a total of 1,633 male body weight ( bwt ) and age measurements , for 232 thoroughbred colts , 
 were collected by the japan bloodhorse breeders association ( jbba ) between 2005 and 2009 . 

the data with extreme values over  4 residual standard deviations were preliminary analyzed 
 by using logistic growth curve equation , and removed from this study .
the scatterplot of the 
 weight - age data is shown with gray dots in fig . 

1.scatterplot of the 1,633 male body weight ( bwt ) data of thoroughbred colts ( light 
 gray dots ) .
thin black line indicates expected ( actual ) data averages of bwt , and the 
 thick gray line indicates estimated richards growth curve ( i.e. , equation 2 ) .. in the figure , the tendency of the seasonal compensatory growth can be recognized 
 between about 300 and 600 days , which is the period of compensatory growth .
scatterplot of the 1,633 male body weight ( bwt ) data of thoroughbred colts ( light 
 gray dots ) .
thin black line indicates expected ( actual ) data averages of bwt , and the 
 thick gray line indicates estimated richards growth curve ( i.e. , equation 2 ) .
seven sigmoid growth curve equations ( logistic , gompertz , von bertalanffy , brody , richards , 
 bridges and janoscheck ; see ) were fit to the 
 weight - age data .
the best - fit equation based on akaike s information criterion ( aic ) was the 
 richards equation   as follows : in this  general  richards equation ( equation 1 ) , body weight ( bwt , kg ) is 
 described as a function of age ( t , day ) .
the biological interpretations of 
 the four parameters ( a , b , k and 
 m ) have been discussed previously .
briefly , a is the asymptotic limit for bwt 
 as t approaches infinity ; b is a scaling parameter that 
 defines the degree of maturity ; k is a rate constant that determines the 
 spread of the curve along the time axis ; m is the point of inflection in 
 the sigmoid curve in relation to age .
the estimated richards equation using the weight - age data was as 
 follows : in fig .
1 , the thick gray line indicates the 
 estimated richards growth curve listed on equation 2 , and the thin black line is a plot of 
 the expected ( actual ) data averages of the bwt , where the averages were computed based on 
 the monthly averages of the bwt data and their monthly daily gain .
this thin black line is 
 identical to that published in the japanese feeding standard for horses .
the deviation of the thin black line ( expected data 
 averages ) from the thick gray line ( richards growth curve ) is clear between about 300 and 
 600 days .
the deviation of the expected data averages from the richards growth curve is 
 shown in fig .
2.deviation of the expected ( actual ) data averages of the male body weight ( bwt ) from 
 the estimated richards growth curve ( i.e. , equation 2 ) .
deviation of the expected ( actual ) data averages of the male body weight ( bwt ) from 
 the estimated richards growth curve ( i.e. , equation 2 ) . to adjust for the deviation caused by compensatory growth shown in fig .
2 , a sigmoid sub - function was designed as follows . a general 
 sigmoid function f(t )
is expressed as : where e is the base of the natural logarithm , t is time , 
 where  < t <  , and 0  f(t )  1
. a sigmoid 
 function is a monotonous increase function , and the shapes of the function depend on the  
 values .
the results around 1  t  1 are 
 shown in fig . 3fig .
3.sigmoid curve with two different values of . black and gray lines are with =10.0 
 and =5.0 , respectively .. sigmoid curve with two different values of . black and gray lines are with =10.0 
 and =5.0 , respectively . the subtraction between two sigmoid functions with different  values leads to a function 
 having only a single wave ( down and up wave ) at t=0 . by using this 
 subtraction and application of several modifications around the t 
 parameter , the following sigmoid sub - function f(t )
can be obtained : and the shape of the sub - function is shown in fig . 

4.constructed sigmoid sub - function f(t ) for adjustment of seasonal 
 compensatory growth .. this sub - function curve crosses the x axis at 432 days with the 
 wave length ( i.e. , distance between the starting and ending point of the wave ) of 268.49 * 2 . 

this sub - function has zero value when t =  or . only when 
 t is around 432 days , do the non - zero f(t ) values 
 appear .
this characteristic of the sub - function is like the shape of the deviation shown in 
 fig . 2 especially for the period of compensatory 
 growth .
therefore , it was assumed that the f(t ) sub - function can 
 empirically adjust the deviation caused by compensatory growth . in the original richards 
 equation
( equation 1 ) the maturity related parameter is b , and the newly 
 developed f(t ) sub - function can be used for the adjustment of the 
 b parameter .
in general , the newly developed f(t ) sigmoid sub - function can be added to 
 the b parameter in equation 1 .
specifically , this leads to the replacement 
 of  0.94513  by  0.94513 + *f(t )  in equation 2 where  is a 
 coefficient of f(t ) . with this replacement and combination of equation 2 
 and equation 4 ,
the growth curve equation adjusting for the seasonal compensatory growth was 
 obtained as : the optimal coefficient value of 0.3582 ( i.e. ,  , the coefficient of 
 f(t ) ) and also the optimal wave length of 268.49 were estimated again by 
 using the weight - age data and sas nlmixed procedure .
equation 5 looks complicated but is a 
 single function of age t in days .
the insertion of f(t ) 
 in equation 5 affects the bwt only when t is within about 432  268.49 
 days .
the growth curve of this new richards equation ( equation 5 ) combined with the sigmoid 
 sub - function is shown with thick gray line in fig . 

5.scatterplot of the 1,633 male body weight ( bwt ) data of thoroughbred colts ( light 
 gray dots ) .
thin black line indicates expected ( actual ) data averages of bwt , and 
 thick gray line indicates estimated richards growth curve with the developed sigmoid 
 sub - function ( i.e. , equation 5 ) . , where the black line and dots are the same as in fig .
the shapes 
 of the two lines were almost identical in the period of compensatory growth . by using this 
 approach ,
the aic value decreased from 13,053 ( equation 2 ) to 12,794 ( equation 5 ) , 
 indicating the better fit of equation 5 to the weight - age data than equation 2 .
scatterplot of the 1,633 male body weight ( bwt ) data of thoroughbred colts ( light 
 gray dots ) .
thin black line indicates expected ( actual ) data averages of bwt , and 
 thick gray line indicates estimated richards growth curve with the developed sigmoid 
 sub - function ( i.e. , equation 5 ) .
the deviation of the expected data averages from the new richards growth curve adjusting 
 for compensatory growth is shown in fig .
6.deviation of the expected ( actual ) data averages of the male body weight ( bwt ) from 
 the estimated richards growth curve with the developed sub - function ( i.e. , equation 
 5 ) .. the deviation in the period of compensatory growth is clearly reduced when compared 
 to the case of non - adjusted ( fig .
these 
 results suggest the usefulness of this proposed approach for handling compensatory growth 
 typical in thoroughbred horses .
deviation of the expected ( actual ) data averages of the male body weight ( bwt ) from 
 the estimated richards growth curve with the developed sub - function ( i.e. , equation 
 5 ) .
as richards   noted , some researchers may 
 consider that the parameter b is unimportant biologically .
we can choose an 
 alternative approach to adjust parameter k by the sub - function .
if 
 k was adjusted by the f(t ) sub - function , the combined 
 equation became more complicated and difficult in numerical computation for parameter 
 estimation because parameter k is the one used in the exponential .
we aimed 
 at simplicity for the combined equation , and chose b parameter to be 
 adjusted in the f(t ) sub - function . accounting for compensatory growth in thoroughbreds , staniar et al
.   assumed that their growth rate consists of baseline 
 and systematic deviation components .
even though they did not propose any unique growth 
 curve equations handling compensatory growth , their idea shows a similarity to our approach 
 of using both standard richards function as the baseline and sigmoid sub - function as the 
 systematic deviation .
.   showed a biphasic growth curve equation by combining two different equations 
 with different phases ( i.e. , two different periods of age ) , and this may be an alternative 
 approach to handle the compensatory growth because it has generally two different phases . 

.   proposed their new 
 sigmoidal growth curve functions but these equations seem not to be applicable for handling 
 compensatory growth . concerning developmental orthopaedic disease in horses , donabedian et al . 
 
investigated the relationship between nutrient 
 intake and growth rate , and raub   discussed the 
 making of a durable equine .
the standard growth curve equation handling compensatory growth 
 would be useful for such experiments that relate secure development of thoroughbred horses 
 in winter and spring time . in conclusion ,
the proposed method in this study is one of the useful approaches for 
 adjusting seasonal compensatory growth in growth curve estimations for thoroughbreds . based 
 on this approach
, the optimal growth curve equations can be estimated also for female bwt 
 data of thoroughbreds or another growth traits such as body height or entire width of chest 
 that are considered to be also affected by seasonal compensatory growth .
this approach is easily applicable to the general sigmoidal 
 growth curve equation families such as those having biological parameters ( e.g. 
 a , b and k parameters in richards 
 equation ) .
 thoroughbred horses are seasonal mating animals , raised in northern regions or countries . 
 foals born yearly in spring generally show a typical seasonal compensatory growth pattern , 
 in which their growth rate declines in the first winter and increases in the next spring . 
 in this study , a new empirical adjustment approach is proposed to adjust for this 
 compensatory growth when growth curve equations are estimated , by using 1,633 male body 
 weights of thoroughbreds as an illustrating example . based on general richards growth 
 curve equation , 
 a new growth curve equation was developed and fit to the weight - age data . 
 
 the new growth curve equation had a sigmoid sub - function that can adjust the compensatory 
 growth , combined with the richards biological parameter responsible for the maturity of 
 animals . 
 the unknown parameters included in the equations were estimated by sas nlmixed 
 procedure . 
 the goodness of fit was examined by using akaike s information criterion ( aic ) . 
 
 the aic values decreased from 13,053 ( general richards equation ) to 12,794 ( the newly 
 developed equation ) , indicating the better fit of the new equation to the weight - age data . 
 
 the shape of the growth curve was improved during the period of compensatory growth . 
 the 
 proposed method is one of the useful approaches for adjusting seasonal compensatory growth 
 in growth curve estimations for thoroughbreds , and for their management during the 
 compensatory period . 
 based on this approach , the optimal growth curve equations can be 
 estimated also for female body weight of thoroughbreds or other growth traits affected by 
 seasonal compensatory growth . 
Materials and Methods
Results and Discussion