---
title: '`refitME`: Tutorial for fitting measurement error models using Monte Carlo Expectation Maximization in `R`'
author: "Jakub Stoklosa, Hwang W-H., & David Warton"
date: "`r format(Sys.time(), '%d %B, %Y')`"
header-includes: # Allows you to add in your own Latex packages.
- \usepackage{float} # Use the 'float' package.
- \floatplacement{figure}{H} # Make every figure with caption = h
- \usepackage{ragged2e}
- \usepackage{geometry}
- \geometry{verbose,tmargin=3cm,bmargin=3cm,lmargin=3cm,rmargin=3cm}
output:
  pdf_document: default
  fig_caption: yes
  html_document: default
  word_document: default
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

```{r global_options, include=FALSE}
knitr::opts_chunk$set(fig.pos = 'H')
```

This tutorial documents fitting an MCEM algorithm via the `refitME` `R`-package. For more specific details see: *`refitME`: Measurement Error Modelling using Monte Carlo Expectation Maximization in `R`.* Also, see ```?refitME``` for further details on the fitting function, input arguments and output.

## Example 1: A simple GLM example taken from Carroll _et al._ (2006). 

We use the Framingham heart study data set. In addition to the naive model, we also fit a `simex` model and compare it with MCEM. Computational times for both models are also reported.

### Load data and `R`-packages.

```{r, echo=TRUE}
suppressWarnings(suppressMessages(library(refitME)))
suppressWarnings(suppressMessages(library(simex)))

set.seed(2020)

data(Framinghamdata)
```

### Fit the naive model.

The first stored variable `w1` is the error contaminated predictor as used in the original Carroll _et al._ (2006) analysis.

```{r, echo=TRUE}
glm_naiv <- glm(Y ~ w1 + z1 + z2 + z3, x = TRUE, family = binomial, 
                 data = Framinghamdata)
```

### Setup measurement error (ME) variance and all tuning parameters.

```{r, echo=TRUE}
sigma.sq.u <- 0.01259/2 # ME variance, as obtained from Carroll et al. (2006).
B <- 100  # The number of Monte Carlo replication values/SIMEX simulations.
```

\clearpage

### Fit the SIMEX model.

```{r, echo=TRUE}
start <- Sys.time()
glm_simex <- simex(glm_naiv, SIMEXvariable = c("w1"),
                  measurement.error = cbind(sqrt(sigma.sq.u)), B = B) # SIMEX.
end <- Sys.time()
t1 <- difftime(end, start, units = "secs")
comp.time <- c(t1)
```

### Fit the MCEM model.

```{r, echo=TRUE}
start <- Sys.time()
glm_MCEM <- refitME(glm_naiv, sigma.sq.u, B)
end <- Sys.time()
t2 <- difftime(end, start, units = "secs")
comp.time <- c(comp.time, t2)
```

### Report model estimates and compare computational times.

```{r, echo=TRUE}
est.beta <- rbind(coef(glm_naiv), coef(glm_simex), coef(glm_MCEM))
est.beta.se <- rbind(sqrt(diag(vcov(glm_naiv))),
                   sqrt(diag(glm_simex$variance.jackknife)), sqrt(diag(vcov(glm_MCEM))))
row.names(est.beta) = row.names(est.beta.se) <- c("Naive GLM", "SIMEX", "MCEM")
colnames(est.beta) = colnames(est.beta.se) <- c("(Intercept)", "SBP", "chol. level", 
                                                "age", "smoke")
round(est.beta, digits = 3)
round(est.beta.se, digits = 3)  # Standard error estimates.

names(comp.time) <- c("SIMEX", "MCEM")
comp.time  # SIMEX and MCEM.
```

## Example 2: A GAM example taken from Ganguli _et al._ (2005).

The Milan mortality air pollution data set is part of the `SemiPar` package. Here, we fit GAM models via the `mgcv` package where one predictor (daily total suspended particles measurements) is error-contaminated.

### Load data and `R`-packages.

```{r, echo=TRUE}
suppressWarnings(suppressMessages(library(refitME)))
suppressWarnings(suppressMessages(library(SemiPar)))
suppressWarnings(suppressMessages(library(mgcv)))

set.seed(2020)

data(milan.mort)
dat.air <- milan.mort
``` 

### Setup all variables.

```{r, echo=TRUE}
Y <- dat.air[, 6]  # Mortality counts.
n <- length(Y)

z1 <- (dat.air[, 1])
z2 <- (dat.air[, 4])
z3 <- (dat.air[, 5])
w1 <- log(dat.air[, 9]) # The error-contaminated predictor (total suspended particles).
dat <- data.frame(cbind(Y, w1, z1, z2, z3))
```

### Fit the naive model.

```{r, echo=TRUE}
gam_naiv <- gam(Y ~ s(w1) + s(z1, k = 25) + s(z2) + s(z3), family = "poisson", data = dat)
```

```{r, echo=FALSE, fig.show='hide'}
plot_gam_naiv <- plot(gam_naiv, select = 1)
```

\clearpage

### Fit the MCEM model.

Note this will take a fair while to run if `B` is large (consider decreasing `B` if you want a rough answer).

```{r, echo=TRUE}
sigma.sq.u <- 0.0915 # This gives a reliability ratio of 0.7.
rel.rat <- round((1 - sigma.sq.u/var(dat$w1))*100, digits = 0)

gam_MCEM1 <- refitME(gam_naiv, sigma.sq.u)
```

```{r fig1, fig.height=5.5, fig.pos='H', echo=FALSE, fig.cap="\\label{fig:fig1} *Plots of smooths against each predictor. TSP (top left is the error contaminated variable).*"}
xlab.names <- c("log(TSP)", "Day", "Temp", "Humidity")

op <- par(mfrow = c(2, 2), las = 1)

for(i in 1:4) {
  if (i == 1) {
    plot(gam_MCEM1, select = i, ylim = c(-0.35, 0.1), xlim = range(plot_gam_naiv[[1]]$x), 
         rug = FALSE, col = "blue", all.terms = TRUE,
         xlab = xlab.names[i], ylab = "s(Mortality counts)", lwd = 2, cex.lab = 1.3, cex.axis = 1.3,
         cex.main = 2, font.lab = 1.1, cex = 1.4, shade = T)
    lines(plot_gam_naiv[[1]]$x, plot_gam_naiv[[1]]$fit, type = "l", col = "red", lwd = 2, lty = 2)
    title(main = bquote("Reliability ratio of predictor is"~.(rel.rat) ~ "%"),
          outer = F, line = 1, cex = 1.4)
    legend("bottomright", c("Naive GAM", "MCEM GAM"), col = c("red", "blue"), 
           lty = c(2, 1), lwd = 2, bty = "n")
    for(j in 1:2) {
      axis(j, labels = FALSE)
    }
  }
  if (i == 2) {
    plot(gam_MCEM1, select = i, ylim = c(-0.25, 0.3), rug = FALSE, col = "blue", all.terms = TRUE,
         xlab = xlab.names[i], ylab = "s(Mortality counts)", lwd = 2, cex.lab = 1.3, cex.axis = 1.3,
         cex.main = 2, font.lab = 1.1, cex = 1.4, shade = T)
    lines(plot_gam_naiv[[2]]$x, plot_gam_naiv[[2]]$fit, type = "l", col = "red", lwd = 2, lty = 2)
    for(j in 1:2) {
      axis(j, labels = FALSE)
    }
  }
  if (i == 3) {
    plot(gam_MCEM1, select = i, ylim = c(-0.2, 0.4), rug = FALSE, col = "blue", all.terms = TRUE,
         xlab = xlab.names[i], ylab = "s(Mortality counts)", lwd = 2, cex.lab = 1.3, cex.axis = 1.3,
         cex.main = 2, font.lab = 1.1, cex = 1.4, shade = T)
    lines(plot_gam_naiv[[3]]$x, plot_gam_naiv[[3]]$fit, type = "l", col = "red", lwd = 2, lty = 2)
    for(j in 1:2) {
      axis(j, labels = FALSE)
    }
  }
  if (i == 4) {
    plot(gam_MCEM1, select = i, ylim = c(-0.06, 0.08), rug = FALSE, col = "blue", all.terms = TRUE,
         xlab = xlab.names[i], ylab = "s(Mortality counts)", lwd = 2, cex.lab = 1.1, cex.axis = 1.1,
         cex.main = 2, font.lab = 1.1, cex = 1.4, shade = T)
    lines(plot_gam_naiv[[4]]$x, plot_gam_naiv[[4]]$fit, type = "l", col = "red", lwd = 2, lty = 2)
    for(j in 1:2) {
      axis(j, labels = FALSE)
    }
  }
}

title(main = "MCEM (Poisson GAM) fitted to the air pollution data.", outer = T, line = -2)
par(op)

detach(package:mgcv)
```

\clearpage

## Example 3: A point-process model using presence-only data

We use the _Corymbia eximia_ presence-only data set from Renner and Warton (2013). Here, we fit a naive models point-process model (PPM) and an MCEM PPM.

### Load data and `R`-packages.

```{r, echo=TRUE}
suppressWarnings(suppressMessages(library(refitME)))
suppressWarnings(suppressMessages(library(caret)))

data(Corymbiaeximiadata)

dat <- Corymbiaeximiadata
``` 

### Setup all variables.

```{r, echo=TRUE}
Y1 <- dat$Y.obs

Rain <- dat$Rain
D.Main <- dat$D.Main
MNT <- dat$MNT

p.wt <- rep(1.e-6, length(Y1))
p.wt[Y1 == 0] <- 1

Y <- Y1/p.wt

X <- cbind(rep(1, length(Y)), poly(MNT, degree = 2, raw = TRUE),
          poly(Rain, degree = 2, raw = TRUE),
            poly(sqrt(D.Main), degree = 2, raw = TRUE))

colnames(X) <- c("(Intercept)", "X1", "X2", "Z1", "Z2", "Z3", "Z4")

dat1 <- data.frame(cbind(Y, X, p.wt))
colnames(dat1)[c(1, ncol(dat1))] <- c("Y", "p.wt")
```

### Fit the naive (PPM) model.

```{r, echo=TRUE}
PPM_naiv1 <- glm(Y ~ X1 + X2 + Z1 + Z2 + Z3 + Z4, family = "poisson",
 weights = p.wt, data = dat1)
```

\clearpage

### Fit the MCEM PPM.

```{r, echo=TRUE}
sigma.sq.u <- 0.25

PPM_MCEM1 <- refitME(PPM_naiv1, sigma.sq.u)
```

```{r fig2, fig.height=5.5, fig.pos='H', echo=FALSE, fig.cap="\\label{fig:fig1} *Plot of predicted presences of Corymbia eximia using presence-only data when fitting the MCEM model. Here the max temperature predictor is assumed to be error-contaminated.*"}
coord.dat <- cbind(Corymbiaeximiadata$X, Corymbiaeximiadata$Y)

colnames(coord.dat) <- c("Longitude", "Latitude")

pred.dats <- as.data.frame(cbind(coord.dat[, 1], coord.dat[, 2], PPM_MCEM1$fitted.values))

colnames(pred.dats) <- c("x", "y", "preds")

levelplot(preds ~ x + y, data = pred.dats, asp = "iso", ylab = "Latitude", 
 xlab = "Longitude", col.regions = heat.colors(1024)[900:1], cuts = 900, 
 main = list("", cex = 5), scales = list(y = list(draw = FALSE), 
 x = list(draw = FALSE), relation = "free"), 
 colorkey = list(labels = list(cex = 0.8)))
```

## Example 4: A `VGAM` example using the Prinia flaviventris capture-recapture data.

We use the _Prinia flaviventris_ capture-recapture data set from Hwang, Huang and Wang (2007). Here, we fit naive `vglm` and `vgam` capture-recapture models, and the MCEM capture-recapture model. For all models we used the `posbinomial()` family provided in `VGAM`.

### Load data and `R`-packages.

```{r, echo=TRUE}
suppressWarnings(suppressMessages(library(refitME)))
suppressMessages(library(VGAM))
data(Priniadata)
```

### Setup all variables.

```{r, echo=TRUE}
tau <- 17   # No. of capture occasions.
w2 <- c(scale(Priniadata$w1)) # Standardize the wing length predictor.
Priniadata$w2 <- w2
```

### Fit the naive `vglm` and `vgam` capture-recapture models.

```{r, echo=TRUE}
CR_naiv1 <- vglm(cbind(cap, noncap) ~ w2,
                 VGAM::posbinomial(omit.constant = TRUE, 
                                   parallel = TRUE ~ w2),
                 data = Priniadata, trace = FALSE)
CR_naiv2 <- vgam(cbind(cap, noncap) ~ s(w2, df = 2),
                 VGAM::posbinomial(omit.constant = TRUE, 
                                   parallel = TRUE ~ s(w2, df = 2)),
                 data = Priniadata, trace = FALSE)
```

### Fit the MCEM capture-recapture model.

```{r, echo=TRUE}
sigma.sq.u <- 0.37/var(Priniadata$w1) # Measurement error variance.
B <- 100
CR_MCEM <- refitME(CR_naiv2, sigma.sq.u, B)

detach(package:VGAM)
```
