Detailed Statistical Methods
This document provides a detailed technical description of the statistical and analytical methods used to generate the indicators of the population status of groups of birds in the State of Canada's Birds, 2012. It is intended for an audience of scientists and technical experts.
In the State of Canada's Birds, we presented indicators of the national population status of groups of bird species, as well as their status in eight regions of Canada (seven terrestrial regions and the Oceans region). In the national overview, the "All birds" indicator included all extant native species that regularly occur in Canada and for which suitable data were available (317 of 451 species). An additional 10 species were included in the national "All seabirds" indicator but the periodic, breeding colony survey data available for these species could not be included in the annual "All birds" indicator. We also presented a series of sub-indicators to represent the bird communities of the dominant, broadly defined landcover types (forest birds, grassland birds, seabirds, etc.). We separated waterfowl from all other water birds (i.e., species associated with water or wetlands), because the status of waterfowl populations is affected by a distinct suite of factors related to hunting and the associated intensive management. In addition to the landcover-based groups of birds, we presented some additional sub-indicators to represent groups of species that were known to have distinct and noteworthy trends (e.g., aerial insectivores and raptors).
The species lists for the indicators in the "Beyond our Borders" section were defined based on the primary wintering area of each of the 317 species included in the national "All birds" indicator. Species were included in one of the four wintering-region indicators only if 50% or more of the species' wintering range or wintering population was contained within the region. Therefore, species with their wintering range spread relatively evenly across three or more regions were excluded from the Beyond our Borders indicators.
We defined the seven terrestrial regions based largely on Canadian portions of the 12 North American Bird Conservation Regions (BCRs, Table 1), developed by the North American Bird Conservation Initiative (NABCI).
Table 1. Definitions of the eight regions used in the State of Canada's Birds.
|Region||Definition||Total Area (km2)|
|Southern Shield and Maritimes||BCRs 12 and 14||595727|
|Lower Great Lakes-St.Lawrence||BCR 13||113210|
|Eastern Boreal|| BCRs 7and 8, east of the|
|Western Boreal|| BCRs 4 and 6, plus 7 and 8, west|
of the Manitoba-Ontario border
|West Coast and Mountains||BCRs 5, 9 and 10||698809|
|Oceans||Canada's territorial waters||N/A|
Characteristic species lists
Indicators of the status of bird groups, which combine information across species in a group, are very sensitive to the list of species included. Therefore, for each of the eight regions, we defined a limited list of species considered "characteristic" of the region, to ensure our indicators would primarily reflect the status of the region's distinctive native bird community. For each region, we aimed to:
- select the broadest possible group of species that regularly occurred in the region, while excluding peripheral species that are more representative of other regions;
- include every native species that regularly occurs in Canada as characteristic in at least one region;
- allow broadly and evenly distributed species to be included in more than one region; and,
- create species lists that are independent of regional area, i.e., large areas should not necessarily have longer lists of species.
We used one of the following two criteria to identify characteristic species in each of the seven terrestrial regions of Canada:
- Relative Density: Species were considered characteristic if the species' average density within the region was 20% or more of their maximum density in any of the other six terrestrial regions, or
- Relative Range: Species were considered characteristic if the proportion of the region overlapped by the species' range was 50% or more of the maximum proportion in any of the other six terrestrial regions
The relative density criterion was used for species with available density data (e.g., relative abundance on Breeding Bird Survey routes). The relative range criterion was used if density data were not available. In the Oceans region, all seabird species that regularly occur in Canada were considered characteristic.
Only characteristic species for a given region were included in the regional indicators and sub-indicators, and the "All birds" indicator in each region included all characteristic species for which suitable data were available.
We calculated indicators for selected sub-groups of each region's characteristic species. Similar to the national analysis, these sub-indicators represented the bird communities of the dominant, broadly defined landcover types in each region (forest birds, grassland birds, etc.). And again, we separated waterfowl from all other water birds (i.e., species associated with water or wetlands).
We presented sub-indicators on a each regional graph if there was a sufficiently large group of characteristic species associated with the habitat type (i.e., we excluded sub-indicators for a region if there were only a few characteristic species in that region, such as grassland birds in the boreal regions). For the Arctic, we defined sub-indicators based primarily on taxonomy (e.g., shorebirds and landbirds) because these groupings provided more relevant summaries of the region's bird communities. For the Oceans, we defined sub-indicators based on Canada's three major coasts (Atlantic, Pacific, Arctic), because the species, population status data, and key human and natural factors affecting bird populations are relatively distinct among the coasts.
In some cases, additional sub-indicators were calculated to represent a group of species that was known to have distinct and noteworthy trends (e.g., aerial insectivores and raptors), or groups of species of particular relevance to a region that were not well represented by the other sub-indicators (e.g., colonial waterbirds in the Lower Great Lakes-St.Lawrence and Pacific Coast birds in the West Coast and Mountains region). The list of species included in these additional sub-indicators overlapped to some degree with the lists for the landcover-based sub-indicators.
Lists of the species considered characteristic for each region and of the species included in each subindicator group are available here as supplementary material (Excel file).
Calculating the indicators
Data sources used
We considered any data source that could provide annual or periodic estimates of species' population status (index or population estimate), at a regional, national, or continental level, over the long-term (i.e., > 20 years). We also included data from the Marsh Monitoring Program, although it only has annual estimates over 15 years because it was the best available data source for one species (Least Bittern). For each analysis and species, we selected the most appropriate data source by giving priority to estimates that:
- most clearly reflected the species' population status in the region being considered,
- provided frequent estimates (i.e., preferring annual over periodic estimates),
- provided a measure of the estimate's precision, and
- provided estimates across the greatest proportion of the full time period.
To meet the first of the above criteria-regional estimates-we selected data sources in the following order, depending on availability:
- regional, breeding season estimates;
- regional, non-breeding season estimates for non-migratory species;
- national, breeding season estimates, if the region represented >50% of the national population or national range;
- national, non-breeding season estimates for non-migratory species, if the region represented >50% of the national population or national range; or
- continental, non-breeding season estimates, if the region represented >50% of the continental population or continental range.
In some cases, we gave priority to certain data sources, if those data were thought to better represent a species' status. Very rarely, we removed a species from an analysis if there was reason to believe the available data did not accurately represent the species' status.
Table 2: Data sources (surveys) that provided annual or periodic estimates of species' population status used in the State of Canada's Birds - see supplementary material for details of which surveys were used for each species in each region.
|Survey||Scale of existing estimates used in our analyses||Survey acronym|
|Breeding Bird Survey||Regional, National||BBS|
|Christmas Bird Count||Regional, Continental||CBC|
|Eastern Breeding Waterfowl Survey||Regional, National||Eastern Waterfowl|
|Western (traditional) Breeding Waterfowl Survey||Regional, National||Western Waterfowl|
|Southern Ontario Ground Waterfowl Survey||Regional||S.Ont Waterfowl|
|Migration surveys for shorebirds||Continental||Shorebird|
|Seabird Colony surveys||Regional||Seabird|
|Great Lakes Colonial Waterbird Surveys||Regional||Colonial Waterbird|
|Mid-Winter Waterfowl Survey||Continental||Winter Waterfowl|
|Staging-area Snow Goose survey||National||Staging Waterfowl|
|Marsh Monitoring Program (Least Bittern)||Regional||MMP|
|Singing Ground Survey (American Woodcock)||Regional, National||SGS|
Standardizing across species and data sources
The original species-specific estimates of population status from each data-source were in different units, e.g., BBS estimates are scaled to the number of birds seen on a single route, CBC estimates are scaled to the number observed in an average count-circle, and waterfowl survey estimates are regional population sizes. To be comparable among species and data-sources, the annual estimates of population status for each species were re-scaled as follows:
Rationale for using 1990 as the base-year
The analysis was conducted using 1990 as the base year for the following reasons:
- All of the main data sources have estimates for 1990 but this is not true for 1970 or for 2010, the beginning and end of the time series of interest for this report. BBS estimates for some regions begin in 1970, 1973, or 1986, the Eastern waterfowl survey began in 1990, and the CBC estimates extend to 2006.
- For many species, annual status estimates are more precise in 1990 than in 1970. For these species, the high variance in 1970 (if 1970 was used as the base year) would affect the variance around all annual estimates, and therefore, down-weight these species' importance in the overall analysis, even if their estimates were relatively precise in later years. This effect is reduced by using 1990 as the base-year.
Accounting for relative precision among species and years
Species-specific estimates of population status vary in their precision among species and among years. To calculate the indicators in the State of Canada's Birds, we used a model that explicitly considers the relative precision of each species' estimate of population status. Accounting for the relative precision ensures that extreme values for species whose status is imprecisely estimated do not dominate our indicators. Precision is defined as the inverse of variance (high precision = low variance and vice versa) and in most statistical literature, all of the equations below and the data-sources used in our analyses variance is the quantity used. Hereafter, both terms are used, but the reader should be aware that conceptually they are measures of the same thing-uncertainty.
We estimated the variance around each re-scaled species-specific annual estimate in a way that accounts for the variance in both a given year (year-y) and in the base-year (i.e., 1990).
We estimated the variance of each re-scaled annual estimate using an approximation of the variance of a ratio of two random variables (Cochran 1977):
We could not apply equation 2 directly, because the covariance of each annual estimate with the estimate in the base year is not known. In addition, it would be very complicated to estimate after the fact; it would depend on numerous factors including: the number of years separating year y and 1990, the population trend and annual fluctuations for each species, and the analytical methods used to estimate the original estimates for each data source. Therefore, we approximated the variance of the re-scaled estimates by assuming that the annual estimates were independent (i.e., that the covariance term in equation 2 equalled zero), such that:
For the base-year, the variance of the re-scaled estimate was derived directly from the coefficient of variation (CV) of the original base-year estimate (the second term inside the brackets of Equation 3).
Species data with no variance estimates
For some species status estimates, variance estimates do not exist e.g., most wintering and staging waterfowl surveys and colony counts for nesting seabirds and waterbirds. Estimates from these surveys are generally regarded as a complete census of a highly concentrated grouping of birds. We derived coarse variance estimates, in order to include these data in models that also include estimates with measured variance. For winter and staging waterfowl surveys, the variances were set so that the CV of the estimates would be equal to the average annual CV from the Quebec, Spring Snow Goose survey estimates-the only survey of this kind for which variance estimates were available. Similarly, the variance estimates for seabird and colonial waterbird colony surveys were set to the average annual CV across all colony surveys for which variances were available.
Statistical model for the indicators
For each year, the standardized individual species estimates ( ) were combined into a composite indicator (Iy) using a Bayesian hierarchical model, described in Sauer and Link (2011). The model generates a Bayesian estimate of the geometric average population status across all species in the composite species group, which is consistent with other approaches to composite species indicators (e.g., Collen et al. 2009 and Gregory et al. 2005). It also accounts for the varying precision of each species' estimate, so that imprecise estimates have less influence on the indicator. The composite indicators represent our best estimate of the group's status, given the distributional assumptions of the model, the species for which we have data and the precision of the data we do have.
As an example, the grassland birds sub-indicator from the national overview section is a compilation of the annual estimates for 21 grassland bird species (Figure 1). The indicator line (thick black line) can be roughly interpreted as the average of the 21 species-specific lines (solid and dotted grey lines), where each of the species lines influences the indicator to a degree dependent on its precision. Hence, the grassland bird indicator runs approximately through the centre of the more precise species lines (solid grey lines). The indicator does not run as closely through the centre of the less precise species lines (dotted grey lines), which are more numerous and follow more extreme trends below the indicator line than above. This apparent imbalance reflects the reduced influence that less precise species estimates have on the indicator.
Figure 1. National grassland birds sub-indicator (thick black line) and lines showing the species-specific annual estimates of population status for the 21 species included in the grassland birds group (grey lines). Dotted lines indicate species with relatively imprecise estimates of population status and solid lines species with more precise estimates. The coloured axis on the right side of the line plot indicates the ranges of the long-term change categories used to group species in the bar-graph on the right. The number of species-specific (grey) lines that end inside each coloured section determines the size of the coloured bars in the bar graph, as described in the section on "Variation among species".
Formally, the composite indicator in a given year (Iy) is exponent of the hyperparameter iy, estimated from the median of the posterior distribution. The hyperparameter iy is the average across species of the lognormally distributed species-specific change parameters (). The following distributional assumptions are adapted from Sauer and Link (2011).
Re-scaling for final presentation
The output from the Bayesian hierarchical model is a composite species indicator that shows the average status of the group with respect to 1990. Because our primary interest was to examine and display long-term changes, we re-scaled all indicators in the report and plotted them as a percent change since the indicator's value in the first year of the time series - in most cases, 1970. This re-scaling is effectively a log-scale shift of the indicator line so that it is anchored at 0 in 1970, from its original position, anchored at 1.0 in 1990.
Open circles were plotted along the indicator lines in years where the perceived increase or decrease in the indicator was relatively uncertain, i.e., years where there was a > 5% probability that the value of the indicator was on the other side of the zero line. To estimate this probability, we calculated the proportion of the posterior distribution of the indicator in each year that was above or below the average value of the indicator in the first year of the time series (usually 1970).
Variation among species
To depict some of the variation among individual species' trends within each indicator line, we presented bar graphs showing the number of species in each of 5 categories based on their long term (~40 year) changes in population (Table 3). The categories were symmetrical in the log-scale and reflect the change in population required to balance out the opposite decrease or increase in each corresponding category. For example, a population that has declined by 50% (reduced to ½ of its original abundance) must then increase by 100% (i.e., double) to return to its original level. The specific categories were adapted from Blancher et al. (2009).
The variation depicted in these bar graphs highlights the appropriate interpretation of these indicators-as indicators of the average or overall status of the group, not as indicators of the status of individual species within the group. Our indicators give the best overall estimate of the group's status, but do not reflect the trends for all species in a group equally well; a stable indicator may reflect a group in which most or all species have stable trends, or it may reflect a group with an equal number of species with large increases and large decreases. Almost all of the indicators in the report, regardless of their overall pattern of change, include both species that are increasing and species that are decreasing. For example, populations of grassland birds as a group have decreased, but not all grassland bird populations have decreased (Figure 1).
Table 3. Categories of long-term population change used to group species for the bar graphs in the State of Canada's Birds.
|Long-term population change category||Range of estimates of total population change over a 40-year period||Colour of long-term population change category in bar graphs|
|Strong Increase||> 100% increase||Dark blue|
|Increase||33% - 100% increase||Light blue|
|Little Change||33% increase - 25% decrease||Grey|
|Decrease||25% - 50% decrease||Light Orange|
|Strong Decrease||> 50% decrease||Dark Orange|
Note: estimates of long-term population change were calculated based on an estimate of the percent annual change in the species' status, i.e., the slope of a regression of the log of each species' population status estimates on year (estimates of percent annual change are available in the supplementary lists of characteristic species and data available here). If population status estimates were not available for the full 40-year time series, the percent annual change was extrapolated to 40 years to be comparable among species and regions. These estimates of total change are therefore approximate and may not agree with similar published estimates for all species, which may vary both in the length of time considered and in their methods of calculation.
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Collen, B., Loh, J., Whitmee, S., Mcrae, L., Amin, R. and Baillie, J. E. M. 2009. Monitoring Change in Vertebrate Abundance: the Living Planet Index. Conservation Biology, 23: 317-327.
Gregory, R. D., A. van Strien, P. Vorisek, A. W. G. Meyling, D. G. Noble, R. P. B. Foppen, and D. W. Gibbons. 2005. Developing indicators for European birds. Philosophical Transactions of the Royal Society of London B, 360:269-288.
Sauer, J.R. and W.A. Link. 2011. Analysis of the North American Breeding Bird Survey Using Hierarchical Models. The Auk, 128:87-98.