All statistical models rely on assumptions and because of this they will never be 100% accurate. They make assumptions on how many people will become infected, the number of cases that will require hospitalization and whether this will exceed NHS intensive care capacity. Weiss (2020) suggests the most basic model to use is the SIR model – where the population are split into three subgroups: susceptible people, i.e. those vulnerable to getting the disease; infected people and removed people (those who have recovered and gained immunity or those who have died) – but with an added category for carriers. This extra category is required as a person can carry the disease for up to 14 days without showing any physical symptoms. This means people can pass the disease on to the vulnerable without knowing they were even sick. This is why social distancing measures are deemed so vital in preventing the spread of this virus. In reality much more complex models are required to fully describe the viral outbreak, however, regardless of the complexity it will never fully describe what will happen in real life. Models are used simply as an “informed prediction” based on the data and the population; they are used to make decisions regarding the interventions to be put in place and when they can be lifted.

As this is a new virus, modelling it’s spread is challenging. This is because assumptions need to be made but there is only a small amount of data available. These assumptions are used for key model parameters (Enserink & Kpferschmidt (2020)). One such parameter is the number of new infections caused by one infected person when no intervention measures are put in place, as well as the time frame in which these infections occur. It takes some time and a large quantity of data for these parameters to be accurately estimated.

Coronavirus and the UK: The Numbers

Modelling by epidemiologists in Imperial College London was used to determine the intervention approach the UK government implemented. It was believed that if a hard lock-down was implemented like in China, Italy and Spain then the infection rate would spike once the intervention was lifted (Enserink & Kupferschmidt (2020)). Because of these beliefs, at first less severe social distancing restrictions were implemented to ensure the peak of infections was flattened and the demand for intensive care beds within hospitals did not exceed capacity. However, taking into account new data, a revised model was created and it was decided that more strict lock-down measures were required to save the NHS from being overwhelmed. These new measures were announced on 23rd March 2020. Due to the nature of the disease, it could take up to a month to determine whether these measures are sufficient.

Some of the data used to model the virus outbreak was obtained from the “BBC Pandemic project” (Klepac et. al (2020)) which collected data from over 36,000 UK citizens in order to create age-specific population contact matrices. This data was used to reduce the amount of social contact in order to reduce the spread of the virus. The details of this paper are listed at the bottom of this page and I highly recommend reading it. The study worked through an app downloaded by participants. The app recorded their approximate location hourly for a day, at the end of which the users recorded all social contacts, providing information on each. This study does come with the flaw of self-reporting and the misinformation that comes alongside that, however it does allow for a massive sample size to more accurately portray the UK population.

Below are three graphs that depict: 1. The cumulative number of cases, 2. The cumulative number of deaths and 3. The daily number of deaths, all of which relate to the UK only. According to Roser et. al (2020), under current death rates, it would take 7 days for the total number of confirmed deaths to double. This is one of the worst growth rates in the world, only behind that of the US and Belgium (both of which have a 6 day doubling rate). The first two graphs imply exponential growth, however we do expect this to level off and reach some sort of peak in the coming weeks.

This data has to be taken with a pinch of salt as it may not all be up to date, for example there is a lag between testing and the results being obtained. To add to this, according to Richarson and Spieglhalter (2020) just over 317,000 have been tested in the UK to date, compared to the 1.3 million tests carried out in Germany, this may mean that a greater number of people have had the virus but have not had it confirmed. There are also the aforementioned carriers who will not yet know they were infected as they do not present with symptoms. So is the increase in growth caused by an increases in cases or an increase in testing? The answer to this is likely both.

It is also thought that the number of deaths is much higher than what is being reported. This is because the numbers released only include those who have died in hospital and have tested positive for Coronavirus, there is often a delay of a few days or more for the death to be recorded as being caused by Covid-19.

Singapore Case-Control Study

In this section of the blog, I’m going to summarize one of many studies currently being carried out around the globe into the Coronavirus pandemic. The study I will focus on was conducted by Sun et. al (2020). It investigated risk factors on the virus using a case-control study in Singapore between 26th January and 16th February 2020. 54 cases of Coronavirus were compared to 734 control cases. The data collected included: demographic, co-morbidity factors, exposure risk, symptoms and vitals (including blood pressure, pulse and temperature). Predictors of the virus were split into four categories:

• Exposure Risk
• Demographic Variables
• Clinical Findings
• Clinical Test Results (some patients presented all clinical tests, others just radiological tests)

From this they created four prediction models, whose variables were selected using stepwise AIC to create logistic regression models:

• Model 1: all covariates from all 4 categories,
• Model 2: demographic variables, clinical findings and all clinical test results,
• Model 3: demographic variables, clinical findings and clinical test results excluding radiology,
• Model 4: only demographic variables and clinical findings.

From this study, they found that positive cases of Covid-19 were more likely to be older compared to the controls (with a p-value less than 0.0001) but they were not more likely to have any of the co-morbidity factors than any of the controls (this is an unusual finding as the UK government listed a set of conditions such as diabetes, asthma and heart disease make a person more vulnerable to the disease; I would have thought this would have shown up in the co-morbidity results). However, the exposure factor was deemed significant with 59.3% of cases having had contact with someone with the virus or having recently traveled to Wuhan, compared to only 17.2% of the controls. Cases were also deemed more likely to have a fever (p-value of 0.003) and signs of pneumonia through radiology results (present in 42.6% of cases compared to 11.1% of controls).

From Model 1 it was deemed that exposure risk was most significant in resulting in a positive Covid-19 result. In the other 3 models, which exclude exposure, a high temperature is deemed the most relevant clinical finding in predicting a positive result apart from in Model 2 where Gastrointestinal symptoms were deemed marginally more significant.

It was concluded that Model 1, which takes into account all risk factors, performs exceptionally well in predicting a positive Coronavirus status but even with an absence of exposure status the models 2 and 3 performed sufficiently. The evidence did however show a reduce in performance in using model 4, where basic clinical tests such as bloods were not used. For more information on this study I would recommend reading the paper by Sun et. al (2020) listed in the references below.

Concluding Remarks

Although modelling is very useful in analyzing the spread of Coronavirus and decision making for intervention practices, there is a lot that these models will not show us such as: the degree to which the public comply to social distancing intervention measures, the introduction of a vaccine as well as the toss-up between saving the economy and reducing the death rate. As all models will contain some degree of uncertainty, they must be analysed for pitfalls and decisions should not be based solely on their findings.

References & Further Reading

• Klepac, P., Kucharski, A., et al. (2020). Contacts in context: large-scale setting-specific social mixing matrices from the BBC Pandemic project. medRxiv
• Enserink, M., Kuperschmidt, K., (2020). With COVID-19, modelling takes on life and death importance. Science (New York, N.Y.) 367(6485)
• Richardson, S., Spiegelhalter, D., (2020). Coronavirus statistics: what can we trust and what should we ignore? The Observer
• Roser, M., Ritchie, H., Ortiz-Ospina, E., (2020). Coronavirus Disease (COVID-2019) – Statistics and Research, Our World In Data
• Sun, Y., Vanessa, K., et. al (2020). Epidemiological and Clinical Predictors of COVID-19, Clinical infectious diseases: an official publication of the Infectious Disease Society of America
• Weiss, S. (2020). Why modelling can’t tell us when the UK’s lockdown will end, Wired

During my research for my MSci dissertation (see the last blog post to find out the basics of my research) I came across the concept of meta-analysis. The overall motivation for conducting meta-analyses is to draw more reliable and precise conclusions on the effect of a treatment. In this blog post I will outline for you both the benefits and costs of conducting a meta-analysis.

Meta-analysis is a statistical method used to merge findings of single, independent studies that investigate the same or similar clinical problem [Shorten (2013)]. Each of these studies should have been carried out using similar procedures, methods and conditions. Data from the individual trials are collected and we calculate a pooled estimate (although data is not usually pooled!) of the treatment effect to determine efficacy.

Effectiveness vs. Efficacy: ‘Efficacy can be defined as the performance of an intervention under ideal and controlled circumstance, whereas effectiveness refers to its performance under real-world conditions.’ [Singal, A. et al. (2014)].

If conducted correctly, efficacy conclusions from a meta-analysis should be more powerful due to the larger sample size created by considering several studies. Often this sample size is far greater than what we could feasibly achieve in a single clinical trial, which is constraint by funds and resources including the availability of patients. This increased sample size also improves the precision of our estimate in terms of how closely the trial results relate to effectiveness in the whole population [Moore (2012)].

Although meta-analysis can be a useful tool to increase sample size and hence statistical power, it does have significant associated methodological issues. The first of these is publication bias. This may be introduced because trials which show significant results in favor of a new treatment are more likely to be published than those which are inconclusive or favor the standard treatment. Another form of publication bias arises when researchers exclude studies that are not published in English [Walker et al. (2008)]. This exclusion of studies may lead to an over/under-estimate of the true treatment effect.

The issue of publication bias in a meta-analysis exploring the effects of breastfeeding on children’s IQ scores was discussed by Ritchie (2017). A funnel plot of the original data set showed a tendency for larger studies to show a smaller treatment effect, indicating publication bias. The original study found that breastfed children had IQ scores that were on average, 3.44 points higher than non-breastfed children. However, after adjusting for publication bias a much lower estimate of 1.38 points higher IQ was given. Although this is still a significant result, this example highlights the issue of overestimation resulting from publication bias.

Funnel Plots: A funnel plot is a method used to assess the role of publication bias. It is a plot of sample size versus treatment effects. As the sample size increases, the effect size is likely to converge to the true effect size [Lee, (2012)]. We will obviously have a scatter of points surrounding this true effect size, however, if we have publication bias, we may have a lack of ‘small effect size’ small studies included in the meta-analysis. This will lead to the funnel plot being skewed.

Another key issue with meta-analysis is heterogeneity. This is defined as the variation in results between studies included in the analysis. Investigators must consider the source of this inconsistency, they may include differences in trial design, study population or inclusion/exclusion criteria between trials but also differences due to chance. High levels of heterogeneity comprises the justification for meta-analysis, as grouping together studies whose results vary greatly will give a questionable pooled treatment effect, thus reducing our confidence in making recommendations about treatments. There are methods to handle heterogeneity, one of which is to fit a random effects model.

A meta-analysis considering strategies to prevent falls and fractures in hospitals and care homes [Oliver et al. (2007)] obtained strong evidence to suggest heterogeneity between studies. This variation was highlighted in forest plots which showed a very wide spread of results. As the investigators believed all trials were similar enough in design and all aimed to trial the same treatment, they felt it was justified to calculate a pooled treatment effect. However, this high variability brings into question the reliability of the estimate, complicating decisions regarding recommendation of treatments.

In summary, Meta-analysis is a very useful tool for combining results of studies in order to boost the precision of our conclusions. We do however need to proceed with caution and ensure heterogeneity and check for publication bias.

References & Further Reading

• Lee, W., Hotopf, M., (2012). Critical appraisal: Reviewing scientific evidence and reading academic papers. Core Psychiatry, 131-142.
• Moore, Z. (2012). Meta-analysis in context. Journal of Clinical Nursing, 21(19):2798-2807.
• Oliver, D., Connelly, J. B., Victor, C. R., Shaw, F. E., Whitehead, A., Genc, Y., Vanoli, A., Martin, F. C., and Gonsey, M. A. (2007). Strategies to prevent falls and fractures in hospitals and care homes and effect of cognitive impairment: systematic review and meta-analyses. BMJ, 334(7584).
• Ritchie, S. J. (2017). Publication bias in a recent meta-analysis on breastfeeding and IQ. Acta Paediatrica, 106(2):345-345.
• Singal, A., Higgins, P., Waljee, A. (2014). A primer on effectiveness and efficacy trials. Clinical and Translational Gastroenterology, 5(1).
• Shorten, A. (2013). What is meta-analysis? Evidence Based Nursing, 16(1).
• Walker, E., Hernandez, A., and Kattan, M. (2008). Meta-analysis: Its strengths and limitations. Cleveland Clinic Journal Of Medicine, 75(6):431-439.

Multi-armed bandits provide a framework for solving sequential decision problems in which an agent learns information regarding some uncertain factor as time progresses to make more informed choices. A common application for this is in slot machines, where a player must select one of K arms to pull.

A K-armed bandit is set up as follows: the player has K arms to choose from (known as actions), each of which yields a reward payout defined by random variables. These rewards are independently and identically distributed (IID) with an unknown mean that the player learns through experimentation.

In order to experiment, the player requires a policy regarding which action to take next. Formally a policy is an algorithm that chooses the next arm to play using information on previous plays and the rewards they obtained (Auer et al., 2002).

When making the decision on which arm to pull next, the policy must weigh up the benefits of exploration vs. exploitation. Exploitation refers to choosing an arm that is known to yield a desirable reward in short-term play. Whereas, exploration is the search for higher payouts in different arms, this can be used to optimize long term reward.

In order to quantify a policy’s success, a common measure known as regret is used. The regret of a policy over \(T\) rounds is defined as the difference between the expected reward if the optimal arm is played in all \(T\) rounds and the sum of the rewards observed over the \(T\) rounds.

Thompson sampling (TS) is one of the policies used for tackling multi-armed bandit problems. For simplicity we will consider a case where the rewards are binary i.e. they take the values 0 (for failure) or 1 (for success), however TS can be extended to rewards with any general distribution.

Consider a multi-armed bandit with \(k\in\{1,\ldots,K\}\) arms each with an unknown probability of a successful reward \(\theta_k\). Our goal is to maximize the number of successes over a set time period.

We begin by defining a prior belief on our success probabilities which we set to be Beta distributed. Each time we gain an observation we update this prior belief by updating the parameters of the beta distribution and use this updated distribution as a prior for the next arm pull.

The process for deciding which arm to pull at each time step using the Thompson Sampling algorithm is given in the following diagram:

Although here we have specified a Beta distribution, a distribution of any kind can be used. However, in this case it is a convenient choice as it’s conjugate to the Bernoulli distribution and hence the posterior is also Beta distributed. This is why we only need to update the parameters at each time step. If the prior chosen was not conjugate, then it would not be so simple. Instead we may be in a situation where we need to specify a completely new prior each time.

The prior chosen plays a large role in the effectivenss of the Thompson Sampling algorithm, and thus care must be chosen over its specification. Ideally it should be chosen to describe all plausible values of the reward success parameter.

In the slot machine case, the player often has no intuition on the values of this success parameter. This is usually described using an uninformative Beta(1,1) prior as it takes uniform value across the entire [0,1] range. This promotes exploration to learn which arm is optimal.

In the opposite case where we have some knowledge of the true values of the success parameter, we may center the priors about these values in order to reduce regret. Here, we require less exploration to identify the optimal arm and so can exploit this to maximize reward.

Although TS tends to be efficient in minimizing regret, there are occasionally outlier cases where the regret is much higher than expected. This may happen if we begin by pulling the sub-optimal arm and receive a successful reward. We are likely to exploit this further and continue to pull this arm, falsely leading us to believe this arm is actually optimal. Or alternatively, we may begin by selecting the optimal arm and observe a failure reward. We then may exploit the sub-optimal arm under the false belief that it will give us a higher reward.

To conclude, Thompson Sampling is a simple but efficient algorithm for solving stochastic multi-armed bandit problems, successfully balancing exploitation and exploration to maximize reward and minimize regret.

References & Further Reading

Auer, P., Cesa-Bianchi, N., and Fischer, P. (2002). Finite-time analysis of the mulitarmed bandit problem. Machine Learning, 47(2):235-236.

Russo, D., Van Roy, B., Kazerouni, A., Osband, I., and Wen, Z. (2017). A tutorial on Thompson Sampling.