First example: A logistic regression model with repeated measurements
| Error types | Likelihood | Response | Covariate with error | Other covariate(s) |
|---|---|---|---|---|
| Classical | Binomial | disease |
sbp1, sbp2 |
smoking |
The data and model in this example was also used in Muff et al (2015), so more
information on the measurement error model can be found there. The model
is identical, this example just shows how it can be implemented in
inlamemi.
In this example, we fit a logistic regression model for whether or
not a patient has heart disease, using systolic blood pressure (SBP) and
smoking status as covariates. SBP is measured with error, but we have
repeated measurements, and so we would like to feed both measurements of
SBP into the model. This can be done easily in the inlamemi
package.
The formula for the main model of interest will be \[ \text{logit}\{\texttt{disease}_i\} = \beta_0 + \beta_{\texttt{sbp}} \texttt{sbp}_i + \beta_{\texttt{smoking}} \texttt{smoking}_i, \] and the formula for the imputation model will be
\[ \texttt{sbp}_i = \alpha_0 + \alpha_{\texttt{smoking}} \texttt{smoking}_i + \varepsilon_i^{\texttt{sbp}}. \] In addition, we of course also have the classical measurement error model that describes the actual error in the SBP measurements, and since we have repeated measurements we actually have two: \[ \begin{align} \texttt{sbp}^1_i = \texttt{sbp}_i + u_i^{1}, \\ \texttt{sbp}^2_i = \texttt{sbp}_i + u_i^{2}, \end{align} \] where \(u_i^{1}, u_i^{2} \sim N(0, \tau_u)\) are the measurement error terms.
We can then call the fit_inlamemi function directly with
the above formulas for the model of interest and imputation model. Also
note the repeated measurements argument, which must be set to
TRUE to ensure that the model is specified correctly. We
give the precision for the error term of the measurement error model a
prior, and the error term for the imputation model a prior. By default
in R-INLA, the fixed effects are given Gaussian priors with mean \(0\) and precision \(0.001\). We re-assign the precisions to be
\(0.01\), but keep the means at 0
(therefore they are not specified in the control.fixed
argument).
framingham_model <- fit_inlamemi(formula_moi = disease ~ sbp + smoking,
formula_imp = sbp ~ smoking,
family_moi = "binomial",
data = framingham,
error_type = "classical",
repeated_observations = TRUE,
prior.prec.classical = c(100, 1),
prior.prec.imp = c(10, 1),
prior.beta.error = c(0, 0.01),
initial.prec.classical = 100,
initial.prec.imp = 10,
control.fixed = list(
prec = list(beta.0 = 0.01,
beta.smoking = 0.01,
alpha.0 = 0.01,
alpha.smoking = 0.01)))Once the model is fit we can look at the summary.
summary(framingham_model)
#> Formula for model of interest:
#> disease ~ sbp + smoking
#>
#> Formula for imputation model:
#> sbp ~ smoking
#>
#> Error types:
#> [1] "classical"
#>
#> Fixed effects for model of interest:
#> mean sd 0.025quant 0.5quant 0.975quant mode
#> beta.0 -2.3607112 0.2687679 -2.8893566 -2.3601356 -1.835308 -2.3601232
#> beta.smoking 0.3989493 0.2976650 -0.1843516 0.3988021 0.983082 0.3987996
#>
#> Coefficient for variable with measurement error and/or missingness:
#> mean sd 0.025quant 0.5quant 0.975quant mode
#> beta.sbp 1.904282 0.5619162 0.8468405 1.888489 3.057103 1.817615
#>
#> Fixed effects for imputation model:
#> mean sd 0.025quant 0.5quant 0.975quant mode
#> alpha.sbp.0 0.01454299 0.01858129 -0.02190450 0.01454299 0.05099049 0.01454299
#> alpha.sbp.smoking -0.01958477 0.02156261 -0.06188018 -0.01958477 0.02271064 -0.01958477
#>
#> Model hyperparameters (apart from beta.sbp):
#> mean sd 0.025quant 0.5quant 0.975quant mode
#> Precision for sbp classical model 75.90183 3.690471 68.89250 75.81344 83.41963 75.63979
#> Precision for sbp imp model 19.89457 1.234048 17.55069 19.86502 22.40737 19.82449
For a comparison, we can also fit a “naive” model, that is, a model that ignores the measurement error in SBP. For this model, we will take an average of the two SBP measurements and use that as the SBP variable.
framingham$sbp <- (framingham$sbp1 + framingham$sbp2)/2
naive_model <- inla(formula = disease ~ sbp + smoking,
family = "binomial",
data = framingham)
naive_model$summary.fixed
#> mean sd 0.025quant 0.5quant 0.975quant mode kld
#> (Intercept) -2.3547258 0.2692636 -2.8824728 -2.3547258 -1.8269788 -2.3547258 0
#> sbp 1.6686897 0.4937443 0.7009686 1.6686897 2.6364107 1.6686897 0
#> smoking 0.3972557 0.2992211 -0.1892069 0.3972557 0.9837183 0.3972557 0Then we can compare the estimated coefficients from both models.
naive_result <- naive_model$summary.fixed
rownames(naive_result) <- c("beta.0", "beta.sbp", "beta.smoking")
naive_result$variable <- rownames(naive_result)
me_result <- rbind(summary(framingham_model)$moi_coef[1:6],
summary(framingham_model)$error_coef[1:6])
me_result$variable <- rownames(me_result)
results <- dplyr::bind_rows(naive = naive_result, me_adjusted = me_result, .id = "model")
ggplot(results, aes(x = mean, y = model, color = variable)) +
geom_point() +
geom_linerange(aes(xmin = `0.025quant`, xmax = `0.975quant`)) +
facet_grid(~ variable, scales = "free_x") +
theme_bw()
