--- title: "Influence of systolic blood pressure on coronary heart disease" output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{Influence of systolic blood pressure on coronary heart disease} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- 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. ``` r library(inlamemi) library(INLA) library(ggplot2) ``` ``` r 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)](https://arxiv.org/abs/1302.3065) and [Reverse attenuation in interaction terms due to covariate measurement error, Muff & Keller (2015)](https://doi.org/10.1002/bimj.201400157). ## 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)](https://arxiv.org/abs/1302.3065), 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 \texttt{Gamma(100, 1)} prior, and the error term for the imputation model a \texttt{Gamma(10, 1)} 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). ``` r 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. ``` r 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 -2.3615593 0.2704869 -2.893447 -2.3610439 -1.8325961 -2.3610384 #> beta.smoking 0.3993937 0.2994675 -0.187501 0.3992729 0.9869775 0.3992727 #> #> Coefficient for variable with measurement error and/or missingness: #> mean sd 0.025quant 0.5quant 0.975quant mode #> beta.sbp 1.897425 0.5598872 0.8404353 1.882702 3.043093 1.816999 #> #> Fixed effects for imputation model: #> mean sd 0.025quant 0.5quant 0.975quant mode #> alpha.sbp.0 0.01454298 0.01858277 -0.02190736 0.01454298 0.05099333 0.01454298 #> alpha.sbp.smoking -0.01958476 0.02156433 -0.06188347 -0.01958476 0.02271395 -0.01958476 #> #> Model hyperparameters (apart from beta.sbp): #> mean sd 0.025quant 0.5quant 0.975quant mode #> Precision for sbp classical model 75.90178 3.690529 68.89246 75.81334 83.41982 75.63953 #> Precision for sbp imp model 19.89452 1.234087 17.55047 19.86501 22.40730 19.82463 ``` ``` r plot(framingham_model) ``` ![plot of chunk unnamed-chunk-6](figure/unnamed-chunk-6-1.png) 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. ``` r 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. ``` r 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$variable <- rownames(me_result) results <- dplyr::bind_rows(naive = naive_result, me_adjusted = me_result, .id = "model") ggplot(results, aes(x = mean, y = model, color = variable)) + geom_point() + geom_linerange(aes(xmin = `0.025quant`, xmax = `0.975quant`)) + facet_grid(~ variable, scales = "free_x") + theme_bw() ``` ![plot of chunk unnamed-chunk-8](figure/unnamed-chunk-8-1.png) ## Second example: Heteroscedastic measurement error and interaction between error variable and error free variable |Error types | Likelihood | Response | Covariate with error | Other covariate(s) | |:-----------|:----------|:--------|:----------|:------------| |Classical (with interaction effects) | Binomial | `disease` | `sbp1`, `sbp2` | `smoking` | In this example, we artificially increase the measurement error for the smoking group in order to study a model with homoscedastic measurement error. The model in this example is identical to that in [Muff & Keller (2015)](https://doi.org/10.1002/bimj.201400157) ``` r framingham2 <- framingham n <- nrow(framingham2) set.seed(1) framingham2$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 ``` ``` r error_scaling <- c(1/(1+9*framingham2$smoking), 1/(1+9*framingham2$smoking)) ``` ``` r # 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) ``` ![plot of chunk unnamed-chunk-11](figure/unnamed-chunk-11-1.png) ``` r plot(framingham_model2.2) ``` ![plot of chunk unnamed-chunk-11](figure/unnamed-chunk-11-2.png) ``` r plot(framingham_model2.3) ``` ![plot of chunk unnamed-chunk-11](figure/unnamed-chunk-11-3.png) ``` r plot(framingham_model2.5) ``` ![plot of chunk unnamed-chunk-11](figure/unnamed-chunk-11-4.png) ``` r framingham_model2.1$summary.hyperpar #> mean sd 0.025quant 0.5quant 0.975quant mode #> Precision for the Gaussian observations[2] 75.941763 3.694984 68.9643744 75.839341 83.509525 75.608321 #> Precision for the Gaussian observations[3] 19.900701 1.237511 17.5888076 19.857868 22.459486 19.763236 #> Beta for beta.smokingsbp -1.549041 1.354832 -4.1265836 -1.578576 1.205354 -1.708567 #> Beta for beta.sbp 3.030323 1.193298 0.6201075 3.050937 5.317642 3.140323 ``` ``` r framingham_model2.2$summary.hyperpar #> mean sd 0.025quant 0.5quant 0.975quant mode #> Precision for the Gaussian observations[2] 156.9883399 7.229709 143.1926400 156.8372549 171.652037 156.5682728 #> Precision for the Gaussian observations[3] 22.1572213 1.655540 19.0896008 22.0913348 25.604402 21.9514968 #> Beta for beta.smokingsbp -0.5231413 1.490996 -3.6739986 -0.4503943 2.171770 -0.1023884 #> Beta for beta.sbp 2.9651518 1.117829 0.8996622 2.9228739 5.292024 2.7260210 ``` ``` r framingham_model2.3$summary.hyperpar #> mean sd 0.025quant 0.5quant 0.975quant mode #> Precision for the Gaussian observations[2] 45.151167 2.190922 40.9948014 45.097007 49.619168 44.986819 #> Precision for the Gaussian observations[3] 19.176661 1.275318 16.7672718 19.141740 21.786114 19.087701 #> Beta for beta.smokingsbp -1.556773 1.452213 -4.1875780 -1.626083 1.507422 -1.956430 #> Beta for beta.sbp 3.240676 1.294155 0.5215732 3.298372 5.600184 3.571394 ``` ``` r framingham_model2.5$summary.hyperpar #> mean sd 0.025quant 0.5quant 0.975quant mode #> Precision for the Gaussian observations[2] 5.053238e-11 2.451925e-12 4.585460e-11 5.048081e-11 5.550617e-11 5.039490e-11 #> Precision for the Gaussian observations[3] 1.880235e+01 1.224319e+00 1.650633e+01 1.876300e+01 2.132478e+01 1.868547e+01 #> Beta for beta.smokingsbp -1.570571e+00 1.390761e+00 -4.317334e+00 -1.567547e+00 1.158593e+00 -1.555004e+00 #> Beta for beta.sbp 3.234344e+00 1.245149e+00 7.929082e-01 3.230958e+00 5.695506e+00 3.216894e+00 ```