Author: Ken Kellner

# Overview of ubms

### Unmarked Bayesian Models with Stan

#### Ken Kellner

# 1 Introduction

## 1.1 What is ubms?

`ubms` is an R package for fitting models of wildlife occurrence and abundance to with datasets where animals are not individually marked. It provides a nearly identical interface to the popular `unmarked` package (Fiske, Chandler, and others 2011). Instead of using maximum likelihood to fit models (as with `unmarked`), models are fit in a Bayesian framework using Stan (Carpenter et al. 2017). It is generally expected that you are already familiar with `unmarked` when using `ubms`. You can download `ubms`, report issues, or help with development on Github.

## 1.2 Why should you use it?

There are several advantages to using `ubms` over `unmarked`. First, it is possible to include random effects in `ubms` models, which is not currently possible in `unmarked`. These are specified using the familiar syntax of lme4 (Bates et al. 2015). Second, `ubms` generates posterior distributions for model parameters, including latent occupancy and abundance parameters. These can be useful for post-hoc analyses and diagnostics. Finally, fitting models with Stan gives you access to the large ecosystem of Stan-related tools, such as LOO (leave-one-out-cross validation; (Vehtari, Gelman, and Gabry 2017)).

Another alternative to `ubms` would be to fit models in an existing modeling language such as BUGS, JAGS, or directly in Stan. `ubms` abstracts away the complex process of defining and writing custom models in these languages which need to be updated each time you make changes to your model. It also provides many useful helper functions (e.g. `predict`) which would otherwise require custom code. Finally, because of Stan’s efficient sampler (Hoffman and Gelman 2014) and because the underlying likelihoods in `ubms` are marginalized, `ubms` will often fit models much faster than equivalent models in BUGS or JAGS (Yackulic et al. 2020).

## 1.3 What are the disadvantages?

Relative to `unmarked`, `ubms` has fewer types of models available. For example, models that incorporate temporary emigration (like `gdistsamp`) (Chandler, Royle, and King 2011) are currently not available in `ubms`. Models should run much faster in `unmarked` as well. If you do not need one of the specific benefits of `ubms` described above, it makes sense to stick with `unmarked`. Even if you do plan to use `ubms`, it makes sense to test the models in `unmarked` first. The similar interface between the two packages makes this very easy, as you will see in the next section.

Relative to BUGS/JAGS/Stan, `ubms` is less flexible because you cannot customize your model structure. You are limited to the provided model types. Furthermore, you cannot currently customize prior distributions (although I plan to add this in the future in some form). Finally, writing your own BUGS/JAGS model can be valuable for gaining a deeper understanding of how a model works; `ubms`, like `unmarked`, is essentially a black box.

To summarize the advantages and disadvantages: I see `ubms` as an intermediate step along the continuum from `unmarked` to custom models in BUGS/JAGS. It is not meant to replace either approach but rather to supplement them, for situations when a Bayesian framework is needed and “off-the-shelf” model structures are adequate.

# 2 Example: Fitting a single-season occupancy model

Occupancy models estimate the probability \(\psi\) that a species occupies a site, while accounting for detection probability \(p < 1\) (MacKenzie et al. 2002). In order to estimate both \(p\) and \(\psi\), repeated observations (detection/non-detection data) at each site are required.

## 2.1 Set up the data

First, load the dataset we’ll be using, which comes with `unmarked`:

    library(unmarked)
    data(crossbill)

The `crossbill` dataset is a `data.frame` with many columns. It contains detection/non-detection data for the European crossbill (Loxia curvirostra) in Switzerland (Schmid, Zbinden, and Keller 2004).

    dim(crossbill)

    ## [1] 267  58

    names(crossbill)

    ##  [1] "id"      "ele"     "forest"  "surveys" "det991"  "det992"  "det993"
    ##  [8] "det001"  "det002"  "det003"  "det011"  "det012"  "det013"  "det021"
    ## [15] "det022"  "det023"  "det031"  "det032"  "det033"  "det041"  "det042"
    ## [22] "det043"  "det051"  "det052"  "det053"  "det061"  "det062"  "det063"
    ## [29] "det071"  "det072"  "det073"  "date991" "date992" "date993" "date001"
    ## [36] "date002" "date003" "date011" "date012" "date013" "date021" "date022"
    ## [43] "date023" "date031" "date032" "date033" "date041" "date042" "date043"
    ## [50] "date051" "date052" "date053" "date061" "date062" "date063" "date071"
    ## [57] "date072" "date073"

Check `?crossbill` for details about each column. The first three columns `id`, `ele`, and `forest` are site covariates.

    site_covs <- crossbill[,c("id", "ele", "forest")]

The following 27 columns beginning with `det` are the binary detection/non-detection data; 9 years with 3 observations per year. For this example we want to fit a single-season occupancy model; thus we will use only the first three columns (year 1) of `det` as our response variable `y`.

    y <- crossbill[,c("det991","det992","det993")]
    head(y)

    ##   det991 det992 det993
    ## 1      0      0      0
    ## 2      0      0      0
    ## 3     NA     NA     NA
    ## 4      0      0      0
    ## 5      0      0      0
    ## 6     NA     NA     NA

Note that missing values are possible. The final 27 columns beginning with `date` are the Julian dates for each observation. As with `y` we want only the first three columns corresponding to year 1.

    date <- crossbill[,c("date991","date992","date993")]

Finally, we build our `unmarkedFrame` object holding our detection/non-detection data, site covariates, and observation covariates. Since we will conduct a single-season occupancy analysis, we need to use `unmarkedFrameOccu` specifically. The resulting `unmarkedFrame` can be used by both `unmarked` and `ubms`.

    umf <- unmarkedFrameOccu(y=y, siteCovs=site_covs, obsCovs=list(date=date))
    head(umf)

    ## Data frame representation of unmarkedFrame object.
    ##   y.1 y.2 y.3 id  ele forest date.1 date.2 date.3
    ## 1   0   0   0  1  450      3     34     59     65
    ## 2   0   0   0  2  450     21     17     33     65
    ## 3  NA  NA  NA  3 1050     32     NA     NA     NA
    ## 4   0   0   0  4  950      9     29     59     65
    ## 5   0   0   0  5 1150     35     24     45     65
    ## 6  NA  NA  NA  6  550      2     NA     NA     NA

## 2.2 Fit a model

### 2.2.1 Fit the model in unmarked

First, we fit a null model (no covariates) in `unmarked` using the `occu` function. The `occu` function requires as input a double formula (for detection and occupancy, respectively) along with our `unmarkedFrame`.

    (fit_unm <- occu(~1~1, data=umf))

    ##
    ## Call:
    ## occu(formula = ~1 ~ 1, data = umf)
    ##
    ## Occupancy (logit-scale):
    ##  Estimate    SE     z P(>|z|)
    ##    -0.546 0.218 -2.51  0.0121
    ##
    ## Detection (logit-scale):
    ##  Estimate    SE     z P(>|z|)
    ##    -0.594 0.208 -2.86 0.00426
    ##
    ## AIC: 511.2538
    ## Number of sites: 245
    ## ID of sites removed due to NA: 3 6 38 63 71 106 116 118 125 126 152 163 168 178 181 197 199 212 218 220 238 257

### 2.2.2 Fit the model in ubms

Next, we fit the same model in `ubms`. The equivalent to `occu` in `ubms` is `stan_occu`. Functions in `ubms` generally use this `stan_` prefix, based on the approach used in package rstanarm for GLMs. We need to provide the same arguments to `stan_occu`. In addition, we will specify that we want 3 MCMC chains (`chains=3`), with 500 iterations per chain (`iter=500`) of which the first half will be warmup iterations. It is beyond the scope of this vignette to discuss the appropriate number or length of chains; see the Stan user’s guide for more details. Generally 4 chains of 2000 iterations each is recommended (of which 1000 per chain are warmups). Thus, 500 iterations per chain is probably not enough, but to keep things running quickly it is sufficient for this vignette. Note that if you are more familiar with BUGS or JAGS, Stan generally requires a smaller number of iterations to reach convergence thanks to its default NUTS sampler (Hoffman and Gelman 2014). If you have a good multi-core CPU, you can run chains in parallel. Tell Stan how many parallel cores you want to use by assigning a value to the `cores` argument.

    library(ubms)

    (fit_stan <- stan_occu(~1~1, data=umf, chains=3, iter=500, cores=3, seed=123))

    ##
    ## Call:
    ## stan_occu(formula = ~1 ~ 1, data = umf, chains = 3, iter = 500,
    ##     refresh = 0, seed = 123)
    ##
    ## Occupancy (logit-scale):
    ##  Estimate    SD   2.5%   97.5% n_eff Rhat
    ##    -0.483 0.234 -0.882 0.00981   236    1
    ##
    ## Detection (logit-scale):
    ##  Estimate    SD  2.5%  97.5% n_eff Rhat
    ##    -0.642 0.226 -1.11 -0.229   285    1
    ##
    ## LOOIC: 511.954
    ## Runtime: 17.876 sec

### 2.2.3 Compare results

The structure of the output from `unmarked` and `ubms` is intentionally similar. Estimates of the occupancy and detection parameters are also similar, but not identical. For a more direct comparison, call the `coef` function on both model objects:

    cbind(unmarked=coef(fit_unm), stan=coef(fit_stan))

    ##            unmarked       stan
    ## psi(Int) -0.5461203 -0.4834253
    ## p(Int)   -0.5939612 -0.6419488

### 2.2.4 Understanding the ubms output summary

Let’s look at the output from our `fit_stan` model again:

    fit_stan

    ##
    ## Call:
    ## stan_occu(formula = ~1 ~ 1, data = umf, chains = 3, iter = 500,
    ##     refresh = 0, seed = 123)
    ##
    ## Occupancy (logit-scale):
    ##  Estimate    SD   2.5%   97.5% n_eff Rhat
    ##    -0.483 0.234 -0.882 0.00981   236    1
    ##
    ## Detection (logit-scale):
    ##  Estimate    SD  2.5%  97.5% n_eff Rhat
    ##    -0.642 0.226 -1.11 -0.229   285    1
    ##
    ## LOOIC: 511.954
    ## Runtime: 17.876 sec

The first part (under `Call:`) is the command we used to get this model output. Underneath are two tables, one per submodel, corresponding to the occupancy and detection parts of the model. Within each table there is one row per parameter in the submodel. Since `fit_stan` had no covariates, there is only an intercept term for each submodel. Model parameters in this summary table are always shown on the appropriate transformed scale, in this case logit. To get the corresponding probabilities, you can use the `predict` function, which we will demonstrate later.

For each parameter, the mean and standard deviation of the posterior distribution are given. Unlike `unmarked` output, there is no \(Z\) or \(p\)-value. Instead, there is a 95% uncertainty interval provided.

The final two columns in each summary table `n_eff` and `Rhat` are MCMC diagnostics. We will discuss their meaning later.

### 2.2.5 Extracting individual parameters

To extract summary values into an R table for easy manipulation, use the `summary` method. Note that you have to specify which submodel you want (`"state"` for occupancy or `"det"` for detection).

    sum_tab <- summary(fit_stan, "state")
    sum_tab$mean[1]

    ## [1] -0.4834253

To extract the entire posterior for a parameter, use the `extract` method. To avoid name collisions you need to use the full name of the parameter (which contains both the submodel and the parameter name) when extracting. To see a list of the available full parameter names, use the `names` method.

    names(fit_stan)

    ## [1] "beta_state[(Intercept)]" "beta_det[(Intercept)]"

    occ_intercept <- extract(fit_stan, "beta_state[(Intercept)]")[[1]]
    hist(occ_intercept, freq=FALSE)
    lines(density(occ_intercept), col='red', lwd=2)

## 2.3 Compare candidate models

Now we’ll fit two candidate models to the `crossbill` data in `ubms` and compare them.

### 2.3.1 Fit the models

Along with our previous null model, we’ll fit a “global” model with both site and observation covariates. This is just an example; perhaps other models should also be considered if we were preparing this analysis for publication. In our model formulas, we have normalized all covariates with `scale` so they have a mean of 0 and a standard deviation of 1. This can help improve model convergence and is generally a good idea.

    fit_null <- fit_stan

    fit_global <- stan_occu(~scale(date)~scale(forest)+scale(ele), data=umf,
                            chains=3, iter=500, seed=123)

    ## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
    ## Running the chains for more iterations may help. See
    ## https://mc-stan.org/misc/warnings.html#bulk-ess

    ## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
    ## Running the chains for more iterations may help. See
    ## https://mc-stan.org/misc/warnings.html#tail-ess

The `fit_global` model gave us some warnings about the effective sample size (`n_eff`) along with a suggested solution. We will ignore this warning for now but normally it is a good idea to pay close attention to these warnings.

### 2.3.2 Compare the models

First we combine the models into a `fitList`:

    mods <- fitList(fit_null, fit_global)

Then we generate a model selection table:

    round(modSel(mods), 3)

    ##                elpd nparam elpd_diff se_diff
    ## fit_global -236.776  5.984       0.0   0.000
    ## fit_null   -255.977  2.524     -19.2   6.618

Instead of AIC, models are compared using leave-one-out cross-validation (LOO) (Vehtari, Gelman, and Gabry 2017) via the `loo` package. Based on this cross-validation, the expected predictive accuracy (`elpd`) for each model is calculated. The model with the largest `elpd` (`fit_global`) performed best. The `elpd_diff` column shows the difference in `elpd` between a model and the top model; if this difference is several times larger than the standard error of the difference (`se_diff`), we are confident the model with the larger `elpd` performed better. We can see that the `fit_global` model is clearly the best performing model.

You can obtain LOO information for a single model using the `loo` method:

    loo(fit_global)

    ##
    ## Computed from 750 by 245 log-likelihood matrix.
    ##
    ##          Estimate   SE
    ## elpd_loo   -236.8 19.2
    ## p_loo         6.0  0.6
    ## looic       473.6 38.5
    ## ------
    ## MCSE of elpd_loo is 0.1.
    ## MCSE and ESS estimates assume MCMC draws (r_eff in [0.3, 1.5]).
    ##
    ## All Pareto k estimates are good (k < 0.65).
    ## See help('pareto-k-diagnostic') for details.

The `looic` value is analogous to AIC.

You can also obtain the WAIC (Widely Applicable Information Criterion) if you prefer (Vehtari, Gelman, and Gabry 2017):

    waic(fit_global)

    ##
    ## Computed from 750 by 245 log-likelihood matrix.
    ##
    ##           Estimate   SE
    ## elpd_waic   -236.7 19.2
    ## p_waic         5.9  0.6
    ## waic         473.5 38.5

## 2.4 Diagnostics and model fit

We’ll define the global model as our top model:

    fit_top <- fit_global
    fit_top

    ##
    ## Call:
    ## stan_occu(formula = ~scale(date) ~ scale(forest) + scale(ele),
    ##     data = umf, chains = 3, iter = 500, refresh = 0, seed = 123)
    ##
    ## Occupancy (logit-scale):
    ##               Estimate    SD    2.5% 97.5% n_eff Rhat
    ## (Intercept)     -0.603 0.359 -1.1896 0.293   208 1.01
    ## scale(forest)    1.077 0.325  0.5595 1.925   211 1.01
    ## scale(ele)       0.566 0.245  0.0924 1.054   533 1.00
    ##
    ## Detection (logit-scale):
    ##             Estimate    SD   2.5%  97.5% n_eff Rhat
    ## (Intercept)   -0.754 0.253 -1.290 -0.316   300 1.01
    ## scale(date)    0.558 0.170  0.233  0.883  1030 1.00
    ##
    ## LOOIC: 473.553
    ## Runtime: 25.354 sec

### 2.4.1 MCMC diagnostics

Again looking at the summary of `fit_top`, we conclude MCMC chains have converged if all \(\hat{R} < 1.05\). To visualize convergence, look at the traceplots:

    traceplot(fit_top, pars=c("beta_state", "beta_det"))

The traceplots look a little messy. We may also get a warning that `n_eff` is lacking for some parameters. The rule of thumb is to have `n_eff` > 100 * number of chains (300). The easy solution to both problems would be to re-run this model with more iterations.

### 2.4.2 Model fit

Calculating residuals is tricky for occupancy models. There isn’t one widely accepted way of doing it. `ubms` implements the approach of Wright (Wright, Irvine, and Higgs 2019) in which residuals are calculated separately for the state and observation processes. To quickly plot residuals against fitted values, use the `plot` method:

    plot_residuals(fit_top, submodel="state")

Note that the residuals are automatically binned, which is appropriate for a binomial response (Gelman and Hill 2007). If the model fits the data well, you would expect 95% of the binned residual points to fall within the shaded area. You can also plot residuals against covariate values using the `plot_residuals` function:

    plot_residuals(fit_top, submodel="state", covariate="forest")

`ubms` also supports goodness-of-fit tests (posterior predictive checks) for some models. In the case of occupancy models, there is support for the MacKenzie-Bailey (M-B) chi-square test (MacKenzie and Bailey 2004) via the `gof` function. For each posterior draw, the M-B statistic is calculated for the actual data and for a simulated dataset. The proportion of draws for which the simulated statistic is larger than the actual statistic should be near 0.5 if the model fits well.

    (fit_top_gof <- gof(fit_top, draws=100, quiet=TRUE))

    ## MacKenzie-Bailey Chi-square
    ## Point estimate = 27.787
    ## Posterior predictive p = 0.03

    plot(fit_top_gof)

Our model does not appear to fit well based on this posterior predictive check. The first step to addressing this would be to run the model for more iterations to make sure that isn’t the reason.

You can use the `posterior_predict` function to simulate new datasets, which you can use to calculate your own fit statistic. The following command generates 100 simulated datasets.

    sim_y <- posterior_predict(fit_top, "y", draws=100)
    dim(sim_y)

    ## [1] 100 801

The output is a matrix with dimensions draws x observations (in site-major order). As an example, we can calculate the proportion of zeros in each simulated dataset

    prop0 <- apply(sim_y, 1, function(x) mean(x==0, na.rm=TRUE))

and compare that to the proportion of zeros in the actual dataset.

    actual_prop0 <- mean(getY(fit_top) == 0, na.rm=TRUE)

    #Compare
    hist(prop0, col='gray')
    abline(v=actual_prop0, col='red', lwd=2)

## 2.5 Model inference

### 2.5.1 Marginal covariate effects

Based on the 95% uncertainty intervals, both forest and elevation have a positive effect on occupancy probability (both intervals do not contain 0). Similarly, Julian date has a positive impact on detection probability. We can quickly visualize these marginal covariate effects with the `plot_effects` function:

    plot_effects(fit_top, "state")

    plot_effects(fit_top, "det")

### 2.5.2 Predict parameter values

As with `unmarked`, we can get the predicted \(psi\) or \(p\) for each site or observation using the `predict` function. For example, to get occupancy:

    head(predict(fit_top, submodel="state"))

    ##    Predicted         SD       2.5%     97.5%
    ## 1 0.08475341 0.04074847 0.02603972 0.1870306
    ## 2 0.15341676 0.06590045 0.05564030 0.3312959
    ## 3 0.30784613 0.07968393 0.18695527 0.5197789
    ## 4 0.14593709 0.04629648 0.07153534 0.2537933
    ## 5 0.35156491 0.08389258 0.22562494 0.5726406
    ## 6 0.08783817 0.04008446 0.02912174 0.1868335

You can also supply newdata as a `data.frame`.

    nd <- data.frame(ele=100, forest=25)
    predict(fit_top, submodel="state", newdata=nd)

    ##   Predicted         SD       2.5%     97.5%
    ## 1 0.1401127 0.07725292 0.03806715 0.3547347

### 2.5.3 Simulate latent occupancy states

One of the advantages of BUGS/JAGS is that you can directly model latent parameters, such as the true unobserved occupancy state of a site \(z\). Using the `posterior_predict` function in `ubms`, you can generate an equivalent posterior distribution of \(z\).

    zpost <- posterior_predict(fit_top, "z", draws=100)
    dim(zpost)

    ## [1] 100 267

The output has one row per posterior draw, and one column per site. The posterior of \(z\) can be useful for post-hoc analyses. For example, suppose you wanted to test for a difference in mean occupancy probability between sites 1-50 and sites 51-100:

    group1 <- rowMeans(zpost[,1:50], na.rm=TRUE)
    group2 <- rowMeans(zpost[,51:100], na.rm=TRUE)

    plot_dat <- rbind(data.frame(group="group1", occ=mean(group1),
                                 lower=quantile(group1, 0.025),
                                 upper=quantile(group1, 0.975)),
                      data.frame(group="group2", occ=mean(group2),
                                 lower=quantile(group2, 0.025),
                                 upper=quantile(group2, 0.975)))

Now plot the posterior distributions of the two means:

    library(ggplot2)

    ggplot(plot_dat, aes(x=group, y=occ)) +
      geom_errorbar(aes(ymin=lower, ymax=upper), width=0.2) +
      geom_point(size=3) +
      ylim(0,1) +
      labs(x="Group", y="Occupancy + 95% UI") +
      theme_bw() +
      theme(panel.grid.major=element_blank(), panel.grid.minor=element_blank(),
            axis.text=element_text(size=12), axis.title=element_text(size=14))

# 3 References

Bates, Douglas, Martin Mächler, Ben Bolker, and Steve Walker. 2015. “Fitting Linear Mixed-Effects Models Using Lme4.” Journal of Statistical Software 67 (1). https://doi.org/10.18637/jss.v067.i01.
Carpenter, Bob, Andrew Gelman, Matthew D. Hoffman, Daniel Lee, Ben Goodrich, Michael Betancourt, Marcus Brubaker, Jiqiang Guo, Peter Li, and Allen Riddell. 2017. “Stan: A Probabilistic Programming Language.” Journal of Statistical Software 76 (1). https://doi.org/10.18637/jss.v076.i01.
Chandler, Richard B., J. Andrew Royle, and David I. King. 2011. “Inference about Density and Temporary Emigration in Unmarked Populations.” Ecology 92 (7): 1429–35. https://doi.org/10.1890/10-2433.1.
Fiske, Ian, Richard Chandler, and others. 2011. “Unmarked: An R Package for Fitting Hierarchical Models of Wildlife Occurrence and Abundance.” Journal of Statistical Software 43 (10): 1–23. https://doi.org/10.18637/jss.v043.i10.
Gelman, Andrew, and Jennifer Hill. 2007. Data Analysis Using Regression and Multilevel/Hierarchical Models. New York, NY: Cambridge University Press.
Hoffman, Matthew D, and Andrew Gelman. 2014. “The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo.” Journal of Machine Learning Research 15 (1): 1593–623.
MacKenzie, Darryl I., and Larissa L. Bailey. 2004. “Assessing the Fit of Site-Occupancy Models.” Journal of Agricultural, Biological, and Environmental Statistics 9 (3): 300–318. https://doi.org/10.1198/108571104x3361.
MacKenzie, Darryl I., James D. Nichols, Gideon B. Lachman, Sam Droege, J. Andrew Royle, and Catherine A. Langtimm. 2002. “Estimating Site Occupancy Rates When Detection Probabilities Are Less Than One.” Ecology 83 (8): 2248–55. https://doi.org/10.1890/0012-9658(2002)083[2248:esorwd]2.0.co;2.
Schmid, Hans, Niklaus Zbinden, and Verena Keller. 2004. “Überwachung Der Bestandsentwicklung Häufiger Brutvögel in Der Schweiz.” Swiss Ornithological Institute Sempach Switzerland.
Vehtari, Aki, Andrew Gelman, and Jonah Gabry. 2017. “Practical Bayesian Model Evaluation Using Leave-One-Out Cross-Validation and WAIC.” Statistics and Computing 27 (5): 1413–32. https://doi.org/10.1007/s11222-016-9696-4.
Wright, Wilson J., Kathryn M. Irvine, and Megan D. Higgs. 2019. “Identifying Occupancy Model Inadequacies: Can Residuals Separately Assess Detection and Presence?” Ecology, e02703. https://doi.org/10.1002/ecy.2703.
Yackulic, Charles B., Michael Dodrill, Maria Dzul, Jamie S. Sanderlin, and Janice A. Reid. 2020. “A Need for Speed in Bayesian Population Models: A Practical Guide to Marginalizing and Recovering Discrete Latent States.” Ecological Applications 30 (5). https://doi.org/10.1002/eap.2112.