7 2011 Iss 4 Art 6 DOI 1022021559 04101307

7 2011 iss 4 art 6 doi 1022021559 04101307

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Journal of Quantitative Analysis in Sports, Vol. 7 [2011], Iss. 4, Art. 6 DOI: 10.2202/1559-0410.1307
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factors. The main difference compared to the results from LVM is the order of the first three factors resulting from the identifiability restriction in LVM estimation. Table 4 shows the estimated results for the direct effect of the covariate year on the ten indicators. The first column (labeled with “mean”) displays the estimated direct effects on the point scale, according to model (2.2). These values were originally estimated for a standardized version of the indicator vari- ables and thus had small absolute values. To get an idea of the effects on the point scale, we transformed these effects back to the point scale by multiplying the mean points from Table 1 with the estimated means of the direct effect for each event. The boundaries of 95% credibility intervals (transformed to point scale) are given by the empirical quantiles of the posterior distribution. The interpretation of Table 4 is as follows. For the 100m race, for example, the mean number of points achieved by the athletes decreases by 13.98 per year. The effects on shot put, high jump, discus, and javelin have a positive sign, meaning that in these events the mean performance on point scale has increased over the years. However, these effects cannot be rated as indicator Factor 2 Factor 3 Factor 1 Factor 4 Day 1 M100 0.91 0.19 0.07 -0.07 LJ 0.40 0.62 0.16 0.02 SP 0.09 0.17 0.84 -0.12 HJ 0.05 0.47 0.13 0.02 M400 0.70 0.09 0.02 0.53 Day 2 MH110 0.46 0.38 0.24 0.04 Disc 0.07 0.11 0.81 -0.06 PV 0.10 0.32 0.32 0.10 Jav 0.04 0.15 0.50 0.02 M1500 0.02 0.04 -0.06 0.74 Table 3: Estimated matrix of factor loadings for a classical factor analysis model without covariates. Factor loadings 0 . 29 are printed in bold. Note that order of factors was changed to compare it with Table 2. significant in a Bayesian point of view since their 95% credibility intervals cover zero. A significant point decrease is only estimated for the 100m and 400m events. At first glance, a possible reason for this might be a general decline in the world’s best performances. Yet, the maximum mean number of points of the athletes was achieved in 2001 (7438.7) followed by a steady decrease until 2005 (7332.1), and an increase until 2009 (7410.9). 13 Wimmer et al.: Latent Variable Models for Decathlon Published by De Gruyter, 2011
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indicator mean 2.5% 97.5% Day 1 M100 - 13.98 - 22.14 - 5.56 LJ - 3.29 - 11.26 5.11 SP 3.41 - 2.76 9.57 HJ 3.42 - 3.94 10.73 M400 - 16.30 - 24.61 - 8.55 Day 2 MH110 - 3.60 - 11.44 5.24 Disc 0.71 - 5.12 6.60 PV - 3.16 - 10.56 4.26 Jav 3.93 - 2.21 10.09 M1500 - 6.11 - 12.52 0.52 Table 4: Estimated direct effects and boundaries of 95% credibility intervals for a LVM. All effects were scaled back to the original point scale. Figure 4 shows the non-linear indirect effects of the covariates age and month on each of the four latent factors. For both, Factor 1 representing“sprint abilities” and Factor 2 standing for “jumping abilities”, one can see an increase by age until the early 30’s followed by a slight decrease, see Figures 4(a) and 4(c). For Factor 2, the athletes reach the maximum 2 to 3 years later than for Factor 1. Figure 4(e) shows that Factor 3 for “throwing abilities” is steadily increasing by age. Furthermore, Factor 4 representing “endurance abilities” is
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  • Statistics, De Gruyter, LVMs, Valentin Wimmer, Journal of Quantitative Analysis

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