Evaluation of Hungarian showjumping results using different measurement variables

Many criteria have been proposed to appreciate the individual performance of horses in jumping competition. As no objective metric scale exists to express the horse’s performance (Hassenstein et al., 1998), the measurement of competition performance is complicated (Bruns, 1981; Tavernier, 1990). Most used criteria are based on transformations of the ranking, earnings or grading scores of the horses (table 1). These quantitative traits can be considered as repeating measurement during the career of the horse. Many complex traits studied in genetics have markedly non-normal distributions (Micceri, 1989; Allison et al., 1999), this often implies that the assumption of normally distributed residuals has been violated (Beasley et al., 2009). However, the validity of many statistical tests depends on the assumption that residuals from a fitted model are normally distributed (Berry, 1993). The residual maximum likelihood (REML) estimation introduced by Patterson and Thompson (1971) has been developed for estimating variance components in linear mixed models (O’Neill, 2010a), which method require the normality prerequisite mentioned above (Oehlert, 2012). Genetic parameters like heritability and repeatability values are derived from estimated variance components. The aim of the study was to compare different fitted models for show-jumping results of sport horses and to estimate heritability and repeatability value.


INTRODUCTION
Many criteria have been proposed to appreciate the individual performance of horses in jumping competition.As no objective metric scale exists to express the horse's performance (Hassenstein et al., 1998), the measurement of competition performance is complicated (Bruns, 1981;Tavernier, 1990).Most used criteria are based on transformations of the ranking, earnings or grading scores of the horses (table 1).These quantitative traits can be considered as repeating measurement during the career of the horse.
Many complex traits studied in genetics have markedly non-normal distributions (Micceri, 1989;Allison et al., 1999), this often implies that the assumption of normally distributed residuals has been violated (Beasley et al., 2009).However, the validity of many statistical tests depends on the assumption that residuals from a fitted model are normally distributed (Berry, 1993).
The residual maximum likelihood (REML) estimation introduced by Patterson and Thompson (1971) has been developed for estimating variance components in linear mixed models (O'Neill, 2010a), which method require the normality prerequisite mentioned above (Oehlert, 2012).Genetic parameters like heritability and repeatability values are derived from estimated variance components.
The aim of the study was to compare different fitted models for show-jumping results of sport horses and to estimate heritability and repeatability value.

MATERIAL AND METHODS
Show-jumping competition results collected between 1996 and 2011 were analyzed.The data used in this study were obtained from the Hungarian Equestrian Federation.The final dataset contained in total 358 342 competition records on 10 199 individual horses after data screening, results were gathered from Hungary and other European countries.Identity number, name and sex of the horse, rider, competition year, the level and location of the competition and placing were recorded in the dataset.Information about pedigree of horses were gathered and set up with help of the National Horse Breeder Information System.The pedigree file contained 39878 animals four generation back.
Competitions were categorized into five groups based on their difficulty level.For the evaluation of show-jumping performance, scores were created using transformations of placing and number of starters.Repeatability animal model proposed by Mrode (1996) was fitted for the traits.The model y ijklmnop = µ + Age i + Gender j + year k + Place l + Level m + Rider n + Perm o + Animal o + e ijklmnop was used for traits without level-transformation (not weighted with the difficulty level of competition), and y ijklmnop = µ + Age i + Gender j + year k + Place l + Rider n + Perm o + Animal o + e ijklmnop for traits with level-transformation (weighted with the difficulty level of competition), where y ijklmnop , y ijklmnop = the score value representing the performance in a particular trait µ = population mean, Age i = fix effect of age, Gender j = fix effect of gender, year k = fix effect of year of competition, Place l = fix effect of place of competition, Level m = fix effect of difficulty level of competition, Rider n = random effects of the rider, Perm o = permanent environmental effect, Animal o = additiv genetic effect of the animal, e ijklmnop = random residual effect.
The given repeatability animal models utilizes all relationships between horses in pedigree during the genetic evaluation.The level of significance was determined using SAS PROC GLM (SAS Institute, 1999) for each fixed effect.
Traits for evaluation of show-jumping performance are shown in Table 2.As the square root function is strictly monotonic, the transformed rank at finish was subtracted from constant 15, thus horses with better placing received higher scores.The constant value can be defined that the final score will be non-negative value (Bugislaus et al., 2005).
Traits based on rank at finish do not take into account the number of starters, thus traits which depend on placing and number of competing horses, were also investigated.The most commonly used rank-based inverse normal transformation entails creating a modified rank variable and then computing a new transformed value (Beasley et al., 2009) of the phenotype for the i-th subject where y i J = the phenotype for the i-th subject, R i = the ordinary rank of the i-th case (rank at finish), N = number of observations (number of starters), Ø -1 = the standard normal probit function, c = a constant value.Tukey (1962) proposed the value 1/3 for c, van der Waerden (1952) suggested c=0.The Tukey and Waerden scores take into account not only the placing of the horse but the number of starters also.The Hungarian grading scores (Díjugrató Szabályzat, 2012) based on rank at finish and number of starters was also included in our investigation.
The traits in 2 nd , 4 th , 6 th , and 8 th measurement were performed with level-transformation, because results of different horses can be compared within a level.Recordings at higher levels need to be upgraded (Ducro, 2011).As placings do not reflect the level at which the result has been obtained, an alternative way for transformation of performance measurement traits is using different weights for different difficulty levels.In this way if two horses obtained the same placing, the horse competing at higher level will receive higher scores.Other option can be performance at different levels can be considered as different traits and analyze in a multivariate analysis (Hassenstein et al., 1998;Huizinga and van der Meij, 1989;Aldridge et al., 2000).
The goodness-of-fit of the models was assessed by using coefficient of determination.Variance components and standard errors were estimated with a repeatability animal model (mentioned before) using the REML method with VCE-6 (Kovac and Groeneveld, 2003) software package.Genetic parameters were predicted from the estimated variance components.
Heritability value (h 2 ): where σ 2 a is the additive genetic variance, σ 2 p is the permanent environmental variance, and σ 2 e is the residual variance.

Repeatability value (R):
where σ 2 a is the additive genetic variance, σ 2 p is the permanent environmental variance, and σ 2 e is the residual variance.

RESULTS
Fixed effects were significant in all fitted models (table 2).The goodness-of-fit in case of the 1 st , 3 rd , 4 th , 5 th and 7 th measurement was low R 2 = 0.07-0.18;while 2 nd , 6 th and 8 th measurements had moderate R 2 = 0.45-0.47value.The goodness-of-fit values were higher in case of weighted measurement variables, where leveltransformation was used.
The Kolmogorov-Smirnov normality test resulted the distribution of the residuals follow normal distribution in all traits (P<0.01).Model assumptions can be checked using histograms of residuals (O'Neill, 2010b).The distributions of residuals are demonstrated in figure 1.
Estimated heritability and repeatability values are represented in table 3. Heritabilities are significantly different from zero, and low 0.01-0.07.The low heritability values show the high impact of various non-genetic (environmental) effects on the showjumping competition performance.The biggest values of heritability and repeatability were in the 2 nd , 6 th and 8 th measurement.

Traits used for evaluation of show-jumping performance and significance level of fixed effects T r a i t ( 1 )
A g e ( 2) G e n d e r ( 3 ) Year( 4) Place( 5) Level(6) R 2 (7) 15 -square root of placings (1 st measurement)( 8  Measuring show-jumping performance as normalized scores, Janssens et al. (1997)  Considering performance at different competition level as different traits, estimated repeatability based on square root of ranking was R=0.14-0.21 in Hassenstein et al. (1998), estimated repeatability based on absolute ranking was R=0.09 in Meinardus (1988).

CONCLUSIONS
During the measurement of show-jumping performance with different traits, it is worth to use competition level as weighting factors.Fitting models for weighted scores had better goodness-of-fit value.The best goodness-of-fit value were in case of weighted square root, weighted Tukey and weighted Waerden transformation, the biggest heritability and repeatability values were estimated in these models also.Estimated heritability for showjumping performance traits were low (h 2 =0.01-0.07)for each measurement variable.The repeatability values were more favourable R=0.09-0.30.The best measure of show-jumping performance was the weighted square root transformation of placing.

Figure 1 :
Figure 1: Distribution of the residuals