Birdwatching morning and evening

The really short version of what follows is that you see more birds in the morning than you do in the early afternoon.

First, look at the data

The Minitab dot-plot below, in conjunction with a little explanation, tells you almost everything you need to know (if it looks a bit pixellated to you, try increasing the size of your browser window). Each dot represents - by its position - the number of different species seen in a chunk of time spent walking and birdwatching. The two-letter codes by each line show the time and route, as follows:

There are four different visits in each of four categories, and it appears that more birds are seen in the morning than in the evening: this isn't news to birdwatchers, but I do like to check things out for myself when I can. I decided ahead of time that I was going to do 4 visits to each of 4 sites, and then used dice to choose an order for these 16 visits at random. Since this series of visits stretched from the beginning of April to the beginning of July, I'd better show them in time order, as well.

It is unfortunate that all the morning walks towards Lacock occurred after all the evening walks towards Lacock, but that's how the dice fell. If you just had the Lacock data alone, you wouldn't be able to distinguish the effects of time of day from the effects of time of year, but if you include the Corsham data, you should be inclined to believe that most of what you are seeing is due to the time of day, not the time of year.

What was I trying to prove?

These data were actually collected to try and prove a theory that turns out to be false. I knew that most birds are more visible in the morning than throughout the day. See, for example, Figure 2.11 in the book "Bird Census Techniques", by Bibby, Burgess, Hill, and Mustoe. I had hoped that this would be much less evident in water birds, such as those seen in the lake at Corsham. I hoped to show that, if you had to choose both a morning visit and an afternoon visit, your best bet would be to visit the countryside in the morning and the waterside in the afternoon. Since I live in Wiltshire, a lake such as Corsham was the best waterside I could find locally.

My plan, therefore, was to add together the counts of different species seen in morning walks towards Lacock and afternoon walks around Corsham Lake, and show that this total is much higher than you would expect if you took 8 visits at random from my collection of 16, subject to the restriction that 4 are in the morning, 4 in the afternoon, 4 walks towards Lacock, and 4 walks around Corsham lake and the surrounding park. Unfortunately this isn't so: if you add together the 4 morning visits to Lacock and the 4 afternoon visits to Corsham, you get a total of 125. A computer program tells me that if you just choose at random, subject to the 4 Corsham, 4 Morning condition, the probability of getting less than 125 is 0.238, equal to 125 is 0.0591, and greater than 125 is 0.703. So, if anything, you would be doing slightly worse than random with this strategy (by the way, the attempt to prove a statement about the sum of two visits also explains why I didn't design the experiment with better protection against changes in the abundance of birds over time: I couldn't find a way of doing this that would still allow me to test what I wanted with a decent chance of finding something, if the effect was there, and a reasonable number of observations).

Looking back on my records, I can see two reasons why Corsham was more sensitive to time than I expected:

• At Corsham, the water birds only accounted for about half of all the bird species seen, so while I saw either 6 or 7 water birds in each visit to Corsham, I saw from 5 to 15 land birds, and the number of different land birds seen did depend on the time of day.
• On two of my four afternoon visits, there were boats sailing on Corsham lake! The birds definately didn't like this, and had moved over to stretches of water too narrow to sail on (or, for all I know flown off). Mind you, this can't be the main reason, because as I say above, I still saw 6 water birds at the very least, and the maximum number of different water birds I ever saw was 7.

What can I actually prove?

The simplest thing to do now is to establish that mornings are better than evenings by testing the two sites separately, but this doesn't produce a significant result: the Mann-Whitney-Wilcoxon test produces tail probabilities of 0.1016 and 0.1441 for Corsham and Lacock respectively, against the null hypothesis that there is no difference in the distribution of the number of birds seen between morning and evening. If we multiply these together we get 14.64E-3. If we take two uniformly random numbers between 0 and 1 and multiply them together the probability of getting this low a number, or less, is 0.0765. (you can do this from first principles by integration, or by noticing that the chi-squared distribution with two degrees of freedom is the exponential distribution with mean 2. -2 times the logarithm of a uniformly random number between 0 and 1 is distributed according to the exponential distribution with mean 2, so if we take -2 times the sum of the logarithm of our two probabilities the result should be distributed according to the chi-squared distribution on 4 degrees of freedom). This wouldn't normally be regarded as a significant result either.

That's not what I actually did, though: what I did follows on naturally from the idea of adding together scores from individual days. If you add up the number of different birds seen on all the mornings, you get a total of 145. If you were to pick 8 days at random, subject to the constraint that 4 should be from Corsham and 4 should be from Lacock, my program tells me that the probability of getting less than 155 is 0.9923, the probability of getting 155 is 0.003673, and the probability of getting more than 155 is 0.00408, so the probability of getting 155 or more at random is 0.0077, and we have good evidence (with a one-tailed test) for the theory that mornings are more productive than afternoons. If we couldn't specify a preference ahead of time we should double this to get a tail probability of 0.0154 for a two tailed test, unfortunately just over the magic 1% threshold, but well under the 5% threshold: we have found something out! One guess at the number of extra birds per day you see by turning up in the morning is the difference between the two means, which is 4.125.

We can extract a rough 95% confidence interval for the number of extra birds seen in the morning by playing about with the data: you add to the number of birds seen in the evening session until the test used stops reporting a signficant result to get a lower bound, and keep on adding until reports a significant result in the other direction to get an upper bound. I get a range of 1-7. That is, I have used a procedure that, with probability 95%, gives me a range that includes the true difference between the number of birds seen in the morning and evening, and it has handed me the range 1-7.

A more conventional way of analysing the data (at the cost of making a few more assumptions) is to run an ANOVA. Minitab does this for me. The usual diagnostic graphs look reasonable, except that the last two observations record fewer birds than it expected (but not outrageously few: it doesn't stand out in the normal plot of residuals). I used two factors, time and site. Time is significant at P = 0.011, but neither site nor the time-site interaction are significant. The mean number of morning birds is 14.00, and the mean number of evening birds is 18.13, with 95% confidence intervals of about 12-16 and 16-20 respectively. Regressing birds seen on Corsham/Lacock and Morning/Evening indicator variables (with no interaction term) gives us a coefficient for the Morning/Evening term of 4.125 with a standard error of 1.343.

The underlying data

The following table gives the observations and the number of birds seen, together with the number of water birds seen at Corsham: I haven't split out water birds for the Lacock walk, although I very often see Mallard, Moorhen, or Swan. All dates are in 2005.