Install github package on safe haven server

I’ve had few enquires about how to install the `summarizer` package on a server without internet access, such as the NHS Safe Havens.

1. Uploadsummarizer-master.zip from here to server.
2. Unzip.
3. Run this:

“`
library(devtools)
source = devtools:::source_pkg(“summarizer-master”)
install(source)

“`

Edit

As per comments, `devtools::install_local()` has previously failed, but may now also work directly.

P-values from random effects linear regression models

lme4::lmer is a useful frequentist approach to hierarchical/multilevel linear regression modelling. For good reason, the model output only includes t-values and doesn’t include p-values (partly due to the difficulty in estimating the degrees of freedom, as discussed here).

Yes, p-values are evil and we should continue to try and expunge them from our analyses. But I keep getting asked about this. So here is a simple bootstrap method to generate two-sided parametric p-values on the fixed effects coefficients. Interpret with caution.

```library(lme4)

# Run model with lme4 example data
fit = lmer(angle ~ recipe + temp + (1|recipe:replicate), cake)

# Model summary
summary(fit)

# lme4 profile method confidence intervals
confint(fit)

# Bootstrapped parametric p-values
boot.out = bootMer(fit, fixef, nsim=1000) #nsim determines p-value decimal places
p = rbind(
(1-apply(boot.out\$t<0, 2, mean))*2,
(1-apply(boot.out\$t>0, 2, mean))*2)
apply(p, 2, min)

# Alternative "pipe" syntax
library(magrittr)

lmer(angle ~ recipe + temp + (1|recipe:replicate), cake) %>%
bootMer(fixef, nsim=100) %\$%
rbind(
(1-apply(t<0, 2, mean))*2,
(1-apply(t>0, 2, mean))*2) %>%
apply(2, min)
```

Prediction is very difficult, especially about the future

As Niels Bohr, the Danish physicist, put it, “prediction is very difficult, especially about the future”. Prognostic models are commonplace and seek to help patients and the surgical team estimate the risk of a specific event, for instance, the recurrence of disease or a complication of surgery. “Decision-support tools” aim to help patients make difficult choices, with the most useful providing personalized estimates to assist in balancing the trade-offs between risks and benefits. As we enter the world of precision medicine, these tools will become central to all our practice.

In the meantime, there are limitations. Overwhelming evidence shows that the quality of reporting of prediction model studies is poor. In some instances, the details of the actual model are considered commercially sensitive and are not published, making the assessment of the risk of bias and potential usefulness of the model difficult.

In this edition of HPB, Beal and colleagues aim to validate the American College of Surgeons National Quality Improvement Program (ACS NSQIP) Surgical Risk Calculator (SRC) using data from 854 gallbladder cancer and extrahepatic cholangiocarcinoma patients from the US Extrahepatic Biliary Malignancy Consortium. The authors conclude that the “estimates of risk were variable in terms of accuracy and generally calculator performance was poor”. The SRC underpredicted the occurrence of all examined end-points (death, readmission, reoperation and surgical site infection) and discrimination and calibration were particularly poor for readmission and surgical site infection. This is not the first report of predictive failures of the SRC. Possible explanations cited previously include small sample size, homogeneity of patients, and too few institutions in the validation set. That does not seem to the case in the current study.

The SRC is a general-purpose risk calculator and while it may be applicable across many surgical domains, it should be used with caution in extrahepatic biliary cancer. It is not clear why the calculator does not provide measures of uncertainty around estimates. This would greatly help patients interpret its output and would go a long way to addressing some of the broader concerns around accuracy.

Cutting-edge liver surgery is often associated with modern technology such as the robot. In this edition of HPB, Torzilli and colleagues provide a fascinating account of 12 years of “radical but conservative” open liver surgery.

This is extreme parenchymal-sparing hepatectomy (PSH) in 169 patients with colorectal liver metastases. In all cases, tumour was touching or infiltrating portal pedicles or hepatic veins, a situation where most surgeons would advocate a major hepatectomy where possible. The PSH by its nature results in a 0 mm resection margin when the vessel is preserved, which was the aim in many of these procedures. Although this is off-putting, the cut-edge recurrence rate was no higher than average.

PSH in the form of “easy atypicals” is performed by all HPB surgeons. There are two main differences here. First is the aim to detach tumours from intrahepatic vascular structures. For instance, hepatic veins in contact with tumour were preserved and only resected if infiltrated. Even then, they were tangentially incised if possible and reconstructed with a bovine pericardial patch. Second is the careful attention paid to identifying and using communicating hepatic veins. This is well described but used extensively here to allow complete resection of segments while avoiding congestion in the draining region.

Short-term mortality and morbidity rates are comparable with other published series. A median survival of 36 months and 5-year overall survival of around 30% is reasonable given some of these patients may not be offered surgery in certain centres. The authors describe the parenchymal sparing approach “failing” in 14 (10%) patients: 7 (5%) has recurrence at the cut edge and 8 (6%) within segments which would have been removed using a standard approach. 44% of the 55 patients with liver-only recurrence underwent re-resection.

This is not small surgery. The average operating time is 8.5 h with the longest taking 18.5 h. The 66% thoracotomy rate is also notable in an era of minimally invasive surgery and certainly differs from my own practice. This study is challenging and I look forward to the debates that should arise from it.

Effect of day of the week on mortality after emergency general surgery

Out latest paper published in the BJS describes short- and long-term outcomes after emergency surgery in Scotland. We looked for a weekend effect and didn’t find one.

• In around 50,000 emergency general surgery patients, we didn’t find an association between day of surgery or day of admission and death rates;
• In around 100,000 emergency surgery patients including orthopaedic and gynaecology procedures, we didn’t find an association between day of surgery or day of admission and death rates;
• In around 500,000 emergency and planned surgery patients, we didn’t find an association between day of surgery or day of admission and death rates.

We also found that emergency surgery performed at weekends, or in those admitted at weekends, was performed a little quicker compared with weekdays.

More details can be found here:

Press coverage

Broadcast: BBC GOOD MORNING SCOTLAND, HEART FM,

Print: DAILY TELEGRAPH, DAILY MIRROR, METRO, HERALD, HERALD (Leader), SCOTSMAN, THE NATIONAL, YORKSHIRE POST, GLASGOW EVENING TIMES

Publishing mortality rates for individual surgeons

This is our new analysis of an old topic.In Scotland, individual surgeon outcomes were published as far back as 2006. It wasn’t pursued in Scotland, but has been mandated for surgeons in England since 2013.

This new analysis took the current mortality data and sought to answer a simple question: how useful is this information in detecting differences in outcome at the individual surgeon level?

Well the answer, in short, is not very useful.

We looked at mortality after planned bowel and gullet cancer surgery, hip replacement, and thyroid, obesity and aneurysm surgery. Death rates are relatively low after planned surgery which is testament to hard working NHS teams up and down the country. This together with the fact that individual surgeons perform a relatively small proportion of all these procedures means that death rates are not a good way to detect under performance.

At the mortality rates reported for thyroid (0.08%) and obesity (0.07%) surgery, it is unlikely a surgeon would perform a sufficient number of procedures in his/her entire career to stand a good chance of detecting a mortality rate 5 times the national average.

Surgeon death rates are problematic in more fundamental ways. It is the 21st century and much of surgical care is delivered by teams of surgeons, other doctors, nurses, physiotherapists, pharmacists, dieticians etc. In liver transplantation it is common for one surgeon to choose the donor/recipient pair, for a second surgeon to do the transplant, and for a third surgeon to look after the patient after the operation. Does it make sense to look at the results of individuals? Why not of the team?

It is also important to ensure that analyses adequately account for the increased risk faced by some patients undergoing surgery. If my granny has had a heart attack and has a bad chest, I don’t want her to be deprived of much needed surgery because a surgeon is worried that her high risk might impact on the public perception of their competence. As Harry Burns the former Chief Medical Officer of Scotland said, those with the highest mortality rates may be the heroes of the health service, taking on patients with difficult disease that no one else will face.

We are only now beginning to understand the results of surgery using measures that are more meaningful to patients. These sometimes get called patient-centred outcome measures. Take a planned hip replacement, the aim of the operation is to remove pain and increase mobility. If after 3 months a patient still has significant pain and can’t get out for the groceries, the operation has not been a success. Thankfully death after planned hip replacement is relatively rare and in any case, might have little to do with the quality of the surgery.

Transparency in the results of surgery is paramount and publishing death rates may be a step towards this, even if they may in fact be falsely reassuring. We must use these data as part of a much wider initiative to capture the success and failures of surgery. Only by doing this will we improve the results of surgery and ensure every patient receives the highest quality of care.

Press coverage

Print:

• New Scientist
• Scotsman
• Daily Mail
• Express
• the I

Online:

All the transplant statistics you need

If you have a hunger for statistics on organ transplantation, check out NHS Blood and Transplant. There are regularly updated and reflect what is actually happening in UK transplant today. We should have a competition for novel ways of presenting these visually. Ideas?!

RStudio and GitHub

Version control has become essential for me keeping track of projects, as well as collaborating. It allows backup of scripts and easy collaboration on complex projects. RStudio works really well with Git, an open source open source distributed version control system, and GitHub, a web-based Git repository hosting service. I was always forget how to set up a repository, so here’s a reminder.

This example is done on RStudio Server, but the same procedure can be used for RStudio desktop. Git or similar needs to be installed first, which is straight forward to do.

Setup Git on RStudio and Associate with GitHub

In RStudio, Tools -> Version Control, select Git.

In RStudio, Tools -> Global Options, select Git//SVN tab. Ensure the path to the Git executable is correct. This is particularly important in Windows where it may not default correctly (e.g. C:/Program Files (x86)/Git/bin/git.exe).
Now hit, Create RSA Key …

Close this window.

Click, View public key, and copy the displayed public key.

If you haven’t already, create a GitHub account. Open your account settings and click the SSH keys tab. Click Add SSH key. Paste in the public key you have copied from RStudio.

Tell Git who you are. Remember Git is a piece of software running on your own computer. This is distinct to GitHub, which is the repository website. In RStudio, click Tools -> Shell … . Enter:

```git config --global user.email "mai[email protected]"
git config --global user.name "ewenharrison"```

Create New project AND git

In RStudio, click New project as normal. Click New Directory.

Name the project and check Create a git repository.

Now in RStudio, create a new script which you will add to your repository.

After saving your new script (test.R), it should appear in the Git tab on the Environment / history panel.

Click the file you wish to add, and the status should turn to a green ‘A’. Now click Commit and enter an identifying message in Commit message.

You have now committed the current version of this file to your repository on your computer/server. In the future you may wish to create branches to organise your work and help when collaborating.

Now you want to push the contents of this commit to GitHub, so it is also backed-up off site and available to collaborators. In GitHub, create a New repository, called here test.

In RStudio, again click Tools -> Shell … . Enter:

```git remote add origin https://github.com/ewenharrison/test.git
git config remote.origin.url [email protected]:ewenharrison/test.git
git pull -u origin master
git push -u origin master```

You have now pushed your commit to GitHub, and should be able to see your files in your GitHub account. The Pull Push buttons in RStudio will now also work. Remember, after each Commit, you have to Push to GitHub, this doesn’t happen automatically.

Clone an existing GitHub project to new RStudio project

In RStudio, click New project as normal. Click Version Control.

In Clone Git Repository, enter the GitHub repository URL as per below. Change the project directory name if necessary.

In RStudio, again click Tools -> Shell … . Enter:

`git config remote.origin.url [email protected]:ewenharrison/test.git`

Interested in international trials? Take part in GlobalSurg.

Journal bans p-values

Editors from the journal Basic and Applied Social Psychology have banned p-values. Or rather null hypothesis significance testing – which includes all the common statistical tests usually reported in studies.

A bold move and an interesting one. In an editorial, the new editor David Trafimow states,

null hypothesis significance testing procedure has been shown to be logically invalid and to provide little information about the actual likelihood of either the null or experimental hypothesis.

He seems to be on a mission and cites his own paper from 12 years ago in support of the position.

So what should authors provide instead to support or refute a hypothesis? Strong descriptive statistics including effect sizesl and the presentation of frequency or distributional data is encouraged. Which sounds reasonable. And larger sample sizes are also required. Ah, were it that easy.

Bayesian approaches are encouraged but not required.

Challenging the dominance of poorly considered p-value is correct. I’d like to see a medical journal do the same.

Bayesian statistics and clinical trial conclusions: Why the OPTIMSE study should be considered positive

Statistical approaches to randomised controlled trial analysis

The statistical approach used in the design and analysis of the vast majority of clinical studies is often referred to as classical or frequentist. Conclusions are made on the results of hypothesis tests with generation of p-values and confidence intervals, and require that the correct conclusion be drawn with a high probability among a notional set of repetitions of the trial.

Bayesian inference is an alternative, which treats conclusions probabilistically and provides a different framework for thinking about trial design and conclusions. There are many differences between the two, but for this discussion there are two obvious distinctions with the Bayesian approach. The first is that prior knowledge can be accounted for to a greater or lesser extent, something life scientists sometimes have difficulty reconciling. Secondly, the conclusions of a Bayesian analysis often focus on the decision that requires to be made, e.g. should this new treatment be used or not.

There are pros and cons to both sides, nicely discussed here, but I would argue that the results of frequentist analyses are too often accepted with insufficient criticism. Here’s a good example.

OPTIMSE: Optimisation of Cardiovascular Management to Improve Surgical Outcome

Optimising the amount of blood being pumped out of the heart during surgery may improve patient outcomes. By specifically measuring cardiac output in the operating theatre and using it to guide intravenous fluid administration and the use of drugs acting on the circulation, the amount of oxygen that is delivered to tissues can be increased.

It sounds like common sense that this would be a good thing, but drugs can have negative effects, as can giving too much intravenous fluid. There are also costs involved, is the effort worth it? Small trials have suggested that cardiac output-guided therapy may have benefits, but the conclusion of a large Cochrane review was that the results remain uncertain.

A well designed and run multi-centre randomised controlled trial was performed to try and determine if this intervention was of benefit (OPTIMSE: Optimisation of Cardiovascular Management to Improve Surgical Outcome).

Patients were randomised to a cardiac output–guided hemodynamic therapy algorithm for intravenous fluid and a drug to increase heart muscle contraction (the inotrope, dopexamine) during and 6 hours following surgery (intervention group) or to usual care (control group).

The primary outcome measure was the relative risk (RR) of a composite of 30-day moderate or major complications and mortality.

OPTIMSE: reported results

Focusing on the primary outcome measure, there were 158/364 (43.3%) and 134/366 (36.6%) patients with complication/mortality in the control and intervention group respectively. Numerically at least, the results appear better in the intervention group compared with controls.

Using the standard statistical approach, the relative risk (95% confidence interval) = 0.84 (0.70-1.01), p=0.07 and absolute risk difference = 6.8% (−0.3% to 13.9%), p=0.07. This is interpreted as there being insufficient evidence that the relative risk for complication/death is different to 1.0 (all analyses replicated below). The authors reasonably concluded that:

In a randomized trial of high-risk patients undergoing major gastrointestinal surgery, use of a cardiac output–guided hemodynamic therapy algorithm compared with usual care did not reduce a composite outcome of complications and 30-day mortality.

A difference does exist between the groups, but is not judged to be a sufficient difference using this conventional approach.

OPTIMSE: Bayesian analysis

Repeating the same analysis using Bayesian inference provides an alternative way to think about this result. What are the chances the two groups actually do have different results? What are the chances that the two groups have clinically meaningful differences in results? What proportion of patients stand to benefit from the new intervention compared with usual care?

With regard to prior knowledge, this analysis will not presume any prior information. This makes the point that prior information is not always necessary to draw a robust conclusion. It may be very reasonable to use results from pre-existing meta-analyses to specify a weak prior, but this has not been done here. Very grateful to John Kruschke for the excellent scripts and book, Doing Bayesian Data Analysis.

The results of the analysis are presented in the graph below. The top panel is the prior distribution. All proportions for the composite outcome in both the control and intervention group are treated as equally likely.

The middle panel contains the main findings. This is the posterior distribution generated in the analysis for the relative risk of the composite primary outcome (technical details in script below).

The mean relative risk = 0.84 which as expected is the same as the frequentist analysis above. Rather than confidence intervals, in Bayesian statistics a credible interval or region is quoted (HDI = highest density interval is the same). This is philosphically different to a confidence interval and says:

Given the observed data, there is a 95% probability that the true RR falls within this credible interval.

This is a subtle distinction to the frequentist interpretation of a confidence interval:

Were I to repeat this trial multiple times and compute confidence intervals, there is a 95% probability that the true RR would fall within these confidence intervals.

This is an important distinction and can be extended to make useful probabilistic statements about the result.

The figures in green give us the proportion of the distribution above and below 1.0. We can therefore say:

The probability that the intervention group has a lower incidence of the composite endpoint is 97.3%.

It may be useful to be more specific about the size of difference between the control and treatment group that would be considered equivalent, e.g. 10% above and below a relative risk = 1.0. This is sometimes called the region of practical equivalence (ROPE; red text on plots). Experts would determine what was considered equivalent based on many factors. We could therefore say:

The probability of the composite end-point for the control and intervention group being equivalent is 22%.

Or, the probability of a clinically relevant difference existing in the composite endpoint between control and intervention groups is 78%

Finally, we can use the 200 000 estimates of the probability of complication/death in the control and intervention groups that were generated in the analysis (posterior prediction). In essence, we can act like these are 2 x 200 000 patients. For each “patient pair”, we can use their probability estimates and perform a random draw to simulate the occurrence of complication/death. It may be useful then to look at the proportion of “patients pairs” where the intervention patient didn’t have a complication but the control patient did:

Using posterior prediction on the generated Bayesian model, the probability that a patient in the intervention group did not have a complication/death when a patient in the control group did have a complication/death is 28%.

Conclusion

On the basis of a standard statistical analysis, the OPTIMISE trial authors reasonably concluded that the use of the intervention compared with usual care did not reduce a composite outcome of complications and 30-day mortality.

Using a Bayesian approach, it could be concluded with 97.3% certainty that use of the intervention compared with usual care reduces the composite outcome of complications and 30-day mortality; that with 78% certainty, this reduction is clinically significant; and that in 28% of patients where the intervention is used rather than usual care, complication or death may be avoided.

```# OPTIMISE trial in a Bayesian framework
# JAMA. 2014;311(21):2181-2190. doi:10.1001/jama.2014.5305
# Ewen Harrison
# 15/02/2015

# Primary outcome: composite of 30-day moderate or major complications and mortality
N1 <- 366
y1 <- 134
N2 <- 364
y2 <- 158
# N1 is total number in the Cardiac Output–Guided Hemodynamic Therapy Algorithm (intervention) group
# y1 is number with the outcome in the Cardiac Output–Guided Hemodynamic Therapy Algorithm (intervention) group
# N2 is total number in usual care (control) group
# y2 is number with the outcome in usual care (control) group

# Risk ratio
(y1/N1)/(y2/N2)

library(epitools)
riskratio(c(N1-y1, y1, N2-y2, y2), rev="rows", method="boot", replicates=100000)

# Using standard frequentist approach
# Risk ratio (bootstrapped 95% confidence intervals) = 0.84 (0.70-1.01)
# p=0.07 (Fisher exact p-value)

# Reasonably reported as no difference between groups.

# But there is a difference, it just not judged significant using conventional
# (and much criticised) wisdom.

# Bayesian analysis of same ratio
# Base script from John Krushcke, Doing Bayesian Analysis

#------------------------------------------------------------------------------
source("~/Doing_Bayesian_Analysis/openGraphSaveGraph.R")
source("~/Doing_Bayesian_Analysis/plotPost.R")
require(rjags) # Kruschke, J. K. (2011). Doing Bayesian Data Analysis, Academic Press / Elsevier.
#------------------------------------------------------------------------------
# Important
# The model will be specified with completely uninformative prior distributions (beta(1,1,).
# This presupposes that no pre-exisiting knowledge exists as to whehther a difference
# may of may not exist between these two intervention.

# Plot Beta(1,1)
# 3x1 plots
par(mfrow=c(3,1))
# Adjust size of prior plot
par(mar=c(5.1,7,4.1,7))
plot(seq(0, 1, length.out=100), dbeta(seq(0, 1, length.out=100), 1, 1),
type="l", xlab="Proportion",
ylab="Probability",
main="OPTIMSE Composite Primary Outcome\nPrior distribution",
frame=FALSE, col="red", oma=c(6,6,6,6))
legend("topright", legend="beta(1,1)", lty=1, col="red", inset=0.05)

# THE MODEL.
modelString = "
# JAGS model specification begins here...
model {
# Likelihood. Each complication/death is Bernoulli.
for ( i in 1 : N1 ) { y1[i] ~ dbern( theta1 ) }
for ( i in 1 : N2 ) { y2[i] ~ dbern( theta2 ) }
# Prior. Independent beta distributions.
theta1 ~ dbeta( 1 , 1 )
theta2 ~ dbeta( 1 , 1 )
}
# ... end JAGS model specification
" # close quote for modelstring

# Write the modelString to a file, using R commands:
writeLines(modelString,con="model.txt")

#------------------------------------------------------------------------------
# THE DATA.

# Specify the data in a form that is compatible with JAGS model, as a list:
dataList =  list(
N1 = N1 ,
y1 = c(rep(1, y1), rep(0, N1-y1)),
N2 = N2 ,
y2 = c(rep(1, y2), rep(0, N2-y2))
)

#------------------------------------------------------------------------------
# INTIALIZE THE CHAIN.

# Can be done automatically in jags.model() by commenting out inits argument.
# Otherwise could be established as:
# initsList = list( theta1 = sum(dataList\$y1)/length(dataList\$y1) ,
#                   theta2 = sum(dataList\$y2)/length(dataList\$y2) )

#------------------------------------------------------------------------------
# RUN THE CHAINS.

parameters = c( "theta1" , "theta2" )     # The parameter(s) to be monitored.
adaptSteps = 500              # Number of steps to "tune" the samplers.
burnInSteps = 1000            # Number of steps to "burn-in" the samplers.
nChains = 3                   # Number of chains to run.
numSavedSteps=200000           # Total number of steps in chains to save.
thinSteps=1                   # Number of steps to "thin" (1=keep every step).
nIter = ceiling( ( numSavedSteps * thinSteps ) / nChains ) # Steps per chain.
# Create, initialize, and adapt the model:
jagsModel = jags.model( "model.txt" , data=dataList , # inits=initsList ,
# Burn-in:
cat( "Burning in the MCMC chain...\n" )
update( jagsModel , n.iter=burnInSteps )
# The saved MCMC chain:
cat( "Sampling final MCMC chain...\n" )
codaSamples = coda.samples( jagsModel , variable.names=parameters ,
n.iter=nIter , thin=thinSteps )
# resulting codaSamples object has these indices:
#   codaSamples[[ chainIdx ]][ stepIdx , paramIdx ]

#------------------------------------------------------------------------------
# EXAMINE THE RESULTS.

# Convert coda-object codaSamples to matrix object for easier handling.
# But note that this concatenates the different chains into one long chain.
# Result is mcmcChain[ stepIdx , paramIdx ]
mcmcChain = as.matrix( codaSamples )

theta1Sample = mcmcChain[,"theta1"] # Put sampled values in a vector.
theta2Sample = mcmcChain[,"theta2"] # Put sampled values in a vector.

# Plot the chains (trajectory of the last 500 sampled values).
par( pty="s" )
chainlength=NROW(mcmcChain)
plot( theta1Sample[(chainlength-500):chainlength] ,
theta2Sample[(chainlength-500):chainlength] , type = "o" ,
xlim = c(0,1) , xlab = bquote(theta[1]) , ylim = c(0,1) ,
ylab = bquote(theta[2]) , main="JAGS Result" , col="skyblue" )

# Display means in plot.
theta1mean = mean(theta1Sample)
theta2mean = mean(theta2Sample)
if (theta1mean > .5) { xpos = 0.0 ; xadj = 0.0
} else { xpos = 1.0 ; xadj = 1.0 }
if (theta2mean > .5) { ypos = 0.0 ; yadj = 0.0
} else { ypos = 1.0 ; yadj = 1.0 }
text( xpos , ypos ,
bquote(
"M=" * .(signif(theta1mean,3)) * "," * .(signif(theta2mean,3))

# Plot a histogram of the posterior differences of theta values.
thetaRR = theta1Sample / theta2Sample # Relative risk
thetaDiff = theta1Sample - theta2Sample # Absolute risk difference

par(mar=c(5.1, 4.1, 4.1, 2.1))
plotPost( thetaRR , xlab= expression(paste("Relative risk (", theta[1]/theta[2], ")")) ,
compVal=1.0, ROPE=c(0.9, 1.1),
main="OPTIMSE Composite Primary Outcome\nPosterior distribution of relative risk")
plotPost( thetaDiff , xlab=expression(paste("Absolute risk difference (", theta[1]-theta[2], ")")) ,
compVal=0.0, ROPE=c(-0.05, 0.05),
main="OPTIMSE Composite Primary Outcome\nPosterior distribution of absolute risk difference")

#-----------------------------------------------------------------------------
# Use posterior prediction to determine proportion of cases in which
# using the intervention would result in no complication/death
# while not using the intervention would result in complication death

chainLength = length( theta1Sample )

# Create matrix to hold results of simulated patients:
yPred = matrix( NA , nrow=2 , ncol=chainLength )

# For each step in chain, use posterior prediction to determine outcome
for ( stepIdx in 1:chainLength ) { # step through the chain
# Probability for complication/death for each "patient" in intervention group:
pDeath1 = theta1Sample[stepIdx]
# Simulated outcome for each intervention "patient"
yPred[1,stepIdx] = sample( x=c(0,1), prob=c(1-pDeath1,pDeath1), size=1 )
# Probability for complication/death for each "patient" in control group:
pDeath2 = theta2Sample[stepIdx]
# Simulated outcome for each control "patient"
yPred[2,stepIdx] = sample( x=c(0,1), prob=c(1-pDeath2,pDeath2), size=1 )
}

# Now determine the proportion of times that the intervention group has no complication/death
# (y1 == 0) and the control group does have a complication or death (y2 == 1))
(pY1eq0andY2eq1 = sum( yPred[1,]==0 & yPred[2,]==1 ) / chainLength)
(pY1eq1andY2eq0 = sum( yPred[1,]==1 & yPred[2,]==0 ) / chainLength)
(pY1eq0andY2eq0 = sum( yPred[1,]==0 & yPred[2,]==0 ) / chainLength)
(pY10eq1andY2eq1 = sum( yPred[1,]==1 & yPred[2,]==1 ) / chainLength)

# Conclusion: in 27% of cases based on these probabilities,
# a patient in the intervention group would not have a complication,
# when a patient in control group did.
```