I have a data set of counts of various bird species in three national parks. My aim is to see if they’re declining in any of those areas. I’m using `brms`

to fit a hierarchical model (I give some more detail of the model below*).

My question is what’s the best way to assess whether my coefficients on the dates are negative which would indicate a decline?

I’ve seen some authors argue for using a ROPE but I’d appreciate your thoughts on the intervals to specify when I’m only interested if there’s a decline. This is complicated because I’m working on the log scale given a negative binomial error structure.

Here’s my fixed effect output:

```
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept -0.13 0.57 -1.21 1.10 1.00 2424 2469
ndate -0.00 0.01 -0.02 0.01 1.00 2684 2657
NPRuaha -0.09 0.69 -1.48 1.32 1.00 2840 2476
NPSelous 0.54 0.86 -1.24 2.14 1.00 3085 2712
SeasonWet 0.14 0.16 -0.17 0.45 1.00 5330 3321
CarcassPres1 0.62 0.27 0.11 1.18 1.00 4709 2861
ndate:NPRuaha -0.01 0.01 -0.03 0.00 1.00 2836 2964
ndate:NPSelous -0.02 0.01 -0.05 0.01 1.00 2702 2574
```

Here’s the model:

```
library(brms)
library(bayestestR)
#' set normal priors on the intercept terms
newprior_1 <-
c(
set_prior("normal(0,1)", class = "b", coef = "Intercept"),
set_prior("normal(0,1)", class = "b", coef = "NPRuaha"),
set_prior("normal(0,1)", class = "b", coef = "NPSelous")
)
#' fit the model with tighter normal priors
nbglm_norm_prior_tight <-
brm(
AWB ~ 0 + Intercept + # this allows control of prior on the intercept
ndate * NP + Season + CarcassPres +
(1 | NP / StandardTransect) +
offset(log(Tlength)),
family = negbinomial,
data = mydata,
chains = 4,
control = list(adapt_delta = 0.999, max_treedepth = 15),
prior = newprior_1,
save_pars = save_pars(all = TRUE)
)
```

My use of ROPE drawn from the `equivalence_test`

function:

```
#' run a test using ROPE
bayestestR::equivalence_test(draws$b_Intercept)
bayestestR::equivalence_test(draws$b_ndate + draws$`b_ndate:NPRuaha`)
bayestestR::equivalence_test(draws$b_ndate + draws$`b_ndate:NPSelous`)
```

*I modelled the counts as a function of time (in months) interacting with park, the presence of carrion, the season (wet or dry), and a random effect of the transect nested within the park. We also included the log length of the transect as an offset to account for unequal sampling effort.

Full data set:

```
mydata <- structure(list(AWB = c(7, 66, 15, 44, 22, 60, 45, 32, 30, 33,
14, 0, 45, 39, 39, 24, 37, 66, 37, 60, 18, 3, 25, 13, 34, 38,
58, 0, 12, 6, 33, 2, 34, 18, 75, 20, 4, 9, 15, 4, 0, 21, 50,
24, 21, 9, 5, 87, 13, 43, 1, 19, 13, 1, 28, 56, 18, 42, 13, 2,
53, 16, 37, 51, 79, 5, 49, 11, 34, 91, 30, 2, 0, 15, 3, 57, 5,
18, 31, 14, 56, 72, 35, 94, 10, 45, 8, 29, 33, 34, 8, 53, 54,
24, 5, 21, 11, 27, 83, 23, 24, 4, 10, 13, 17, 11, 51, 6), ndate = c(0,
0, 0, 0, 0, 14, 14, 14, 14, 14, 14, 19, 19, 19, 19, 20, 20, 25,
25, 25, 25, 25, 25, 32, 32, 33, 33, 33, 33, 33, 38, 38, 38, 38,
38, 38, 39, 39, 39, 39, 39, 41, 41, 42, 42, 42, 42, 43, 43, 43,
44, 46, 46, 46, 46, 51, 51, 51, 51, 51, 51, 54, 54, 54, 55, 56,
56, 56, 58, 58, 61, 61, 61, 61, 61, 63, 63, 66, 66, 67, 67, 67,
68, 68, 68, 68, 68, 71, 72, 73, 78, 78, 79, 79, 89, 89, 89, 90,
90, 91, 91, 91, 96, 96, 96, 96, 97, 97), nyear = c(2013, 2013,
2013, 2013, 2013, 2014, 2014, 2014, 2014, 2014, 2014, 2015, 2015,
2015, 2015, 2015, 2015, 2015, 2015, 2015, 2015, 2015, 2015, 2016,
2016, 2016, 2016, 2016, 2016, 2016, 2016, 2016, 2016, 2016, 2016,
2016, 2016, 2016, 2016, 2016, 2016, 2017, 2017, 2017, 2017, 2017,
2017, 2017, 2017, 2017, 2017, 2017, 2017, 2017, 2017, 2017, 2017,
2017, 2017, 2017, 2017, 2018, 2018, 2018, 2018, 2018, 2018, 2018,
2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2018, 2019, 2019,
2019, 2019, 2019, 2019, 2019, 2019, 2019, 2019, 2019, 2019, 2019,
2020, 2020, 2020, 2020, 2021, 2021, 2021, 2021, 2021, 2021, 2021,
2021, 2021, 2021, 2021, 2021, 2021, 2021), NP = structure(c(1L,
1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 1L, 2L, 2L, 2L, 2L, 1L, 1L,
1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
2L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 2L, 2L, 2L, 2L, 1L, 1L, 2L, 2L, 2L, 2L, 1L, 1L, 3L, 3L,
2L, 2L, 2L, 3L, 3L, 2L, 2L, 2L, 2L, 2L, 1L, 1L, 1L, 1L, 3L, 3L,
3L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 1L, 1L, 3L, 3L, 2L, 2L, 2L,
1L, 1L, 3L, 3L, 3L, 2L, 2L, 2L, 3L, 3L, 1L), .Label = c("Katavi",
"Ruaha", "Selous"), class = "factor"), Season = c("Dry", "Dry",
"Dry", "Dry", "Dry", "Dry", "Dry", "Dry", "Dry", "Dry", "Dry",
"Wet", "Wet", "Wet", "Wet", "Wet", "Wet", "Dry", "Dry", "Dry",
"Dry", "Dry", "Dry", "Wet", "Wet", "Wet", "Wet", "Wet", "Wet",
"Wet", "Wet", "Dry", "Dry", "Dry", "Dry", "Dry", "Dry", "Dry",
"Dry", "Dry", "Dry", "Wet", "Wet", "Wet", "Wet", "Wet", "Wet",
"Wet", "Wet", "Wet", "Wet", "Dry", "Dry", "Dry", "Dry", "Dry",
"Dry", "Dry", "Dry", "Dry", "Dry", "Wet", "Wet", "Wet", "Wet",
"Wet", "Wet", "Wet", "Dry", "Dry", "Dry", "Wet", "Dry", "Dry",
"Dry", "Dry", "Dry", "Wet", "Wet", "Wet", "Wet", "Wet", "Wet",
"Wet", "Wet", "Wet", "Wet", "Dry", "Dry", "Dry", "Wet", "Wet",
"Wet", "Wet", "Wet", "Wet", "Wet", "Wet", "Wet", "Wet", "Wet",
"Wet", "Dry", "Dry", "Dry", "Dry", "Dry", "Dry"), CarcassPres = structure(c(1L,
1L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 2L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 1L), .Label = c("0",
"1"), class = "factor"), StandardTransect = structure(c(4L, 3L,
1L, 5L, 7L, 5L, 1L, 8L, 6L, 4L, 3L, 8L, 5L, 6L, 1L, 4L, 3L, 4L,
3L, 5L, 1L, 8L, 6L, 8L, 6L, 5L, 1L, 8L, 6L, 1L, 5L, 5L, 1L, 6L,
4L, 3L, 8L, 6L, 5L, 1L, 8L, 4L, 3L, 5L, 1L, 8L, 6L, 5L, 1L, 6L,
8L, 5L, 1L, 8L, 6L, 4L, 3L, 5L, 1L, 8L, 6L, 3L, 4L, 10L, 2L,
1L, 5L, 6L, 2L, 10L, 5L, 8L, 8L, 1L, 6L, 3L, 4L, 4L, 3L, 9L,
2L, 10L, 5L, 1L, 7L, 5L, 7L, 2L, 10L, 9L, 4L, 3L, 10L, 2L, 5L,
1L, 7L, 4L, 3L, 2L, 10L, 9L, 5L, 7L, 1L, 2L, 9L, 4L), .Label = c("Jongomero",
"Kidai", "LakeChada", "LakeKatavi", "Lunda", "Magangwe", "Mbagi-Mdonya",
"Mpululu", "Msolwa", "Mtemere"), class = "factor"), Tlength = c(35.2,
86.7, 93, 75, 27.2, 74.4, 93, 10.3, 45.8, 35.2, 78.2, 10.3, 71,
45.8, 93, 35.2, 63.9, 35.2, 77.9, 86.6, 93, 10.3, 45.8, 10.3,
45.8, 68.9, 93, 10.3, 45.8, 93, 86.7, 90.5, 93, 45.8, 35.2, 81.6,
10.3, 45.8, 88.2, 93, 10.3, 35.2, 64.6, 82.3, 93, 10.3, 45.8,
77.9, 93, 45.8, 10.3, 90.3, 93, 10.3, 45.8, 35.2, 77.4, 87.5,
93, 10.3, 45.8, 66, 35.2, 71.2, 85.7, 93, 87.5, 45.8, 85.5, 69.6,
97.8, 10.3, 10.3, 93, 45.8, 86.6, 35.2, 35.2, 71.9, 77.9, 88.5,
80, 85.2, 93, 56.1, 85.5, 56.1, 97.6, 81.8, 79.7, 35.2, 71.1,
81.9, 53.8, 86.5, 68.7, 60.7, 78.7, 56.6, 66.9, 71.8, 79.2, 82.6,
71.8, 92.4, 85.6, 78.5, 77.3)), row.names = c(NA, -108L), class = "data.frame")
```