Statistical preview, World Cup final, New Zealand vs England

Here is a brief statistical preview.

Recent head to head:

In the past 5 years England lead 8-5.
In the past 2 years England lead 6-3.

At Lord's the ball tends to bounce a bit more. As a result it tends to not suit England as much as their other home grounds. It is the only ground that England have a losing record at over the past few years, with 3 wins and 4 losses in their last 6 years.

It is also a ground where scores have been defended quite regularly.

The slope, large straight boundaries and the bounce combine to make a more bowler friendly ground than most in England, but grounds in this world cup have not exactly gone to type.

Adding in times where New Zealand bat first, and where England bowl first, gives the following result:

New Zealand had a clear plan to use the pressure of the situation as a weapon to help them defeat India, and the pressure from playing at home may do the same against England.

The model that I used to build my simulation has England at 69.8%, while New Zealand are at 30.2%. That feels about right too, New Zealand have a realistic chance, but England are certainly favorites.

The bookies have England at 73%, CricViz have England at 68%, New Zealand at 30% and a tie at a fairly high 2%.

The two teams are close enough that nobody can say exactly who will win, but it is a World Cup final - that's exactly how it should be.

World Cup Simulation update - 5 July

Here's the latest update for the world cup simulation. I have New Zealand at 100%, but that's simply due to the probability of Pakistan getting the required run-rate being so low that that possibility never eventuated in the 50000 trials that I used. The probability of Pakistan going through is slightly lower than the probability of someone being shot accidentally by a dog running along a beach while holding a handgun in it's mouth during the next week,

The next graph is the expected points. The simulation has had the correct top 4 from the second match on, however, the expected points and the order of the teams have changed considerably

The top 4 was looking fairly likely from about match number 6 on. There was some excitement from the two upset losses by England, but Pakistan never got beyond 40% on the simulation.

The complete make up of the semi-finalists has not yet been decided, nor has the team in 5th place. Pakistan, Bangladesh and Sri Lanka could all end up 5th. 

Next I looked at the winning probability. This is getting close to the point where it can be calculated analytically without much trouble.

 The next thing to look at is the rankings. A thing to remember here is that it is all relative to Afghanistan, so everybody going up is more an indication that Afghanistan has gone down.

The order that the teams are in here is the same as David Kendix' official rankings order, with one exception - I have India ahead of England, rather than the other way round.

Finally, a little graph to show what Pakistan needs to do to make the semi-finals. They need to keep Bangladesh below the green line.

World Cup simulation update - 1 July

Here's the latest update to the simulation. The first two graphs disagree slightly, and that's because I have two different methods to calculate the expected net run rate. The first one seemed to be slightly more accurate than the second, but there was not a big difference when I tested them. (The margin of victory in cricket matches is actually really difficult to estimate - teams batting second tend to cruise to victory rather than try to win by as big a margin as possible) I decided to use both when doing the calculations. With the first method, New Zealand and India both have a higher than 99.98% probability of going through, while it's 99% for India and 97.7% for New Zealand with the second method. These seem more realistic.

The big thing to notice is the change to England's probability, and how England beating India damaged the chances of both Pakistan and Bangladesh. Pakistan's probability went down by slightly more than Bangladesh's probability because the ranking of India dropped slightly, and Bangladesh need to beat India to get through.

This graph shows expected value - not the most likely value. Those are actually different things. The expected value is the mean of all the expected outcomes. As a result, none of the teams will actually end up with the points that this shows, but they should mostly get close to it.

 It's now looking like there's a roughly 45% chance that net run rate will be a deciding factor in who goes through to the semi-finals.

If Bangladesh beat India (which is admittedly a fairly unlikely outcome), we could then see a situation where Pakistan and Bangladesh are playing for the opportunity to be level on points with New Zealand and India on 11 points. If that is the case, then (in all likelihood) the rained out match between New Zealand and India will have allowed both to progress at the expense of the winner of Pakistan vs Bangladesh.

The most likely semi-finals at this point are Australia vs New Zealand and England vs India, but these are by no means confirmed yet.

In individual matches, England effectively has a higher ranking than that, because teams playing at home get a ranking boost of 0.86 over their opponent. That's why I have England back on top in the next graph:
This one is quite different to what the book-makers have. I have England as favourites, while they have India and Australia both tied for favourite on roughly 30%. They also have Pakistan and Bangladesh at about double the probability that I do.

I used the first net run rate model for the winning probability, but the difference in numbers suggests that the bookies are possibly using a model that is more similar to the second one.

World Cup simulation update - 26 June

Are the wheels falling off?

England have now got a 4 win, 3 loss record, and, with 2 difficult matches coming up, have a genuine chance of not going through to the semi-finals. They are still not relying on other results, but they're getting close to the point where they are.

There's been a significant change, with Australia going up, and England going down. England are now expected to get to 10 points. That might still be enough. But it also might not be.

England's ranking has now dropped well below India's, to the point where the expected probability of England winning against India has dropped by almost 10%. They're still ahead due to home advantage, but the difference is decreasing.

There's about a 15% chance that a tie-breaker (total wins or net run rate) will be required. This may count out Sri Lanka, who have had two rain affected matches, and so will probably be on fewer wins than anyone else with the same number of points.

We see a huge drop in the semi-final probability of England, and a resultant increase in Bangladesh, Pakistan and Sri Lanka. Australia have qualified now, and there are fewer options now for New Zealand to be knocked out also (only 35 out of 50000 trials saw New Zealand miss the semi-finals.)

The decrease in England, and increase in probability of lower ranked teams making the semi-finals has meant that there are a lot more semi-final combinations with more than a 0.5% chance of happening. West Indies vs New Zealand was an epic match in the pool play, and that's now a reasonable possibility for a semi-final. The ICC and Star Sports will be licking their lips at the prospect of the 8th most likely outcome - an India Pakistan semi-final would be absolute ratings gold.

This is the first time that England has dipped below India on the winning probability graph, but it's hard to win the final if you don't get out of the group stage.

World Cup Simulation update 24th June

 Here's the update after the South Africa vs Pakistan match

Firstly, this pushed Pakistan's ranking back above Bangladesh's ranking, although they are both so close that the match between them is now predicted as 50.2% to 49.8%.

 Looking at the expected points, Pakistan have now jumped ahead of Sri Lanka and Bangladesh.

It's looking fairly likely that 5th place will be on 9 or 10 points, while 4th will be on 10, 11 or 12 points.

My simulation only uses net run rate as the tie breaker. Accordingly, there's actually a slightly higher probability of Sri Lanka and Pakistan getting through than this shows, and a slightly lower chance of England and Bangladesh.

It's takes a lot of processor time to improve the simulation, and it's likely to be less than 1% difference, but I might have a go at improving it once we get to the last 5 matches.

England are still the overwhelming favourite to be the 4th team to go through. There were still 41 out of the 50000 trials where New Zealand hadn't made it. So nobody is guaranteed through just yet.

If you have semi-final tickets - this is who you're likely to see.

The probabilities for Bangladesh and Pakistan being so low here are understandable. They both have about a 5% chance of making the semi-final, but, given that they both have about a 1/3 chance of winning each match against the top teams, it gives them a roughly 0.5% chance of winning the tournament from here. However, if Bangladesh, Australia and Pakistan win the next 3 matches, that number will rise.

It's starting to look like England's style that is so effective in series may not be so effective in one off matches. It will be interesting to see if that trend continues.

World Cup Simulation Update, 23 June

Here's the latest outputs from the simulation.

England's loss to Sri Lanka opened the door somewhat, but we can still be fairly confident in who the semi-finalists are.

 England's ranking has gone down, after two losses to fairly ordinary sides.

It's looking like 10 points will be the magic number. Roughly a 10% chance that we'll rely on a tie-breaker.

The average points expected certainly favour England on that count to be in fourth


Accordingly, they have a much higher chance of making it through.

What the likely match ups are. (Teams in alphabetical order, rather than placings)

England are still firm favourites by my model. Home advantage is massive.

World Cup simulation update

The group stage of the World Cup is now roughly half way through, and there are 4 clear favourites to be the semi-finalists.

Afghanistan is the first team to be eliminated (they may have a mathematical possibility, but they don't have a statistical one). At this point, Sri Lanka are not far behind.

The rankings of the teams have remained fairly consistent, suggesting that the extra weighting for world cup matches is about right.

The fact that almost all the teams seem to have gone up is due to them all being relative to Afghanistan. Afghanistan do not seem to be quite as good as they were seeming to be and so they have dropped, but as they are set to 0, it's pushed everyone else up slightly.

The semi-final probability is the most interesting. 

I personally feel that this is underestimating the chances of South Africa, but we will see as the tournament progresses.

The key point on this graph is match 5, where Bangladesh overcame South Africa. If South Africa had won that match, they would be on about 40% and New Zealand and Australia would both be a lot lower.

The simulation also puts out the points for 4th, 5th and the difference between them. This suggests at the moment that there's only a fairly low chance that net run rate will come into play. However, one more rained out match, or a Bangladesh upset of Australia, and this could change dramatically. This makes the expected lines to be 9 points for 5th place, and 11 points for 4th place.

So far of the teams that I've had as favourite to win, 14 out of the 17 have won. Given the probabilities that the models assigned them, that's slightly higher than I would have expected - I would have expected there to have been 4 upsets rather than 3, but it's still telling me that my model is working quite well. That may be due to teams not always playing their best combinations in every match between the world cup, adding extra uncertainty to the results than exist inside a world cup.

It will be interesting to see if it continues to have the same success rate after the cup is finished.

Finally, applying the same system to find the probable winner gets the following results:

England are still favourites, but India are not far behind them.

Preview - Match 6 - England vs Pakistan - Trent Bridge

England and Pakistan return to the scene of a recent run-fest, but this time there is more on the line.

To say that Trent Bridge tends to be batting friendly is like saying that Elton John tends to play the piano. However, the pitches so far have not exactly been typical of the grounds, and this may prove to be another incident of that.

England start as heavy favourites - Bet365 have them at 82%, Google gives them a 79% chance of winning, and my model gives them 84%. But the favourites don't always win ODI matches.

Here's the historical first innings score chart:

Pakistan have a reputation as an unpredictable team, but the reality is that they are one of the more predictable sides. They very rarely beat teams that are better than them, and very rarely lose to teams that they are better than. England should win this one reasonably comfortably.

A simulation to see who will win the World Cup

One of the main purposes of statistics is to help inform decisions. Cricket statistics are often used when deciding on selection of players, or (more often) arguments about who is the best at a particular aspect. They can help decide which strategies are best, what an equivalent score is in a reduced match (with a particular case of Duckworth Lewis Stern) or which teams should automatically qualify for the World Cup (David Kendix). They are often also used by bookmakers (both the reputable, legal variety and the more dubious underworld version) to set odds about who is going to win.

I decided to attempt to build a model to calculate the probability of each team winning, based on their previous form. This was going to allow me (hopefully) to predict the probabilities of each outcome of the world cup, by using a simulation. It didn’t prove to be as easy as I had hoped.

My first thought was to look at each team’s net run rate in each match, adjust for home advantage, and then average it out. That seemed sensible, and the first attempt at doing that looked like it would be perfect. Most teams (all except Zimbabwe) had roughly symmetrical net run rates, and they fitted a normal curve really well. The only problem was that Afghanistan was miles ahead of everyone else. The fact that they had mostly played lower quality opponents in the past 4 years meant that they had recorded a lot more convincing wins than anyone else.

This was clearly a problem. India and England both had negative net run rates, while Afghanistan, Bangladesh and West Indies were all expected to win most of their matches.

I then tried a different approach, based off David Kendix’s approach of using each result to adjust a ranking. But rather than having a ranking that was based off wins, I based it off net run rate. So if a team had an expected net run rate of 0.5, and another had an expected net run rate of 0.6, the first team would have an expected net run rate of -0.1 for their match. If they did better than that, they went up, and if they did worse than that, they went down.

However, I found that some results ended up having too much bearing. If I made it sensitive to a change in the results, it ended up changing way too much based off one big loss/win. England dropped almost a whole net run per over based on the series in the West Indies. So this was clearly not a good option.

Next, I decided to try using logistic regression, and seeing how that turned out. Logistic regression is a way of determining probabilities of events happening if there are only two outcomes. To do that, I removed every tie or match with no result, and set to work building the models.

My initial results were exciting. By just using the team, opposition and home/away status, I was able to predict the results of the previous three world cups quite accurately using the data from the preceding 4 years. (I could not go back further than that, as they included teams making their ODI debut, and there was accordingly no data to use to build the model.

The results were really pleasing. I graphed them here, grouped to the nearest 0.2 (ie the point at 0.6 represents all matches that the model gave between 0.5 and 0.7 as the chance for a team to win), compared to the actual result for that match. It seems that they slightly overstate the chance of an upset (possibly due to upsets being more common outside world cups, where players tend to be rested against smaller nations), but overall they were fairly reliable, and (most importantly) the team that the model predicted would win, generally won.

I could then use this to give a ranking of each team that directly related to their likelihood of winning against each other. The model gave everything in relation to Afghanistan, with the being 0, and any number higher than 0 being how much more likely a team was to win against the same opponent as Afghanistan. (Afghanistan was the reference simply because they were first in the alphabet).

This turns out to be fairly close to the ICC rankings. So that was encouraging.

I tried adding a number of things to the model (ground types, continents, interactions, weighting the more recent matches more highly) but the added complexity did not result in better predictions when I tested them, so I stuck to a fairly simple model, only really controlling for home advantage.

Next I applied the probabilities to every match and found the probabilities of each team making the semi-finals.

The next step was to then extend the simulation past the group stage, and find the winner.

After running through the simulation a few more times, I came out with this:

A couple of points to remember here: every simulation is an estimate. The model is almost certainly going to estimate the probabilities incorrectly, but it will get them close, and they will be close enough to give a good estimate of the actual final probabilities. It is also likely to overstate Bangladesh’s ability due to their incredible home record; overstate Pakistan’s ability as a lot of neutral matches for them they have had a degree of home advantage in UAE; and understate West Indies, due to them having not played their best players in a lot of matches in the past 4 years. But these are not likely to make a massive difference to the semi-finalist predictions.

Given this, I’d suggest that if you are wanting to bet on the winner of the world cup, these are the odds that I would consider fair for each team:

I will try to update these probabilities periodically throughout the world cup, and report on their accuracy.

World-cup quarter finals simulation

After Pakistan's tremendous win over South Africa, and Ireland's remarkable victory over Zimbabwe, the make up of the quarter finals is not really much clearer.

They question as to who is likely to be going through, and who will play whom has been the subject of many, many twitter conversations.

I thought it might be helpful to run a simulation to look at some of the possibilities.

I used Microsoft Excel as it's quite convenient. I used the scores already made in this tournament to decide the probable scores. For each team I got their average rpo scored in relation to the overall group run rate, and their average conceded in relation to the overall. Hence if a team in group A averaged scoring 5.5 rpo and conceded 5.3 rpo, they got values of +0.4 for batting and +0.2 for bowling (as the average rpo in group A has been 5.1 so far). From that point I then used an inverse normal, with a random number between 0 and 1 for the area, the group run rate plus the batting run rate modifier and the other team's bowling run rate modifier as the mean. For the standard deviation, I used the smallest of one third of the mean and 1.6. This allowed me to make sure there was (almost) no chance of a team getting a negative score, but that the scores weren't going to blow out too much.  I used 1.6 as that's the standard deviation of all innings run rates this tournament..  This gave me a 50 over score for each team, and so which ever was ahead got the points for the win.

There are a few limitations with this method. I didn't take into account the quality of the teams that each side had faced. England has played Australia, New Zealand and Sri Lanka, but has yet to play Bangladesh or Afghanistan. Their numbers are not going to necessarily show how well they will do against less fancied opponents. Likewise no adjustments were made for the pitch that the match is being played on. We know that South Africa have tended to favour playing on bouncier tracks, so an innings at the 'Gaba won't necessarily tell us much about how they would go in Dunedin. I also haven't taken into account player strengths. Bangladesh's batsmen tend to struggle against tall bowlers, such as Finn and Woakes. England can expect that those two bowlers will perform better than average against Bangladesh, and hence their team is likely to do better than the numbers would suggest.

Another major limitation is that I haven't made provision for rain. That would obviously throw off all calculations. However, given the limited information I felt that a more simple model was best.

I decided to do 2000 trials, so that I could feel that the major source of uncertainly was the assumptions rather than the natural sampling variability.

First I found the probability of the different teams making the quarter finals with my simulation:

Team	Probabiity
New Zealand	100%
Australia	100%
Sri Lanka	99.95%
Bangladesh	82.51%
England	17.54%
-	-
India	100%
South Africa	100%
Pakistan	74.71%
Ireland	61.82%
West Indies	63.47%

We can see that Pool A has one crucial match (England vs Bangladesh)
Pool B, however, is still wide open. Ireland vs Pakistan is the last game of the round robin, and it's shaping up to potentially be one that has 3 team's fortunes riding on the result.

If West Indies make the final 8, they will almost definitely face New Zealand. It's very unlikely that New Zealand will not end up on top of Pool A, and impossible that West Indies will end up 3rd or higher in pool B.

Here's the full results for all possible matchups

Pool A	Pool B	Probability
New Zealand	Pakistan	14.99%
New Zealand	South Africa	0.35%
New Zealand	Ireland	21.23%
New Zealand	West Indies	63.44%
Australia	India	2.30%
Australia	Pakistan	43.11%
Australia	South Africa	27.57%
Australia	Ireland	27.02%
Sri Lanka	India	18.18%
Sri Lanka	Pakistan	15.83%
Sri Lanka	South Africa	53.75%
Sri Lanka	Ireland	12.19%
Bangladesh	India	64.74%
Bangladesh	Pakistan	0.75%
Bangladesh	South Africa	15.88%
Bangladesh	Ireland	1.15%
England	India	14.79%
England	Pakistan	0.05%
England	South Africa	2.45%
England	Ireland	0.25%

I'll redo this after tomorrow's results, and then again on Monday.

The most likely scenario at the moment is India to play Bangladesh, Australia to play Pakistan, South Africa to play Sri Lanka and New Zealand to play West Indies.

I've updated this here

Winning an ODI from the front

Is Cook the captain to correct England's ODI ship?

There have been a lot of debates recently about the role of the top order in an ODI. Is it more effective to come out swinging or is a more cautious approach more appropriate?

Until 1992 the expectation was that the top order's job in an ODI was to see off the new ball, and scoring at 3 an over was fine. If you look at the great opening bowlers of the 80's you will see that most of them had economy rates near 3.5 rpo. Then in the 1992 World Cup something wonderful happened. In the 10th match, New Zealand were playing South Africa, and one of New Zealand's premier batsmen, John Wright, got injured. In came Mark Greatbatch. Rather than playing the traditional openers role, he took advantage of the fielding restrictions and scored 68 off 60, including hitting Allan Donald back over his head for a 6 that landed on the roof of the stand. From this point onwards, New Zealand's approach changed, and the first 15 overs were seen as the best time to score quick runs.

In the 1995/6 Australian tri-series, Sri Lanka took that tactic to the next level. Kaluwitharana and Jayasuriya batted like whirlwinds and not long later they helped take their team to victory in the World Cup. At this point the world really stood up and took notice. The dashing opener was now the in thing. New Zealand had Astle, Australia had Gilchrist, India used Sehwag, Pakistan opened for a while with Afridi. An attacking opener had become as much a part of the game as using a spin bowler in the "boring middle overs."

But recent changes to the game, such as the different fielding restrictions at the end of the game and 2 new balls have meant that some people have questioned if going for it at the outset is such a good tactic now. Is it a better idea to keep wickets hand and "go harder, later."

The majority of this discussion has come out of England, where the roles of captain Cook, Bell and (before he dropped out of the game) Trott have been coming under increasing scrutiny. Are they batting too slow? Are they putting too much pressure on the players coming in after them?

This led me to have a look at the role of the top 3, in the past 2 years. What I wanted to know was what had the biggest impact on a team winning, the openers batting a long time, them scoring a lot of runs or them scoring at a quick run rate.

The first thing that I did was I got a list of the scores when the second wicket fell in each match in the past 2 years, and the outcomes of the matches.

The next step was to graph it, and see what came out.

First I looked to see if there was a different relationship between the overs taken and runs scored for the first 2 wickets for teams that won and teams that lost.

We can see that there is a fairly strong relationship with both graphs.

I set the intercept of the trend lines to 0, so that the gradients are effectively the run rates.

There are a few noticeable differences.  Firstly the teams that win tend to score at a higher run rate for the first 2 wickets than the teams that lose. However when I overlapped the two graphs, this difference was not as striking visually as it is numerically. (After 40 overs the trend lines are actually only 28 runs apart)

More significantly, there's a lot more data above 15 overs on the winning graph than on the losing graph. There is also a lot more instances on the winning graph where the first two wickets have contributed 150 or more runs.

I ran a quick bootstrap analysis and found that there was a statistically significant difference between both the median number of runs scored at the fall of the second wicket by the winning teams and the number of overs that they batted for.

To highlight that difference, I drew a cumulative frequency graph. The difference in the distribution of the number of overs faced by the winning and losing teams is quite striking.

That gave me some reason to search further.  It seems that there is a statistical evidence to say that there is a difference between the performances of the top 3 batsmen of teams that win and lose.

The next question was to see which made more of an impact: the number of runs, the number of overs or the rate that the runs were scored at.

To do this I ordered the innings by each of these three variables, and then looked at how many of the 9 innings surrounding each one were won and lost. This is not particularly intuitive, but it did give me some idea about the impact an increase in the variables would have on the likelihood of winning.

First I looked at run rate:

There is a trend here, but it's certainly not a strong relationship.

It's clear that increasing the run rate that the first two partnerships score at can contribute to increasing the likelihood of winning.

The important number here is the R² value of 0.35914. The closer to one this value is, the more linear the relationship is. While it is not a perfect measure of how strong a relationship is, it is a good indication.

The next graph that I looked at was the total runs scored.

This is a similar relationship, but it is clearly stronger.

The points are generally closer to the trend line and the R² value is higher (0.44074).

The data (expectedly) thins out as the number of runs increases, as it's quite rare for teams to get to 200 for the loss of only one wicket.

The R² value is less than 0.5, so there's still more of the variation that is unexplained than explained by this relationship, but again that is to be expected, as there are a lot more factors in a game than the first 3 batsmen, and it is always possible for a game to change suddenly.

The total runs scored for the first two partnerships seems so far to be a better predictor of success than the rate that they scored at.

The third factor is the one that I found the most interesting.


The relationship between the overs batted and wins looks (unsurprisingly) quite similar to the relationship between runs and wins.

The R² value for overs is lower than the corresponding value for runs, but both are higher than the relationship with run rates.

The point where the teams are winning more than they are losing are roughly 14 overs, 75 runs and 4.7 rpo respectively. These numbers start to give us an idea about what we should be looking for in an opener.

However, I wasn't totally convinced by these graphs.  I wondered if they would have turned out the same if I had chosen to look at 15 innings or 5 innings, or some other slightly different way of looking at it.  So I decided to try looking at the winning probability for individual points.  To do this I rounded the run rates to the nearest 0.1 rpo, the overs faced to the nearest over and the runs to the nearest 5 runs.

Again these graphs were quite interesting.

Visually, the strength of the relationship is indicated by the degree to which the top of the colours bars get close to trend line.

The runs and overs graphs look like they are a better fit than the run rates graph. But we can get extra evidence for the strength of fit from the R² value again.

This time the R² value for overs was significantly stronger than it was for either the run rate or the total runs.

The lower (green) graph suggests that for every extra over before the second wicket falls, the probability of winning increases by about 1.7%

Likewise, for every extra 5 runs scored, the probability of winning increases by about 1.7%

At this point I started to feel like there was some fairly significant evidence that having a top 3 that can see off the new ball is definitely the way to go.

The rate that the top order score at is important, but it doesn't seem to be as important as the number of overs that they bat for.

This makes sense for a couple of reasons. Firstly, a cricket ball is at it's most hittable when it is about 15 overs old. At this point it's still hard enough to go quickly off the bat, it has normally stopped swinging conventionally and hasn't yet started to swing unorthodoxly. The ball is the hardest to play when it is less than 10 overs old, because the ball will swing and seam, and the edges will go to hand, rather than dying as quickly as they do later on.

When the balls are 15 overs old, the match is 30 overs in.  At this point it's sensible for teams to have their best hitters at the crease.  If they are in too much before the 20th over, the hitters are being exposed to a swinging ball that makes it difficult to time their shots.

The next bit of analysis I did was to look at those 3 marks from above (14 overs, 75 runs and 4.7 rpo) and look at the difference in the results between teams that reached these milestones and teams that did not.

The outcomes were again quite interesting:

Here fast was any innings where the first two partnerships scored at more than 4.7 runs per over, big was where the second wicket fell when the score was over 75 and long was where the wicket fell after the 14th over.

The column titled "Relative" is the relative probability.  This means that teams that have their first two wickets score at more than 4.7 rpo are 37% more likely to win than teams who score slower than that.

Again we see that the biggest advantage is when the first two wickets last more than 14 overs.  It isn't a panacea, but it is important.

The final thing I did was to look at two similar skilled options, with different approaches, and see which was best.  To do this I looked at every innings where the second wicket fell between the 7th and 14th over, and the run rate was between 4.7 and 5.3 (this puts them roughly in the 2nd quartile for scoring rates and the 3rd quartile for length). I then also looked at times where the second wicket fell between the 14th and 19th over, with a run rate between 3.8 and 4.7 (which puts them roughly in the 3rd quartile for scoring rates and the second quartile for length).  Again the more cautious approach paid dividends, although this time with a much smaller sample size.

The more attacking start yielded 9 wins and 13 losses (40.9% winning record.) The more cautious start also yielded 9 wins, but only 8 losses (52.9% winning record). The difference here, however, is not statistically significant, given the low sample size. 

Putting this all together, an ideal opener is a player who averages at least 37.5 (half of 75), averages lasting 42 deliveries (half of 14 overs) and has a strike rate of at least 78.3 (equivalent to 4.7 runs per over).  If they a compromise has to be made on one of these it should be the strike rate, as that's the least important to help the team win.

So, finally, how does Alastair Cook stack up to these criteria?

At the time of writing Alastair Cook averages 37.51 runs from 48 balls at a strike rate of 77.66.

He (just) makes two of the 3 criteria, and the one that he misses out on he is very close to, and is the least important.

England have not been going well in ODI cricket recently, but Cook's batting is not the right thing to blame. The issues are clearly elsewhere.