Fit a generalized linear model.
stan_glm(
formula,
slx,
re,
data,
C,
family = gaussian(),
prior = NULL,
ME = NULL,
centerx = FALSE,
prior_only = FALSE,
censor_point,
chains = 4,
iter = 2000,
refresh = 1000,
keep_all = FALSE,
slim = FALSE,
drop = NULL,
pars = NULL,
control = NULL,
quiet = FALSE,
...
)Donegan, Connor and Chun, Yongwan and Griffith, Daniel A. (2021). Modeling community health with areal data: Bayesian inference with survey standard errors and spatial structure. Int. J. Env. Res. and Public Health 18 (13): 6856. DOI: 10.3390/ijerph18136856 Data and code: https://github.com/ConnorDonegan/survey-HBM.
Donegan, Connor (2021). Building spatial conditional autoregressive (CAR) models in the Stan programming language. OSF Preprints. doi:10.31219/osf.io/3ey65 .
A model formula, following the R formula syntax. Binomial models are specified by setting the left hand side of the equation to a data frame of successes and failures, as in cbind(successes, failures) ~ x.
Formula to specify any spatially-lagged covariates. As in, ~ x1 + x2 (the intercept term will be removed internally). When setting priors for beta, remember to include priors for any SLX terms.
To include a varying intercept (or "random effects") term, alpha_re, specify the grouping variable here using formula syntax, as in ~ ID. Then, alpha_re is a vector of parameters added to the linear predictor of the model, and:
alpha_re ~ N(0, alpha_tau)
alpha_tau ~ Student_t(d.f., location, scale).A data.frame or an object coercible to a data frame by as.data.frame containing the model data.
Spatial connectivity matrix which will be used to calculate residual spatial autocorrelation as well as any user specified slx terms. See shape2mat.
The likelihood function for the outcome variable. Current options are poisson(link = "log"), binomial(link = "logit"), student_t(), and the default gaussian().
A named list of parameters for prior distributions (see priors):
The intercept is assigned a Gaussian prior distribution (see normal
Regression coefficients are assigned Gaussian prior distributions. Variables must follow their order of appearance in the model formula. Note that if you also use slx terms (spatially lagged covariates), and you use custom priors for beta, then you have to provide priors for the slx terms. Since slx terms are prepended to the design matrix, the prior for the slx term will be listed first.
For family = gaussian() and family = student_t() models, the scale parameter, sigma, is assigned a (half-) Student's t prior distribution. The half-Student's t prior for sigma is constrained to be positive.
nu is the degrees of freedom parameter in the Student's t likelihood (only used when family = student_t()). nu is assigned a gamma prior distribution. The default prior is prior = list(nu = gamma2(alpha = 3, beta = 0.2)).
The scale parameter for random effects, or varying intercepts, terms. This scale parameter, tau, is assigned a half-Student's t prior. To set this, use, e.g., prior = list(tau = student_t(df = 20, location = 0, scale = 20)).
To model observational uncertainty (i.e. measurement or sampling error) in any or all of the covariates, provide a list of data as constructed by the prep_me_data function.
To center predictors on their mean values, use centerx = TRUE. If the ME argument is used, the modeled covariate (i.e., latent variable), rather than the raw observations, will be centered. When using the ME argument, this is the recommended method for centering the covariates.
Draw samples from the prior distributions of parameters only.
Integer value indicating the maximum censored value; this argument is for modeling censored (suppressed) outcome data, typically disease case counts or deaths. For example, the US Centers for Disease Control and Prevention censors (does not report) death counts that are nine or fewer, so if you're using CDC WONDER mortality data you could provide censor_point = 9.
Number of MCMC chains to estimate.
Number of samples per chain.
Stan will print the progress of the sampler every refresh number of samples; set refresh=0 to silence this.
If keep_all = TRUE then samples for all parameters in the Stan model will be kept; this is required if you want to do model comparison with Bayes factors and the bridgesampling package.
If slim = TRUE, then the Stan model will not collect the most memory-intensive parameters (including n-length vectors of fitted values, log-likelihoods, and ME-modeled covariate values). This will disable many convenience functions that are otherwise available for fitted geostan models, such as the extraction of residuals, fitted values, and spatial trends, WAIC, and spatial diagnostics, and ME diagnostics; many quantities of interest, such as fitted values and spatial trends, can still be calculated manually using given parameter estimates. The "slim" option is designed for data-intensive routines, such as regression with raster data, Monte Carlo studies, and measurement error models. For more control over which parameters are kept or dropped, use the drop argument instead of slim.
Provide a vector of character strings to specify the names of any parameters that you do not want MCMC samples for. Dropping parameters in this way can improve sampling speed and reduce memory usage. The following parameter vectors can potentially be dropped from GLM models:
The N-length vector of fitted values
Vector of 'random effects'/varying intercepts.
N-length vector of 'latent'/modeled covariate values created for measurement error (ME) models.
Using drop = c('fitted', 'alpha_re', 'x_true') is equivalent to slim = TRUE. If slim = TRUE, then drop will be ignored.
Specify any additional parameters you'd like stored from the Stan model.
A named list of parameters to control the sampler's behavior. See stan for details.
Controls (most) automatic printing to the console. By default, any prior distributions that have not been assigned by the user are printed to the console. If quiet = TRUE, these will not be printed. Using quiet = TRUE will also force refresh = 0.
Other arguments passed to sampling.
An object of class class geostan_fit (a list) containing:
Summaries of the main parameters of interest; a data frame
Residual spatial autocorrelation as measured by the Moran coefficient.
an object of class stanfit returned by rstan::stan
a data frame containing the model data
the user-provided or default family argument used to fit the model
The model formula provided by the user (not including ESF component)
The slx formula
The spatial weights matrix, if one was provided by the user.
A list containing re, the random effects (varying intercepts) formula if provided, and
Data a data frame with columns id, the grouping variable, and idx, the index values assigned to each group.
Prior specifications.
If covariates are centered internally (centerx = TRUE), then x_center is a numeric vector of the values on which covariates were centered.
The ME data list, if one was provided by the user for measurement error models.
NA, slot is maintained for use in geostan_fit methods.
Fit a generalized linear model using the R formula interface. Default prior distributions are designed to be weakly informative relative to the data. Much of the functionality intended for spatial models, such as the ability to add spatially lagged covariates and observational error models, are also available in stan_glm. All of geostan's spatial models build on top of the same Stan code used in stan_glm.
In spatial statistics, Poisson models are often used to calculate incidence rates (mortality rates, or disease incidence rates) for administrative areas like counties or census tracts. If \(y\) are counts of cases, and \(P\) are populations at risk, then the crude rates are \(y/P\). The purpose is to model risk \(\eta\) for which crude rates are a (noisy) indicator. Our analysis should also respect the fact that the amount of information contained in the observations \(y/P\) increases with \(P\). Hierarchical Poisson models are often used to incorporate all of this information.
For the Poisson model, \(y\) is specified as the outcome and the log of the population at risk log(P) needs to be provided as an offset term. For such a case, disease incidence across the collection of areas could be modeled as:
$$y \sim Poisson(e^{log(P) + \eta})$$ $$ \eta = \alpha + A$$ $$ A \sim Gauss(0, \tau)$$ $$\tau \sim Student(20, 0, 2)$$ where \(\alpha\) is the mean log-risk (incidence rate) and \(A\) is a vector of (so-called) random effects, which enable partial pooling of information across observations. Covariates can be added to the model for the log-rates, such that \(\eta = \alpha + X \beta + A\).
Note that the denominator for the rates is specified as a log-offset to provide a consistent, formula-line interface to the model. Using the log-offest (as above) is equivalent to the following: $$ y \sim Poisson(P * e^{\eta}) $$ where \(P\) is still the population at risk and it is multiplied by \(e^{\eta}\), the incidence rate (risk).
The slx argument is a convenience function for including SLX terms. For example,
$$
y = W X \gamma + X \beta + \epsilon
$$
where \(W\) is a row-standardized spatial weights matrix (see shape2mat), \(WX\) is the mean neighboring value of \(X\), and \(\gamma\) is a coefficient vector. This specifies a regression with spatially lagged covariates. SLX terms can specified by providing a formula to the slx argument:
stan_glm(y ~ x1 + x2, slx = ~ x1 + x2, \...),which is a shortcut for
stan_glm(y ~ I(W \%*\% x1) + I(W \%*\% x2) + x1 + x2, \...)SLX terms will always be prepended to the design matrix, as above, which is important to know when setting prior distributions for regression coefficients.
For measurement error (ME) models, the SLX argument is the only way to include spatially lagged covariates since the SLX term needs to be re-calculated on each iteration of the MCMC algorithm.
The ME models are designed for surveys with spatial sampling designs, such as the American Community Survey (ACS) estimates. For a tutorial, see vignette("spatial-me-models", package = "geostan").
Given estimates \(x\), their standard errors \(s\), and the target quantity of interest (i.e., the unknown true value) \(z\), the ME models have one of the the following two specifications, depending on the user input. If a spatial CAR model is specified, then:
$$x \sim Gauss(z, s^2)$$ $$z \sim Gauss(\mu_z, \Sigma_z)$$ $$\Sigma_z = (I - \rho C)^{-1} M$$ $$\mu_z \sim Gauss(0, 100)$$ $$\tau_z \sim Student(10, 0, 40), \tau > 0$$ $$\rho_z \sim uniform(l, u)$$
where \(\Sigma\) specifies the covariance matrix of a spatial conditional autoregressive (CAR) model with scale parameter \(\tau\) (on the diagonal of \(M\)), autocorrelation parameter \(\rho\), and \(l\), \(u\) are the lower and upper bounds that \(\rho\) is permitted to take (which is determined by the extreme eigenvalues of the spatial connectivity matrix \(C\)). \(M\) contains the inverse of the row sums of \(C\) on its diagonal multiplied by \(\tau\) (following the "WCAR" specification).
For non-spatial ME models, the following is used instead: $$x \sim Gauss(z, s^2)$$ $$z \sim student_t(\nu_z, \mu_z, \sigma_z)$$ $$\nu_z \sim gamma(3, 0.2)$$ $$\mu_z \sim Gauss(0, 100)$$ $$\sigma_z \sim student(10, 0, 40)$$
For strongly skewed variables, such as census tract poverty rates, it can be advantageous to apply a logit transformation to \(z\) before applying the CAR or Student-t prior model. When the logit argument is used, the first two lines of the model specification become:
$$x \sim Gauss(z, s^2)$$
$$logit(z) \sim Gauss(\mu_z, \Sigma_z) $$
and similarly for the Student t model:
$$x \sim Gauss(z, s^2)$$
$$logit(z) \sim student(\nu_z, \mu_z, \sigma_z)$$
For most geostan models, missing (NA) observations are allowed in the outcome variable. However, there cannot be any missing covariate data. Models that can handle missing data are: any Poisson or binomial model (GLM, SAR, CAR, ESF, ICAR), all GLMs and ESF models. The only models that cannot handle missing outcome data are the CAR and SAR models when the outcome is a continuous variable (auto-normal/Gaussian models).
When observations are missing, they will simply be ignored when calculating the likelihood in the MCMC sampling process (reflecting the absence of information). The estimated model parameters (including any covariates and spatial trend) will then be used to produce estimates or fitted values for the missing observations. The fitted and posterior_predict functions will work as normal in this case, and return values for all rows in your data.
Vital statistics systems and disease surveillance programs typically suppress case counts when they are smaller than a specific threshold value. In such cases, the observation of a censored count is not the same as a missing value; instead, you are informed that the value is an integer somewhere between zero and the threshold value. For Poisson models (family = poisson())), you can use the censor_point argument to encode this information into your model.
Internally, geostan will keep the index values of each censored observation, and the index value of each of the fully observed outcome values. For all observed counts, the likelihood statement will be:
$$
p(y_i | data, model) = poisson(y_i | \mu_i),
$$
as usual, where \(\mu_i\) may include whatever spatial terms are present in the model.
For each censored count, the likelihood statement will equal the cumulative Poisson distribution function for values zero through the censor point: $$ p(y_i | data, model) = \sum_{m=0}^{M} Poisson( m | \mu_i), $$ where \(M\) is the censor point and \(\mu_i\) again is the fitted value for the \(i^{th}\) observation.
For example, the US Centers for Disease Control and Prevention's CDC WONDER database censors all death counts between 0 and 9. To model CDC WONDER mortality data, you could provide censor_point = 9 and then the likelihood statement for censored counts would equal the summation of the Poisson probability mass function over each integer ranging from zero through 9 (inclusive), conditional on the fitted values (i.e., all model parameters). See Donegan (2021) for additional discussion, references, and Stan code.
##
## Linear regression model
##
N = 100
x <- rnorm(N)
y <- .5 * x + rnorm(N)
dat <- cbind(y, x)
# no. of MCMC samples
iter = 600
# fit model
fit <- stan_glm(y ~ x, data = dat, iter = iter, quiet = TRUE)
# see results with MCMC diagnostics
print(fit)
##
## Custom prior distributions
##
PL <- list(
intercept = normal(0, 1),
beta = normal(0, 1),
sigma = student_t(10, 0, 2)
)
fit2 <- stan_glm(y ~ x, data = dat, prior = PL, iter = iter,
quiet = TRUE)
print(fit2)
##
## Poisson model for count data
## with county 'random effects'
##
data(sentencing)
# note: 'name' is county identifier
head(sentencing)
# denominator in standardized rate Y/E
# (observed count Y over expected count E)
# (use the log-denominator as the offest term)
sentencing$log_e <- log(sentencing$expected_sents)
# fit model
fit.pois <- stan_glm(sents ~ offset(log_e),
re = ~ name,
family = poisson(),
data = sentencing,
iter = iter, quiet = TRUE)
# Spatial autocorrelation/residual diagnostics
sp_diag(fit.pois, sentencing)
# summary of results with MCMC diagnostics
print(fit.pois)
# MCMC diagnostics plot: Rhat values should all by very near 1
rstan::stan_rhat(fit.pois$stanfit)
# effective sample size for all parameters and generated quantities
# (including residuals, predicted values, etc.)
rstan::stan_ess(fit.pois$stanfit)
# or for a particular parameter
rstan::stan_ess(fit.pois$stanfit, "alpha_re")
##
## Visualize the posterior predictive distribution
##
# plot observed values and model replicate values
yrep <- posterior_predict(fit.pois, S = 65)
y <- sentencing$sents
ltgray <- rgb(0.3, 0.3, 0.3, 0.5)
plot(density(yrep[1,]), col = ltgray,
ylim = c(0, 0.014), xlim = c(0, 700),
bty = 'L', xlab = NA, main = NA)
for (i in 2:nrow(yrep)) lines(density(yrep[i,]), col = ltgray)
lines(density(sentencing$sents), col = "darkred", lwd = 2)
legend("topright", legend = c('Y-observed', 'Y-replicate'),
col = c('darkred', ltgray), lwd = c(1.5, 1.5))
# plot replicates of Y/E
E <- sentencing$expected_sents
# set plot margins
old_pars <- par(mar=c(2.5, 3.5, 1, 1))
# plot yrep
plot(density(yrep[1,] / E), col = ltgray,
ylim = c(0, 0.9), xlim = c(0, 7),
bty = 'L', xlab = NA, ylab = NA, main = NA)
for (i in 2:nrow(yrep)) lines(density(yrep[i,] / E), col = ltgray)
# overlay y
lines(density(sentencing$sents / E), col = "darkred", lwd = 2)
# legend, y-axis label
legend("topright", legend = c('Y-observed', 'Y-replicate'),
col = c('darkred', ltgray), lwd = c(1.5, 1.5))
mtext(side = 2, text = "Density", line = 2.5)
# return margins to previous settings
par(old_pars)