An octopus named Paul has been making the news due to his alleged ability to correctly predict the winner of Germany's international football matches. It started with Euro 2008, where he supposedly called 4 of Germany's 6 games correctly. The BBC reported this as "nearly 70%", which is perhaps being a little generous, as a 70% success rate sounds rather more impressive than correctly making 4 out of 6 50/50 guess.
For the World Cup Paul has (apparently) correctly picked the results of Germany's 5 games up until tonight's semi-final, where he has controversially chosen Spain to triumph. So is Paul a Predicting Phenomenon, or just lucky?
We'll start with his World Cup picks where he's got 5 out of 5 right (so far). Our null hypothesis is that Paul is merely picking at random, and since each pick is a 50/50 choice this is the same as saying the probability he picks correctly is 0.5. The probability of getting 5 correct selections is then the same as tossing a coin 5 times and getting 5 heads. This is easy to compute, as we just multiply the probabilities together to get 0.5 x 0.5 x 0.5 x 0.5 x 0.5 = (0.5)5 = 1/32 or about 3%. That seems pretty unlikely (although not too astronomical).
Amusingly, were we performing a statistical hypothesis test, we would in fact be likely to say that the data are not consistent with the null hypothesis that Paul is picking at random. This is because the probability that he would have got all 5 predictions correct is less than 5%, the standard cut-off used in hypothesis testing (we would say "the data are significant at the 5% level"). Of course, this highlights the danger of the common practice of just looking at a p-values (which is what our probability above is) and concluding that the null hypothesis must be true or false - it would take a rather stronger run of successes to convince most people that an octopus could really correctly predict football results. A 5% significance level means that even if our null hypothesis is true, an outcome will appear 'significant' (and we would question the null hypothesis) if the chances of it happening are less than 1 in 20. This really isn't that unlikely.
We do have more data, however, thanks to Paul's Euro 2008 picks. This takes his record to 9 correct out of 11 - is this statistically significant as well? Once again we want to calculate the probability that Paul would get this success rate picking at random, but it's slightly harder to work out this time. What we want to know is the probability that Paul would be at least this successful were he picking at random. So whereas before we just had to calculate the probability of 5 heads from 5 tosses, here we need to calculate the probability of 9 heads from 11 tosses, 10 heads from 11 tosses, and 11 heads from 11 tosses; adding these three probabilities up will tell us how 'lucky' Paul is.
So 11 heads from 11 tosses is easy, like the case with 5 out of 5, it's just 0.5 multiplied by itself 11 times. What about 10 heads, or 9? Things get a little trickier. Whilst there is only one way to get 11 heads from 11 tosses, there are several ways to get 9 heads. This might sound odd, but if you imagine tossing a coin twice, you can get either:
1) Tails followed by tails (TT)
2) Heads followed by heads (HH)
3) Tails followed by heads (TH)
4) Heads followed by tails (HH)
All of these outcomes are equally likely, but two of them (HT and TH) correspond to getting one head and one tail, and it's this which makes computing the probability of 9 heads from 11 tosses a bit tricky. Fortunately there's a simple formula for calculating this, known as the binomial coefficient. I'll spare you the details (since it's mathsy, and you can read Wikipedia if you like), and tell you how to use Google to get the number you want. Just type in "x choose y" and Google will tell you how many ways there are to get y heads from x coin tosses. Here, we want 11 choose 9, which gives us 55 ways to get 9 heads from 11 tosses. The probability of getting any one particular combination of 9 heads and 2 tails is just 0.5 multiplied by itself 11 times; once for each 50/50 coin toss. Since there are 55 different ways of doing this we then want 55 times this to allow for each possibility. So the final probability that one would get 9 heads from 11 tosses is 55 x (0.5)11, about 2.7% or 1 in 37.
Similarly, we see there are 11 ways to get 10 heads from 11 coin tosses, so the probability of exactly 10 heads is 11 x (0.5)11, about 1 in 186.
We can now put these three probabilities together and add them up to give Paul's prediction p-value as (1 + 11 + 55) x (0.5)11 = 3.3% or about 1 in 30.
So even taking Paul's two mistakes into account, his four extra correct picks mean the chances of him managing his record at random have only increased marginally, and his punditry powers which remain statistically significant.
So how important is the match tonight? He's selected Germany's opponents Spain to progress, and of course if he's right it will be further evidence that his powers are not merely down to chance, so what if he's wrong? It would take his world cup prediction record to 5 right out of 6, would this still be statistically significant? The probability of getting at least 5 out of 6 right is calculated the same as our example above with 9 out of 11. The probability of 6 out of 6 is just (0.5)6, and there are 6 ways of getting 5 heads from 6 tosses, so the probability of exactly 5 out of 6 is 6 x (0.5)6. Adding these together we get a probability of about 11%, or 1 in 9. With just one wrong choice his picking would stop being statistically significant.
What about his lifetime record? That would go to 9 out of 12. I'll spare you the maths now and just tell you that the probability of getting at least 9 out of 12 right is 7.3%, or 1 in 14. Again, statisticians would stop heralding Paul as the mussel eating messiah.
So if he's wrong tonight he'll seem unremarkable (to p-value cultists, at least), whilst if he's right he'll be pushed further towards probabilistic stardom. This does of course demonstrate the dangers of trying to perform statistics entirely through p-values (which many practitioners do), and how susceptible they can be towards even a single result one way or the other.
Now I'm off to watch the football where I'll be rooting for the Germans. If they win the World Cup it means England come joint second, right?
Showing posts with label moderate. Show all posts
Showing posts with label moderate. Show all posts
Wednesday, 7 July 2010
Thursday, 13 May 2010
Doing it by Degrees
When it comes to looking at university education, many do not have to think too hard about what they want to study, the bigger dilemma is where to study it. That said, university prospectuses will often tout various statistics to try and lure potential students into plumping for a particular course, with employability one of the more commonly seen figures. But is this a reliable metric of how 'valuable' a degree is? Or is it yet another example of STATISTICS ABUSE? (roll opening credits)
Every year AGCAS produces a report looking at destinations of graduates six months after graduation. The latest report, from 2009, is the result of questionnaires sent to all graduates from the 2007/8 academic year, and can be downloaded here. The report itself contains quite a lot of interesting data, with destination breakdown (how many graduates are employed, unemployed, or studying for further degrees) as well as stats on the types of job those in employment have found themselves in. With these statistics available for a number of subjects (or subject areas), we can get a feel for which subjects seem the most or least valuable.
The report provides details on the following subjects/subject areas:
Science
Biology; Chemistry; Environmental, Physical Geographical and Terrestrial Sciences; Physics; Sports Science
Mathematics, IT and Computing
Computer Science and Information Technology; Mathematics
Engineering and Building Management
Architecture and Building; Civil Engineering; Electric and Electronic Engineering; Mechanical Engineering
Social Sciences
Economics; Geography; Law; Politics; Psychology; Sociology
Arts, Creative Arts and Humanities
Art and Design; English; History; Media Studies; Languages; Performing Arts
Business and Administrative Studies
Accountancy; Business and Management; Marketing
So let's start with employment, surely a perfectly good benchmark of how 'good' a degree is. The AGCAS report splits graduates into those in UK employment, overseas employment, as well as those working and studying. We add these three together to give us our employment figures:
Top 5 for Employment:
Civil Engineering (78.3% employed)
Marketing (74.6%)
Business and Management (73.6%)
Architecture and Building (73.4%)
Accountancy (73.0%)
Bottom 5 for Employment:
Law (35.2%)
Physics (37.9%)
Chemistry (44.0%)
Biology (58.0%)
History (58.7%)
I think it's fair to say there are some surprises here. Marketing, and Business and Management two subject areas often cited as housing archetypal 'Mickey Mouse' degrees make the top 5, whilst historically 'tough' subjects like chemistry and physics are at the opposite end. Are people really better off studying business over biology? Or is there something wrong with our metric?
Naturally, I'm inclined to believe the latter, and with good reason. As is so often the case, one statistic does not tell the whole story; whilst these numbers tell us what proportion of graduates were employed six months after graduation, it is not simply the case that everyone else was unemployed. AGCAS reports a number of 'studying' statistics as well, such as those studying for a higher degree, a PGCE, or professional qualifications. Perhaps then, unemployment is a better way of assessing degrees, as this takes people who are 'employed' with study into accout. Let's see what happens:
Top 5 for Unemployment:
Law (5.5% unemployed)
Sports Science (5.6%)
Geography (6.4%)
Civil Engineering (7.0%)
Psychology (7.4%)
Bottom 5 for Unemployment
Computer Science and Information Technology (13.7%)
Media Studies (12.3%)
Art and Design (12.2%)
Electrical and Electronic Engineering (11%)
Accountancy (10.9%)
Quite a big change. Law jumps from worst for employment to best for unemployment (as you might expect, they're all studying), and accountancy has done the opposite. There are still some surprises, such as Computer Science and IT having the highest rate of unemployment, and another 'Mickey Mouse' course in the form of Sports Science being second best. However, this seems a much less debatable statistic than employment, and so it seems reasonable to take these figures at face value.
There is, of course, an issue we have yet to discuss, which will be a rather pressing one for many new graduates: money. What good is being employed if you're only getting paid £5 an hour for those fancy letters after your name?
The salary data in the AGCAS report are a little harder to find, let alone digest. Whilst we get nice pie charts and percentage breakdowns for destinations, discussion of salaries is restricted to an introductory paragraph. If we trawl through these, however, we do get some numbers, and merging them all together we can do another top and bottom 5, this time based on the average salary of respondents.
Top 5 for Salary
Economics (£24065)
Civil Engineering (£24006)
Architecture and Building (£23689)
Mechanical Engineering (£23683)
Electrical and Electronic Engineering (£22372)
Bottom 5 for Salary
Art and Design (£15656)
Media Studies (£16295)
Psychology (£16500)
Sports Science (£16627)
English (£16642)
Once again, a rather marked change. Media Studies keeps the bottom 5 place it enjoyed under the unemployment stats, but it is joined by Sports Science, which was second best for unemployment. There are no real surprises in our top 5, however, all these subjects having a fairly substantial pedigree.
For the sake of argument, then, let's suppose that you are most interested in average salary. As I mentioned, the AGCAS report makes it much easier to find the employment/unemployment figures for a subject than it does to find average salaries. Do these provide an adequate indicator of average salary? Our top/bottom 5s above would suggest not, but these only cover 10 of 26 subjects. Let's plot some graphs!
First up, average salary against employment, is there a strong link between the two?
Hmm, no obvious pattern there, then. How about unemployment, does that give us a better fit?
There doesn't seem to be any sort of pattern there either.
We can in fact calculate a number that gives us an idea of how closely related two sets of numbers are. The correlation coefficient between two sets of (x,y) points (like our (employment %, salary) points on our graph) varies from -1 to 1. If it's close to 0 that means our numbers are not closely related, whilst if it is close to +1 or -1 it suggests a strong relationship. For example, if in our plot of employment against salary above all our points seemed to be on a straight line, this would suggest a correlation of around 1 or -1. The sign indicates the direction of the correlation. If it's positive this means as salary increases, so does employment. If it's negative, then as salary increases, employment decreases. This doesn't mean the two are related - "correlation does not imply causation" is one of a statistician's many mantras - it just shows that these data happen to have an association (which we may go on to convince ourselves is a causal one).
So that diversion aside, what correlations do we get in our two plots above? Looking at them, we'd expect it to be close to zero; there doesn't seem to be much of a pattern in either of them. For the first plot, of employment against salary, we find a correlation coefficient of 0.12 - so not much of a surprise there. For unemployment it's even worse: 0.06. In short, neither employment nor unemployment is a good indicator of average salary.
There is one area of the AGCAS report we haven't discussed, however, which might prove useful. Whilst each subject has a page of percentages of those in employment, studying, and so on, it also has a page showing what types of jobs are held by those who are employed. These range from a variety of 'Professionals' down to 'Numerical Clerks and Cashiers', and 'Retail, Catering, Waiting and Bar Staff'. This last one doesn't sound too glamarous; you've just spent 3 years earning a degree and you're still working in a bar? More to the point, these jobs are going to be low paying, so hopefully they're a better indicator of average salary. Let's see:
There definitely seems to be a pattern there, and the correlation between the two variables is -0.88 - that's a pretty strong negative correlation. The higher the proportion of those employed in retail, the lower the average salary. Not a surprising result, but it's always worth checking these things.
Is this at all useful, though? The salary data are in the document, you just have to dig for them a bit more. There is, however, one thing we've not mentioned. Because the report doesn't give average salaries the same prominent treatment as the employment data, some numbers are, in fact, missing. Whilst we can see what proportion of history graduates are studying in the UK for a teaching qualification, we can't find their average salary six months after graduation (and the same goes for performing arts). However, because we've identified the percentage of those working in retail as a useful indicator of average salary, we can use this knowledge to predict the average salaries of history and performing arts graduates. (In statistics, we'd call our retail statistic an 'instrument' for salary.)
So how do we turn our retail employment data into a prediction of salary? If you read my previous post about the times goals are scored in football matches, you should already know where I'm going with this. If not, then go and read it now, and come back when you're ready to apologise for such an oversight.
So anyway, it's time for some more linear regression. We're looking to fit the model S = a + bR, where S is salary, and R is the percentage of those employed who are employed in retail. If we can estimate a and b, then we can use this equation to estimate S when we only know R, as is the case for history and performing arts degrees. We can also plot a cool line on our graph to show the trend. Running the numbers, we find a = 25014 and b = -468, and plotting the line this generates onto our graph gives us:
We can now either use our equation S = a + bR with a and b replaced with 25014 and -468, or read straight off the line on our graph. For both history and performing arts, retail employment was 17.4%, so plugging R = 17.4 into this equation gives S = 25014 - 468*17.4 = £16,870.40. Our model suggests that both subjects seem to lead to (relatively) low average salaries, something which would not have been easy to discern from the report alone.
Alas, this all assumes our model is accurate, and with a relatively small number of observations I wouldn't be inclined to place too much confidence in these conclusions. Here I've taken a single report to base rather a lot of analysis on. However, it does illustrate a couple of interesting points. Firstly, mere 'employability' figures seem a rather dubious metric on which to base the value of a degree. Perhaps more surprisingly, unemployment doesn't seem to be a particularly good one either, at least in terms of indicating average salary. Whilst this report did have salary data in it, they weren't as clearly laid out as the other data, and were in fact missing for some subjects. This has allowed us to demonstrate how you can use another variable (if you think it's a good enough surrogate) to estimate missing data. Whilst for this particular problem you're probably better off just trying to hunt down the data you want in another report, our way is clearly much more fun.
Every year AGCAS produces a report looking at destinations of graduates six months after graduation. The latest report, from 2009, is the result of questionnaires sent to all graduates from the 2007/8 academic year, and can be downloaded here. The report itself contains quite a lot of interesting data, with destination breakdown (how many graduates are employed, unemployed, or studying for further degrees) as well as stats on the types of job those in employment have found themselves in. With these statistics available for a number of subjects (or subject areas), we can get a feel for which subjects seem the most or least valuable.
The report provides details on the following subjects/subject areas:
Science
Biology; Chemistry; Environmental, Physical Geographical and Terrestrial Sciences; Physics; Sports Science
Mathematics, IT and Computing
Computer Science and Information Technology; Mathematics
Engineering and Building Management
Architecture and Building; Civil Engineering; Electric and Electronic Engineering; Mechanical Engineering
Social Sciences
Economics; Geography; Law; Politics; Psychology; Sociology
Arts, Creative Arts and Humanities
Art and Design; English; History; Media Studies; Languages; Performing Arts
Business and Administrative Studies
Accountancy; Business and Management; Marketing
So let's start with employment, surely a perfectly good benchmark of how 'good' a degree is. The AGCAS report splits graduates into those in UK employment, overseas employment, as well as those working and studying. We add these three together to give us our employment figures:
Top 5 for Employment:
Civil Engineering (78.3% employed)
Marketing (74.6%)
Business and Management (73.6%)
Architecture and Building (73.4%)
Accountancy (73.0%)
Bottom 5 for Employment:
Law (35.2%)
Physics (37.9%)
Chemistry (44.0%)
Biology (58.0%)
History (58.7%)
I think it's fair to say there are some surprises here. Marketing, and Business and Management two subject areas often cited as housing archetypal 'Mickey Mouse' degrees make the top 5, whilst historically 'tough' subjects like chemistry and physics are at the opposite end. Are people really better off studying business over biology? Or is there something wrong with our metric?
Naturally, I'm inclined to believe the latter, and with good reason. As is so often the case, one statistic does not tell the whole story; whilst these numbers tell us what proportion of graduates were employed six months after graduation, it is not simply the case that everyone else was unemployed. AGCAS reports a number of 'studying' statistics as well, such as those studying for a higher degree, a PGCE, or professional qualifications. Perhaps then, unemployment is a better way of assessing degrees, as this takes people who are 'employed' with study into accout. Let's see what happens:
Top 5 for Unemployment:
Law (5.5% unemployed)
Sports Science (5.6%)
Geography (6.4%)
Civil Engineering (7.0%)
Psychology (7.4%)
Bottom 5 for Unemployment
Computer Science and Information Technology (13.7%)
Media Studies (12.3%)
Art and Design (12.2%)
Electrical and Electronic Engineering (11%)
Accountancy (10.9%)
Quite a big change. Law jumps from worst for employment to best for unemployment (as you might expect, they're all studying), and accountancy has done the opposite. There are still some surprises, such as Computer Science and IT having the highest rate of unemployment, and another 'Mickey Mouse' course in the form of Sports Science being second best. However, this seems a much less debatable statistic than employment, and so it seems reasonable to take these figures at face value.
There is, of course, an issue we have yet to discuss, which will be a rather pressing one for many new graduates: money. What good is being employed if you're only getting paid £5 an hour for those fancy letters after your name?
The salary data in the AGCAS report are a little harder to find, let alone digest. Whilst we get nice pie charts and percentage breakdowns for destinations, discussion of salaries is restricted to an introductory paragraph. If we trawl through these, however, we do get some numbers, and merging them all together we can do another top and bottom 5, this time based on the average salary of respondents.
Top 5 for Salary
Economics (£24065)
Civil Engineering (£24006)
Architecture and Building (£23689)
Mechanical Engineering (£23683)
Electrical and Electronic Engineering (£22372)
Bottom 5 for Salary
Art and Design (£15656)
Media Studies (£16295)
Psychology (£16500)
Sports Science (£16627)
English (£16642)
Once again, a rather marked change. Media Studies keeps the bottom 5 place it enjoyed under the unemployment stats, but it is joined by Sports Science, which was second best for unemployment. There are no real surprises in our top 5, however, all these subjects having a fairly substantial pedigree.
For the sake of argument, then, let's suppose that you are most interested in average salary. As I mentioned, the AGCAS report makes it much easier to find the employment/unemployment figures for a subject than it does to find average salaries. Do these provide an adequate indicator of average salary? Our top/bottom 5s above would suggest not, but these only cover 10 of 26 subjects. Let's plot some graphs!
First up, average salary against employment, is there a strong link between the two?
Hmm, no obvious pattern there, then. How about unemployment, does that give us a better fit?
There doesn't seem to be any sort of pattern there either.
We can in fact calculate a number that gives us an idea of how closely related two sets of numbers are. The correlation coefficient between two sets of (x,y) points (like our (employment %, salary) points on our graph) varies from -1 to 1. If it's close to 0 that means our numbers are not closely related, whilst if it is close to +1 or -1 it suggests a strong relationship. For example, if in our plot of employment against salary above all our points seemed to be on a straight line, this would suggest a correlation of around 1 or -1. The sign indicates the direction of the correlation. If it's positive this means as salary increases, so does employment. If it's negative, then as salary increases, employment decreases. This doesn't mean the two are related - "correlation does not imply causation" is one of a statistician's many mantras - it just shows that these data happen to have an association (which we may go on to convince ourselves is a causal one).
So that diversion aside, what correlations do we get in our two plots above? Looking at them, we'd expect it to be close to zero; there doesn't seem to be much of a pattern in either of them. For the first plot, of employment against salary, we find a correlation coefficient of 0.12 - so not much of a surprise there. For unemployment it's even worse: 0.06. In short, neither employment nor unemployment is a good indicator of average salary.
There is one area of the AGCAS report we haven't discussed, however, which might prove useful. Whilst each subject has a page of percentages of those in employment, studying, and so on, it also has a page showing what types of jobs are held by those who are employed. These range from a variety of 'Professionals' down to 'Numerical Clerks and Cashiers', and 'Retail, Catering, Waiting and Bar Staff'. This last one doesn't sound too glamarous; you've just spent 3 years earning a degree and you're still working in a bar? More to the point, these jobs are going to be low paying, so hopefully they're a better indicator of average salary. Let's see:
There definitely seems to be a pattern there, and the correlation between the two variables is -0.88 - that's a pretty strong negative correlation. The higher the proportion of those employed in retail, the lower the average salary. Not a surprising result, but it's always worth checking these things.
Is this at all useful, though? The salary data are in the document, you just have to dig for them a bit more. There is, however, one thing we've not mentioned. Because the report doesn't give average salaries the same prominent treatment as the employment data, some numbers are, in fact, missing. Whilst we can see what proportion of history graduates are studying in the UK for a teaching qualification, we can't find their average salary six months after graduation (and the same goes for performing arts). However, because we've identified the percentage of those working in retail as a useful indicator of average salary, we can use this knowledge to predict the average salaries of history and performing arts graduates. (In statistics, we'd call our retail statistic an 'instrument' for salary.)
So how do we turn our retail employment data into a prediction of salary? If you read my previous post about the times goals are scored in football matches, you should already know where I'm going with this. If not, then go and read it now, and come back when you're ready to apologise for such an oversight.
So anyway, it's time for some more linear regression. We're looking to fit the model S = a + bR, where S is salary, and R is the percentage of those employed who are employed in retail. If we can estimate a and b, then we can use this equation to estimate S when we only know R, as is the case for history and performing arts degrees. We can also plot a cool line on our graph to show the trend. Running the numbers, we find a = 25014 and b = -468, and plotting the line this generates onto our graph gives us:
We can now either use our equation S = a + bR with a and b replaced with 25014 and -468, or read straight off the line on our graph. For both history and performing arts, retail employment was 17.4%, so plugging R = 17.4 into this equation gives S = 25014 - 468*17.4 = £16,870.40. Our model suggests that both subjects seem to lead to (relatively) low average salaries, something which would not have been easy to discern from the report alone.
Alas, this all assumes our model is accurate, and with a relatively small number of observations I wouldn't be inclined to place too much confidence in these conclusions. Here I've taken a single report to base rather a lot of analysis on. However, it does illustrate a couple of interesting points. Firstly, mere 'employability' figures seem a rather dubious metric on which to base the value of a degree. Perhaps more surprisingly, unemployment doesn't seem to be a particularly good one either, at least in terms of indicating average salary. Whilst this report did have salary data in it, they weren't as clearly laid out as the other data, and were in fact missing for some subjects. This has allowed us to demonstrate how you can use another variable (if you think it's a good enough surrogate) to estimate missing data. Whilst for this particular problem you're probably better off just trying to hunt down the data you want in another report, our way is clearly much more fun.
Thursday, 8 April 2010
Football - Goal times in brief
I just had a quick look at the times goals were scored in the Premier League dataset:
If you'll excuse the somewhat poorly labelled x-axis (R is being fussy, and I'm not particularly inclined to try and fix it for something so trivial), there are a few interesting points.
Most obvious are the two huge bars at around half time and full time. Of course, as you might have already guessed, this is due to how goal times have been reported - goals in injury time in the first half are reported as a 45th minute goal, and in the second half as a 90th minute goal. The fact the 9oth minute bar is so much taller than the 45th minute one demonstrates something most of us have probably observed: second half injury time is almost always longer than first half injury time.
If we filter out these two anomolies, and look at the goal times without the 46th of 91st minute goals, we get the following:
A couple of things to notice here. The first is that goals in the first minute really do seem quite rare, occurring in just 82 games (once in over 200 games). The second is that there looks like there might be a slight pattern to goal times - it seems goals become a bit more likely as the match wears on. Is this the case, or are our eyes just deceiving us?
Fortunately, we can test this hypothesis using a statistical technique called linear modelling. What this means is that we assume that the number of goals scored for a particular minute have a linear relationship with the time at which they're scored. In other words, if we just plotted the above bar graph but with points instead of bars, the points would lie on a roughly straight line. In fact, let's do this and see how it looks. One thing we'll change is from goal frequency to percentage, the data are the same, but it will make more sense to talk about percentages later on.
It looks a lot clearer when we plot the data this way, with most of the points seeming to lie (very roughly) on a straight line. We also note that each minute seems responsible for around 1% of goals. With 90 minutes in a game we'd expect something like this, so we've provided ourselves with a useful 'sanity check' - never a bad thing when playing with data.
Having observed this pattern, can we make any use of it? It might be nice to be able to fit a model that could tell us how likely a goal in, say, the 10th minute of a match would be, if it's true that there is a simple, straight line relationship between time in the game and likelihood of a goal. One option would be to just put a ruler on the graph and try and draw a straight line that seemed to best fit the data (indeed, I can remember doing this when I was at school, back when I would call this a 'line of best fit'). Fortunately, we can use statistical software to do this for us, and in arguably a much more reliable way.
To put our model mathematically, suppose the percentage of goals scored in the Xth minute of the game is G, then we'd assume that G = a + bX, where a and b are some numbers we want to find out. Using statistical software we can fit this model through a method called least squares). What this method does is a bit like what you do when you put a ruler on the graph and try and move it around until it looks about right, with the same number of points above and below the line you want to draw. What your eye is doing when you do this is probably trying to minimise the total distance all of your points are from the line you're drawing; what least squares does is calculate the line that minimises the square of these distances. In other words, if you imagine a line drawn on the graph, and measure the distance from the line to each of the points, square these distances and add them all up, least squares will find you the line that makes this total the smallest.
Applying this method we find that a = 0.895 and b = 0.005, which we can then plug back into our model equation: instead of G = a + bX we now get G = 0.895 + 0.005X. This tells us that for every extra minute in the game, the percentage of goals scored in that minute goes up by 0.005. Admittedly, this isn't very much at all, but over the course of the game that works out to around a 0.45% increase from start to finish, which when you consider that the average minute will only have 1.11% of goals, seems a little more dramatic. For example, a goal in the 80th minute is 40% more likely than a goal in the 10th.
Finally, let's plot the line onto the previous graph, like so:
You might well think that it looks about as good as something you could have done by eye with a ruler, and you're probably right. However, another thing using a computational method can tell us is how 'significant' the effect of time on goal probability is. In other words, to what extent is there really an underlying relationship between the time in a game and the probability of a goal, and to what extent is this just the result of our data randomly falling into a pattern. We find that, mostly thanks to the sheer size of our dataset, that this pattern is not likely to be down to chance; there really does seem to be a linear relationship between time in a game and the probability of a goal going in.
Perhaps something to bear in mind after a goalless first half.
If you'll excuse the somewhat poorly labelled x-axis (R is being fussy, and I'm not particularly inclined to try and fix it for something so trivial), there are a few interesting points.
Most obvious are the two huge bars at around half time and full time. Of course, as you might have already guessed, this is due to how goal times have been reported - goals in injury time in the first half are reported as a 45th minute goal, and in the second half as a 90th minute goal. The fact the 9oth minute bar is so much taller than the 45th minute one demonstrates something most of us have probably observed: second half injury time is almost always longer than first half injury time.
If we filter out these two anomolies, and look at the goal times without the 46th of 91st minute goals, we get the following:
A couple of things to notice here. The first is that goals in the first minute really do seem quite rare, occurring in just 82 games (once in over 200 games). The second is that there looks like there might be a slight pattern to goal times - it seems goals become a bit more likely as the match wears on. Is this the case, or are our eyes just deceiving us?
Fortunately, we can test this hypothesis using a statistical technique called linear modelling. What this means is that we assume that the number of goals scored for a particular minute have a linear relationship with the time at which they're scored. In other words, if we just plotted the above bar graph but with points instead of bars, the points would lie on a roughly straight line. In fact, let's do this and see how it looks. One thing we'll change is from goal frequency to percentage, the data are the same, but it will make more sense to talk about percentages later on.
It looks a lot clearer when we plot the data this way, with most of the points seeming to lie (very roughly) on a straight line. We also note that each minute seems responsible for around 1% of goals. With 90 minutes in a game we'd expect something like this, so we've provided ourselves with a useful 'sanity check' - never a bad thing when playing with data.
Having observed this pattern, can we make any use of it? It might be nice to be able to fit a model that could tell us how likely a goal in, say, the 10th minute of a match would be, if it's true that there is a simple, straight line relationship between time in the game and likelihood of a goal. One option would be to just put a ruler on the graph and try and draw a straight line that seemed to best fit the data (indeed, I can remember doing this when I was at school, back when I would call this a 'line of best fit'). Fortunately, we can use statistical software to do this for us, and in arguably a much more reliable way.
To put our model mathematically, suppose the percentage of goals scored in the Xth minute of the game is G, then we'd assume that G = a + bX, where a and b are some numbers we want to find out. Using statistical software we can fit this model through a method called least squares). What this method does is a bit like what you do when you put a ruler on the graph and try and move it around until it looks about right, with the same number of points above and below the line you want to draw. What your eye is doing when you do this is probably trying to minimise the total distance all of your points are from the line you're drawing; what least squares does is calculate the line that minimises the square of these distances. In other words, if you imagine a line drawn on the graph, and measure the distance from the line to each of the points, square these distances and add them all up, least squares will find you the line that makes this total the smallest.
Applying this method we find that a = 0.895 and b = 0.005, which we can then plug back into our model equation: instead of G = a + bX we now get G = 0.895 + 0.005X. This tells us that for every extra minute in the game, the percentage of goals scored in that minute goes up by 0.005. Admittedly, this isn't very much at all, but over the course of the game that works out to around a 0.45% increase from start to finish, which when you consider that the average minute will only have 1.11% of goals, seems a little more dramatic. For example, a goal in the 80th minute is 40% more likely than a goal in the 10th.
Finally, let's plot the line onto the previous graph, like so:
You might well think that it looks about as good as something you could have done by eye with a ruler, and you're probably right. However, another thing using a computational method can tell us is how 'significant' the effect of time on goal probability is. In other words, to what extent is there really an underlying relationship between the time in a game and the probability of a goal, and to what extent is this just the result of our data randomly falling into a pattern. We find that, mostly thanks to the sheer size of our dataset, that this pattern is not likely to be down to chance; there really does seem to be a linear relationship between time in a game and the probability of a goal going in.
Perhaps something to bear in mind after a goalless first half.
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