---
title: "geeCRT package"
author: "Hengshi Yu, Fan Li, Paul Rathouz, Elizabeth L. Turner, John Preisser"
date: "`r Sys.Date()`"
bibliography: references.bib
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{geeCRT package for the design and analysis of cluster randomized trials}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---
```{r start,echo=FALSE,results="hide"}
library(geeCRT)
```


## Overview
geeCRT is an R package for implementing the bias-corrected generalized estimating equations in analyzing cluster randomized trials.

Population-averaged models have been increasingly used in the design and analysis of cluster randomized trials (CRTs). To facilitate the applications of population-averaged models in CRTs, we implement the generalized estimating equations (GEE) and matrix-adjusted estimating equations (MAEE) approaches to jointly estimate the marginal mean models correlation models both for general CRTs and stepped wedge CRTs.

Despite the general GEE/MAEE approach, we also implement a fast cluster-period GEE method specifically for stepped wedge CRTs with large and variable cluster-period sizes. The individual-level GEE/MAEE approach becomes computationally infeasible in this setting due to inversion of high-dimensional covariance matrices and the enumeration of a high-dimensional design matrix for the correlation estimation equations. The package gives a simple and efficient estimating equations approach based on the cluster-period means to estimate the intervention effects as well as correlation parameters.

In addition, the package also provides functions for generating correlated binary data with specific mean vector and correlation matrix based on the multivariate probit method [@emrich1991method] or the conditional linear family method [@qaqish2003family]. These two functions facilitate generating correlated binary data in future simulation studies. 

## `geemaee()` examples: matrix-adjusted GEE for estimating the mean and correlation parameters in SW-CRTs

The `geemaee()` function implements the matrix-adjusted GEE or regular GEE developed for analyzing cluster randomized trials (CRTs). It provides valid estimation and inference for the treatment effect and intraclass correlation parameters within the population-averaged modeling framework. The program allows for flexible marginal mean model specifications. The program also offers bias-corrected intraclass correlation coefficient (ICC) estimates as well as bias-corrected sandwich variances for both the treatment effect parameter and the ICC parameters. The technical details of the matrix-adjusted GEE approach are provided in [@preisser2008finite] and [@li2018sample].

For the individual-level data, we use the `geemaee()` function to estimate the marginal mean and correlation parameters in CRTs. We use two simulated stepped wedge CRT datasets with true nested exchangeable correlation structure to illustrate the `geemaee()` function examples. 

### Simulated dataset with 12 clusters and 4 periods

```{r, echo=FALSE, results='asis'}
knitr::kable(head(sampleSWCRTSmall))
```

We first create an auxiliary function `createzCrossSec()` to help create the design matrix for the estimating equations of the correlation parameters assuming that the correlations follow a nested exchangeable structure. We then collect design matrix `X` for the mean parameters with five period indicators and the treatment indicator.

```{r createz, fig.keep="all", fig.width = 7, fig.height=4}
# Z matrix for nested exchangeable correlation structure
createzCrossSec <- function(m) {
  Z <- NULL
  n <- dim(m)[1]

  for (i in 1:n) {
    alpha_0 <- 1
    alpha_1 <- 2
    n_i <- c(m[i, ])
    n_length <- length(n_i)
    POS <- matrix(alpha_1, sum(n_i), sum(n_i))
    loc1 <- 0
    loc2 <- 0

    for (s in 1:n_length) {
      n_t <- n_i[s]
      loc1 <- loc2 + 1
      loc2 <- loc1 + n_t - 1

      for (k in loc1:loc2) {
        for (j in loc1:loc2) {
          if (k != j) {
            POS[k, j] <- alpha_0
          } else {
            POS[k, j] <- 0
          }
        }
      }
    }

    zrow <- diag(2)
    z_c <- NULL

    for (j in 1:(sum(n_i) - 1)) {
      for (k in (j + 1):sum(n_i)) {
        z_c <- rbind(z_c, zrow[POS[j, k], ])
      }
    }

    Z <- rbind(Z, z_c)
  }

  return(Z)
}
``` 

We implement the `geemaee()` function on continuous, binary and count outcomes using the nested exchangeable correlation structure, and consider both matrix-adjusted estimating equations (MAEE) with `alpadj = TRUE` and uncorrected generalized estimating equations (GEE) with `alpadj = FALSE`. We use the binary outcome for the count outcome type of the `geemaee()` function, and consider both `poisson` and `quasipoisson` distributions for the count outcome. For the `shrink` argument, we use the `"ALPHA"` method to tune step sizes and focus on using estimated variances in the correlation estimating equations rather than using unit variances by specifying `makevone = FALSE`. 


```{r geemaee_small, fig.keep="all", fig.width = 7, fig.height=4}
########################################################################
### Example 1): simulated SW-CRT with small cluster-period sizes (5~10)
########################################################################
sampleSWCRT <- sampleSWCRTSmall

### Individual-level id, period, outcome, and design matrix
id <- sampleSWCRT$id
period <- sampleSWCRT$period
X <- as.matrix(sampleSWCRT[, c("period1", "period2", "period3", "period4", "treatment")])
m <- as.matrix(table(id, period))
n <- dim(m)[1]
t <- dim(m)[2]

### design matrix for correlation parameters
Z <- createzCrossSec(m)

### (1) Matrix-adjusted estimating equations and GEE
### on continuous outcome with nested exchangeable correlation structure

### MAEE
est_maee_ind_con <- geemaee(
  y = sampleSWCRT$y_con,
  X = X, id = id, Z = Z,
  family = "continuous",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = TRUE,
  shrink = "ALPHA", makevone = FALSE
)

### GEE
est_uee_ind_con <- geemaee(
  y = sampleSWCRT$y_con,
  X = X, id = id, Z = Z,
  family = "continuous",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = FALSE,
  shrink = "ALPHA", makevone = FALSE
)

### (2) Matrix-adjusted estimating equations and GEE
### on binary outcome with nested exchangeable correlation structure

### MAEE
est_maee_ind_bin <- geemaee(
  y = sampleSWCRT$y_bin,
  X = X, id = id, Z = Z,
  family = "binomial",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = TRUE,
  shrink = "ALPHA", makevone = FALSE
)

### GEE
est_uee_ind_bin <- geemaee(
  y = sampleSWCRT$y_bin,
  X = X, id = id, Z = Z,
  family = "binomial",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = FALSE,
  shrink = "ALPHA", makevone = FALSE
)


### (3) Matrix-adjusted estimating equations and GEE
### on count outcome with nested exchangeable correlation structure
### using Poisson distribution

### MAEE
est_maee_ind_cnt_poisson <- geemaee(
  y = sampleSWCRT$y_bin,
  X = X, id = id, Z = Z,
  family = "poisson",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = TRUE,
  shrink = "ALPHA", makevone = FALSE
)

### GEE
est_uee_ind_cnt_poisson <- geemaee(
  y = sampleSWCRT$y_bin,
  X = X, id = id, Z = Z,
  family = "poisson",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = FALSE,
  shrink = "ALPHA", makevone = FALSE
)


### (4) Matrix-adjusted estimating equations and GEE
### on count outcome with nested exchangeable correlation structure
### using Quasi-Poisson distribution

### MAEE
est_maee_ind_cnt_quasipoisson <- geemaee(
  y = sampleSWCRT$y_bin,
  X = X, id = id, Z = Z,
  family = "quasipoisson",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = TRUE,
  shrink = "ALPHA", makevone = FALSE
)

### GEE
est_uee_ind_cnt_quasipoisson <- geemaee(
  y = sampleSWCRT$y_bin,
  X = X, id = id, Z = Z,
  family = "quasipoisson",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = FALSE,
  shrink = "ALPHA", makevone = FALSE
)
```


Then we have the following output for continuous and binary outcomes: 
```{r set-options1, echo=FALSE, fig.keep="all", fig.width = 7, fig.height=4}
options(width = 100)
```

```{r geemaee_small_nex, fig.keep="all", fig.width = 7, fig.height=4}
# MAEE for continuous outcome
print(est_maee_ind_con)

# GEE for continuous outcome
print(est_uee_ind_con)

# MAEE for binary outcome
print(est_maee_ind_bin)

# GEE for binary outcome
print(est_uee_ind_bin)
```


In addition, we have the following output for count outcomes using Poisson and Quasi-Poisson distributions: 
```{r geemaee_small_nex_cnt, fig.keep="all", fig.width = 7, fig.height=4}
# MAEE for count outcome using Poisson distribution
print(est_maee_ind_cnt_poisson)

# GEE for count outcome using Poisson distribution
print(est_uee_ind_cnt_poisson)

# MAEE for count outcome using Quasi-Poisson distribution
print(est_maee_ind_cnt_quasipoisson)

# GEE for count outcome using Quasi-Poisson distribution
print(est_uee_ind_cnt_quasipoisson)
```

We also consider the simple exchangeable correlation structure for the cluster-period responses, where we specify the `Z` matrix as a single column of 1's. We implement the `geemaee()` function on both the continuous outcome and binary outcome, and consider both matrix-adjusted estimating equations (MAEE) with `alpadj = TRUE` and uncorrected generalized estimating equations (GEE) with `alpadj = FALSE`. For the `shrink` argument, we use the `"ALPHA"` method to tune step sizes and focus on using estimated variances in the correlation estimating equations rather than using unit variances by specifying `makevone = FALSE`. 


```{r geemaee_small_exc, fig.keep="all", fig.width = 7, fig.height=4}
sampleSWCRT <- sampleSWCRTSmall

### Individual-level id, period, outcome, and design matrix
id <- sampleSWCRT$id
period <- sampleSWCRT$period
X <- as.matrix(sampleSWCRT[, c("period1", "period2", "period3", "period4", "treatment")])
m <- as.matrix(table(id, period))
n <- dim(m)[1]
t <- dim(m)[2]

### design matrix for correlation parameters (simple exchangeable correlation structure)
Z <- createzCrossSec(m)
Z <- matrix(1, dim(Z)[1], 1)

### (1) Matrix-adjusted estimating equations and GEE
### on continuous outcome with nested exchangeable correlation structure

### MAEE
est_maee_ind_con <- geemaee(
  y = sampleSWCRT$y_con,
  X = X, id = id, Z = Z,
  family = "continuous",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = TRUE,
  shrink = "ALPHA", makevone = FALSE
)

### GEE
est_uee_ind_con <- geemaee(
  y = sampleSWCRT$y_con,
  X = X, id = id, Z = Z,
  family = "continuous",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = FALSE,
  shrink = "ALPHA", makevone = FALSE
)

### (2) Matrix-adjusted estimating equations and GEE
### on binary outcome with nested exchangeable correlation structure

### MAEE
est_maee_ind_bin <- geemaee(
  y = sampleSWCRT$y_bin,
  X = X, id = id, Z = Z,
  family = "binomial",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = TRUE,
  shrink = "ALPHA", makevone = FALSE
)


### GEE
est_uee_ind_bin <- geemaee(
  y = sampleSWCRT$y_bin,
  X = X, id = id, Z = Z,
  family = "binomial",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = FALSE,
  shrink = "ALPHA", makevone = FALSE
)
```

Then we have the following output: 
```{r set-options2, echo=FALSE, fig.keep="all", fig.width = 7, fig.height=4}
options(width = 100)
```

```{r , fig.keep="all", fig.width = 7, fig.height=4}
# MAEE for continuous outcome
print(est_maee_ind_con)

# GEE for continuous outcome
print(est_uee_ind_con)

# MAEE for binary outcome
print(est_maee_ind_bin)

# GEE for binary outcome
print(est_uee_ind_bin)
```



### Simulated dataset with 12 clusters and 5 periods


```{r, echo=FALSE, results='asis'}
knitr::kable(head(sampleSWCRTLarge))
```

Using the nested exchangeable correlation structure, we have the following results.
```{r geemaee_large, fig.keep="all", fig.width = 7, fig.height=4}
########################################################################
### Example 2): simulated SW-CRT with larger cluster-period sizes (20~30)
########################################################################

## This will elapse longer.
sampleSWCRT <- sampleSWCRTLarge

### Individual-level id, period, outcome, and design matrix
id <- sampleSWCRT$id
period <- sampleSWCRT$period
X <- as.matrix(sampleSWCRT[, c("period1", "period2", "period3", "period4", "period5", "treatment")])
m <- as.matrix(table(id, period))
n <- dim(m)[1]
t <- dim(m)[2]
### design matrix for correlation parameters
Z <- createzCrossSec(m)

### (1) Matrix-adjusted estimating equations and GEE
### on continous outcome with nested exchangeable correlation structure

### MAEE
est_maee_ind_con <- geemaee(
  y = sampleSWCRT$y_con,
  X = X, id = id, Z = Z,
  family = "continuous",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = TRUE,
  shrink = "ALPHA", makevone = FALSE
)
print(est_maee_ind_con)

### GEE
est_uee_ind_con <- geemaee(
  y = sampleSWCRT$y_con,
  X = X, id = id, Z = Z,
  family = "continuous",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = FALSE,
  shrink = "ALPHA", makevone = FALSE
)
print(est_uee_ind_con)

### (2) Matrix-adjusted estimating equations and GEE
### on binary outcome with nested exchangeable correlation structure

### MAEE
est_maee_ind_bin <- geemaee(
  y = sampleSWCRT$y_bin,
  X = X, id = id, Z = Z,
  family = "binomial",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = TRUE,
  shrink = "ALPHA", makevone = FALSE
)
print(est_maee_ind_bin)

### GEE
est_uee_ind_bin <- geemaee(
  y = sampleSWCRT$y_bin,
  X = X, id = id, Z = Z,
  family = "binomial",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = FALSE,
  shrink = "ALPHA", makevone = FALSE
)
print(est_uee_ind_bin)
```

Using the simple exchangeable correlation structure, we have the following results.
```{r geemaee_large_exc, fig.keep="all", fig.width = 7, fig.height=4}
sampleSWCRT <- sampleSWCRTLarge

### Individual-level id, period, outcome, and design matrix
id <- sampleSWCRT$id
period <- sampleSWCRT$period
X <- as.matrix(sampleSWCRT[, c("period1", "period2", "period3", "period4", "period5", "treatment")])
m <- as.matrix(table(id, period))
n <- dim(m)[1]
t <- dim(m)[2]
### design matrix for correlation parameters
Z <- createzCrossSec(m)
Z <- matrix(1, dim(Z)[1], 1)

### (1) Matrix-adjusted estimating equations and GEE
### on continous outcome with nested exchangeable correlation structure

### MAEE
est_maee_ind_con <- geemaee(
  y = sampleSWCRT$y_con,
  X = X, id = id, Z = Z,
  family = "continuous",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = TRUE,
  shrink = "ALPHA", makevone = FALSE
)
print(est_maee_ind_con)

### GEE
est_uee_ind_con <- geemaee(
  y = sampleSWCRT$y_con,
  X = X, id = id, Z = Z,
  family = "continuous",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = FALSE,
  shrink = "ALPHA", makevone = FALSE
)
print(est_uee_ind_con)

### (2) Matrix-adjusted estimating equations and GEE
### on binary outcome with nested exchangeable correlation structure

### MAEE
est_maee_ind_bin <- geemaee(
  y = sampleSWCRT$y_bin,
  X = X, id = id, Z = Z,
  family = "binomial",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = TRUE,
  shrink = "ALPHA", makevone = FALSE
)
print(est_maee_ind_bin)

### GEE
est_uee_ind_bin <- geemaee(
  y = sampleSWCRT$y_bin,
  X = X, id = id, Z = Z,
  family = "binomial",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = FALSE,
  shrink = "ALPHA", makevone = FALSE
)
print(est_uee_ind_bin)
```

## `geemaee()` examples: matrix-adjusted GEE for estimating the mean and correlation parameters in SW-CRTs when some clusters had only 1 observation 
We can  use the `geemaee()` function for SW-CRTs when some clusters had only one observation. 

### Simulated dataset with 12 clusters and small cluster sizes
```{r geemaee_small_cluOneObsCount_swcrt, fig.keep="all", fig.width = 7, fig.height=4}
## let the cluster 5 and cluster 10 be with only 1 observation
sampleSWCRT <- sampleSWCRTSmall[sampleSWCRTSmall$id != 5 & sampleSWCRTSmall$id != 10, ]
sampleSWCRT5 <- sampleSWCRTSmall[sampleSWCRTSmall$id == 5, ][1, ]
sampleSWCRT10 <- sampleSWCRTSmall[sampleSWCRTSmall$id == 10, ][1, ]
sampleSWCRT <- rbind(sampleSWCRT, sampleSWCRT5)
sampleSWCRT <- rbind(sampleSWCRT, sampleSWCRT10)
sampleSWCRT <- sampleSWCRT[order(sampleSWCRT$id), ]

id <- sampleSWCRT$id
period <- sampleSWCRT$period
X <- as.matrix(sampleSWCRT[, c("period1", "period2", "period3", "period4", "treatment")])
m <- as.matrix(table(id, period))
n <- dim(m)[1]
t <- dim(m)[2]

## function to generate Z matrix (ignore the clusters with only 1 observation)
createzCrossSecIgnoreOneObsCount <- function(m) {
  Z <- NULL
  n <- dim(m)[1]

  for (i in 1:n) {
    alpha_0 <- 1
    alpha_1 <- 2
    n_i <- c(m[i, ])
    n_i <- c(n_i[n_i > 0])
    n_length <- length(n_i)
    POS <- matrix(alpha_1, sum(n_i), sum(n_i))
    loc1 <- 0
    loc2 <- 0

    for (s in 1:n_length) {
      n_t <- n_i[s]
      loc1 <- loc2 + 1
      loc2 <- loc1 + n_t - 1

      for (k in loc1:loc2) {
        for (j in loc1:loc2) {
          if (k != j) {
            POS[k, j] <- alpha_0
          } else {
            POS[k, j] <- 0
          }
        }
      }
    }

    zrow <- diag(2)
    z_c <- NULL

    if (sum(n_i) > 1) {
      for (j in 1:(sum(n_i) - 1)) {
        for (k in (j + 1):sum(n_i)) {
          z_c <- rbind(z_c, zrow[POS[j, k], ])
        }
      }
    }

    Z <- rbind(Z, z_c)
  }

  return(Z)
}
Z <- createzCrossSecIgnoreOneObsCount(m)

### (1) Matrix-adjusted estimating equations and GEE
### on continuous outcome with nested exchangeable correlation structure

### MAEE
est_maee_ind_con <- geemaee(
  y = sampleSWCRT$y_con,
  X = X, id = id, Z = Z,
  family = "continuous",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = TRUE,
  shrink = "ALPHA", makevone = FALSE
)

### GEE
est_uee_ind_con <- geemaee(
  y = sampleSWCRT$y_con,
  X = X, id = id, Z = Z,
  family = "continuous",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = FALSE,
  shrink = "ALPHA", makevone = FALSE
)

### (2) Matrix-adjusted estimating equations and GEE
### on binary outcome with nested exchangeable correlation structure

### MAEE
est_maee_ind_bin <- geemaee(
  y = sampleSWCRT$y_bin,
  X = X, id = id, Z = Z,
  family = "binomial",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = TRUE,
  shrink = "ALPHA", makevone = FALSE
)

### GEE
est_uee_ind_bin <- geemaee(
  y = sampleSWCRT$y_bin,
  X = X, id = id, Z = Z,
  family = "binomial",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = FALSE,
  shrink = "ALPHA", makevone = FALSE
)
```

Then we have the following output: 
```{r set-options3, echo=FALSE, fig.keep="all", fig.width = 7, fig.height=4}
options(width = 100)
```

```{r , fig.keep="all", fig.width = 7, fig.height=4}
# MAEE for continuous outcome
print(est_maee_ind_con)

# GEE for continuous outcome
print(est_uee_ind_con)

# MAEE for binary outcome
print(est_maee_ind_bin)

# GEE for binary outcome
print(est_uee_ind_bin)
```


### Simulated dataset with 12 clusters and large cluster sizes

```{r geemaee_large_cluOneObsCount_swcrt, fig.keep="all", fig.width = 7, fig.height=4}
## let the cluster 2, 3 be with only 1 observation
sampleSWCRT <- sampleSWCRTLarge[sampleSWCRTLarge$id != 2 & sampleSWCRTLarge$id != 3, ]
sampleSWCRT2 <- sampleSWCRTLarge[sampleSWCRTLarge$id == 2 & sampleSWCRTLarge$period == 4, ][1, ]
sampleSWCRT3 <- sampleSWCRTLarge[sampleSWCRTLarge$id == 3 & sampleSWCRTLarge$period == 5, ][1, ]
sampleSWCRT <- rbind(sampleSWCRT, sampleSWCRT2)
sampleSWCRT <- rbind(sampleSWCRT, sampleSWCRT3)
sampleSWCRT <- sampleSWCRT[order(sampleSWCRT$id, sampleSWCRT$period), ]

id <- sampleSWCRT$id
period <- sampleSWCRT$period
X <- as.matrix(sampleSWCRT[, c("period1", "period2", "period3", "period4", "treatment")])
m <- as.matrix(table(id, period))
n <- dim(m)[1]
t <- dim(m)[2]
Z <- createzCrossSecIgnoreOneObsCount(m)

### (1) Matrix-adjusted estimating equations and GEE
### on continous outcome with nested exchangeable correlation structure

### MAEE
est_maee_ind_con <- geemaee(
  y = sampleSWCRT$y_con,
  X = X, id = id, Z = Z,
  family = "continuous",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = TRUE,
  shrink = "ALPHA", makevone = FALSE
)
print(est_maee_ind_con)

### GEE
est_uee_ind_con <- geemaee(
  y = sampleSWCRT$y_con,
  X = X, id = id, Z = Z,
  family = "continuous",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = FALSE,
  shrink = "ALPHA", makevone = FALSE
)
print(est_uee_ind_con)

### (2) Matrix-adjusted estimating equations and GEE
### on binary outcome with nested exchangeable correlation structure

### MAEE
est_maee_ind_bin <- geemaee(
  y = sampleSWCRT$y_bin,
  X = X, id = id, Z = Z,
  family = "binomial",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = TRUE,
  shrink = "ALPHA", makevone = FALSE
)
print(est_maee_ind_bin)

### GEE
est_uee_ind_bin <- geemaee(
  y = sampleSWCRT$y_bin,
  X = X, id = id, Z = Z,
  family = "binomial",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = FALSE,
  shrink = "ALPHA", makevone = FALSE
)
print(est_uee_ind_bin)
```


## `geemaee()` examples: matrix-adjusted GEE for estimating the mean and correlation parameters in general CRTs

We can also use the `geemaee()` function for general CRTs with only one time period. We need to specify a single column of 1's for the `Z` matrix. 


### Simulated dataset with 12 clusters and small cluster sizes

```{r geemaee_small_crt, fig.keep="all", fig.width = 7, fig.height=4}
### use only the second period data to get a general CRT data
sampleSWCRT <- sampleSWCRTSmall[sampleSWCRTSmall$period == 2, ]

### Individual-level id, period, outcome, and design matrix
id <- sampleSWCRT$id
period <- sampleSWCRT$period
X <- as.matrix(sampleSWCRT[, c("period2", "treatment")])
m <- as.matrix(table(id, period))
n <- dim(m)[1]
t <- dim(m)[2]

### design matrix for correlation parameters
Z <- createzCrossSec(m)
Z <- matrix(1, dim(Z)[1], 1)

### (1) Matrix-adjusted estimating equations and GEE
### on continuous outcome with nested exchangeable correlation structure

### MAEE
est_maee_ind_con <- geemaee(
  y = sampleSWCRT$y_con,
  X = X, id = id, Z = Z,
  family = "continuous",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = TRUE,
  shrink = "ALPHA", makevone = FALSE
)

### GEE
est_uee_ind_con <- geemaee(
  y = sampleSWCRT$y_con,
  X = X, id = id, Z = Z,
  family = "continuous",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = FALSE,
  shrink = "ALPHA", makevone = FALSE
)

### (2) Matrix-adjusted estimating equations and GEE
### on binary outcome with nested exchangeable correlation structure

### MAEE
est_maee_ind_bin <- geemaee(
  y = sampleSWCRT$y_bin,
  X = X, id = id, Z = Z,
  family = "binomial",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = TRUE,
  shrink = "ALPHA", makevone = FALSE
)

### GEE
est_uee_ind_bin <- geemaee(
  y = sampleSWCRT$y_bin,
  X = X, id = id, Z = Z,
  family = "binomial",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = FALSE,
  shrink = "ALPHA", makevone = FALSE
)
```


Then we have the following output: 
```{r set-options4, echo=FALSE, fig.keep="all", fig.width = 7, fig.height=4}
options(width = 100)
```

```{r , fig.keep="all", fig.width = 7, fig.height=4}
# MAEE for continuous outcome
print(est_maee_ind_con)

# GEE for continuous outcome
print(est_uee_ind_con)

# MAEE for binary outcome
print(est_maee_ind_bin)

# GEE for binary outcome
print(est_uee_ind_bin)
```


### Simulated dataset with 12 clusters and large cluster sizes

```{r geemaee_large_crt, fig.keep="all", fig.width = 7, fig.height=4}
### use only the second period data to get a general CRT data
sampleSWCRT <- sampleSWCRTLarge[sampleSWCRTLarge$period == 2, ]

### Individual-level id, period, outcome, and design matrix
id <- sampleSWCRT$id
period <- sampleSWCRT$period
X <- as.matrix(sampleSWCRT[, c("period2", "treatment")])
m <- as.matrix(table(id, period))
n <- dim(m)[1]
t <- dim(m)[2]
### design matrix for correlation parameters
Z <- createzCrossSec(m)
Z <- matrix(1, dim(Z)[1], 1)

### (1) Matrix-adjusted estimating equations and GEE
### on continous outcome with nested exchangeable correlation structure

### MAEE
est_maee_ind_con <- geemaee(
  y = sampleSWCRT$y_con,
  X = X, id = id, Z = Z,
  family = "continuous",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = TRUE,
  shrink = "ALPHA", makevone = FALSE
)
print(est_maee_ind_con)

### GEE
est_uee_ind_con <- geemaee(
  y = sampleSWCRT$y_con,
  X = X, id = id, Z = Z,
  family = "continuous",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = FALSE,
  shrink = "ALPHA", makevone = FALSE
)
print(est_uee_ind_con)

### (2) Matrix-adjusted estimating equations and GEE
### on binary outcome with nested exchangeable correlation structure

### MAEE
est_maee_ind_bin <- geemaee(
  y = sampleSWCRT$y_bin,
  X = X, id = id, Z = Z,
  family = "binomial",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = TRUE,
  shrink = "ALPHA", makevone = FALSE
)
print(est_maee_ind_bin)

### GEE
est_uee_ind_bin <- geemaee(
  y = sampleSWCRT$y_bin,
  X = X, id = id, Z = Z,
  family = "binomial",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = FALSE,
  shrink = "ALPHA", makevone = FALSE
)
print(est_uee_ind_bin)
```

## `geemaee()` examples: matrix-adjusted GEE for estimating the mean and correlation parameters in SW-CRTs with all cluster-period sizes being 1

We consider an edge case when all cluster period sizes are 1, the `geemaee()` function can still estimate the intra-period correlation parameter.


```{r geemaee_large_cpsize1, fig.keep="all", fig.width = 7, fig.height=4}
### Simulated dataset with 12 clusters and 5 periods
# use the first observation for each cluster-period combination
sampleSWCRT <- NULL
for (i in 1:12) {
  for (j in 1:5) {
    row <- sampleSWCRTLarge[sampleSWCRTLarge$id == i &
      sampleSWCRTLarge$period == j, ][1, , drop = FALSE]
    sampleSWCRT <- rbind(sampleSWCRT, row)
  }
}

### Individual-level id, period, outcome, and design matrix
id <- sampleSWCRT$id
period <- sampleSWCRT$period
X <- as.matrix(sampleSWCRT[, c("period2", "treatment")])
m <- as.matrix(table(id, period))
n <- dim(m)[1]
t <- dim(m)[2]
### design matrix for correlation parameters
Z <- createzCrossSec(m)
Z <- matrix(1, dim(Z)[1], 1)

### (1) Matrix-adjusted estimating equations and GEE
### on continous outcome with nested exchangeable correlation structure

### MAEE
est_maee_ind_con <- geemaee(
  y = sampleSWCRT$y_con,
  X = X, id = id, Z = Z,
  family = "continuous",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = TRUE,
  shrink = "ALPHA", makevone = FALSE
)
print(est_maee_ind_con)

### GEE
est_uee_ind_con <- geemaee(
  y = sampleSWCRT$y_con,
  X = X, id = id, Z = Z,
  family = "continuous",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = FALSE,
  shrink = "ALPHA", makevone = FALSE
)
print(est_uee_ind_con)

### (2) Matrix-adjusted estimating equations and GEE
### on binary outcome with nested exchangeable correlation structure

### MAEE
est_maee_ind_bin <- geemaee(
  y = sampleSWCRT$y_bin,
  X = X, id = id, Z = Z,
  family = "binomial",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = TRUE,
  shrink = "ALPHA", makevone = FALSE
)
print(est_maee_ind_bin)

### GEE
est_uee_ind_bin <- geemaee(
  y = sampleSWCRT$y_bin,
  X = X, id = id, Z = Z,
  family = "binomial",
  maxiter = 500, epsilon = 0.001,
  printrange = TRUE, alpadj = FALSE,
  shrink = "ALPHA", makevone = FALSE
)
print(est_uee_ind_bin)
```


## `cpgeeSWD()` examples: cluster-period GEE for estimating the marginal mean and correlation parameters in cross-sectional SW-CRTs

The `cpgeeSWD()` function implements the cluster-period GEE developed for cross-sectional stepped wedge cluster randomized trials (SW-CRTs). It provides valid estimation and inference for the treatment effect and intraclass correlation parameters within the GEE framework, and is computationally efficient for analyzing SW-CRTs with large cluster sizes. The program currently only allows for a marginal mean model with discrete period effects and the intervention indicator without additional covariates. The program offers bias-corrected ICC estimates as well as bias-corrected sandwich variances for both the treatment effect parameter and the ICC parameters. The technical details of the cluster-period GEE approach are provided in [@li2022marginal].

### Simulated dataset with 12 clusters and 4 periods

We summarize the individual-level simulated SW-CRT data to cluster-period data and use the `cpgeeSWD()` function to estimate the marginal mean and correlation parameters on cluster-period means of binary outcome. We first transform the variables to get the cluster-period mean outcome `y_cp`, mean parameters' design matrix `X_cp` as well as other arguments.  

```{r cpgee_gather_small, fig.keep="all", fig.width = 7, fig.height=4}
########################################################################
### Example 1): simulated SW-CRT with smaller cluster-period sizes (5~10)
########################################################################

sampleSWCRT <- sampleSWCRTSmall

### cluster-period id, period, outcome, and design matrix
### id, period, outcome
id <- sampleSWCRT$id
period <- sampleSWCRT$period
y <- sampleSWCRT$y_bin
X <- as.matrix(sampleSWCRT[, c("period1", "period2", "period3", "period4", "treatment")])

m <- as.matrix(table(id, period))
n <- dim(m)[1]
t <- dim(m)[2]
clp_mu <- tapply(y, list(id, period), FUN = mean)
y_cp <- c(t(clp_mu))

### design matrix for correlation parameters
trt <- tapply(X[, t + 1], list(id, period), FUN = mean)
trt <- c(t(trt))

time <- tapply(period, list(id, period), FUN = mean)
time <- c(t(time))
X_cp <- matrix(0, n * t, t)

s <- 1
for (i in 1:n) {
  for (j in 1:t) {
    X_cp[s, time[s]] <- 1
    s <- s + 1
  }
}
X_cp <- cbind(X_cp, trt)
id_cp <- rep(1:n, each = t)
m_cp <- c(t(m))
```

We implement the `cpgeeSWD()` function on all the three choices of the correlation structure including `"exchangeable"`, `"nest_exch"` and `"exp_decay"`. We consider both matrix-adjusted estimating equations (MAEE) with `alpadj = TRUE` and uncorrected generalized estimating equations (GEE) with `alpadj = FALSE`. 


```{r cpgee_small, fig.keep="all", fig.width = 7, fig.height=4}
### cluster-period matrix-adjusted estimating equations (MAEE)
### with exchangeable, nested exchangeable and exponential decay correlation structures
# exponential
est_maee_exc <- cpgeeSWD(
  y = y_cp, X = X_cp, id = id_cp,
  m = m_cp, corstr = "exchangeable",
  alpadj = TRUE
)
print(est_maee_exc)

# nested exchangeable
est_maee_nex <- cpgeeSWD(
  y = y_cp, X = X_cp, id = id_cp,
  m = m_cp, corstr = "nest_exch",
  alpadj = TRUE
)
print(est_maee_nex)

# exponential decay
est_maee_ed <- cpgeeSWD(
  y = y_cp, X = X_cp, id = id_cp,
  m = m_cp, corstr = "exp_decay",
  alpadj = TRUE
)
print(est_maee_ed)


### cluster-period GEE
### with exchangeable, nested exchangeable and exponential decay correlation structures

# exchangeable
est_uee_exc <- cpgeeSWD(
  y = y_cp, X = X_cp, id = id_cp,
  m = m_cp, corstr = "exchangeable",
  alpadj = FALSE
)
print(est_uee_exc)

# nested exchangeable
est_uee_nex <- cpgeeSWD(
  y = y_cp, X = X_cp, id = id_cp,
  m = m_cp, corstr = "nest_exch",
  alpadj = FALSE
)
print(est_uee_nex)

# exponential decay
est_uee_ed <- cpgeeSWD(
  y = y_cp, X = X_cp, id = id_cp,
  m = m_cp, corstr = "exp_decay",
  alpadj = FALSE
)
print(est_uee_ed)
```

### Simulated dataset with 12 clusters and 5 periods

```{r cpgee_large, fig.keep="all", fig.width = 7, fig.height=4}
########################################################################
### Example 2): simulated SW-CRT with larger cluster-period sizes (20~30)
########################################################################

sampleSWCRT <- sampleSWCRTLarge

### cluster-period id, period, outcome, and design matrix
### id, period, outcome
id <- sampleSWCRT$id
period <- sampleSWCRT$period
y <- sampleSWCRT$y_bin
X <- as.matrix(sampleSWCRT[, c("period1", "period2", "period3", "period4", "period5", "treatment")])

m <- as.matrix(table(id, period))
n <- dim(m)[1]
t <- dim(m)[2]
clp_mu <- tapply(y, list(id, period), FUN = mean)
y_cp <- c(t(clp_mu))

### design matrix for correlation parameters
trt <- tapply(X[, t + 1], list(id, period), FUN = mean)
trt <- c(t(trt))

time <- tapply(period, list(id, period), FUN = mean)
time <- c(t(time))
X_cp <- matrix(0, n * t, t)

s <- 1
for (i in 1:n) {
  for (j in 1:t) {
    X_cp[s, time[s]] <- 1
    s <- s + 1
  }
}
X_cp <- cbind(X_cp, trt)
id_cp <- rep(1:n, each = t)
m_cp <- c(t(m))

### cluster-period matrix-adjusted estimating equations (MAEE)
### with exchangeable, nested exchangeable and exponential decay correlation structures
# exponential
est_maee_exc <- cpgeeSWD(
  y = y_cp, X = X_cp, id = id_cp,
  m = m_cp, corstr = "exchangeable",
  alpadj = TRUE
)
print(est_maee_exc)

# nested exchangeable
est_maee_nex <- cpgeeSWD(
  y = y_cp, X = X_cp, id = id_cp,
  m = m_cp, corstr = "nest_exch",
  alpadj = TRUE
)
print(est_maee_nex)

# exponential decay
est_maee_ed <- cpgeeSWD(
  y = y_cp, X = X_cp, id = id_cp,
  m = m_cp, corstr = "exp_decay",
  alpadj = TRUE
)
print(est_maee_ed)


### cluster-period GEE
### with exchangeable, nested exchangeable and exponential decay correlation structures

# exchangeable
est_uee_exc <- cpgeeSWD(
  y = y_cp, X = X_cp, id = id_cp,
  m = m_cp, corstr = "exchangeable",
  alpadj = FALSE
)
print(est_uee_exc)

# nested exchangeable
est_uee_nex <- cpgeeSWD(
  y = y_cp, X = X_cp, id = id_cp,
  m = m_cp, corstr = "nest_exch",
  alpadj = FALSE
)
print(est_uee_nex)

# exponential decay
est_uee_ed <- cpgeeSWD(
  y = y_cp, X = X_cp, id = id_cp,
  m = m_cp, corstr = "exp_decay",
  alpadj = FALSE
)
print(est_uee_ed)
```
## `simbinPROBIT()` example: generating correlated binary data using the multivariate probit method

The `simbinPROBIT()` function generates correlated binary data using the multivariate Probit method [@emrich1991method]. It simulates a vector of binary outcomes according the specified marginal mean vector and correlation structure. Constraints and compatibility between the marginal mean and correlation matrix are checked.

We use the `simbinPROBIT()` function to generate correlated binary data with different correlation structures. We consider simulating a cross-sectional SW-CRT dataset with 2 clusters, 3 periods with the same cluster-period size of 5. We use two mean vectors for the two clusters and specify the `mu` argument.  

For the exchangeable correlation structure, we specify both the within-period and inter-period correlation parameters to be `0.015`. We use `0.03` and `0.015` for the within-period and inter-period correlations, respectively. The exponential decay correlation structure has an decay parameter `0.8` with the within-period correlation parameter `0.03`.

```{r sim_ep, fig.keep="all", fig.width = 7, fig.height=4}
#### Simulate 2 clusters, 3 periods and cluster-period size of 5

t <- 3
n <- 2
m <- 5
# means of cluster 1
u_c1 <- c(0.4, 0.3, 0.2)
u1 <- rep(u_c1, c(rep(m, t)))
# means of cluster 2
u_c2 <- c(0.35, 0.25, 0.2)
u2 <- rep(u_c2, c(rep(m, t)))

# List of mean vectors
mu <- list()
mu[[1]] <- u1
mu[[2]] <- u2
# List of correlation matrices

## correlation parameters
alpha0 <- 0.03
alpha1 <- 0.015
rho <- 0.8

## (1) exchangeable
Sigma <- list()
Sigma[[1]] <- diag(m * t) * (1 - alpha1) + matrix(alpha1, m * t, m * t)

Sigma[[2]] <- diag(m * t) * (1 - alpha1) + matrix(alpha1, m * t, m * t)

y_exc <- simbinPROBIT(mu = mu, Sigma = Sigma, n = n)

## (2) nested exchangeable
Sigma <- list()
cor_matrix <- matrix(alpha1, m * t, m * t)
loc1 <- 0
loc2 <- 0
for (t in 1:t) {
  loc1 <- loc2 + 1
  loc2 <- loc1 + m - 1
  for (i in loc1:loc2) {
    for (j in loc1:loc2) {
      if (i != j) {
        cor_matrix[i, j] <- alpha0
      } else {
        cor_matrix[i, j] <- 1
      }
    }
  }
}

Sigma[[1]] <- cor_matrix
Sigma[[2]] <- cor_matrix

y_nex <- simbinPROBIT(mu = mu, Sigma = Sigma, n = n)

## (3) exponential decay

Sigma <- list()

### function to find the period of the ith index
region_ij <- function(points, i) {
  diff <- i - points
  for (h in 1:(length(diff) - 1)) {
    if (diff[h] > 0 & diff[h + 1] <= 0) {
      find <- h
    }
  }
  return(find)
}

cor_matrix <- matrix(0, m * t, m * t)
useage_m <- cumsum(m * t)
useage_m <- c(0, useage_m)

for (i in 1:(m * t)) {
  i_reg <- region_ij(useage_m, i)
  for (j in 1:(m * t)) {
    j_reg <- region_ij(useage_m, j)
    if (i_reg == j_reg & i != j) {
      cor_matrix[i, j] <- alpha0
    } else if (i == j) {
      cor_matrix[i, j] <- 1
    } else if (i_reg != j_reg) {
      cor_matrix[i, j] <- alpha0 * (rho^(abs(i_reg - j_reg)))
    }
  }
}

Sigma[[1]] <- cor_matrix
Sigma[[2]] <- cor_matrix

y_ed <- simbinPROBIT(mu = mu, Sigma = Sigma, n = n)
```

## `simbinCLF()` example: generating correlated binary data using the conditional linear family method

The `simbinCLF()` function generates correlated binary data using the conditional linear family method [@qaqish2003family]. It simulates a vector of binary outcomes according the specified marginal mean vector and correlation structure. Natural constraints and compatibility between the marginal mean and correlation matrix are checked.

We use the `simbinCLF()` function to generate correlated binary data with different correlation structures. We consider simulating a cross-sectional SW-CRT dataset with 2 clusters, 3 periods with the same cluster-period size of 5. We use two mean vectors for the two clusters and specify the `mu` argument.  

For the exchangeable correlation structure, we specify both the within-period and inter-period correlation parameters to be `0.015`. We use `0.03` and `0.015` for the within-period and inter-period correlations, respectively. The exponential decay correlation structure has an decay parameter `0.8` with the within-period correlation parameter `0.03`. 


```{r sim_clf, fig.keep="all", fig.width = 7, fig.height=4}
##### Simulate 2 clusters, 3 periods and cluster-period size of 5

t <- 3
n <- 2
m <- 5

# means of cluster 1
u_c1 <- c(0.4, 0.3, 0.2)
u1 <- rep(u_c1, c(rep(m, t)))
# means of cluster 2
u_c2 <- c(0.35, 0.25, 0.2)
u2 <- rep(u_c2, c(rep(m, t)))

# List of mean vectors
mu <- list()
mu[[1]] <- u1
mu[[2]] <- u2
# List of correlation matrices

## correlation parameters
alpha0 <- 0.03
alpha1 <- 0.015
rho <- 0.8

## (1) exchangeable
Sigma <- list()
Sigma[[1]] <- diag(m * t) * (1 - alpha1) + matrix(alpha1, m * t, m * t)
Sigma[[2]] <- diag(m * t) * (1 - alpha1) + matrix(alpha1, m * t, m * t)
y_exc <- simbinCLF(mu = mu, Sigma = Sigma, n = n)

## (2) nested exchangeable
Sigma <- list()
cor_matrix <- matrix(alpha1, m * t, m * t)
loc1 <- 0
loc2 <- 0
for (t in 1:t) {
  loc1 <- loc2 + 1
  loc2 <- loc1 + m - 1
  for (i in loc1:loc2) {
    for (j in loc1:loc2) {
      if (i != j) {
        cor_matrix[i, j] <- alpha0
      } else {
        cor_matrix[i, j] <- 1
      }
    }
  }
}

Sigma[[1]] <- cor_matrix
Sigma[[2]] <- cor_matrix
y_nex <- simbinCLF(mu = mu, Sigma = Sigma, n = n)

## (3) exponential decay

Sigma <- list()

### function to find the period of the ith index
region_ij <- function(points, i) {
  diff <- i - points
  for (h in 1:(length(diff) - 1)) {
    if (diff[h] > 0 & diff[h + 1] <= 0) {
      find <- h
    }
  }
  return(find)
}

cor_matrix <- matrix(0, m * t, m * t)
useage_m <- cumsum(m * t)
useage_m <- c(0, useage_m)

for (i in 1:(m * t)) {
  i_reg <- region_ij(useage_m, i)
  for (j in 1:(m * t)) {
    j_reg <- region_ij(useage_m, j)
    if (i_reg == j_reg & i != j) {
      cor_matrix[i, j] <- alpha0
    } else if (i == j) {
      cor_matrix[i, j] <- 1
    } else if (i_reg != j_reg) {
      cor_matrix[i, j] <- alpha0 * (rho^(abs(i_reg - j_reg)))
    }
  }
}

Sigma[[1]] <- cor_matrix
Sigma[[2]] <- cor_matrix

y_ed <- simbinCLF(mu = mu, Sigma = Sigma, n = n)
```


# Session Information

```{r info, results='markup', echo=FALSE}
sessionInfo()
```

# References