---
title: "Getting Started with `SteppedPower`"
author:
- name: Philipp Mildenberger^[pmildenb@uni-mainz.de]
  affiliation: Institute of Medical Biostatistics, Epidemiology and Informatics ([IMBEI, Mainz](https://www.unimedizin-mainz.de/imbei/imbei/welcome-page/))
- name: Federico Marini^[marinif@uni-mainz.de]
  affiliation: Institute of Medical Biostatistics, Epidemiology and Informatics ([IMBEI, Mainz](https://www.unimedizin-mainz.de/imbei/imbei/welcome-page/))
date: "`r Sys.Date()`"
output: 
  rmarkdown::html_vignette:
    toc: true
    number_sections: true
bibliography: references.bib
vignette: >
  %\VignetteIndexEntry{Getting Started with `SteppedPower`}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.width = 7,
  warning = FALSE
)
library(knitr)
library(SteppedPower)
library(Matrix)
library(plotly)
```

# About `SteppedPower`

SteppedPower offers tools for power and sample size 
calculation as well as design diagnostics for 
longitudinal mixed model settings, with a focus on stepped wedge designs.
Other implemented study design types are parallel, 
parallel with baseline period(s) and 
crossover designs. Further design types can be 
easily defined by the user.  

Currently, normal outcomes and binomial outcomes with 
identity link are implemented. 
The following random effects can be specified:
random cluster intercept, random treatment effect, 
random subject specific intercept and random time effect.
The covariance structure can be compound symmetry or autoregressive.

This package is modularised in order to be flexible and easy to use 
(and hopefully to maintain as well). 
At the same time, the use of sparse matrix classes of the `matrix` 
package makes computation of large designs feasible. 

# Methods and Notation

A common approach to model the correlation in 
longitudinal studies are random effects [@hussey2007design; @li2020mixed].
Such a model has the form

$$y_{ijk}= T_{ij} \theta + c_i + \mu_j + e_{ijk}$$ 

with

* $y_{ijk}$ the response in cluster $i$ at time $j$ for individual $k$
* $\mu_j$ the time trend at time $j$
* $T_{ij}$ indicates the treatment status (0 = control, 1 = interventional treatment)
* $\theta$ the treatment effect 
* $c_i$ a random cluster effect for cluster $i$ with $c_i \sim N(0,\tau^2)$
* $e_{ijk}$ a normal random error term with $e_{ijk}\sim N(0,\sigma^2)$

For power calculation, the standard deviation of random effects is assumed to be known.
Let's define $\beta:=(\theta,\mu')'$ and $\omega_{ijk}:=c_i+e_{ijk}$. 
This leads to a compact and more general notation of the above equation:

$$\begin{align}
y_{ijk}&= X_{ij}\beta + \omega_{ijk}\\
\text{or, in matrix notation:} \qquad \\
y&=X\beta + \omega
\end{align}$$ 

Where $X$ is the corresponding design matrix and $\omega\sim N(0,\Omega)$,
where $\Omega$ is a compound-symmetry (syn. exchangeable) variance matrix defined 
by $\tau$ and $\sigma$. We are thus in a weighted least squares setting, 
so the variance of $\beta$ is

$$ \text{Var}(\hat\beta) =  {(X'\Omega^{-1}X)^{-1}}$$

We can then calculate the power of a z-test 

$$ \text{power} = \Phi\left(\frac{\theta_A-\theta_0}{\sqrt{\text{Var}(\hat \theta)}}- Z_{1-\frac{\alpha}{2}}\right) $$

where $\text{Var}(\hat \theta)$ is the diagonal element of $\Omega$ 
that corresponds to $\hat\theta$.

Extensions to the above formula implemented in this package are  


with leads to the following extended model formula:

$$y_{ijk}= T (\theta_{ij} + d_i) + c_i + \mu_j + t_{ij} + s_{ijk} + e_{ijk}$$ 
with  

* $d_i$ a random treatment effect
* $c_i$ and $d_i$ jointly distributed with $\left(\begin{smallmatrix} \tau^2 & \rho\tau\eta \\ \rho\tau\eta & \eta^2\end{smallmatrix}\right)$
* $s_{ijk}$ a random subject specific effect
* $t_{ij}$ a random time effect




## Specification of the correlation structure


<!-- The variances of the random effects define a correlation structure -->


<!-- Besides to the random effect specification -->

<!-- $\left(\operatorname{corr}\left(y_{i j k}, y_{i j k^{\prime}}\right)=\alpha_{0} \text { for } k \neq k^{\prime}\right)$  -->

<!-- $\left(\operatorname{corr}\left(y_{i j k}, y_{i j^{\prime} k^{\prime}}\right)=\alpha_{1}\right. \text{ for } j \neq j^{\prime} \text{ for } k \neq k^{\prime}$  -->

<!-- $\left(\operatorname{corr}\left(y_{i j k}, y_{i j^{\prime} k}\right)=\alpha_{2} \text { for } j \neq j^{\prime}\right)$ -->

<!-- $$\begin{align} -->
<!-- \sigma_M^2 &:= \tau^2+\gamma^2 + \psi^2 + \sigma^2   \\ -->
<!-- \alpha_0 &:= \frac{\tau^2 + \gamma^2}{\sigma_M^2}  \\ -->
<!-- \alpha_1 &:= \frac{\tau^2}{\sigma_M^2} \\ -->
<!-- \alpha_2 &:= \frac{\tau^2 + \psi^2}{\sigma_M^2} -->
<!-- \end{align}$$ -->

A different approach to establish a covariance structure is to directly assume cluster cell correlation as proposed by \cite{Preisser2003integrated} and introduced to the context of stepped wedge designs in [@li2018sample]. They use $\boldsymbol{\alpha}:=(\alpha_0, \alpha_1, \alpha_2)$ where 
		$\alpha_0:=\operatorname{corr}({y_{ijk},y_{ijk'}})$ for $ k\neq k'$ defines the within-period correlation; $\alpha_1:=\operatorname{corr}({y_{ijk},y_{ij'k'}})$ for $j\neq j', \, k\neq k'$ defines the inter-period correlation and $\alpha_0:=\operatorname{corr}({y_{ijk},y_{ij'k}})$ for $j\neq j'$ defines the within-individual correlation.
		
One receives an easily interpretable correlation matrix, albeit at the cost of some flexibility. We can translate $\boldsymbol{\alpha}$ into a representation with random effects, given a marginal variance $\sigma^2_{\text{marg}}$.
		
$$
	\begin{align*}
		\tau^2   		&:= {\sigma^2_{\text{marg}} \alpha_1} \\
		\gamma^2 		&:= {\sigma^2_{\text{marg}} (\alpha_0-\alpha_1)} \\
		\psi^2   	 	&:= {\sigma^2_{\text{marg}} (\alpha_2-\alpha_2)} \\
		\sigma^2_{\text{res}} 	&:= {\sigma^2_{\text{marg}} (1-\alpha_0 - \alpha_2 + \alpha_1)}
	\end{align*}
$$ 

Note that $\boldsymbol{\alpha}$ as defined above cannot specify a random treatment effect.


# A quick tour

For most users, the probably most important function is `glsPower`. 
It calls several auxiliary functions which will be shortly discussed here.
This section is not essential for the usage of `SteppedPower`, it might be
helpful to design non-standard user defined settings. 

`glsPower` is essentially just a flexible wrapper for the function 
`compute_glsPower`, which does the actual computation.  
`compute_glsPower` then calls `construct_DesMat` and `construct_CovMat`.

`construct_DesMat` builds the design matrix which consists of the treatment 
status, usually built by`construct_trtMat` and the time adjustment, usually
built by `construct_timeadjust`. 
There is also the option to pass a user defined definition of the treatment 
status to `construct_DesMat`.
If not specified, the number of timepoints is guessed as 
the fewest number of periods (timepoints) possible with the given design, 
i.e. two for cross-over designs or the number of waves plus one
for stepped wedge designs. 

`construct_CovMat` builds the covariance matrix (explicitly). It uses 
`construct_CovBlk` to construct the blocks for each cluster which are then combined
to a block diagonal matrix.

# Features 

## Plot Method

The `plot.glsPower` method makes up to four plots, visualising the influence 
of cluster-period cells, the information content of cluster-period cells, 
the treatment matrix $T$ and the covariance matrix 
(selectable by `which=`). 

By default, only the first plot is returned. The information content is also
plotted if it the object to be plotted contains the corresponding matrix 
(which is usually the case). It is constructed by exploiting the 
approximation to a weighted least squares setting. Therefore, 
the estimator $\hat \beta$ is a linear function of the data $y$

$$\hat\beta = \underbrace{(X'\Omega^{-1}X)^{-1}(X'\Omega^{-1})}_{=:\text{M}}\cdot y$$

with $X$ the design matrix and $\Omega$ the covariance matrix as above.
The matrix $H$ gives an impression of the importance of clusters and 
time periods with regard to the estimated coefficients $\hat\beta$. 
The first row of $H$ corresponds to the coefficient of the treatment status, 
i.e. the treatment effect.  
The `plot.glsPower` method visualises this first row of $M$ as a matrix 
where rows and columns correspond to clusters and time periods, respectively.

Furthermore, to give a rough comparison of importance between clusters 
(or between time periods), the sum of absolute weights per row (or per column) 
is also shown. 


```{r}
glsPwr <- glsPower(Cl=c(3,2,3), mu0=0, mu1=1, sigma=1, tau=.5, verbose=2)
plot(glsPwr,which=1, show_colorbar=FALSE)$WgtPlot 
```

The information content, as described in [@kasza2019information], looks like this:

```{r}
plot(glsPwr,which=2, show_colorbar=FALSE)$ICplot
```


>CAVE: The influence plot (above) yields out-of-bag estimates for the influence of 
observations on $\hat\theta$, but not for $\text{Var}(\hat\theta)$. 
The information content (below) visualises the relative change in 
$\text{Var}(\hat\theta)$ when one particular cluster period is omitted.

## Find Sample Size for given Power

When the argument `power` is passed to `glsPower`, 
the sample size needed is calculated, under the assumption of equally sized 
clusters and periods. 

```{r}
glsPower(Cl=c(3,3,3), mu0=0, mu1=.2, sigma=1, tau=0, power=.8)
```

So in this setting, you would need $50$ individuals per period in each cluster to 
achieve a target power of $80\%$

# Use cases and examples

## Comparison of two groups -- Z-Test

This might be a proof of concept rather than an example with 
practical relevance, but let's try to compare the mean in two groups. 
For two groups of 10 observations each, the power of a Z-test 
can be calculated as follows:

```{r}
glsPower(Cl=c(10,10), mu0=0,mu1=1.2,sigma=1, tau=0, N=1, 
              dsntype="parallel", timepoints=1)$power

## the same:
glsPower(Cl=c(1,1), mu0=0,mu1=1.2, sigma=1, tau=0, N=10,
              dsntype="parallel", timepoints=1)$power
```

```{r}
pwr::pwr.norm.test(.6,n=20)$power
```

>A quick note on t-tests: It is much more challenging to use `SteppedPower` 
to reproduce settings in which the variance is assumed to be unknown, 
most prominently the well known t-test.
In this package, you find implemented some (experimental) heuristics 
for guessing the denominator degrees of freedom, but they yield rather 
scaled Wald tests than t-tests. 
The main difference is that the distribution under 
the alternative is assumed to be symmetric,
whereas the t-test assumes a non-central (hence skewed) t-distribution.

## Longitudinal study -- parallel groups

```{r}
glsPower(Cl=c(10,10),timepoints=5,mu0=0,mu1=.25,
         sigma=.5,dsntype="parallel")

glsPower(Cl=c(10,10),timepoints=5,mu0=0,mu1=.25,
         sigma=.5,tau=.2,dsntype="parallel")
```


## Stepped Wedge designs with empty sequences (i.e. waves)


Periods in which no cluster switches to the intervention 
are specified by inserting zeros into 
the `Cl` argument, i.e. `Cl=c(4,4,4,0)`.

```{r, warning=FALSE}
mod1 <- glsPower(Cl=c(1,1,1,0), mu0=0, mu1=1, 
                 sigma=0.4, tau=0, verbose=2)
```

The treatment matrix is then stored under `mod1$DesignMatrix$trtMat`:
```{r, echo=FALSE}
knitr::kable(mod1$DesignMatrix$trtMat)
```

## Autocorrelated random effects

In longitudinal studies, it can be sensible to assume that correlation within 
clusters decreases with increasing time lag.
The argument `AR` enables the user to specify a AR-1 correlation. `AR`
must be any value between `0` and `1`. 
The former corresponds to i.i.d. observations, 
the latter to the usual compound symmetry covariance type.  

Optionally, you can pass a vector of up to length three to `AR`. Then, the first element 
applies to the cluster intercept, the second to the treatment effect (i.e. slope)
and the third to the subject specific intercept. This is especially useful for 
open cohort designs (see below).

An example of a stepped wedge design with 8 clusters in 4 waves, 
once with medium autocorrelation (`AR=0.6`) and 
once with high autocorrelation (`AR=0.95`):

```{r}
mod2 <- glsPower(Cl=c(2,2,2,2), mu0=0, mu1=1, 
              sigma=1, N=100, tau=1, AR=.6, verbose=2)

mod3 <- glsPower(Cl=c(2,2,2,2), mu0=0, mu1=1, 
              sigma=1, N=100, tau=1, AR=.95, verbose=2)
```

For `AR=0.6`, the covariance matrix within one cluster then looks like this:

```{r, echo=FALSE}
suppressWarnings(knitr::kable(as.matrix(mod2$CovarianceMatrix[1:5,1:5])))
```

For `AR=0.95` it takes the following shape

```{r, echo=FALSE}
suppressWarnings(knitr::kable(as.matrix(mod3$CovarianceMatrix[1:5,1:5])))
rm(mod1,mod2,mod3)
```


## Unequal cluster sizes

The argument `N` defines the cluster size. `N` can be 

* a scalar, if all clusters have the same assumed size, 
  which is also constant over time
* a vector, if the size differs between clusters but 
  is assumed to be constant over time
* a matrix where each row corresponds to either 
  a cluster or a wave of clusters and each column corresponds to a timepoint

```{r}
mod4 <- glsPower(Cl=c(1,1,1), mu0=0, mu1=1, N=c(1,3,10), 
                 sigma=1, tau=.5, verbose=2)
plot(mod4, which=2, show_colorbar=FALSE)$ICplot
```

```{r, echo=FALSE}
rm(mod4)
```

## Incomplete Stepped Wedge Designs

Suppose you do not plan to observe all clusters over the whole study period.
Rather, clusters that switch early to the intervention 
are not observed until the end.
Analogous, observation starts later in clusters 
that switch towards the end of the study.
This is sometimes called 'incomplete SWD' [@hemming2015stepped].

There are two ways to achieve this in `SteppedPower`, both by using the `incomplete` argument.
One can either scalar, which then defines the number of observed periods before and 
after the switch from control to intervention in each cluster.

If for example the study consists of eight clusters 
in four sequences (i.e. five timepoints), 
and we observe two timepoints before and after the switch, then we receive

```{r}
incompletePwr <- glsPower(Cl=rep(2,4), sigma=2, tau=.6, mu0=0,mu1=.5, N=80, 
                             incomplete=2, verbose=2)
incompletePwr
```

A slightly more tedious, but more flexible way is to define a matrix 
where each row corresponds to either a cluster or a wave of clusters 
and each column corresponds to a timepoint. 
If a cluster is not observed at a specific timepoint, 
set the value in the corresponding cell to `0`. 
For the example above, such a matrix would look like this:

```{r}
TM  <- toeplitz(c(1,1,0,0))
incompleteMat1 <- cbind(TM[,1:2],rep(1,4),TM[,3:4])
incompleteMat2 <- incompleteMat1[rep(1:4,each=2),]
```

A matrix where each row represents a wave of clusters

```{r, echo=FALSE}
suppressWarnings(knitr::kable(incompleteMat1))
```

or each row represents a cluster

```{r, echo=FALSE}
suppressWarnings(knitr::kable(incompleteMat2))
```

Now all that's left to do is to plug that into the main function:

```{r}
incompletePwr1 <- glsPower(Cl=rep(2,4), sigma=2, tau=.6, mu0=0, mu1=.5, N=80, 
                        incomplete=incompleteMat1, verbose=2)
incompletePwr2 <- glsPower(Cl=rep(2,4), sigma=2, tau=.6, mu0=0, mu1=.5, N=80, 
                        incomplete=incompleteMat2, verbose=2)

all.equal(incompletePwr,incompletePwr1)
all.equal(incompletePwr,incompletePwr2)
```

We can also have a quick look at the projection matrix where we see that the 
clusters have a weight of exactly zero at the timepoints 
where they are not observed

```{r}
plot(incompletePwr, show_colorbar=FALSE)$WgtPlot
```


```{r, echo=FALSE}
rm(incompletePwr,incompletePwr1,incompletePwr2,incompleteMat1,incompleteMat2)
```

> The argument `incomplete` with matrix input works also for other design types, 
but makes (supposedly) most sense in the context of stepped wedge designs

## Adjustment for Secular Trends 

The most usual method for the modelling of potential secular trends is
to take time period as a factor into the analysis model
[@hussey2007design; @hemming2015stepped; @hemming2020reflection]. 

For diagnostic purposes (or for bold users), some other adjustment options are 
implemented.

```{r}
TimeAdj1 <- glsPower(Cl=rep(2,4), mu0=0, mu1=1, sigma=1, tau=0, 
                     timeAdjust="linear", verbose=2)

TimeAdj2 <- glsPower(Cl=rep(2,4), mu0=0, mu1=1, sigma=1, tau=0, 
                     timeAdjust="factor", verbose=2)
```

Design matrix of the first cluster with linear adjustment for secular trend:
```{r, echo=FALSE}
knitr::kable(head(TimeAdj1$DesignMatrix$dsnmatrix, 5))
```

Design matrix of the first cluster with categorical adjustment for secular trend:
```{r, echo=FALSE}
knitr::kable(head(TimeAdj2$DesignMatrix$dsnmatrix, 5))
```


## Closed cohort SWD

In a closed cohort the patients are observed over the whole study period. 
The same correlation structure arises in cross sectional stepped wedge designs 
if subclusters exist (such as wards within clinics). 
The argument `psi` denotes the standard deviation of a 
random subject (or subcluster) specific intercept.
  
The power is calculated on aggregated cluster means:
```{r}
Closed1 <- glsPower(mu0=0, mu1=5, Cl=rep(3,3), 
                    sigma=5, tau=1, psi=3,
                    N=3, verbose=2)
Closed1
```

With `INDIV_LVL=TRUE`, the calculation is done on the individual level. 
This yields the same results but is far more computationally expensive and is 
mainly intended for diagnostic purposes.

```{r}
Closed2 <- glsPower(mu0=0, mu1=5, Cl=rep(3,3), 
                    sigma=5, tau=1, psi=3,
                    N=3, verbose=2, INDIV_LVL = TRUE)
Closed2
plot(Closed2, annotations=FALSE, show_colorbar=FALSE)$WgtPlot
```

```{r}
Closed1$power - Closed2$power
```


## Open cohort designs 

The `AR` argument mentioned above specifies the autocorrelation of random effects.
It offers a convenient way to model open cohort designs. The third element of `AR` 
can be seen as the estimated probability that a subject included in period $j$ 
is also included in period $j+1$, instead of being replaced by a new subject. 

Expanding on the closed cohort example above, one could assume that subjects 
have a 75% chance to reappear in the next study period. 
```{r}
Open1 <- glsPower(mu0=0, mu1=5, Cl=rep(3,3), 
                  sigma=5, tau=1, psi=3, AR=c(1,1,.75), N=3)

Closed1$power
Open1$power
```
So this would result in a power loss of roughly 
`r round( Closed1$power - Open1$power,3)*100`% 
compared to a closed cohort design in this case. 

### Covariance structure for open cohort designs

This proposed covariance structure is then a mixture of exchangeable (on cluster level) 
and autoregressive (on subject level). Let's have a look at a toy example with
three clusters, each consisting of three subjects. The probability that an 
individual reappears in two consecutive periods is estimated to be 60%. 
The covariance matrix (on subject level) could be visualised as follows:

```{r, fig.height=6}
Open2Indiv <- glsPower(mu0=0, mu1=10, Cl=c(1,1,1,0), 
                       sigma=1, tau=5, psi=10, AR=c(1,1,.60),
                       N=3, verbose=2, INDIV_LVL=TRUE)

plot(Open2Indiv, which=4, show_colorbar=FALSE)$CMplot
```

Note that you can zoom to get a better look at the structure. The autoregressive 
structure on the subject level in the small boxes along the main diagonal and the 
block exchangeable structure on cluster level becomes evident.


## Reproducing common closed formulae

There exist some formulae that do not explicitly construct design and 
covariance matrices, most notably [@kasza2020sample] and [@li2020design]. 
Both include the one presented in [@hussey2007design] as a special case.

These formulae are also implemented in `SteppedPower`. 
As a showcase, here is an example that is loosely based 
on the example in [@hussey2007design]:

```{r}
trtMat <- construct_DesMat(c(6,6,6,6))$trtMat
mu0 <- 0.05 ; mu1 <- 0.032 ; N <- 100 

tau <- .025 ; sigma <- sqrt(.041*.959) 
gamma <- 0.01 ; psi <- .1 ; chi <- .5 ; AR <- .5
```

### Closed formula for proportional decay structure

`VarClosed_Li` computes the treatment variance for stepped wedge designs with a 
autoregressive (syn. proportional decay) structure. It can be used to compute 
a variance for the treatment effect and thus the power for the setting in question:

```{r}
tmp <- VarClosed_Li(trtMat, tau=tau, psi=psi, N=N, AR=AR)
tTestPwr(mu0-mu1, se=sqrt(tmp), df=Inf)
```

Explicit calculation yields the same:
```{r}
a <- SteppedPower::glsPower(Cl=rep(6,4), mu0=mu0, mu1=mu1, AR=AR,
                       sigma=0, tau=tau, N=N, psi=psi, verbose=1, INDIV_LVL = TRUE)
a
```


### Closed formula for open cohort designs 

Kasza et al. offer an extension to open cohort designs by the use of an attrition 
or churn rate `chi` [@kasza2020sample]. It can be understood as the estimated probability 
that a subject included in period $j$ 
is also included in period $j'$ for $j\ne j'$. Note that this differs from the
approach mentioned above, as it results in a mixture of exchangeable covariance matrices.

Since it differs from the way `SteppedPower` intents to model open cohort designs, 
it requires a bit of tinkering to reproduce the results of the closed formula with 
this package. It can be done with the help of the random time effect (`gamma`).

```{r}
tmp <- VarClosed_Kasza(trtMat, sigma=sigma, tau=tau, gamma=gamma, psi=psi, N=N, chi=0)
tTestPwr(mu0-mu1, se=sqrt(tmp), df = Inf)
glsPower(Cl = rep(6,4), N=N, mu0=mu0, mu1=mu1, verbose=0,
         sigma=sigma, tau=tau, gamma=gamma, psi=psi)

tmp <- VarClosed_Kasza(trtMat, sigma=sigma, tau=tau, gamma=gamma, psi=psi, N=N, chi=1)
tTestPwr(mu0-mu1, sqrt(tmp), df = Inf)
glsPower(Cl = rep(6,4), N=N, mu0=mu0, mu1=mu1, verbose=0,
         sigma=sigma, tau=tau, 
         gamma=sqrt(gamma^2+psi^2/N), psi=0)

tmp <- VarClosed_Kasza(trtMat, sigma=sigma, tau=tau, gamma=gamma, psi=psi, N=N, chi=chi)
tTestPwr(mu0-mu1, sqrt(tmp), df = Inf)
glsPower(Cl = rep(6,4), N=N, mu0=mu0, mu1=mu1, verbose=0,
         sigma=sigma, tau=tau, 
         gamma=sqrt(gamma^2+chi*psi^2/N), psi=sqrt(1-chi)*psi)
```


# Session Info {-}

```{r, echo=FALSE}
print(sessionInfo(),locale=FALSE)
```

# References {-}
