Combining event scores to estimate the ability of competitors

Simulation was used to investigate the validities of nine measures of ability derived from scores of two or more competitive events. The measures were: raw means and least-squares means of raw scores, z scores, and normal scores; two measures derived from ranked scores; and the "personal-best" raw score. Simulations were performed for different numbers of competitors, events, and event entries, each for a range of validity of performance in a single event. A complete set of simulations was repeated for each of the following conditions: normal distribution of competitors' ability; skewed distribution of ability; event validity related to ability; validity, ability, and spread of scores differing between events; and events differing in difficulty. The raw mean of raw scores was generally the most valid measure. The personal best was comparable to the mean only when the number of entries approached one per competitor. The least-squares mean of raw scores had highest validity when events differed substantially in difficulty; it should therefore be used when events differ in length, or when event scores are affected by environmental conditions, judging bias, or by uneven matching of competitors in match-play sports.