---
title: "Introduction to rsimsum"
author: "Alessandro Gasparini"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Introduction to rsimsum}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
editor_options:
  chunk_output_type: console
---

```{r setup, include = FALSE}
options(width = 150)
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.align = "center",
  fig.height = 6,
  fig.width = 6,
  out.width = "80%"
)
```

# rsimsum

`rsimsum` is an R package that can compute summary statistics from simulation studies.
It is inspired by the user-written command `simsum` in Stata (White I.R., 2010).

The aim of `rsimsum` is helping reporting of simulation studies, including understanding the role of chance in results of simulation studies.
Specifically, `rsimsum` can compute Monte Carlo standard errors of summary statistics, defined as the standard deviation of the estimated summary statistic; these are reported by default.

Formula for summary statistics and Monte Carlo standard errors are presented in the next section.
Note that the terms _summary statistic_ and _performance measure_ are used interchangeably.

# Notation

We will use th following notation throughout this vignette:

* $\theta$: an estimand, and its true value
* $n_{\text{sim}}$: number of simulations
* $i = 1, \dots, n_{\text{sim}}$: indexes a given simulation
* $\hat{\theta}_i$: the estimated value of $\theta$ for the $i^{\text{th}}$ replication
* $\widehat{\text{Var}}(\hat{\theta}_i)$: the estimated variance $\text{Var}(\hat{\theta}_i)$ of $\hat{\theta}_i$ for the $i^{\text{th}}$ replication
* $\text{Var}(\hat{\theta})$: the empirical variance of $\hat{\theta}$
* $\alpha$: the nominal significance level

# Performance measures

The first performance measure of interest is **bias**, which quantifies whether the estimator targets the true value $\theta$ on average. Bias is calculated as:

$$\text{Bias} = \frac{1}{n_{\text{sim}}} \sum_{i = 1} ^ {n_{\text{sim}}} \hat{\theta}_i - \theta$$

The Monte Carlo standard error of bias is calculated as:

$$\text{MCSE(Bias)} = \sqrt{\frac{\frac{1}{n_{\text{sim}} - 1} \sum_{i = 1} ^ {n_{\text{sim}}} (\hat{\theta}_i - \bar{\theta}) ^ 2}{n_{\text{sim}}}}$$

`rsimsum` can also compute **relative bias** (_relative_ to the true value $\theta$), which can be interpreted similarly as with bias, but in relative terms rather than absolute.
This is calculated as:

$$\text{Relative Bias} = \frac{1}{n_{\text{sim}}} \sum_{i = 1} ^ {n_{\text{sim}}} \frac{\hat{\theta}_i - \theta}{\theta}$$

Its Monte Carlo standard error is calculated as:

$$
\text{MCSE(Relative Bias)} =  \sqrt{\frac{1}{n_{\text{sim}} (n_{\text{sim}} - 1)} \sum_i^{n_{\text{sim} \left[ \frac{\hat{\theta}_i - \theta}{\theta} - \widehat{\text{Relative Bias}} \right]^2}
$$

The **empirical standard error** of $\theta$ depends only on $\hat{\theta}$ and does not require any knowledge of $\theta$.
It estimates the standard deviation of $\hat{\theta}$ over the $n_{\text{sim}}$ replications:

$$\text{Empirical SE} = \sqrt{\frac{1}{n_{\text{sim}} - 1} \sum_{i = 1} ^ {n_{\text{sim}}} (\hat{\theta}_i - \bar{\theta}) ^ 2}$$

The Monte Carlo standard error is calculated as:

$$\text{MCSE(Emp. SE)} = \frac{\widehat{\text{Emp. SE}}}{\sqrt{2 (n_{\text{sim}} - 1)}}$$

When comparing different methods, the **relative precision of a given method B against a reference method A** is computed as:

$$\text{Relative % increase in precision} = 100 \left[ \left( \frac{\widehat{\text{Emp. SE}}_A}{\widehat{\text{Emp. SE}}_B} \right) ^ 2 - 1 \right]$$

Its (approximated) Monte Carlo standard error is:

$$\text{MCSE(Relative % increase in precision)} \simeq 200 \left( \frac{\widehat{\text{Emp. SE}}_A}{\widehat{\text{Emp. SE}}_B} \right)^2 \sqrt{\frac{1 - \rho^2_{AB}}{n_{\text{sim}} - 1}}$$

$\rho^2_{AB}$ is the correlation of $\hat{\theta}_A$ and $\hat{\theta}_B$.


A measure that takes into account both precision and accuracy of a method is the **mean squared error**, which is the sum of the squared bias and variance of $\hat{\theta}$:

$$\text{MSE} = \frac{1}{n_{\text{sim}}} \sum_{i = 1} ^ {n_{\text{sim}}} (\hat{\theta}_i - \theta) ^ 2$$

The Monte Carlo standard error is:

$$\text{MCSE(MSE)} = \sqrt{\frac{\sum_{i = 1} ^ {n_{\text{sim}}} \left[ (\hat{\theta}_i - \theta) ^2 - \text{MSE} \right] ^ 2}{n_{\text{sim}} (n_{\text{sim}} - 1)}}$$

The **model based standard error** is computed by averaging the estimated standard errors for each replication:

$$\text{Model SE} = \sqrt{\frac{1}{n_{\text{sim}}} \sum_{i = 1} ^ {n_{\text{sim}}} \widehat{\text{Var}}(\hat{\theta}_i)}$$

Its (approximated) Monte Carlo standard error is computed as:

$$\text{MCSE(Model SE)} \simeq \sqrt{\frac{\text{Var}[\widehat{\text{Var}}(\hat{\theta}_i)]}{4 n_{\text{sim}} \widehat{\text{Model SE}}}}$$

The model standard error targets the empirical standard error. Hence, the **relative error in the model standard error** is an informative performance measure:

$$\text{Relative % error in model SE} = 100 \left( \frac{\text{Model SE}}{\text{Empirical SE}} - 1\right)$$

Its Monte Carlo standard error is computed as:

$$\text{MCSE(Relative % error in model SE)} = 100 \left( \frac{\text{Model SE}}{\text{Empirical SE}} \right) \sqrt{\frac{\text{Var}[\widehat{\text{Var}}(\hat{\theta}_i)]}{4 n_{\text{sim}} \widehat{\text{Model SE}} ^ 4} + \frac{1}{2(n_{\text{sim}} - 1)}}$$

**Coverage** is another key property of an estimator. It is defined as the probability that a confidence interval contains the true value $\theta$, and computed as:

$$\text{Coverage} = \frac{1}{n_{\text{sim}}} \sum_{i = 1} ^ {n_{\text{sim}}} I(\hat{\theta}_{i, \text{low}} \le \theta \le \hat{\theta}_{i, \text{upp}})$$

where $I(\cdot)$ is the indicator function. The Monte Carlo standard error is computed as:

$$\text{MCSE(Coverage)} = \sqrt{\frac{\text{Coverage} \times (1 - \text{Coverage})}{n_{\text{sim}}}}$$

Under coverage is to be expected if:

1. $\text{Bias} \ne 0$, or
2. $\text{Models SE} < \text{Empirical SE}$, or
3. the distribution of $\hat{\theta}$ is not normal and intervals have been constructed assuming normality, or
4. $\widehat{\text{Var}}(\hat{\theta}_i)$ is too variable

Over coverage occurs as a result of $\text{Models SE} > \text{Empirical SE}$.

As under coverage may be a result of bias, another useful summary statistic is **bias-eliminated coverage**:

$$\text{Bias-eliminated coverage} = \frac{1}{n_{\text{sim}}} \sum_{i = 1} ^ {n_{\text{sim}}} I(\hat{\theta}_{i, \text{low}} \le \bar{\theta} \le \hat{\theta}_{i, \text{upp}}) $$

The Monte Carlo standard error is analogously as coverage:

$$\text{MCSE(Bias-eliminated coverage)} = \sqrt{\frac{\text{Bias-eliminated coverage} \times (1 - \text{Bias-eliminated coverage})}{n_{\text{sim}}}}$$

Finally, **power** of a significance test at the $\alpha$ level is defined as:

$$\text{Power} = \frac{1}{n_{\text{sim}}} \sum_{i = 1} ^ {n_{\text{sim}}} I \left[ |\hat{\theta}_i| \ge z_{\alpha/2} \times \sqrt{\widehat{\text{Var}}(\hat{\theta_i})} \right]$$

The Monte Carlo standard error is analogously as coverage:

$$\text{MCSE(Power)} = \sqrt{\frac{\text{Power} \times (1 - \text{Power})}{n_{\text{sim}}}}$$

Further information on summary statistics for simulation studies can be found in White (2010) and Morris, White, and Crowther (2019).

# Example 1: Simulation study on missing data

With this example dataset included in `rsimsum` we aim to summarise a simulation study comparing different ways to handle missing covariates when fitting a Cox model (White and Royston, 2009). One thousand datasets were simulated, each containing normally distributed covariates $x$ and $z$ and time-to-event outcome. Both covariates has $20\%$ of their values deleted independently of all other variables so the data became missing completely at random (Little and Rubin, 2002). Each simulated dataset was analysed in three ways. A Cox model was fit to the complete cases (`CC`). Then two methods of multiple imputation using chained equations (van Buuren, Boshuizen, and Knook, 1999) were used. The `MI_LOGT` method multiply imputes the missing values of $x$ and $z$ with the outcome included as $\log(t)$ and $d$, where $t$ is the survival time and $d$ is the event indicator. The `MI_T` method is the same except that $\log(t)$ is replaced by $t$ in the imputation model.

We load the data in the usual way:

```{r ex1-load-data}
library(rsimsum)
data("MIsim", package = "rsimsum")
```

Let's have a look at the first 10 rows of the dataset:

```{r ex1-inspect-data}
head(MIsim, n = 10)
```

The included variables are:

```{r ex1-included-variables}
str(MIsim)
```

* `dataset`, the number of the simulated dataset;

* `method`, the method used (`CC`, `MI_LOGT` or `MI_T`);

* `b`, the point estimate;

* `se`, the standard error of the point estimate.

We summarise the results of the simulation study by method using the `simsum` function:

```{r ex1-simsum}
s1 <- simsum(data = MIsim, estvarname = "b", true = 0.50, se = "se", methodvar = "method", ref = "CC")
```

We set `true = 0.50` as the true value of the point estimate `b` - under which the data was simulated - is 0.50. We select `CC` as the reference method as we consider the complete cases analysis the reference method to benchmark against; if we do not set a reference method, `simsum` picks one automatically.

Using the default settings, Monte Carlo standard errors are computed and returned.

Summarising a `simsum` object, we obtain the following output:

```{r ex1-summary-simsum}
ss1 <- summary(s1)
ss1
```

The output begins with a brief overview of the setting of the simulation study (e.g. the method variable, unique methods, etc.), and continues with each summary statistic by method (if defined, as in this case).
The values that are reported are point estimates with Monte Carlo standard errors in brackets; however, it is also possible to require confidence intervals based on Monte Carlo standard errors to be reported instead:

```{r ex1-summary-simsum-ci}
print(ss1, mcse = FALSE)
```

Highlighting some points of interest from the summary results above:

1. The `CC` method has small-sample bias away from the null (point estimate 0.0168, with 95% confidence interval: 0.0074 - 0.0261);
2. `CC` is inefficient compared with `MI_LOGT` and `MI_T`: the relative gain in precision for these two methods is 1.3105% and 1.2637% compared to `CC`, respectively;
3. Model-based standard errors are close to empirical standard errors;
4. Coverage of nominal 95% confidence intervals also seems fine, which is not surprising in view of the generally low (or lack of) bias and good model-based standard errors;
5. `CC` has lower power compared with `MI_LOGT` and `MI_T`, which is not surprising
in view of its inefficiency.

## Tabulating summary statistics

It is straightforward to produce a table of summary statistics for use in an R Markdown document:

```{r ex1-table}
library(knitr)
kable(tidy(ss1))
```

Using `tidy()` in combination with R packages such as [xtable](https://cran.r-project.org/package=xtable), [kableExtra](https://cran.r-project.org/package=kableExtra), [tables](https://cran.r-project.org/package=tables) can yield a variety of tables that should suit most purposes.

More information on producing tables directly from R can be found in the [CRAN Task View on Reproducible Research](https://CRAN.R-project.org/view=ReproducibleResearch).

## Plotting summary statistics

In this section, we show how to plot and compare summary statistics using the popular R package [ggplot](https://CRAN.R-project.org/package=ggplot2).

Plotting bias by method with $95\%$ confidence intervals based on Monte Carlo standard errors:

```{r ex1-plot-bias}
library(ggplot2)
ggplot(tidy(ss1, stats = "bias"), aes(x = method, y = est, ymin = lower, ymax = upper)) +
  geom_hline(yintercept = 0, color = "red", lty = "dashed") +
  geom_point() +
  geom_errorbar(width = 1 / 3) +
  theme_bw() +
  labs(x = "Method", y = "Bias")
```

Conversely, say we want to visually compare coverage for the three methods compared with this simulation study:

```{r ex1-plot-cov}
ggplot(tidy(ss1, stats = "cover"), aes(x = method, y = est, ymin = lower, ymax = upper)) +
  geom_hline(yintercept = 0.95, color = "red", lty = "dashed") +
  geom_point() +
  geom_errorbar(width = 1 / 3) +
  coord_cartesian(ylim = c(0, 1)) +
  theme_bw() +
  labs(x = "Method", y = "Coverage")
```

## Dropping large estimates and standard errors

`rsimsum` allows to automatically drop estimates and standard errors that are larger than a predefined value. Specifically, the argument of `simsum` that control this behaviour is `dropbig`, with tuning parameters `dropbig.max` and `dropbig.semax` that can be passed via the `control` argument.

Set `dropbig` to `TRUE` and standardised estimates larger than `max` in absolute value will be dropped; standard errors larger than `semax` times the average standard error will be dropped too.
By default, robust standardisation is used (based on median and inter-quartile range); however, it is also possible to request regular standardisation (based on mean and standard deviation) by setting the control parameter `dropbig.robust = FALSE`.

For instance, say we want to drop standardised estimates larger than $3$ in absolute value and standard errors larger than $1.5$ times the average standard error:

```{r ex1-dropbig-simsum}
s1.2 <- simsum(data = MIsim, estvarname = "b", true = 0.50, se = "se", methodvar = "method", ref = "CC", dropbig = TRUE, control = list(dropbig.max = 4, dropbig.semax = 1.5))
```

Some estimates were dropped, as we can see from the number of non-missing point estimates, standard errors:

```{r ex1-dropbig-nsim}
summary(s1.2, stats = "nsim")
```

Everything else works analogously as before; for instance, to summarise the results:

```{r ex1-dropbig-summary-simsum}
summary(s1.2)
```

# Example 2: Simulation study on survival modelling

```{r ex2-load-data}
data("relhaz", package = "rsimsum")
```

Let's have a look at the first 10 rows of the dataset:

```{r ex2-inspect-data}
head(relhaz, n = 10)
```

The included variables are:

```{r ex2-included-variables}
str(relhaz)
```

* `dataset`, simulated dataset number;

* `n`, sample size of the simulate dataset;

* `baseline`, baseline hazard function of the simulated dataset;

* `model`, method used (Cox model or Royston-Parmar model with 2 degrees of freedom);

* `theta`, point estimate for the log-hazard ratio;

* `se`, standard error of the point estimate.

`rsimsum` can summarise results from simulation studies with several data-generating mechanisms. For instance, with this example we show how to compute summary statistics by baseline hazard function and sample size.

In order to summarise results by data-generating factors, it is sufficient to define the "by" factors in the call to `simsum`:

```{r ex2-simsum}
s2 <- simsum(data = relhaz, estvarname = "theta", true = -0.50, se = "se", methodvar = "model", by = c("baseline", "n"))
s2
```

The difference between `methodvar` and `by` is as follows: `methodvar` represents methods (e.g. the two models, in this example) compared with this simulation study, while `by` represents all possible data-generating factors that varied when simulating data (in this case, sample size and the true baseline hazard function).

Summarising the results will be printed out for each method and combination of data-generating factors:

```{r ex2-summary-simsum}
ss2 <- summary(s2)
ss2
```

## Plotting summary statistics

Tables could get cumbersome when there are many different data-generating mechanisms. Plots are generally easier to interpret, and can be generated as easily as before.

Say we want to compare bias for each method by baseline hazard function and sample size using faceting:

```{r ex2-plot-bias}
ggplot(tidy(ss2, stats = "bias"), aes(x = model, y = est, ymin = lower, ymax = upper)) +
  geom_hline(yintercept = 0, color = "red", lty = "dashed") +
  geom_point() +
  geom_errorbar(width = 1 / 3) +
  facet_grid(baseline ~ n) +
  theme_bw() +
  labs(x = "Method", y = "Bias")
```

# References

* White, I.R. 2010. _simsum: Analyses of simulation studies including Monte Carlo error_. The Stata Journal 10(3): 369-385
* Morris, T.P., White, I.R. and Crowther, M.J. 2019. _Using simulation studies to evaluate statistical methods_. Statistics in Medicine 38:2074-2102
* White, I.R., and P. Royston. 2009. _Imputing missing covariate values for the Cox model_. Statistics in Medicine 28(15):1982-1998
* Little, R.J.A., and D.B. Rubin. 2002. _Statistical analysis with missing data_. 2nd ed. Hoboken, NJ: Wiley
* van Buuren, S., H.C. Boshuizen, and D.L. Knook. 1999. _Multiple imputation of missing blood pressure covariates in survival analysis_. Statistics in Medicine 18(6):681-694 
