RPI Raptor Population Trends: Interpreting Trend Graphs and Maps

Our goal was to calculate annual indices and population trends for all stations that had at least 10 years of hourly data, but we also calculated annual indices for some stations that started compiling their hourly data when HawkCount.org was launched around 2002, in order to include more sites. However, sites with less than 10 years are considered to have insufficient data to estimate a trend reliably, so graphs for those sites do not show a trend line or an estimated rate of change. At each RPI site, species included in analyses were those detected at least once in at least half of the years available.

Data are presented in two formats, “long-term” which represents the full range of years of hourly data available in HawkCount (see trend graph), and “recent 10-years” representing the trend for the period 2001 to 2010 (displayed on trend map).

Because the analysis is now highly automated and depends on direct access to the data in HawkCount, several key sites that were included in the previous 2008 analysis were not included in this analysis due to incomplete datasets in the HawkCount database. We are in the process of updating these online records and will add the analyses as they become available.

How to read the population graphs

This following example provides the long-term population trend for Osprey as calculated for Hawk Mountain Sanctuary based on fall migration data, as indicated in the chart title. The linear trend value shown below the title is based on all years analyzed for that station (e.g. 1966-2010), and is expressed as annual percent change in population index size. This trend value assumes a linear and discrete change in population size among years over the time period examined. The significance level is expressed by the p value: values of 0.05 or less are generally considered statistically significant (calculation methods for the trend and p value are provided below).

The black dots on the chart are estimated indices of annual population size as calculated by the statistical model. They are expressed on a linear scale and represent the average of the predicted number of individual birds detected per hour of survey. If present, the blue line represents the linear trend.

Trend maps

Trend maps combine the calculations from all RPI stations to easily visualize the overall population trend of a species over a large area. Having several stations with the same population trend strongly re-enforces our confidence that the results we see are representative of the underlying population trends. To ensure that all trends are directly comparable, we only used the trend based on the most recent 10 years for each station, and only for stations with adequate coverage during that period. The example below represents the population trends of Bald Eagle based on fall migration data in eastern North America. Green arrows going up represent all cases where the 10-year population trend is significantly positive, red arrows going down represent significant population declines and blue dots represent non-significant changes. Blue arrows (up or down) represent near-significant trends (p value between 0.05 and 0.10). Note that the size of the green and red arrows is also proportional to the magnitude of the increase of decline, as explained in the map legend. You can click on any of the markers to see more details about the station name, the trend and period covered, as well as the significance level. You can also zoom in or out to better see the details in areas where there are several nearby stations, such as along the Appalachian Mountains.

Analysis methods

Annual population indices and long-term population trends were estimated using Generalized Additive Models with Poisson distribution and log link (mgcv package, R-Project 2010). Additive Models were first described in the 1980’s (e.g., see Hastie and Tibshirani 1986), and are an extension of Generalized Linear Models, first described in the early 1970s (e.g., see Nelder and Wedderburn 1972).

Annual indices of population size were calculated as the mean of predicted hourly counts each year based on the following regression:

g(E(Yijk)) = αk + ƒ1(X1ijk)+ ƒ2(X2ijk) , (eq. 1)

where g() indicates a log-link between the mean of Yijk and the predictor function for the Poisson model; Yijk = hourly observation count (number of hawks per hour) in hour i, on day j, in year k; αk is the intercept (year factor) for year k; X1ijk = day of year (Julian date); X2ijk =, hour of the day; f1 and f2 are smoothing functions for day of the year and hour of the day, respectively. Cases were weighted by W = the ratio of observation length (typically one hour, but sometimes less than one hour) to total number of observation hours each year.

A trend in population index was estimated using the same model structure, but with year as a continuous variable X0 to replace the αk factors:

g(E(Yijk)) = β*X0ijk + ƒ1(X1ijk)+ ƒ2(X2ijk) , (eq. 2)

The trend was calculated as the % rate of change per year = 100*(exp(β )-1). This converts the year coefficient, which is an estimate of the instantaneous rate of change per year, to an estimate of the discrete rate of change per year between any year and the next following year.

Because the p-value for the year coefficient is overly-optimistic using generalized linear and additive models (degrees of freedom are based on the number of observations as opposed to the number of years, incorrectly resulting in highly significant effects), we used an alternative method based on Monte Carlo simulation to estimate a p-value for the trend. To do so, each dataset was re-analyzed 1000 times, with the year term randomized each time without replacement. This tested how many times out of 1000 a trend larger than the year coefficient from eq.2 was achieved, essentially providing a probability that the trend was significantly different from zero.

T. Hastie and R. Tibshirani. 1986. Generalized Additive Models. Statistical Science. 1(3): 297-318.

J.A. Nelder and R.W.M. Wedderburn. 1972. Generalized Linear Models. Journal of the Royal Statistical Society, Series A (General). 135(3): 370-384.

Availability of data

All graphs and maps are available from BSC’s NatureCounts web site: http://www.naturecounts.ca/rpi/, as well as from http://hawkcount.org. RPI stations with a bit of technical skills can easily copy a small line of script and embed the population trends into their own web site if they wish. To do so, simply define the population trends that you would like to see on NatureCounts and look for the link that says “To insert the graphs below in your own web site using our web service click here.”

Annual indices and trends can also be downloaded from NatureCounts in an Excel compatible format, if you wish to create your own graphs for presentation. Simply look for the Download button of the population trend tool on NatureCounts.