Mapping in health economics

Last Friday, I went to one of the health economics seminars that are organised at UCL; the format that is used is that one of the people in the group suggests a paper (typically something that they are working on) but instead of having them leading the discussion, one of the others takes the responsibility of preparing a few slides to highlight what they think are the main points. The author/person who suggested the paper is usually in the room and they respond to the short presentation and then the discussion is open to the group at large.

I missed a couple since they started last summer, but the last two I've been to have really been interesting. Last time the main topic was mapping of utility measures; in a nutshell, the idea is that there are some more or less standardised measures of "quality of life" (QoL $-$ the most common probably being the EQ5D and the SF6D). 

However, they are not always reported. For example, you may have a trial that you want to analyse in which data have been collected on a different scale (and I'm told that there are plenty); or, and that's perhaps even more interesting, as Rachael pointed out at the seminar, sometimes you're interested in a disease area that is not quite covered by the standard QoL measures and therefore you want to derive some induced measure by what is actually observed.

In the paper that was discussed on Friday, the authors had used a Beta-Binomial regression and were claiming that the results were more reasonable than when using standard linear regression $-$ which is probably sensible, given that these measures are far from symmetrical or "normally distributed" (in fact the EQ5D is defined between $-\infty$ and 1).

I don't know much about mapping (so it is likely that what I'm about to say has been thoroughly investigated already $-$ although it didn't come out in the seminar, where people were much more clued up than I am), but this got me thinking that this is potentially a problem that one can solve using (Bayesian) hierarchical models.

The (very raw) way I see it is that effectively there are two compartments to this model: the first one (typically observed) is made by data on some non-standard QoL measure and possibly some relevant covariates; then one can think of a second compartment, which can be build separately to start with, in which the assumptions underlying the standard measure of QoL are spelt out (eg in terms of the impact of some potential covariates, or something).

The whole point, I guess, is to find a way to connecting these two compartments, for example by assuming (in a more or less confident way) that each of them is used to estimate some relevant parameter, representing some form of QoL. These in turns have to be linked in some (theory-based, I should think) way. A Bayesian approach would allow for the exchange of information and "feed-back" between the two components, which would be potentially very helpful, for example if there was a subset of individuals on which observations on both the compartment were available.

I'll try to learn more on this $-$ but I think this could be interesting...

Hand picking

Yesterday we were out with our friends; we met for a drink late in the afternoon and spent quite some time trying to figure out what we wanted to eat.

First, we approached the problem by area, trying to think where we wanted to go and then looking on the internet on our phones (well, at least those of us who have modern phones, ie not Christian or Vale), but that didn't work out.

So we decided to browse by type of food and I think it was Marta who found an Eritrean place close to where we were. Because of the bonfire, it took us nearly twice as long as normal to get there, but in the end we made it. And it was really good!

They bring you a big tray covered with a flat pancake over which they then put all the food; and the fun part is that you are supposed to pick it with your hands, eventually eating also the pancake. At the end of the dinner, they also offer a typical coffee ceremony, which we tried. It takes for ever (in all fairness, they roast and ground the coffee), but to keep you busy the also bring some popcorn, which is unusual as well as actually nice.

(B)AIES

Francesco and Andrea have asked me to join them in doing a short course before the conference of the Italian Health Economics Association (AIES). The course is about Bayesian statistics and health economics and will be in Rome on November 14th. 

I think we have quite a lot to cover and whether we'll actually manage to do everything depends on how people are familiar with Bayesian statistics. But potentially we can talk about quite a few interesting things, including how to do Bayesian statistics in health economics. I think I'll show at least some of the problems from the book, but there's no lab, which is a shame. 

On second thoughts, however, if they had their computer to do the exercises, then we'd definitely not going to make it on time, so probably it's just as well...

Cox & Mayo

One of my favourite Friends episodes is when Joey finally has a breakthrough and gets his first starring role in the show "Mac & Cheese" (in fact the robot was called C.H.E.E.S.E. $-$ "Computerised Humanoid Electronically Enhanced Secret Enforcer"). 

To me, and I know this is veeeeery mature of me, this has the same comic effect of when I first read a paper by David Cox and Deborah Mayo. These two are of course serious people, doing amazing work in their respective fields (of course, statistics for Sir David; and philosophy of science for Deborah Mayo). But, then again, as Sheldon Cooper, PhD would put it: "what's life without whimsy?"

Anyway, I've had a bit of Cox & Mayo this week; first, I've seen this post on Christian Robert's blog, in which he discusses some of Mayo's position on Bayesian statistics. Mayo works in the field of philosophy of science and is a proposer of the frequentist approach. In fact, as Christian also mentions, her position is quite critical of the Bayesian approach and I too have noticed (although I have to say I have not read as much of her work) that she has a sort of "military resistance" attitude to it.

Perhaps in philosophy of science there is a presumption that only the Bayesian approach has philosophically sound foundations; I can see that we Bayesians may sometimes see ourselves as the "righteous ones" (mainly because we are $-$ only kidding, of course), although I think this probably was a real issue quite a while back; certainly not any more, at least from where I stand. 

If this is indeed the case, maybe her position is justifiable and she serves the purpose of keeping a balanced perspective in the field. In my very limited experience and interaction with that environment, I've been lucky enough to talk quite a few times with people like Hasok Chang (who at some point was my joint boss) and Donald Gillies. The impression I had was that there was no perception that from the philosophical point of view, "most of Statistics is under threat of being overcome (or “inundated”) by the Bayesian perspective" (as Christian put it). I really liked Christian's post and essentially agreed on all accounts.

The second part of this story is about this morning's lecture by David Cox (which was organised by Bianca and Rhian). He talked about the principles of statistics, in what he said was a condensed version of a 5-hours lecture. Of course, it was interesting. He's a very good speaker and it's amazing to see how energetic he still is (he's 88 $-$ I was about to complain that my back is hurting today, but that put me in perspective!).

There were a few points I quite liked and a few more I quite didn't. First, I liked that he slipped in an example in which he implicitly said he's an Aston Villa fan; now I feel justified about putting a bit about Sampdoria in chapter 2 of the book $-$ I can always said I did it like Cox, which to a statistician is quite something...

Also, I liked the distinction he made between what he called "phenomenological" and "substantive" models. The first term indicates all those models that are general enough to be widely applicable (including, as he put it "those strange models that people use to analyse survival data"), but not directly related to the science underlying the problem at hand. Something like: you can "always" use regression analysis, and effectively the same formulation can apply to medicine, agriculture or social science. The maths behind the model is the same. The second term indicates models that are specifically built to describe a specific bit of science; they are thus very much specific, although of course you may not be able to apply them in many cases. A bit like decision models (as opposed to individual data models) in health economics.

What I didn't quite like (but that was to be expected) was his take on Bayesian statistics, and specifically on the subjective/personalistic approach. I think I've heard him talk about it on another occasion, and that time it was even worse (from my Bayesian perspective), but the point is that he basically said that there's no room for such an approach in science, with which I respectfully (but totally) disagree. In fact, as Tom pointed out while we were walking back, at some point he was even a bit contradictory when he framed the significance of RCTs data in terms of the problem a doctor faces when deciding what's the best course of action for the next patient $-$ pretty much a personalistic decision problem!

More football

Given that Sampdoria have lost their fourth game in a row, I am not really interested in football any more (as of yesterday, I actually find it a very boring game and think we should focus on real sports $-$ of course if we manage to break the crappy run, I'll be back to liking it...). But, because Kees has done some more work using R2jags (here and here $-$ I've actually only seen this morning, although I think some of it at least is from a couple weeks back), I thought I would spend a few minutes on football anyway.

In the first, he's modified his original model and (I guess) linked to our formulation; in his new Bayesian model, he specifies the number of goals scored by the two competing teams as Poisson variables; the log-linear predictor is then specified depending on the "home-effect", the attack strength for that team and the defence strength of the opponents.

In our formulation, the attack and defence effects were modelled as exchangeable, which induced correlation among the different teams and therefore between the two observed variables. Kees uses a slightly different model where they both depend on an "overall" strength for each team (modelled as exchangeable) plus another exchangeable, team-specific "attack/defence balance" effect.

This is interesting, but the main problem seems to remain the fact that you tend to see clear mixtures of effects (ie "bad" teams having quite different behaviour from "good" teams); he tried to then use the mix module in JAGS to directly model the effects as mixtures of exchangeable Normal distributions, but seems to have run in some problems. 

We did it slightly differently (ie using informative priors on the chance each team belongs to one of three categories: "top" teams; "strugglers"; and "average" teams). We only run it in BUGS, but because of our specification, we didn't have problems with convergence or the interpretation (only slightly slower). I've not tried the JAGS version with the mix module (I suppose we should fiddle a bit with the original code $-$ which is here, by the way, is someone has time/interest in doing so). But it shouldn't be a problem, I would think. 

In Kees model, he needs to fix some of the effects to avoid convergence and identifiability issues, if I understand it correctly. Marta and I had always wanted to explore the mixture issue further, but never got around to have time to actually do it. If I only I could like football again...

Ben pharma

Quite regularly, Marta comes up with some idea to reshuffle something around the house (eg move furniture around, or change the way things are). Normally, mostly because I am a bit lazy, but sometimes because I genuinely don't really see the point, my reaction is to find a logical and rational explanation to why we shouldn't do it. Most of the times, I miserably fail to do so and then we do whatever she plans (in fact, even if I manage to find a reason not to, usually the best I can get is to delay the implementation of the Marta-policy).

At the weekend I ran out of excuses not to move the furniture in the living room (with subsequent ebay bonanza of most of the stuff we have to make way for new exciting things) and so I capitulated. Previously, we had the TV in the top-right corner of the room, with a sofa facing it and an armchair (which I normally sit on) to its left. The new version of the room has the TV in the top-left corner with the armchair now facing it and the sofa to its right.

Obviously, neither I nor the cat liked the changes at first, but the worst thing is that now the two of us are fighting over who gets to sit on the armchair (before, he would sit on the sofa $-$ evidently, like Sheldon Cooper PhD, he just loves that spot). So now, while I sit on (allegedly) my armchair watching telly, or reading my book, I also have to deal with him, patiently looking for his way in to jump on the chair and start pushing me away.

Anyway: the book I'm reading at the moment is Bad Pharma by Ben Goldacre (I've already briefly mentioned it here). It's interesting and I agree with quite a lot that he's arguing. For example, he rants about the fact that trials are often "hidden" by pharmaceutical companies who don't want "unflattering evidence" about their products out for researchers, patients and clinicians to see. This obviously makes like complicated when one tries to put all the evidence in perspective (given that one doesn't have all the evidence to work with...).

But, I have to say (and I'm only in about 100 pages, so just about 1/4 of the book), I don't quite like some of the things he says, or rather the way in which he says them. It seems to me that the underlying assumption is that there is some absolute truth out there (in this case about whether drugs work or not, or have side effects, etc). The problem is that people either hide some evidence, or use biased methods to analyse their data.

For instance, there is an example about a study on duloxetine, in which the industry-led research used last observation carried forward in their analysis. Now, of course we know that this is not ideal and opens up the possibility of crucial bias in the results. But, as far as I'm aware, dealing with missing data is much an art as is a science and there are all sorts of untestable assumptions that have to be included in the model, even when you use a better, more robust, more accepted one. I think this consideration is missing (or at least not clear enough) in Ben's discussion. 

I'm not quite of the persuasion that life is 0/1; wrong/correct; all/nothing. To me, the objective of statistical analysis is not to provide facts, but rather to quantify uncertainty. As Lindley puts it, uncertainty is a fact of life, and we just have to deal with it. And statistical modelling is hardly something that goes without uncertainty. Yes we all use Cox's model for survival data, or logistic regression for binary outcomes; but who's to say (in general) that these are the only good ways of modelling these kind of data?

I think that Andrew Gelman makes this point quite often, when he defends Bayesian models vs frequentist ones. Often people tend to criticise the choice of priors, while the data model is considered as an uncontroversial piece of the model, but I agree with Gelman that the prior is only one side of the story. Christian Robert has an interesting post on his blog, in which he quotes a professor of Astrophysics discussing about the role of stats in science. 

Anyway, as I said, other than this aspect, I agree with Ben's general message and I'll keep reading the book.

By the way: the living room does look better in its new version.

Bayes for President!

I couldn't resist getting sucked into the hype associated with the US election and debates, and so I thought I had a little fun of my own and played around a bit with the numbers. [OK: you may disagree with the definition of "fun" $-$ but then again, if you're reading this you probably don't...]

So, I looked on the internet to find reasonable data on the polls. Of course there are a lot of limitations to this strategy. First, I've not bothered doing some sort of proper evidence synthesis, taking into account the different polls and pooling them in a suitable way. There are two reasons why I didn't: the first one is that not all the data are publicly available (as far as I can tell), so you have to make do with what you can find; second, I did find some info here, which seems to have accounted for this issue anyway. In particular, this website contains some sort of pooled estimates for the proportion of people who are likely to vote for either candidate, by state, together with a "confidence" measure (more on this later). Because not all the states have data, I have also looked here and found some additional info. 

Leaving aside representativeness issues (which I'm assuming are not a problem, but may well be, if this were a real analysis!), the second limitation is of course that voting intentions may not directly translate into actual votes. I suppose there are some studies out there to quantify this, but again, I'm making life (too) easy and discount this effect.

The data on the polls that I have collected in a single spreadsheet look like this
ID State Dem Rep   State_Name Voters Confidence
1    AK  38  62       Alaska      3    99.9999
2    AL  36  54      Alabama      9    99.9999
3    AR  35  56     Arkansas      6    99.9999
4    AZ  44  52      Arizona     11    99.9999
5    CA  53  38   California     55    99.9999
...      ...       ...      ...       ...

The columns Dem and Rep represent the pooled estimation of the proportion of voters for the two main parties (of course, they may not sum to 100%, due to other possible minor candidates or undecided voters). The column labelled Voters gives the number of Electoral Votes (EVs) in each state (eg if you win at least 50% of the votes in Alaska, this is associated with 3 votes overall, etc). Finally, the column Confidence indicates the level of (lack of) uncertainty associated with the estimation. States with high confidence are "nearly certain" $-$ for example, the 62% estimated for Republicans in Alaska is associated with a very, very low uncertainty (according to the polls and expert opinions). In most states, the polls are (assumed to be) quite informative, but there are some where the situation is not so clear cut.

I've saved the data in a file, which can be imported in R using the command

polls <- read.csv("http://www.statistica.it/gianluca/Polls.csv")

At this point, I need to compute the 2-party share for each state (which I'm calling $m$) and fix the number of states at 51

attach(polls)

m <- Dem/(Dem+Rep)

Nstates <- 51

Now, in truth this is not a "proper" Bayesian model, since I'm only assuming informative priors (which are supposed to reflect all the available knowledge on the proportion of voters, without any additional observed data). Thus, all I'm doing is a relatively easy analysis. The idea is to first define a suitable informative prior distribution based on the point estimation of the democratic share and with uncertainty defined in terms of the confidence level. Then I can use Monte Carlo simulations to produce a large number of "possible futures"; in each future and for each state, the Democrats will have an estimated share of the popular vote. If that is greater than 50%, Obama will have won that state and the associated EVs. I can then use the induced predictive distribution on the number of EVs to assess the uncertainty underlying an Obama win (given that at least 272 votes are necessary to become president).

In their book, Christensen et al show a simple way of deriving a Beta distribution based on an estimation of the mode, an upper limit and a confidence level that the variable is below that upper threshold. I've coded this in a function betaPar2, which I've made available from here (so you need to download it, if you want to replicate this exercise).

Using this bit of code, one can estimate the parameters of a Beta distribution centered on the point estimate and for which the probability of exceeding the threshold 0.5 is given by the level of confidence.

a <- b <- numeric()

for (s in 1:Nstates) {

if (m[s] < .5) {

bp <- betaPar2(m[s],.499999,Confidence[s]/100)

a[s] <- bp$res1

b[i] <- bp$res2

}

if (m[s] >=.5) {

bp <- betaPar2(1-m[s],.499999,Confidence[s]/100)

a[s] <- bp$res2

b[s] <- bp$res1

}

}

The function betaPar2 has several outputs, but the main ones are res1 and res2, which store the values of the parameters $\alpha$ and $\beta$, which define the suitable Beta distribution. In fact, the way I'm modelling is to say that if the point estimate is below 0.5 (a state $s$ where Romney is more likely to win), then I want to derive a suitable pair $(\alpha_s,\beta_s)$ so that the resulting Beta distribution is centered on $m_s$ and for which the probability of not exceeding 0.5 is given by $c_s$ (which is defined as the level of confidence for state $s$, reproportioned in [0;1]). However, for states in which Obama is more likely to win ($m_s\geq 0.5$), I basically do it the other way around (ie working with 1$-m_s$). In these cases, the correct Beta distribution has the two parameters swapped (notice that I assign the element res2 to $\alpha_s$ and the element res1 to $\beta_s$).

For example, for Alaska (the first state), the result is an informative prior like this.

In line with the information from the polls, the estimated average proportion of Democratic votes is around 38% and effectively there's no chance of Obama getting a share that is greater than 50%.

Now, I can simulate the $n_{\rm{sims}}=10000$ "futures", based on the uncertainty underlying the estimations using the code

nsims <- 10000

prop.dem <- matrix(NA,nsims,Nstates)

for (i in 1:Nstates) {

prop.dem[,i] <- rbeta(nsims,a[i],b[i])

}

The matrix prop.dem has 10000 rows (possible futures) and 51 columns (one for each state).

I can use the package coefplot2 and produce a nice summary graph

library(coefplot2)

means <- apply(prop.dem,2,mean)

sds <- apply(prop.dem,2,sd)

low <- means-2*sds

upp <- means+2*sds

reps <- which(upp<.5)         # definitely republican states

dems <- which(low>.5)         # definitely democratic states

m.reps <- which(means<.5 & upp>.5) # most likely republican states

m.dems <- which(means>.5 & low<.5) # most likely democratic states

cols <- character()

cols[reps] <- "red"

cols[dems] <- "blue"

cols[m.reps] <- "lightcoral"

cols[m.dems] <- "deepskyblue"

vn <- paste(as.character(State)," (",Voters,")",sep="")

coefplot2(means,sds,varnames=vn,col.pts=cols,main="Predicted probability of democratic votes")

abline(v=.5,lty=2,lwd=2)

This gives me the following graph showing the point estimate (the dots), 95% and 50% credible intervals (the light and dark lines, respectively). Those in dark blue and bright red are the "definites" (ie those that are estimated to be definitely Obama or Romney states, respectively). Light blues and reds are those undecided (ie for which the credible intervals cross 0.5).

Finally, for each simulation, I can check that the estimated proportion of votes for the Democrats exceeds 0.5 and if so allocate the EVs to Obama, to produce a distribution of possible futures for this variable.

obama <- numeric()

for (i in 1:nsims) {

obama[i] <- (prop.dem[i,]>=.5)%*%Voters

}

hist(obama,30,main="Predicted number of Electoral Votes for Obama",xlab="",ylab="")

abline(v=270,col="dark grey",lty=1,lwd=3)

So, based on this (veeeeery simple and probably not-too-realistic!!) model, Obama has a pretty good chance of being re-elected. In almost all the simulations his share of the votes guarantees he gets at least the required 272 EVs $-$ in fact, in many possible scenarios he actually gets many more than that.

Well, I can only hope I'm not jinxing it!

Thank you for smoking (and also doing take-away)

Sometimes you watch a movie from the 1980s or something and they have this weird "hidden adverts", mostly about cigarettes $-$ things like when the camera indulges on the star of the movie fiddling with the packet and lighting up the cigarette. I suppose back then it was just another form of advert and in fact it wasn't much worse than the "normal" advert they were doing on TV, newspapers or Formula 1 cars. Like in the brilliant Thank you for smoking. 

Then, finally, somebody figured out that smoking actually kills (for example, David Spiegelhalter said in his documentary that "two cigarettes cost half an hour". By the way, I thought that the documentary was good, but not great $-$ may be my expectations were too high...) and governments decided it was time to ban tobacco advertisements.

I guess I was too young when they actually did that across Europe, so I don' exactly remember what the public and industry reaction was $-$ I seem to remember I didn't really care; but then again, I've never being fascinated with smoking; plus, back then I was still convinced I'd end up playing for Sampdoria and overtaking Roberto Mancini as their best player ever... 

Anyway, it all seems sensible: once we know (beyond any reasonable doubt and based on a series of scientific studies supporting each other) that something is a public health concern, then to limit its impact on the general population is the first thing a government should do, right?

Well, I'm not too sure: last week I saw this advert on TV. Apparently, there's this company that has a website which links together many take-away places, so all you need to do is give your postcode, select a type of food and choose your dinner from a wide range of menus. 

So far, (almost) so good. I suppose that there are things that are not OK with take-away dinners: for example, probably they increase the level of pollution (eg plastic tupperwares, plastic/paper bags, car fuel, etc). Still, on the other hand, they can't be all bad and if someone did a full economic analysis then they should factor in possible benefits (eg reduced level of pollution due to the fact that you don't do the washing up afterwards?). And to be perfectly honest, I quite fancy the occasional take-away too.

But the advert I linked above really stretches it too far! I think they are meant to be funny and ironic in telling people that "The tyranny of home cooking has gone on for too long" and that "Cooking is burnt fingers; cooking is complicated recipes you need a PhD in Advanced Foodolgy to understand; cooking is celeb chef endorsed gadgets you buy and never use; cooking is hours in the kitchen to end up with mush that looks nothing like the recipe photo (and probably tastes like the paper the recipe was printed on)".

If anything, I think that the advert is in bad taste [see where I'm going with this one? Food, bad taste... Hilarious, isn't it? I won't be a professional footballer, but may be there's still hope for a stand up career?]. 
I think that there is a real danger that the message comes up as serious, rather than a funny (?) joke. And I don't think that's something we want $-$ if there are concerns about obesity as a serious public health concern, probably putting in people's mind that cooking is bad and you should only eat take-away dinners is probably not the smartest idea...

Now, I think the company should be allowed to market its product $-$ after all, everybody should be free to decide what to eat. But equally, doesn't the government have a duty to control over the way in which people are bombarded with adverts on how to live their lives?