kuroic daa modelled using he Box-Cox power exponenial disribuion. Saisics in Medicine, 23: WHO Mulicenre Growh Reference Sudy Group (2006. WHO Child Growh Sandards: Lengh/heigh-for-age, weigh-for-age, weigh-for-lengh, weigh-for-heigh and body mass ex-for-age: Mehods and developmen. Geneva: World Healh Organiaion; pp 32. Page 5 of 5

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