Sunday, January 04, 2009


Russ in a post here suggests a way of measuring form (go and read that post, it's full of goodies). The broad question is, if there are two batsmen with the same true talent, and one has better recent form, will that batsman tend to average more in his next innings?

To start with, Russ weights more recent innings more heavily than earlier innings. Specifically, the k-th innings for a batsmen with N innings in his career is weighted 0.95N-k. (This is, I believe, similar to what the ICC rankings do.)

Now, one problem with assessing "form" statistically is that a batsman will usually play a series against one team, followed by a series against another, etc. Since one of those two opposition teams can have a much stronger bowling attack than the other, what may appear to be good form and bad form may simply be a result of playing against weak bowlers and then strong bowlers. So, for everything I do in this post, I'll adjust the batting averages by the quality of the attack, as explained here.

So, when I talk about a regular average, I really mean an adjusted average. When I talk about a weighted average, I'll mean Russ's weighting by how recent the innings was (each innings also being adjusted for the quality of the attack).

Before continuing about form, I'll just look at the weighted average as a predictive tool. For all batsmen with at least 50 innings, I calculated career-to-date averages and weighted averages, as well as a 10-innings moving average. Then, from the 11th innings of each batsman's career, I calculated the absolute difference between his next innings and each of those three measures. (If the innings was a not-out, I used the not-out score as the absolute difference for each measure.) Then I averaged these errors. I did the same for all batsmen, and then found the "average average" error. The regular average was the best predictor, about 1% better than the weighted average, and 4% better than the moving average. The weighted average becomes more accurate if the 0.95 in the formula is increased towards 1, but it is always worse than the regular average.

So, as a measure of true talent of a batsman, I'll use the regular average rather than the weighted average.

Now to the question of defining form. Russ does this by taking a weighted log average. Defining Ri as the runs scored in the i-th innings, and wi as the weight of that innings, this weighted log average is:

/ SUM wi log(Ri) \
exp | ------------- |
\ SUM wi /

I've actually modified this a bit. If the i-th innings is a not-out, I didn't include it in the sum in the denominator. I hope this isn't too great a crime against statistics.

The measure of form is then the ratio of the weighted log average to the weighted average. Now, if scores are distributed exponentially, then this ratio is about 0.56 (well, it is with equally weighted innings at least). If a batsman makes the same score every innings (and gets out!), the ratio is 1. If a batsman recently has one big score and a bunch of little scores, the ratio is down towards 0.3. So, good form is a high ratio, bad form is a low ratio.

Because I exclude not-outs in the denominator, it's possible to get ratios greater than 1. I'm not really sure how to interpret these, but let's carry on anyway.

Russ's hypotheses are (I hope I've got this right):

a) If there are two batsmen with a similar average, one with a typical ratio and one with a low ratio, then the one with the low ratio will tend to average more in his next innings. The logic here is that the batsman with the low ratio is capable of larger scores, whereas the other batsman is just not so good.

b) Given two batsmen with the same average, one with a high ratio will tend to do better in his next innings than one with a typical ratio.

Both of these are correct, somewhat to my surprise. I went through all batsmen, and for each innings (after the tenth in their career), calculated the career-to-date average, and the ratio-to-date, and binned them as in the table below. I then calculated the overall average for each bin.

Ratios are down the left-hand side, averages across the top. The figures are the low end of the bin. So, e.g., the '5' means that the bin is for averages 5 to 9.99, the '10' is for averages 10-14.99, etc. Only bins with at least 50 innings are shown; bold is used for at least 100 innings.

r/a 5 10 15 20 25 30 35 40 45 50 55
0.35 23.4 28.4 28.7 40.8 41.3
0.40 12.6 18.8 21.4 31.7 30.5 34.9 40.3 40.7 55.3
0.45 7.4 12.1 15.9 20.9 26.7 30.1 34.0 46.9 39.9 50.2
0.50 9.6 11.1 17.6 22.2 26.4 31.7 36.3 34.9 43.5 46.0
0.55 8.4 11.9 16.5 21.0 26.1 31.6 35.2 40.1 42.6 53.3
0.60 7.6 12.2 18.3 24.4 27.9 33.5 38.2 42.7 46.6 58.0 40.4
0.65 7.8 12.3 18.1 24.7 28.0 33.4 39.2 43.3 48.7 48.1 46.6
0.70 8.2 12.4 19.3 25.7 27.5 34.4 40.4 46.1 44.0 56.7
0.75 9.3 15.3 17.4 23.1 29.3 38.9 41.0 47.4 51.8 50.1
0.80 9.8 12.0 17.7 24.4 30.5 35.1 48.2 54.2 51.8 58.9
0.85 15.0 22.1 25.8 38.8 40.5 44.6 51.3 61.9 46.8
0.90 15.6 26.7 42.1 42.6 59.2 54.1
0.95 46.5
1.00 16.6 35.9 32.5 44.7 53.9 68.5 73.2 62.9

When the ratio is very low, the batsman does indeed tend to average much more in his next innings. (Since I've used regular averages to define the true talent, the top row may be full of players early in their career. I'm not sure.) Going down each column, the minimum is usually somewhere around 0.5 to 0.6, which seems to correspond to the 0.56 that you'd expect from the exponential distribution.

Really good recent form seems to give a 20% boost and sometimes more. This is a lot more than I had expected.

(My thinking on this issue seems to have been confused — in my last post I said that Johnson was good because he kept getting starts, which is consistent with this analysis.)

Great post.

Johnson today has justified your faith.

I'm grappling with some of this (my weak and unattuned brain rather than the analysis), but is it possible to arrive at an average length of time that a form 'streak' is likely to last? I wonder how cyclical it is.
David, thanks for doing this. I've been pondering the results for a few days, because I am - and was when I wrote my post - confused about the overlap of competing trends.

Firstly though, a clarification, is the form ration in the table weighted towards current form, or the ratio of the player to date?

Secondly, on averages, I am not surprised to see the regular average performed better at predicting the next innings. It carries more information, while the weighted one is affected by shifts in form. The only query I have over it, is whether the regular average loses value towards the end of a player's career, or is affected by apparent shifts in a player's true talent. To the extent that a weighted/moving average and career to date average measure different things, I suspect the table would look slightly different.

I have no issue with how you handled not outs. I didn't do that because it was technically easier to just average them, and because I doubt think it makes a lot of difference.

Thirdly, on the hypotheses. In some ways it is only one hypotheses: that a player doing well/badly should do well/badly in the next game relative to their true talent. However, relative to their recent average they will, in general, regress back to their true talent. And that the form ratio gives a way of differentiating players with similar recent averages but different form.

Going back to predictions, part of the reason the recent average gives inconclusive results in terms of next innings averages, is because it mixes form and average together. This, appears to distinguish them, though I'm slightly surprised by the size of the effect amongst players with poor form.

Addressing OB's question, I am curious what the form ratios are for each of the bins in your table? The reason being, that, if form is persistent, then we should see numbers similar to the left column, but slightly regressed towards 0.56. The amount of regression would give an indication of the length of a form cycle.

The ratio I used was weighted towards recent form.

Next time I am at home and have some time to spare, I'll look at how the career average and weighted averages predict the next innings late in a career. Of course there will be a selection effect going on - players are usually dropped after making low scores.

I don't have birth dates in my database, but I should be able to get them off Statsguru, maybe I'll use them and do the comparison for, say, batsmen past the age of 35 or something.

I am curious what the form ratios are for each of the bins in your table?
It took me a while to work out what you're saying here, but that's a good idea. I'll add that to the to-do list... probably for the weekend.

Though I'll play around a bit to tidy up the 0.56 value - my guess is that it should be lower than that for a typical distribution of cricket scores, rather than exponentially distributed scores.
Can you do a comparison between Hayden and Harvey?

My opinion says Harvey is better
On my adjusted averages, Harvey's at 48.8, Hayden 43.6.

Easy win to Harvs. I will leave a similar comment at your blog.
Your formula must be correct then!
I never thought Cricket & Mathematics could go together but your post changes things.

Hello dear blogging friend,

Cricket with balls has now moved, true story.

We are now at, so if you could update our address in your blogroll that would be great.

Ofcourse if you already had us down as .com, never mind.

Cool post.

Did you look at the individual residuals here?
Just thinking that some sort of autoregressive procedure (with lags) might give you even cooler results.

Also, you allude to true talent levels in this post. I am working on a true talent index for ODI batsmen at the moment and just making sure that that has not been done already!
Hey Professor. I didn't look at anything at an individual level, my feeling being that it would take a lot of effort to work out how much is noise.

No-one has come up with what I would call a "good" true talent level for batting in limited-overs cricket. I made a tentative first step here, which at least gives the framework of how I'd go about it - you need to work out how average and strike rate fit together.

Only rather than generating exponentially-distributed scores as in that post, doing this properly would require a large number of ball-by-ball simulations to get the final scores to roughly mach reality. Then you can take a batsman's average and strike rate, put him in a team of "typical" international batsman and see how much better the new team does compared to a team of "typical" batsmen.

There would still be wrinkles, most notably that the number six batsman in a weaker side will come in earlier in the innings than his opposite number in a stronger side, and therefore will need to bat differently. But that sort of thing is a lesser problem, can be ironed out using fall-of-wicket data, etc.

That is a rather detailed project. All the other things I've seen are arbitrary guesses as what is good. Multiplying average and strike rate must be a decent short-cut to use in the meantime before someone does a thorough job.

I see from your blog that you have the right attitude - follow the baseball model of openness!
David, you could probably achieve a similar effect to what you are suggesting there for ODI analysis by the modifying the Duckworth/Lewis par score. It effectively already takes into account wickets and run-rate, as well as variations in the total during chases.

That is, each ball the batsman faces adjusts the expected score (in the first innings) and the par score (in the second innings). The accumulated changes should reflect the ability of the batsmen and bowlers, taking into account the state of the game.
That's a good idea Russ. I recall someone (in NZ?) writing a Masters thesis on rating ODI players based on DL tables like that, but I don't have the thesis on this computer.

It should be online somewhere.
David, found it: The Best Batsmen and Bowlers in One-Day Cricket by David Beaudoin.

He calculates an average by dividing the runs scored for/against by the D/L resources used. I'm not entirely sure how different that is to what I suggested. It doesn't account for differences between innings, but suggests that they should be analysed separately.
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