In this vignette, we show how to build the measurement error and
missingness models that inlamemi can fit, but without using
inlamemi. This is so that if you have a model that for some
reason does not work within inlamemi, you can see some
examples of how to fit these models “manually”.
There are two ways to structure the data for the models, the first
option is the one taken in Muff et al (2015) and Skarstein et al
(2023), where the matrices are constructed directly by hand. The
other approach is the one that is used internally in
inlamemi, where we use the function
inla.stack() to define the data structure for each
sub-model, and then join these together. This latter approach is a bit
more modular, and so this is the approach we will show here.
inla.stack() for hierarchical
modellingThe inla.stack() function is most commonly used for
spatial modelling with stochastic partial differential equations (SPDE),
since the data then is a bit more complex to structure. A description of
using stacks in that context can be found in chapter
7.3.3 of Bayesian inference with INLA. However, they can
also be quite useful when defining hierarchical models with multiple
likelihoods, which traditionally need to be defined by setting up
matrices where the structure of the matrices encode the structure of the
model, as explained in chapter
6.4 of Bayesian inference with INLA. The stacks are
eventually converted to these matrices by INLA, but for the modeller it
may be more convenient to define the model through the stacks. one
reason is that when setting up the matrices you need to know from the
beginning how many levels you will have in the model, since this
determines the number of columns in the matrices. However, with the
stacks, each model level has its own stack, and previous stacks do not
change if you add additional stacks. That is, the definition of stacks
for each model level can be done independently.
The inla.stack function takes four arguments:
data: a list of the data on the left side of the
formula (the response)A: a list of projector matrices, in our case for
non-spatial models this is just set to list(1)effects: a list of the SPDE index and covariates, in
our case, just a list of a list containing the covariates and random
effects.tag: a character vector labelling this group, which can
be useful for extracting certain parts of the stack later.In the first example, we use the dataset simple_data,
generated in the vignette Simulated
examples. This dataset has classical and Berkson measurement error,
and missing data, and so the main model we want to fit is
\[ y_i = \beta_0 + \beta_x x_i + \beta_z z_i + \varepsilon_i, \] the Berkson measurement error is \[ 0 = -x_{true, i} + r_i + u_{b,i}, \] the classical measurement error model is \[ x_i = r_i + u_{c,i}, \] and the imputation model is
\[ 0 = -r_i + \alpha_0 + \alpha_z z_i + \varepsilon_{r,i}. \]
We begin by loading the data and defining the number of observations.
Next, we define all the priors and initial values.
# Prior for beta.x
prior.beta <- c(0, 1/1000) # N(0, 10^3)
# Priors for y, measurement error and true x-value precision
prior.prec.y <- c(10, 9)
prior.prec.u_b <- c(10, 9)
prior.prec.u_c <- c(10, 9)
prior.prec.r <- c(10, 9)
# Initial values
initial.prec.y <- 1
initial.prec.u_b <- 1
initial.prec.u_c <- 1
initial.prec.r <- 1In inlamemi, we could then fit the model like this:
# Fit the model
simple_model <- fit_inlamemi(data = data,
formula_moi = y ~ x + z,
formula_imp = x ~ z,
family_moi = "gaussian",
error_type = c("berkson", "classical"),
prior.prec.moi = prior.prec.y,
prior.prec.berkson = prior.prec.u_b,
prior.prec.classical = prior.prec.u_c,
prior.prec.imp = prior.prec.r,
initial.prec.moi = initial.prec.y,
initial.prec.berkson = initial.prec.u_b,
initial.prec.classical = initial.prec.u_c,
initial.prec.imp = initial.prec.r)If we instead want to do this without inlamemi, we start
by defining the stacks for each model level.
For the main regression model, we have the model \[ y_i = \beta_0 + \beta_x x_i + \beta_z z_i + \varepsilon_i, \] which is defined in a stack like this:
stk_moi <- inla.stack(data = list(y_moi = data$y),
A = list(1),
effects = list(
list(beta.0 = rep(1, n),
beta.x = 1:n,
beta.z = data$z)),
tag = "moi")Next, the Berkson measurement error model is \[ 0 = -x_{true, i} + r_i + u_{b,i}, \] and the corresponding stack is:
stk_b <- inla.stack(data = list(y_berkson = rep(0, n)),
A = list(1),
effects = list(
list(id.x = 1:n,
weight.x = -1,
id.r = 1:n,
weight.r = 1)),
tag = "berkson")The classical measurement error model is \[ x_i = r_i + u_{c,i}, \]
and the stack is:
stk_c <- inla.stack(data = list(y_classical = data$x),
A = list(1),
effects = list(
list(id.r = 1:n,
weight.r = 1)),
tag = "classical")Finally, the imputation model is \[ 0 = -r_i + \alpha_0 + \alpha_z z_i + \varepsilon_{r,i}. \] and the corresponding stack is
stk_imp <- inla.stack(data = list(y_imp = rep(0, n)),
A = list(1),
effects = list(
list(id.r = 1:n,
weight.r = rep(-1, n),
alpha.0 = rep(1, n),
alpha.z = data$z)),
tag = "imputation")We then stack these together, which will give us the matrix formulation that we also could have specified manually.
For this model, we have two latent effects, x and
r, which are both specified in the formula. For further
details on the formula, see the supplementary material of Skarstein et al
(2023), which can be found online here: https://emmaskarstein.github.io/Missing-data-and-measurement-error/simulation_example.html
formula <- list(y_moi, y_berkson, y_classical, y_imp) ~ - 1 + beta.0 + beta.z +
f(beta.x, copy = "id.x",
hyper = list(beta = list(param = c(0, 1/1000), fixed = FALSE))) +
f(id.x, weight.x, model = "iid", values = 1:n,
hyper = list(prec = list(initial = -15, fixed = TRUE))) +
f(id.r, weight.r, model="iid", values = 1:n,
hyper = list(prec = list(initial = -15, fixed = TRUE))) +
alpha.0 + alpha.zFinally, we call the inla() function. This model
consists of four Gaussian sub-models, and in the
control.family argument we give the priors for each of
these.
model_sim <- inla(formula, data = inla.stack.data(stk_full),
family = c("gaussian", "gaussian", "gaussian", "gaussian"),
control.family = list(
list(hyper = list(prec = list(initial = log(initial.prec.y),
param = prior.prec.y,
fixed = FALSE))),
list(hyper = list(prec = list(initial = log(initial.prec.u_b),
param = prior.prec.u_b,
fixed = FALSE))),
list(hyper = list(prec = list(initial = log(initial.prec.u_c),
param = prior.prec.u_c,
fixed = FALSE))),
list(hyper = list(prec = list(initial = log(initial.prec.r),
param = prior.prec.r,
fixed = FALSE)))
),
control.predictor = list(compute = TRUE)
)
summary(model_sim)
#> Time used:
#> Pre = 1.87, Running = 2.34, Post = 0.176, Total = 4.38
#> Fixed effects:
#> mean sd 0.025quant 0.5quant 0.975quant mode kld
#> beta.0 1.030 0.219 0.612 1.029 1.435 1.025 0
#> beta.z 1.911 0.388 1.225 1.901 2.560 1.910 0
#> alpha.0 1.033 0.051 0.934 1.033 1.132 1.033 0
#> alpha.z 2.025 0.052 1.922 2.025 2.127 2.025 0
#>
#> Random effects:
#> Name Model
#> id.x IID model
#> id.r IID model
#> beta.x Copy
#>
#> Model hyperparameters:
#> mean sd 0.025quant 0.5quant 0.975quant mode
#> Precision for the Gaussian observations 1.128 0.365 0.556 1.080 1.98 0.992
#> Precision for the Gaussian observations[2] 1.127 0.355 0.581 1.076 1.96 0.983
#> Precision for the Gaussian observations[3] 0.927 0.111 0.729 0.920 1.17 0.905
#> Precision for the Gaussian observations[4] 0.977 0.128 0.749 0.969 1.25 0.954
#> Beta for beta.x 1.972 0.207 1.559 1.974 2.38 1.981
#>
#> Marginal log-Likelihood: -20699.96
#> is computed
#> Posterior summaries for the linear predictor and the fitted values are computed
#> (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')In this example, dealing with only missing data, we show how to include another level in the model to potentially capture how the missingness mechanism may depend on available covariates.
In the dataset, simulated below, we let the value of the covariate
x with missingness depend on another covariate
z1 (corresponding to the imputation model), while the
probability of an entry in x missing depends on another
covariate z2 (corresponding to the missingness model).
set.seed(2024)
n <- 1000
z1 <- rnorm(n, mean = 0, sd = 1)
z2 <- rnorm(n, mean = 0, sd = 1)
alpha.0 <- 1; alpha.z1 <- 0.3
x <- rnorm(n, mean = alpha.0 + alpha.z1*z1, sd = 1)
gamma.0 <- -1.5; gamma.z2 <- -0.5
m_pred <- gamma.0 + gamma.z2*z2
m_prob <- exp(m_pred)/(1 + exp(m_pred))
m <- as.logical(rbinom(n, 1, prob = m_prob))
x_obs <- x
x_obs[m] <- NA
sum(is.na(x_obs))/length(x_obs)
#> [1] 0.183
beta.0 <- 1; beta.x <- 2; beta.z1 <- 2; beta.z2 <- 2
y <- beta.0 + beta.x*x + beta.z1*z1 + beta.z2*z2 + rnorm(n)
missing_data <- data.frame(y = y, x = x_obs, x_true = x, z1 = z1, z2 = z2)The main model of interest is in this case \[ y_i = \beta_0 + \beta_x x_{true,i} + \beta_{z_1} z_{1,i} + \beta_{z_2} z_{2,i} + \varepsilon_i , \] and the stack is:
stk_moi <- inla.stack(data = list(y_moi = missing_data$y),
A = list(1),
effects = list(
list(beta.0 = rep(1, n),
beta.x = 1:n,
beta.z1 = missing_data$z1,
beta.z2 = missing_data$z2)),
tag = "moi")The classical measurement error is just functioning as a link in this case, but the model is \[ x_{obs, i} = x_{true, i} + u_{c,i}, \] where the precision of \(u_{c,i}\) is set to be very high for those \(i\) where \(x_{obs,i}\) is observed. The stack for this level is:
stk_c <- inla.stack(data = list(y_classical = missing_data$x),
A = list(1),
effects = list(
list(id.x = 1:n,
weight.x = 1)),
tag = "classical")Next we have the imputation model \[ 0 = -x_{true,i} + \alpha_0 + \alpha_{z_1} z_{1,i} + \varepsilon_{x,i}, \] and the stack is
stk_imp <- inla.stack(data = list(y_imp = rep(0, n)),
A = list(1),
effects = list(
list(id.x = 1:n,
weight.x = -1,
alpha.0 = rep(1, n),
alpha.z1 = missing_data$z1)),
tag = "imputation")The missingness model is a binomial model describing how the probability of missing, \(p_m\), may depend on other covariates. The model is \[ \text{logit}(p_{m,i}) = \gamma_0 + \gamma_x x_{true,i} + \gamma_{z_2} z_{2,i}, \] and the stack is:
stk_mis <- inla.stack(data = list(y_mis = as.numeric(is.na(missing_data$x))),
A = list(1),
effects = list(
list(gamma.x = 1:n,
gamma.0 = rep(1, n),
gamma.z2 = missing_data$z2)),
tag = "missingness")We join the stacks together:
In defining the formula, we now need to define the hyperparameter
gamma.x, the same way as we define beta.x.
This is a scaled copy of id.x.
formula <- list(y_moi, y_classical, y_imp, y_mis) ~ - 1 +
beta.0 + beta.z1 + beta.z2 +
f(beta.x, copy = "id.x",
hyper = list(beta = list(param = c(0, 1/1000), fixed = FALSE))) +
f(id.x, weight.x, model = "iid", values = 1:n,
hyper = list(prec = list(initial = -15, fixed = TRUE))) +
f(gamma.x, copy = "id.x",
hyper = list(beta = list(param = c(0, 1/1000), fixed = FALSE))) +
alpha.0 + alpha.z1 + gamma.0 + gamma.z2Since we don’t have any measurement error in this case, we set the
precision to be very high for the classical error model in the argument
scale.
model_mnar <- inla(formula, data = inla.stack.data(stk_full),
family = c("gaussian", "gaussian", "gaussian", "binomial"),
scale = c(rep(1, n), rep(10^8, n), rep(1, n), rep(1, n)),
control.family = list(
list(hyper = list(prec = list(initial = log(1),
param = c(10, 9),
fixed = FALSE))),
list(hyper = list(prec = list(initial = log(1),
param = c(10, 9),
fixed = FALSE))),
list(hyper = list(prec = list(initial = log(1),
param = c(10, 9),
fixed = FALSE))),
list()),
control.predictor = list(compute = TRUE, link = 1)
)
summary(model_mnar)
#> Time used:
#> Pre = 1.9, Running = 3.8, Post = 0.175, Total = 5.88
#> Fixed effects:
#> mean sd 0.025quant 0.5quant 0.975quant mode kld
#> beta.0 1.038 0.050 0.940 1.038 1.136 1.038 0
#> beta.z1 2.004 0.036 1.933 2.004 2.075 2.004 0
#> beta.z2 1.985 0.036 1.914 1.985 2.057 1.985 0
#> alpha.0 1.011 0.032 0.947 1.011 1.074 1.011 0
#> alpha.z1 0.292 0.033 0.228 0.292 0.355 0.292 0
#> gamma.0 -1.510 0.122 -1.754 -1.509 -1.275 -1.509 0
#> gamma.z2 -0.425 0.088 -0.597 -0.425 -0.252 -0.425 0
#>
#> Random effects:
#> Name Model
#> id.x IID model
#> beta.x Copy
#> gamma.x Copy
#>
#> Model hyperparameters:
#> mean sd 0.025quant 0.5quant 0.975quant mode
#> Precision for the Gaussian observations 0.983 0.048 0.890 0.983 1.079 0.984
#> Precision for the Gaussian observations[2] 1.130 0.357 0.590 1.077 1.981 0.977
#> Precision for the Gaussian observations[3] 1.023 0.047 0.932 1.022 1.119 1.020
#> Beta for beta.x 1.953 0.035 1.884 1.953 2.021 1.955
#> Beta for gamma.x -0.036 0.089 -0.213 -0.036 0.138 -0.034
#>
#> Marginal log-Likelihood: -11663.19
#> is computed
#> Posterior summaries for the linear predictor and the fitted values are computed
#> (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')Looking at the summary, most importantly, note that the model picks
up gamma.0 and gamma.z2, which were defined to
be -1.5 and -0.5, respectively. In the data simulation, we did not
construct the missingness of x to depend on the value of
x, which would correspond to a “missing not at random”
mechanism. The fact that the missingness of x depends on
other observed covariates indicates a “missing at random” mechanism (as
we simulated it to be). If the missingness of x did not
depend on any other covariates, it would be “missing completely at
random”.