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In this tutorial, we re-do the analysis from two previous
publications that use the same data set. The data set contains the heart
disease status for a number of patients, along with repeated
measurements of their systolic blood pressure and their smoking status.
The data is also included in the package, simply type
?framingham
for more details.
head(framingham)
#> disease sbp1 sbp2 smoking sbp
#> 1 0 0.043556319 -0.24144440 0 -0.09894404
#> 2 0 -0.088249125 -0.03871185 1 -0.06348049
#> 3 0 0.063634980 0.02971774 1 0.04667636
#> 4 0 -0.331791943 -0.44704230 1 -0.38941712
#> 5 0 0.006502333 -0.09760337 1 -0.04555052
#> 6 0 0.124884993 0.09793482 0 0.11140991
The two publications are Bayesian analysis of measurement error models using INLA, Muff et al (2015) and Reverse attenuation in interaction terms due to covariate measurement error, Muff & Keller (2015).
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 0
Then 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$variable <- rownames(me_result)
results <- dplyr::bind_rows(naive = naive_result, me_adjusted = me_result, .id = "model")
ggplot(results, aes(x = mean, y = model, color = variable)) +
geom_point() +
geom_linerange(aes(xmin = `0.025quant`, xmax = `0.975quant`)) +
facet_grid(~ variable, scales = "free_x") +
theme_bw()
Error types | Likelihood | Response | Covariate with error | Other covariate(s) |
---|---|---|---|---|
Classical (with interaction effects) | Binomial | disease |
sbp1 , sbp2 |
smoking |
In this example, we artificially increase the measurement error for the smoking group in order to study a model with homoscedastic measurement error. The model in this example is identical to that in Muff & Keller (2015)
framingham2 <- framingham
n <- nrow(framingham2)
set.seed(1)
framingham2$sbp1 <- framingham$sbp1 +
ifelse(framingham2$smoking == 1, rnorm(n, 0, 0.117), 0)
framingham2$sbp2 <- framingham$sbp2 +
ifelse(framingham2$smoking == 1, rnorm(n, 0, 0.117), 0)
# Or:
#framingham2 <- read.table("../data-raw/fram_data_case2.txt", header=T)
#names(framingham2) <- c("disease", "sbp1", "sbp2", "smoking")
#framingham2$sbp <- (framingham2$sbp1 + framingham2$sbp2)/2
# Homoscedastic ME modeled as homoscedastic ME (correct)
framingham_model2.1 <- 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)))
# Heteroscedastic ME modeled as heteroscedastic ME (correct)
framingham_model2.2 <- fit_inlamemi(formula_moi = disease ~ sbp:smoking,
formula_imp = sbp ~ smoking,
family_moi = "binomial",
data = framingham2,
error_type = "classical",
repeated_observations = TRUE,
classical_error_scaling = error_scaling,
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)))
# Heteroscedastic ME modeled as homoscedastic ME (incorrect)
framingham_model2.3 <- fit_inlamemi(formula_moi = disease ~ sbp:smoking,
formula_imp = sbp ~ smoking,
family_moi = "binomial",
data = framingham2,
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)))
framingham_model2.5 <- fit_inlamemi(formula_moi = disease ~ sbp:smoking,
formula_imp = sbp ~ smoking,
family_moi = "binomial",
data = framingham2,
error_type = "classical",
repeated_observations = TRUE,
classical_error_scaling = c(rep(10^(-12), 2*n)),
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)))
plot(framingham_model2.1)
framingham_model2.1$summary.hyperpar
#> mean sd 0.025quant 0.5quant 0.975quant mode
#> Precision for the Gaussian observations[2] 75.924697 3.691755 68.9231297 75.832746 83.455384 75.644635
#> Precision for the Gaussian observations[3] 19.898609 1.235568 17.5745308 19.861274 22.437435 19.788861
#> Beta for beta.smokingsbp -1.581615 1.360291 -4.1917069 -1.604255 1.163213 -1.702103
#> Beta for beta.sbp 3.044022 1.187946 0.6632897 3.058256 5.340477 3.118959
framingham_model2.2$summary.hyperpar
#> mean sd 0.025quant 0.5quant 0.975quant mode
#> Precision for the Gaussian observations[2] 157.116140 7.218347 143.230428 157.003324 171.644168 156.89258604
#> Precision for the Gaussian observations[3] 22.110110 1.647325 19.055748 22.045219 25.538246 21.90841768
#> Beta for beta.smokingsbp -1.579268 1.872695 -5.994113 -1.309470 1.023051 -0.01175282
#> Beta for beta.sbp 3.445319 1.421107 1.447729 3.245299 6.784787 2.06839603
framingham_model2.3$summary.hyperpar
#> mean sd 0.025quant 0.5quant 0.975quant mode
#> Precision for the Gaussian observations[2] 45.168891 2.194701 41.0375093 45.103655 49.676929 44.947917
#> Precision for the Gaussian observations[3] 19.176318 1.276858 16.7852215 19.134095 21.810352 19.050388
#> Beta for beta.smokingsbp -1.613992 1.410432 -4.1837629 -1.677446 1.350463 -1.977202
#> Beta for beta.sbp 3.277247 1.263836 0.6311494 3.330524 5.593007 3.580511
framingham_model2.5$summary.hyperpar
#> mean sd 0.025quant 0.5quant 0.975quant mode
#> Precision for the Gaussian observations[2] 7.411261e+02 27.2248319 688.7987773 7.406866e+02 795.9767592 7.399389e+02
#> Precision for the Gaussian observations[3] 1.012029e+01 3.2274084 5.0628791 9.692601e+00 17.6274785 8.911691e+00
#> Beta for beta.smokingsbp 4.336929e-04 0.2996034 -0.5895143 4.765140e-04 0.5901333 6.533428e-04
#> Beta for beta.sbp 4.231469e-04 0.2821837 -0.5552294 4.654132e-04 0.5558305 6.399530e-04
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
Health stats visible at Monitor.