Select best detection function model by Chi-squared Goodness-of-fit
Source:R/CTDS.R
ct_chi2_select.RdCompares detection function models with different key functions using the ratio of the chi-squared statistic to its degrees of freedom. This method selects the best model among different key functions after the best adjustment term model is chosen for each key function.
Arguments
- models
A list of fitted detection function models (objects returned by
Distance::ds()orct_fit_ds()).
Value
A tibble with one row per model containing:
key: The key function of the model.model: The model name.criteria: The chi-squared goodness-of-fit statistic divided by its degrees of freedom, i.e. \(\chi^2/\mathrm{df}\). Lower values indicate better fit.
Details
If only one model is supplied, the function returns the chi-squared
goodness-of-fit ratio for that model and issues a warning that model
selection cannot be performed. For multiple models, each must have a unique key function.
This step is designed to be applied after selecting the best model within
each key function family using QAIC (see ct_QAIC()).
The model with the smallest chi-squared/df ratio is typically preferred.
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)
library(dplyr)
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= 15016.7
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= 15017.437
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= 15014.516
# 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= 15028.793
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= 15002.329
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= 15000.258
# 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= 17381.336
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= 17383.336
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= 17385.336
# 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 52.2
#> 2 hazard-rate key function with cosine(2) adjustments 3 54.2
#> 3 hazard-rate key function with cosine(2,4) adjustments 4 56.2
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 51.3
#> 2 half-normal key function with cosine(2) adjustments 2 53.2
#> 3 half-normal key function with cosine(2,4) adjustments 3 55.2
# 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 308.
#> 2 hazard-rate hazard-rate key function 315.
#> 3 uniform uniform key function 462.
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 318.
#> 2 hazard-rate hazard-rate key function with cosine(2) adjustments 319.
#> 3 uniform uniform key function 462.
# 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 51.3 TRUE
#> 2 2 half-normal half-normal key function with cosine(2) a… 2 53.2 FALSE
#> 3 3 half-normal half-normal key function with cosine(2,4)… 3 55.2 FALSE
#> 4 4 hazard-rate hazard-rate key function 2 52.2 TRUE
#> 5 5 hazard-rate hazard-rate key function with cosine(2) a… 3 54.2 FALSE
#> 6 6 hazard-rate hazard-rate key function with cosine(2,4)… 4 56.2 FALSE
#> 7 7 uniform uniform key function 0 38.2 TRUE
#> 8 8 uniform uniform key function with cosine(2) adjus… 1 40.2 FALSE
#> 9 9 uniform uniform key function with cosine(2,4) adj… 2 42.2 FALSE
#>
#> $`Best QAIC models`
#> $`Best QAIC models`[[1]]
#>
#> Distance sampling analysis object
#>
#> Detection function:
#> Half-normal key function
#>
#> Estimated abundance in covered region: 11870.29
#>
#> $`Best QAIC models`[[2]]
#>
#> Distance sampling analysis object
#>
#> Detection function:
#> Hazard-rate key function
#>
#> Estimated abundance in covered region: 8205.621
#>
#> $`Best QAIC models`[[3]]
#>
#> Distance sampling analysis object
#>
#> Detection function:
#> Uniform key function
#>
#> Estimated abundance in covered region: 3139.819
#>
#>
#> $Chi2
#> # A tibble: 3 × 4
#> key model criteria best
#> <chr> <chr> <dbl> <lgl>
#> 1 half-normal half-normal key function 308. TRUE
#> 2 hazard-rate hazard-rate key function 315. FALSE
#> 3 uniform uniform key function 462. FALSE
#>
#> $`Final model`
#>
#> Distance sampling analysis object
#>
#> Detection function:
#> Half-normal key function
#>
#> Estimated abundance in covered region: 11870.29
#>
# }