An athletes overall throwing ability cannot be me2 Journal of Quantitative

An athletes overall throwing ability cannot be me2

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“throwing ability”. An athlete’s overall throwing ability cannot be measured 2 Journal of Quantitative Analysis in Sports, Vol. 7 [2011], Iss. 4, Art. 6 DOI: 10.2202/1559-0410.1307
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individual level in “throwing ability”. Assuming latent factors also makes sense from the statistical point of view. The point performances in the ten events form a multivariate response consisting of ten continuous, correlated response variables. The underlying correlation structure can be modeled by latent factors, which also involve the advantage of reducing the data complexity from ten variables to, e.g., four latent factors. Using latent factor models in order to describe underlying processes is a widely-used approach in many statistical applications, see, e.g., Skrondal and Rabe-Hesketh (2004). However, classical factor analysis neglects any additional covariate in- formation. Its limitations become apparent when relationships between latent factors and covariates are of interest or when responses or latent factors have to be adjusted for covariates. In our subsequent analysis, the age of an athlete as well as year and month of the competition will be treated as continuous covariates, and their relationship to decathlon performance will be of interest. At first glance one would think of multivariate regression to handle this task. Yet, multivariate regression would, in turn, not allow for latent factors. To sum up, semi-parametric LVMs involve the advantages of (i) tak- ing the correlation structure of a multivariate response variable into account by assuming latent factors, (ii) estimating the effects of covariates, both on the response variables directly and on the latent factors in a non-linear way, (iii) providing a clear and interpretable structure by introducing two separate regression models for response variables and latent factors. In Section 3, we apply semi-parametric LVMs to the decathlon data set, and thereby detect four latent factors standing for sprint, jumping, throw- ing, and endurance abilities. We also obtain interesting results for covariate effects. For example, the estimated non-linear effects of age on the latent fac- tors indicate that older athletes still perform well in comparison to younger athletes. 2 Latent Variable Models Latent variable models (LVMs) are applied to the same type of data structures as in multivariate regression: For each observation i within a sample of size n , the data consists of a p -dimensional vector y i = ( y i 1 ,..., y ip ) 0 of observed values of response variables y 1 ,..., y p and a vector z i of covariates. In our application, the index i corresponds to an athlete’s performance in a certain competition and the response variables are the points y i 1 ,..., y i 10 he obtained in the ten track and field events, beginning with the 100m race and ending directly but the results in shot put, discus, and javelin indicate the athlete’s 3 Wimmer et al.: Latent Variable Models for Decathlon Published by De Gruyter, 2011
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with the 1500m race. Covariates are the age of an athlete, and the date (year and month only) when the competition took place. In contrast to multivariate
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  • Statistics, De Gruyter, LVMs, Valentin Wimmer, Journal of Quantitative Analysis

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