Calculates the quasi-Akaike Information Criterion (QAIC) for one or more detection function models within the same key function family. If multiple models are provided, all must have the same key function. This function is typically used as the first step of a two-step model selection approach (Howe et al., 2019).
Arguments
- models
A list of fitted detection function models (objects returned by
Distance::ds()orct_fit_ds()).- chat
Optional numeric value of overdispersion (\(\hat{c}\)). If not provided, it is estimated from the most parameterised model in each key function set.
- k
Numeric. The penalty term used in QAIC (default is
2).
Value
A tibble with one row per model containing:
model: The model namedf: The degrees of freedom for the model.QAIC: The computed QAIC value.
Details
If only one model is supplied and chat is not provided, the function
estimates \(\hat{c}\) using the provided model and issues a warning that
model selection cannot be performed. For multiple models, All models must use the same key function.
QAIC is calculated as: $$QAIC = -2 \times \log(L) / \hat{c} + 2k$$ where \(L\) is the likelihood, \(\hat{c}\) is the estimated overdispersion, and \(k\) is the number of parameters.
References
Howe, E. J., Buckland, S. T., Després‐Einspenner, M., & Kühl, H. S. (2019). Model selection with overdispersed distance sampling data. Methods in Ecology and Evolution, 10(1), 38-47. doi:10.1111/2041-210X.13082
Examples
# \donttest{
library(Distance)
#> Loading required package: mrds
#> This is mrds 3.0.1
#> Built: R 4.5.0; ; 2025-07-06 04:17:12 UTC; unix
#>
#> Attaching package: ‘Distance’
#> The following object is masked from ‘package:mrds’:
#>
#> create.bins
library(dplyr)
#>
#> Attaching package: ‘dplyr’
#> The following objects are masked from ‘package:stats’:
#>
#> filter, lag
#> The following objects are masked from ‘package:base’:
#>
#> intersect, setdiff, setequal, union
data("duiker")
#> Warning: data set ‘duiker’ not found
duiker_data <- duikers$DaytimeDistances %>%
dplyr::slice_sample(prop = .3) # sample 30% of rows
truncation <- list(left = 2, right = 15) # Keep only distance between 2-15 m
# fit hazard-rate key models
w3_hr0 <- ds(duiker_data, transect = "point", key = "hr", adjustment = NULL,
truncation = truncation)
#> Warning: Unknown or uninitialised column: `distbegin`.
#> Warning: Unknown or uninitialised column: `distend`.
#> Warning: Unknown or uninitialised column: `distbegin`.
#> Fitting hazard-rate key function
#> AIC= 15007.689
w3_hr1 <- ds(duiker_data, transect = "point", key = "hr", adjustment = "cos",
order = 2, truncation = truncation)
#> Warning: Unknown or uninitialised column: `distbegin`.
#> Warning: Unknown or uninitialised column: `distend`.
#> Warning: Unknown or uninitialised column: `distbegin`.
#> Fitting hazard-rate key function with cosine(2) adjustments
#> AIC= 15009.689
w3_hr2 <- ds(duiker_data, transect = "point", key = "hr", adjustment = "cos",
order = c(2, 4), truncation = truncation)
#> Warning: Unknown or uninitialised column: `distbegin`.
#> Warning: Unknown or uninitialised column: `distend`.
#> Warning: Unknown or uninitialised column: `distbegin`.
#> Fitting hazard-rate key function with cosine(2,4) adjustments
#> AIC= 15011.693
# fit half-normal key models
w3_hn0 <- ds(duiker_data, transect = "point", key = "hn", adjustment = NULL,
truncation = truncation)
#> Warning: Unknown or uninitialised column: `distbegin`.
#> Warning: Unknown or uninitialised column: `distend`.
#> Warning: Unknown or uninitialised column: `distbegin`.
#> Fitting half-normal key function
#> AIC= 15034.041
w3_hn1 <- ds(duiker_data, transect = "point", key = "hn", adjustment = "cos",
order = 2, truncation = truncation)
#> Warning: Unknown or uninitialised column: `distbegin`.
#> Warning: Unknown or uninitialised column: `distend`.
#> Warning: Unknown or uninitialised column: `distbegin`.
#> Fitting half-normal key function with cosine(2) adjustments
#> AIC= 15009.335
w3_hn2 <- ds(duiker_data, transect = "point", key = "hn", adjustment = "cos",
order = c(2, 4), truncation = truncation)
#> Warning: Unknown or uninitialised column: `distbegin`.
#> Warning: Unknown or uninitialised column: `distend`.
#> Warning: Unknown or uninitialised column: `distbegin`.
#> Fitting half-normal key function with cosine(2,4) adjustments
#> AIC= 15004.757
# fit uniform key models
w3_u0 <- ds(duiker_data, transect = "point", key = "unif", adjustment = NULL,
truncation = truncation)
#> Warning: Unknown or uninitialised column: `distbegin`.
#> Warning: Unknown or uninitialised column: `distend`.
#> Warning: Unknown or uninitialised column: `distbegin`.
#> Fitting uniform key function
#> AIC= 17311.325
w3_u1 <- ds(duiker_data, transect = "point", key = "unif", adjustment = "cos",
order = 2, truncation = truncation)
#> Warning: Unknown or uninitialised column: `distbegin`.
#> Warning: Unknown or uninitialised column: `distend`.
#> Warning: Unknown or uninitialised column: `distbegin`.
#> Fitting uniform key function with cosine(2) adjustments
#> AIC= 17313.325
w3_u2 <- ds(duiker_data, transect = "point", key = "unif", adjustment = "cos",
order = c(2, 4), truncation = truncation)
#> Warning: Unknown or uninitialised column: `distbegin`.
#> Warning: Unknown or uninitialised column: `distend`.
#> Warning: Unknown or uninitialised column: `distbegin`.
#> Fitting uniform key function with cosine(2,4) adjustments
#> AIC= 17315.325
# Create model list
model_list <- list(w3_hn0, w3_hn1, w3_hn2,
w3_hr0, w3_hr1, w3_hr2,
w3_u0, w3_u1, w3_u2)
# Compute model QAICs
ct_QAIC(list(w3_hr0, w3_hr1, w3_hr2)) # All key functions must be the same
#> # A tibble: 3 × 3
#> model df QAIC
#> <chr> <int> <dbl>
#> 1 hazard-rate key function 2 52.3
#> 2 hazard-rate key function with cosine(2) adjustments 3 54.3
#> 3 hazard-rate key function with cosine(2,4) adjustments 4 56.3
ct_QAIC(list(w3_hn0, w3_hn1, w3_hn2)) # All key functions must be the same
#> # A tibble: 3 × 3
#> model df QAIC
#> <chr> <int> <dbl>
#> 1 half-normal key function 1 51.7
#> 2 half-normal key function with cosine(2) adjustments 2 53.6
#> 3 half-normal key function with cosine(2,4) adjustments 3 55.6
# Compute Chi-squared Goodness-of-fit
ct_chi2_select(list(w3_hn0, w3_hr0, w3_u0)) # All key functions must be different
#> # A tibble: 3 × 3
#> key model criteria
#> <chr> <chr> <dbl>
#> 1 half-normal half-normal key function 305.
#> 2 hazard-rate hazard-rate key function 312.
#> 3 uniform uniform key function 455.
ct_chi2_select(list(w3_hn2, w3_hr1, w3_u0)) # All key functions must be different
#> # A tibble: 3 × 3
#> key model criteria
#> <chr> <chr> <dbl>
#> 1 half-normal half-normal key function with cosine(2,4) adjustments 315.
#> 2 hazard-rate hazard-rate key function with cosine(2) adjustments 318.
#> 3 uniform uniform key function 455.
# Two-step model selection
ct_select_model(model_list)
#> $QAIC
#> # A tibble: 9 × 6
#> id key model df QAIC best
#> <int> <chr> <chr> <int> <dbl> <lgl>
#> 1 1 half-normal half-normal key function 1 51.7 TRUE
#> 2 2 half-normal half-normal key function with cosine(2) a… 2 53.6 FALSE
#> 3 3 half-normal half-normal key function with cosine(2,4)… 3 55.6 FALSE
#> 4 4 hazard-rate hazard-rate key function 2 52.3 TRUE
#> 5 5 hazard-rate hazard-rate key function with cosine(2) a… 3 54.3 FALSE
#> 6 6 hazard-rate hazard-rate key function with cosine(2,4)… 4 56.3 FALSE
#> 7 7 uniform uniform key function 0 38.7 TRUE
#> 8 8 uniform uniform key function with cosine(2) adjus… 1 40.7 FALSE
#> 9 9 uniform uniform key function with cosine(2,4) adj… 2 42.7 FALSE
#>
#> $`Best QAIC models`
#> $`Best QAIC models`[[1]]
#>
#> Distance sampling analysis object
#>
#> Detection function:
#> Half-normal key function
#>
#> Estimated abundance in covered region: 11630.7
#>
#> $`Best QAIC models`[[2]]
#>
#> Distance sampling analysis object
#>
#> Detection function:
#> Hazard-rate key function
#>
#> Estimated abundance in covered region: 8213.104
#>
#> $`Best QAIC models`[[3]]
#>
#> Distance sampling analysis object
#>
#> Detection function:
#> Uniform key function
#>
#> Estimated abundance in covered region: 3133.71
#>
#>
#> $Chi2
#> # A tibble: 3 × 4
#> key model criteria best
#> <chr> <chr> <dbl> <lgl>
#> 1 half-normal half-normal key function 305. TRUE
#> 2 hazard-rate hazard-rate key function 312. FALSE
#> 3 uniform uniform key function 455. FALSE
#>
#> $`Final model`
#>
#> Distance sampling analysis object
#>
#> Detection function:
#> Half-normal key function
#>
#> Estimated abundance in covered region: 11630.7
#>
# }