Recall, these are data from Colorado, described by Knopf et al. (1988). The question here was whether to include pasture as a covariate in the detection function. The biological question being, “does detectability of Savannah sparrows differ between the pastures in which the survey was conducted.”
Region.Label Area Sample.Label Effort object distance Study.Area
1 PASTURE 1 1 POINT 1 1 NA NA SASP 1980
2 PASTURE 1 1 POINT 2 1 NA NA SASP 1980
3 PASTURE 1 1 POINT 3 1 NA NA SASP 1980hist(Savannah_sparrow_1980$distance,
nclass=20, xlab="Distance (m)",
main="Savannah sparrow radial distances '80")
A truncation distance of 55m was chosen. The half normal and hazard rate functions were tried in turn, allowing AIC selection of cosine adjustment terms, then pasture was included as a covariate in the detection function.
Savannah_sparrow_1980.hn <- ds(data=Savannah_sparrow_1980, key="hn",
adjustment="cos", truncation=55,
transect="point",
convert_units=conversion.factor)
Savannah_sparrow_1980.hr <- ds(data=Savannah_sparrow_1980, key="hr",
adjustment="cos", truncation=55,
transect="point",
convert_units=conversion.factor)
Savannah_sparrow_1980.hn.region <- ds(data=Savannah_sparrow_1980, key="hn",
truncation=55,
transect="point",
convert_units=conversion.factor,
formula=~Region.Label)
Savannah_sparrow_1980.hr.region <- ds(data=Savannah_sparrow_1980, key="hr", truncation=55,
transect="point", convert_units=conversion.factor,
formula=~Region.Label)
AIC(Savannah_sparrow_1980.hn, Savannah_sparrow_1980.hr,
Savannah_sparrow_1980.hn.region, Savannah_sparrow_1980.hr.region) df AIC
Savannah_sparrow_1980.hn 2 2126.228
Savannah_sparrow_1980.hr 2 2129.961
Savannah_sparrow_1980.hn.region 3 2125.033
Savannah_sparrow_1980.hr.region 4 2130.447The half normal model with pasture as a covariate had a marginally smaller AIC than the half normal model without pasture. The plots and estimates are shown below.
Summary for distance analysis
Number of observations : 271
Distance range : 0 - 55
Model : Half-normal key function
AIC : 2125.033
Optimisation: mrds (nlminb)
Detection function parameters
Scale coefficient(s):
estimate se
(Intercept) 3.17237666 0.09850468
Region.LabelPASTURE 2 -0.20459862 0.11098924
Region.LabelPASTURE 3 -0.03286901 0.12333031
Estimate SE CV
Average p 0.2896553 0.01973518 0.06813332
N in covered region 935.5946771 79.89055349 0.08539013
Summary statistics:
Region Area CoveredArea Effort n k ER se.ER cv.ER
1 PASTURE 1 1 117.8411 124 59 124 0.4758065 0.07009015 0.14730811
2 PASTURE 2 1 119.7418 126 121 126 0.9603175 0.09521957 0.09915427
3 PASTURE 3 1 116.8908 123 91 123 0.7398374 0.07467322 0.10093193
4 Total 3 354.4738 373 271 373 0.7253204 0.04661364 0.06426628
Abundance:
Label Estimate se cv lcl ucl df
1 PASTURE 1 1.430170 0.3084008 0.21563923 0.9403572 2.175117 353.1228
2 PASTURE 2 4.116182 0.5642114 0.13707154 3.1472623 5.383394 329.1061
3 PASTURE 3 2.345631 0.3717609 0.15849077 1.7208931 3.197169 374.9983
4 Total 7.891983 0.7427326 0.09411229 6.5625680 9.490706 537.6070
Density:
Label Estimate se cv lcl ucl df
1 PASTURE 1 1.430170 0.3084008 0.21563923 0.9403572 2.175117 353.1228
2 PASTURE 2 4.116182 0.5642114 0.13707154 3.1472623 5.383394 329.1061
3 PASTURE 3 2.345631 0.3717609 0.15849077 1.7208931 3.197169 374.9983
4 Total 2.630661 0.2475775 0.09411229 2.1875227 3.163569 537.6070A similar process was conducted for the 1981 data: a truncation distance of 55m was again used.
Savannah_sparrow_1981.hn <- ds(data=Savannah_sparrow_1981, key="hn",
adjustment="cos", truncation=55,
transect="point", convert_units=conversion.factor)
Savannah_sparrow_1981.hr <- ds(data=Savannah_sparrow_1981, key="hr",
adjustment="cos", truncation=55,
transect="point", convert_units=conversion.factor)
Savannah_sparrow_1981.hn.region <- ds(data=Savannah_sparrow_1981,
key="hn", truncation=55, transect="point",
convert_units=conversion.factor,
formula=~Region.Label)
Savannah_sparrow_1981.hr.region <- ds(data=Savannah_sparrow_1981, key="hr",
truncation=55, transect="point",
convert_units=conversion.factor,
formula=~Region.Label)
AIC(Savannah_sparrow_1981.hn, Savannah_sparrow_1981.hr,
Savannah_sparrow_1981.hn.region, Savannah_sparrow_1981.hr.region) df AIC
Savannah_sparrow_1981.hn 1 1266.358
Savannah_sparrow_1981.hr 2 1267.335
Savannah_sparrow_1981.hn.region 4 1261.684
Savannah_sparrow_1981.hr.region 5 1260.638For 1981, there was a clear preference for including pasture as a covariate in the detection function but little to choose from between the half normal and hazard rate key function. For comparability with 1980, the plots and results below are for the half normal model although AIC showed a slight preference for the hazard rate model. The differences in detection between pastures can easily be seen and this is reflected in the estimated densities (birds per hectare).
pastures <- unique(Savannah_sparrow_1981$Region.Label)
plot(Savannah_sparrow_1981.hn.region, showpoints=FALSE,
main="Savannah sparrows with pasture covariate", pdf=TRUE)
k <- 1
for (i in pastures) {
k <- k+1
add_df_covar_line(Savannah_sparrow_1981.hn.region,
data=data.frame(Region.Label=as.character(i)),
lty=1, col=k, lwd=3, pdf=TRUE)
}
legend("topright", legend=tolower(as.character(pastures)),
col=2:k, lwd=2, title = "Pastures")
text(-2,0.038, cex=0.9, pos=4,
expression(widehat(sigma[p])==plain(exp)(widehat(beta[0]) + widehat(beta[1]) * p[1] + widehat(beta[2]) * p[2] + widehat(beta[3]) * p[3])))
library(plotrix)
parms <- data.frame(est=c(2.944, 0.736, 0.166, 0.271),
se=c(0.111, 0.373, 0.153, 0.179))
rownames(parms) <- c("b0", "b1", "b2", "b3")
addtable2plot(2, 0.027, parms, bty="o",
display.rownames=TRUE,hlines=TRUE, cex=0.8,
xpad=0.4, vlines=FALSE,title="Parameter estimates")
Summary for distance analysis
Number of observations : 162
Distance range : 0 - 55
Model : Half-normal key function
AIC : 1261.684
Optimisation: mrds (nlminb)
Detection function parameters
Scale coefficient(s):
estimate se
(Intercept) 2.9440471 0.1110272
Region.LabelPASTURE 1 0.7362681 0.3726357
Region.LabelPASTURE 2 0.1660949 0.1524426
Region.LabelPASTURE 3 0.2703034 0.1790810
Estimate SE CV
Average p 0.3435487 0.03715842 0.1081606
N in covered region 471.5489313 59.62606186 0.1264472
Summary statistics:
Region Area CoveredArea Effort n k ER se.ER cv.ER
1 PASTURE 0 1 95.03318 100 31 100 0.310 0.05448566 0.17576019
2 PASTURE 1 1 95.03318 100 32 100 0.320 0.06175874 0.19299605
3 PASTURE 2 1 95.03318 100 51 100 0.510 0.08225975 0.16129363
4 PASTURE 3 1 95.03318 100 48 100 0.480 0.07174590 0.14947063
5 Total 4 380.13271 400 162 400 0.405 0.03418422 0.08440547
Abundance:
Label Estimate se cv lcl ucl df
1 PASTURE 0 1.3887466 0.3779428 0.2721467 0.8203808 2.350880 255.9017
2 PASTURE 1 0.5241867 0.1817587 0.3467442 0.2699503 1.017861 250.9726
3 PASTURE 2 1.6980708 0.4057104 0.2389243 1.0676527 2.700732 251.7139
4 PASTURE 3 1.3509360 0.3544528 0.2623757 0.8127396 2.245526 253.0658
5 Total 4.9619401 0.6827259 0.1375925 3.7903971 6.495586 357.0057
Density:
Label Estimate se cv lcl ucl df
1 PASTURE 0 1.3887466 0.3779428 0.2721467 0.8203808 2.350880 255.9017
2 PASTURE 1 0.5241867 0.1817587 0.3467442 0.2699503 1.017861 250.9726
3 PASTURE 2 1.6980708 0.4057104 0.2389243 1.0676527 2.700732 251.7139
4 PASTURE 3 1.3509360 0.3544528 0.2623757 0.8127396 2.245526 253.0658
5 Total 1.2404850 0.1706815 0.1375925 0.9475993 1.623896 357.0057In these models, the detection functions have been fitted to all the detections within the study region (for each year). An alternative would be to fit separate detection functions within each pasture (specified in Region.Label), provided there are enough detections. This would allow different shape detection functions to be fitted in each pasture (providing this is a reasonable thing to do).