a total of 5,594 and 5,680 body weight ( bw ; kg ) and age ( day ) measurements of 271 
 thoroughbred colts and 237 thoroughbred fillies , were collected by hidaka training and 
 research center , japan racing association ( jra ) and the japan bloodhorse breeders 
 association ( jbba ) between 1999 and 2009 .
in addition , a total of 3,770 withers height 
 ( wh ; cm ) and age ( day ) measurements of 422 thoroughbred colts and fillies was also 
 collected .
the maximum age in the bw data was about 1,100 days , an age which covers the 
 two winter seasons with cg periods before foals debut in horseracing .
the maximum age in 
 the wh data was about 800 days , which covers only the first winter season with a cg 
 period .
the data collection was mainly organized in the hidaka region of hokkaido , the northern 
 island of japan .
the hidaka region is famous for the intensive production of racehorses , 
 and it has about 815 racehorse stud farms , corresponding to 82% of all japanese racehorse 
 stud farms ( hidaka subprefectural bureau , 2012 ) .
the ground is covered by snow in hidaka 
 in winter , and green pastures only begin to grow in the middle of may . due to the coldness 
 in the winter , the typical cg phenomenon appears in the growth of the thoroughbred foals 
 raised in the hidaka region . for body weights of colt and filly thoroughbreds ,
the following mixed model equations 
 ( equations 1 and 2 ) were used based on the growth curves derived by onoda et 
 al . [ 16 , 17 ] , in which the traditional richards growth curve equation   was modified to incorporate cg effects with sigmoid 
 sub - functions f(t ) and f(t ) .
the sigmoid sub - functions 
 f(t ) and f(t ) adjust the first and second year cgs at 
 432 and 797 ( = 432 + 365 ) days of age , respectively , and were adapted to the biological 
 parameters responsible for maturity in the richards equation . the age
432 days was determined by the crossing points of the 
 actual data averages and the value of the traditional richards equation ( see onoda 
 et al . ) . in the mixed model equations ( equations 1 and 2 ) , both individual horses 
 ( ai ) and residuals ( eij ) for 
 the individual i and age ij
were included as random 
 effects with their variance components , a and 
 e .
the population averages of 
 ai and eij were both zero and 
 they were independent of each other .
the individual weight factor ai , was added to the maturity 
 weight , 575.0 kg .
thus , the maturity weight of each individual was 575.0 + 
 ai , indicating that the final maturity weight differs 
 according to each individual .
[ 16 , 17 ] , the 
 equations for bw for male and female thoroughbreds are : ( equation 2 ) , where , , , and , where bw_maleij and bw_famaleij 
 are the body weights of the individual i measured at age ( time ) 
 tij .
the unknown variance components ( i.e. , 
 a and 
 e ) were estimated for each sex by the sas 
 nlmixed procedure . for wh of thoroughbreds
, the growth curve equation was constructed by following the 
 method of the growth curve constructions for bw .
we combined both sexes in the wh 
 analysis , because the difference of wh between sexes was small in our preliminary analyses 
 ( about 0.5 cm at the most ) , which corresponded to the small gender difference of wh of 2 
 cm at the most , and the total number of data 
 available for wh was small .
furthermore , only the wh data of less than about 800 days 
 which corresponds to the first seasonal cg period was available .
thus , the sub - function 
 f(t ) handling the first cg was only included in the equation ( equation 
 3 ) .
the f(t ) was identical to that used in equations 1 and 2 assuming 
 that the cg affects wh at the same time as bw .
the unknown variance components were also estimated by the sas nlmixed procedure . in the definition of 
percentile , the value of 50 percentile is the median of the data , 
 and not the average . if the data is not normally distributed , the data value of the 50 
 percentile is not the mean of the data , and if the data is normally distributed , the data 
 value of the 50 percentile is identical to the mean of the data . to compute the exact 50 
 percentile curve for body weight , for example ,
numerical medians of the bodyweight data 
 must be computed at all ages , according to kocher and staniar . for the computation of other percentile curves ( e.g. 25 percentile 
 curves ) ,
the same manner of computation is necessary . even though this approach to get 
 exact percentile curves is useful for the illustration of data variations , it is generally 
 difficult to obtain continuous single variable mathematical growth curve function of age 
 ( time ) .
the 
 z - score is a standardized score that indicates how many standard deviations a data point 
 is apart from the population mean , based on the assumption of normal distributions .
based 
 on the z - scores of the estimated variance components , the empirical percentile growth 
 curves are easily constructed .
combined with the equations 1 , 2 and 3 and the information 
 of the estimated variance components and their z - scores , the equations for the empirical 
 percentile growth curves for bw ( equations 4 and 5 ) and wh ( equation 6 ) were constructed 
 as follows : these equations are the continuous single variable function of age ( t ) . 
 in these equations
we 
 assigned the empirical percentile curves for body weight and withers height of 
 thoroughbreds as 3 , 10 , 25 , 50 , 75 , 90 and 97% .
each empirical percentile corresponds to 
 z - scores of 1.881 , 1.282 , 0.675 , 0.000 , 0.675 , 1.282 and 1.881 , respectively .
a z - score 
 of zero corresponds to the empirical 50 percentile curve , the equations 4 and 5 become 
 identical to the equations proposed by onoda et al . .
for the validation of the constructed empirical percentile growth curves , the data 
 percentages between these percentile curves were counted . for the validation analyses , the 
 age data periods were from zero in multiples of 180 days .
the percentages of the data 
 positioned between each percentile curve intervals ( i.e. 0 to 3 , 3 to 10% and so on ) were 
 counted from 0 to each age period ( i.e. 0 to 180 , 360 , 540 , 720 , 900 and all days ) .
a total of 5,594 and 5,680 body weight ( bw ; kg ) and age ( day ) measurements of 271 
 thoroughbred colts and 237 thoroughbred fillies , were collected by hidaka training and 
 research center , japan racing association ( jra ) and the japan bloodhorse breeders 
 association ( jbba ) between 1999 and 2009 .
in addition , a total of 3,770 withers height 
 ( wh ; cm ) and age ( day ) measurements of 422 thoroughbred colts and fillies was also 
 collected .
the maximum age in the bw data was about 1,100 days , an age which covers the 
 two winter seasons with cg periods before foals debut in horseracing .
the maximum age in 
 the wh data was about 800 days , which covers only the first winter season with a cg 
 period .
the data collection was mainly organized in the hidaka region of hokkaido , the northern 
 island of japan .
the hidaka region is famous for the intensive production of racehorses , 
 and it has about 815 racehorse stud farms , corresponding to 82% of all japanese racehorse 
 stud farms ( hidaka subprefectural bureau , 2012 ) .
the ground is covered by snow in hidaka 
 in winter , and green pastures only begin to grow in the middle of may . due to the coldness 
 in the winter , the typical cg phenomenon appears in the growth of the thoroughbred foals 
 raised in the hidaka region .
for body weights of colt and filly thoroughbreds , the following mixed model equations 
 ( equations 1 and 2 ) were used based on the growth curves derived by onoda et 
 al .
[ 16 , 17 ] , in which the traditional richards growth curve equation   was modified to incorporate cg effects with sigmoid 
 sub - functions f(t ) and f(t ) .
the sigmoid sub - functions 
 f(t ) and f(t ) adjust the first and second year cgs at 
 432 and 797 ( = 432 + 365 ) days of age , respectively , and were adapted to the biological 
 parameters responsible for maturity in the richards equation . the age 432 days was determined by the crossing points of the 
 actual data averages and the value of the traditional richards equation ( see onoda 
 et al . ) . in the mixed model equations ( equations 1 and 2 ) , both individual horses 
 ( ai ) and residuals ( eij ) for 
 the individual
i and age ij were included as random 
 effects with their variance components , a and 
 e .
the population averages of 
 ai and eij were both zero and 
 they were independent of each other .
the individual weight factor ai , was added to the maturity 
 weight , 575.0 kg .
thus , the maturity weight of each individual was 575.0 + 
 ai , indicating that the final maturity weight differs 
 according to each individual . based on onoda et al .
[ 16 , 17 ] , the 
 equations for bw for male and female thoroughbreds are : ( equation 2 ) , where , , , and , where bw_maleij and bw_famaleij 
 are the body weights of the individual i measured at age ( time ) 
 tij .
the unknown variance components ( i.e. , 
 a and 
 e ) were estimated for each sex by the sas 
 nlmixed procedure .
for wh of thoroughbreds , the growth curve equation was constructed by following the 
 method of the growth curve constructions for bw .
we combined both sexes in the wh 
 analysis , because the difference of wh between sexes was small in our preliminary analyses 
 ( about 0.5 cm at the most ) , which corresponded to the small gender difference of wh of 2 
 cm at the most , and the total number of data 
 available for wh was small .
furthermore , only the wh data of less than about 800 days 
 which corresponds to the first seasonal cg period was available .
thus , the sub - function 
 f(t ) handling the first cg was only included in the equation ( equation 
 3 ) .
the f(t ) was identical to that used in equations 1 and 2 assuming 
 that the cg affects wh at the same time as bw .
in the definition of  percentile , the value of 50 percentile is the median of the data , 
 and not the average .
if the data is not normally distributed , the data value of the 50 
 percentile is not the mean of the data , and if the data is normally distributed , the data 
 value of the 50 percentile is identical to the mean of the data . to compute the exact 50 
 percentile curve for body weight , for example
, numerical medians of the bodyweight data 
 must be computed at all ages , according to kocher and staniar . for the computation of other percentile curves ( e.g. 25 percentile 
 curves ) ,
the same manner of computation is necessary . even though this approach to get 
 exact percentile curves is useful for the illustration of data variations , it is generally 
 difficult to obtain continuous single variable mathematical growth curve function of age 
 ( time ) .
the 
 z - score is a standardized score that indicates how many standard deviations a data point 
 is apart from the population mean , based on the assumption of normal distributions .
based 
 on the z - scores of the estimated variance components , the empirical percentile growth 
 curves are easily constructed .
combined with the equations 1 , 2 and 3 and the information 
 of the estimated variance components and their z - scores , the equations for the empirical 
 percentile growth curves for bw ( equations 4 and 5 ) and wh ( equation 6 ) were constructed 
 as follows : these equations are the continuous single variable function of age ( t ) . 
 in these equations , z is the z - score that corresponds to an empirical percentile .
we 
 assigned the empirical percentile curves for body weight and withers height of 
 thoroughbreds as 3 , 10 , 25 , 50 , 75 , 90 and 97% .
each empirical percentile corresponds to 
 z - scores of 1.881 , 1.282 , 0.675 , 0.000 , 0.675 , 1.282 and 1.881 , respectively
. a z - score 
 of zero corresponds to the empirical 50 percentile curve , the equations 4 and 5 become 
 identical to the equations proposed by onoda et al . .
for the validation of the constructed empirical percentile growth curves , the data 
 percentages between these percentile curves were counted . for the validation analyses , the 
 age data periods were from zero in multiples of 180 days .
the percentages of the data 
 positioned between each percentile curve intervals ( i.e. 0 to 3 , 3 to 10% and so on ) were 
 counted from 0 to each age period ( i.e. 0 to 180 , 360 , 540 , 720 , 900 and all days ) .
based on equations 1 and 2 , the estimates of the variance components for the body weights 
 of thoroughbred colts are a=1,431.14 and 
 e=191.51 , and those for the body weights of 
 thoroughbred fillies are a=1,431.01 and 
 e=228.95 . for withers heights of both sexes , 
 based on equation 3 ,
. the square root of these values of the 
 estimated variances were introduced to equations 4 , 5 and 6 for the construction of the 
 empirical percentile growth curves .
the scatterplots of the weight- and height - age data and the estimated empirical percentile 
 growth curves are shown in figs . 1 and 2fig . 1.the scatterplot of the weight - age data and the empirical percentile growth curves for 
 colt and filly body weights of japanese thoroughbred horses .
the weight - age data is 
 shown with gray dots and the percentile curves are shown with black lines : from the 
 bottom line , 3 , 10 , 25 , 50 , 75 , 90 and 97 percentile curves are shown .
dashed lines 
 indicate the centers of the first and second cg periods ( 432 and 797 days , 
 respectively).fig .
2.the scatterplot of the height - age data and the empirical percentile growth curves for 
 withers height of japanese thoroughbred horses of both sexes .
the height - age data is 
 shown with gray dots and the percentile curves are shown with black lines : from the 
 bottom line , 3 , 10 , 25 , 50 , 75 , 90 and 97 percentile curves are shown .
dashed line 
 indicates the center of the first cg period ( 432 days ) .. these percentile curves express both the changes of the growth rate in the first and 
 second cg periods , and the distribution of the data in the analyzed population . in these 
 figures
, the tendency of the seasonal cg can be recognized around 432 and 797 days , 
 corresponding to the first and second cg periods , respectively .
our former studies provided 
 clear evidence of the presence of both the first and second cg for body weight , based on 
 model comparison of with and without cg [ 16 , 17 ] . concerning the withers height
, we also confirmed the 
 presence of the cg in the same manner , using a model comparison procedure .
the scatterplot of the weight - age data and the empirical percentile growth curves for 
 colt and filly body weights of japanese thoroughbred horses .
the weight - age data is 
 shown with gray dots and the percentile curves are shown with black lines : from the 
 bottom line , 3 , 10 , 25 , 50 , 75 , 90 and 97 percentile curves are shown .
dashed lines 
 indicate the centers of the first and second cg periods ( 432 and 797 days , 
 respectively ) .
the scatterplot of the height - age data and the empirical percentile growth curves for 
 withers height of japanese thoroughbred horses of both sexes .
the height - age data is 
 shown with gray dots and the percentile curves are shown with black lines : from the 
 bottom line , 3 , 10 , 25 , 50 , 75 , 90 and 97 percentile curves are shown .
dashed line 
 indicates the center of the first cg period ( 432 days ) .
the results of validation analyses for estimated percentile growth curves are shown in 
 tables 1 , 
 2 and 3table 1.data distributions of thoroughbred colts body weights at different periods of 
 ageage ( days)percentile intervals ( % ) 0331010252550507575909097971001800.060.646.0728.5843.4117.074.160.003600.090.555.4127.1040.6520.435.410.375400.070.405.3625.2640.3822.185.700.677200.060.394.9725.1640.0922.335.911.109000.060.414.8525.3639.1522.756.231.16all0.070.394.7425.0139.0122.996.421.29table 2.data distributions of thoroughbred fillies body weights at different periods of 
 ageage ( days)percentile intervals ( % ) 0331010252550507575909097971001800.070.355.1034.7842.8812.923.840.073600.070.777.1234.4241.0413.113.330.105400.070.786.3432.6241.2714.583.850.487200.060.766.3931.5941.4715.153.960.619000.060.766.8731.6340.9114.904.240.65all0.050.087.2231.3740.8114.824.260.69table 3.data distributions of thoroughbred colts and fillies withers heights at 
 different periods of ageage ( days)percentile intervals ( % ) 0331010252550507575909097971001800.000.353.4912.1523.8125.9110.542.233600.000.505.3919.9930.6020.856.751.105400.000.395.0721.7329.9218.646.000.81all0.030.456.0027.2235.8122.337.220.96 for colt and filly body weights and the withers height for both sexes . in 
 these tables , the majority of the data approximated to the over 50% curves at all ages and 
 for all traits .
the growth curve equation models used in this study were based on our former studies [ 16 , 17 ] . in the 
 present study
the data from both jbba and hidaka training and research center were combined 
 in order to estimate the variance components with a larger data set .
interestingly , the 
 estimated variances of the individual horses ( a ) 
 for colt and filly body weights were quite similar to each other in this study ( i.e. 
 1,431.14 vs. 1,431.01 ) .
for the application of this result to other horse populations , 
 however , other studies will be necessary .
slightly higher deviations of the data variation from the estimated percentile growth 
 curves can be recognized ( figs . 1 and 2 ) . in tables 
 1 , 2 and 3 ,
our growth 
 curves considering cg [ 16 , 17 ] were based on accumulated knowledge of japanese thoroughbreds , and 
 the assumptions of a mature weight of 575.0 kg and a mature withers height of 161.0 cm 
 .
the higher deviations of body weight and 
 withers height in the current thoroughbred population suggest the possibility that the body 
 size of recent colts and fillies are larger than those of former japanese thoroughbred 
 populations on the japanese feeding standard for horses .
the potential problems of using the current underestimated percentile curves obtained in 
 this study would relate to misunderstanding of the absolute superiority of the superior 
 growing foals .
in the recent japanese thoroughbred population , the 5075% percentile 
 intervals seem to be ordinary ( see tables 
 1 , 2 and 3 ) .
racehorse managers in the hidaka region should recognize this 
 information when using these underestimated percentile curves for the practical management 
 of their horses .
even though the percentile curves proposed by this study are currently 
 underestimates , the personal growing profile each foal can be precisely obtained and it can 
 be used for the safe feeding management of a foal during cg periods . as more data of the 
 mature weights and heights of
the recent japanese thoroughbred population becomes available , 
 more accurate growth curve models and variances will be estimated . a simple way for the construction of the exact percentile growth curves is to compute the 
 data summarizing statistics ( e.g. ordered data percentages or medians ) at all ages and to 
 plot them on a graph visually , as noted .
this approach gives us percentile graphs that are 
  data dependent , and the obtained graphs tended to have sharp notches ( i.e. not smooth 
 lines ) .
a variety of empirical methods have been used to develop smoothing curves using the 
 actual data plots [ 4 , 7 ] .
methods for smoothing include cubic splines , kernel regression , locally 
 weighted regression and the lambda - mu - sigma ( lms ) method . the lms method is often used for estimation of smoothed percentiles [ 4 , 8 , 22 ] .
these methods are alternatives to deriving other 
 empirical percentile growth curve equations for thoroughbreds .
as discussed by wang and chen 
 , the use of z - scores for the development of 
 empirical percentile growth curves is still useful because of its ability to quantify the 
 extreme data values based on parametric statistical estimations and hypothesis testing . 

based on the assumption of normal distribution of the analyzed data , the estimated empirical 
 percentile growth curves using z - scores give us standardized scales for the analyzed data 
 distributions .
practical control of withers height would be a challenging task compared with the practical 
 control of body weight .
the estimated empirical percentile curves of withers height , 
 however , will still be useful as a reference for the development of young horses when 
 combined with the percentile information of body weight .
the integrated percentile 
 information of many growth traits gives us valuable references for the sound development of 
 young thoroughbred horses .
a useful property of our approach is the introduction of continuous single variable growth 
 curve equations considering cg , which can be easily used to obtain empirical percentile 
 curve equations , as shown in this study .
the developed empirical percentile growth curves 
 using z - scores are computationally feasible and have value for understanding the aspect of 
 body weight and withers height distributions and for managing young thoroughbred horses , 
 especially during compensatory growth periods .
 percentile growth curves are often used as a clinical indicator to evaluate variations of 
 children s growth status . in this study 
 , we propose empirical percentile growth curves 
 using z - scores adapted for japanese thoroughbred horses , with considerations of the 
 seasonal compensatory growth that is a typical characteristic of seasonal breeding 
 animals . 
 we previously developed new growth curve equations for japanese thoroughbreds 
 adjusting for compensatory growth . 
 individual horses and residual effects were included as 
 random effects in the growth curve equation model and their variance components were 
 estimated . 
 based on the z - scores of the estimated variance components , empirical 
 percentile growth curves were constructed . 
 a total of 5,594 and 5,680 body weight and age 
 measurements of male and female thoroughbreds , respectively , and 3,770 withers height and 
 age measurements were used in the analyses . 
 the developed empirical percentile growth 
 curves using z - scores are computationally feasible and useful for monitoring individual 
 growth parameters of body weight and withers height of young thoroughbred horses , 
 especially during compensatory growth periods . 
Materials and Methods
Data description
Growth curves for body weight
Growth curve for withers height
Empirical percentile growth curves using Z-scores
Validation of the estimated percentile growth curves
Results
Discussion