Goal-Kicking Accuracy After Wins and Losses : A Footnote
/Though it wasn't possible to entirely rule out a deterministic component to this phenomenon, a significant proportion of it could be explained, I argued, as simple regression to the mean.
Thinking a little more about the situation, I realised that, were regression to the mean the major cause, we'd expect to see the phenomenon appear most strongly when a team followed a win with a loss, or a loss with a win, and probably disappear entirely when a team won or lost both of the two games we're analysing.
That's the hypothesis we investigate in the following chart, in which four trios of bars are now shown for each season, the leftmost trio providing the information for situations where teams lost the two games being analysed, the rightmost trio where they won the two games, and the two innermost trios the situations where they lost in the second week after winning in the first, or won in the second week after losing in the first.
The pattern we observe here is the pattern we hypothesised, with the phenomenon mostly disappearing for the win/win and lose/lose scenarios, and appearing strongly for the win/lose and lose/win scenarios.
In the previous blog we conditioned only on the outcome of the previous game, which meant that the patterns we saw there were a mixture of two of the four trios of bars seen here. When we conditioned on a previous loss, we'd have been seeing a mix of the first and third trios, and when we conditioned on a win, the second and fourth trios.
So, consider the situation where we're conditioning on a loss in the previous week. The proportion of each first and third trio that we saw will depend on how common it was across all teams for a loss to be followed by another loss or, instead, to be followed by a win. If losses were more commonly followed by another loss, the mixture would contain more of the first trios where conversion rates are virtually identical, and the phenomenon of higher conversion rates conditioned on a previous loss would be weaker.
Conversely, if losses were more commonly followed by wins, the mixture would contain more of the third set of bars where conversion rates are more likely to increase, and the phenomenon of higher conversion rates conditioned on a previous loss would be stronger. It's important to note, however, that a mixture that contains any proportion of the third set of bars will exhibit some tendency for losses to be followed by higher conversion rates.
A similar line or argument can be put forward for the situation where we're conditioning on a previous win.
So, the last piece of the analysis is to look at the relative proportions of loss/loss versus loss/win in the last two games, and of win/win versus win/loss.
What we find is that, even in those years where non-alternating results are relatively more common (eg 2006), it's still the case that losses are followed by wins about 40% of the time, and wins followed by losses about 40% of the time. As a result, we still see compelling evidence of regression to the mean when we condition on a previous loss or a previous win.
Enough.
