Archive for the ‘R’ Category

Robust measurement from a 2-way table


I work in a university.  My department runs degree courses that allow students a lot of flexibility in their choice of course “modules”.  (A typical student takes 8–10 modules per year, and is assessed separately on each module).

After the exams are finished each year, we promise our students to look carefully at the exam marks for each module — to ensure that students taking a “hard” module are not penalized for doing that, and that students taking an “easy” module are not unduly advantaged.

The challenge in this is to separate module difficulty from student ability: we need to be able to tell the difference between (for example) a hard module and a module that was chosen by weaker-than-average students.  This necessitates analysis of the exam marks for all modules together, rather than separately.

The data to be analysed are each student’s score (expressed as a percentage) in each module they took.  It is convenient to arrange those scores in a 2-way table, whose rows are indexed by student IDs, and whose columns correspond to all the different possible modules that were taken.  The task is then to analyse the (typically incomplete) 2-way table, to determine a numerical “module effect” for each module (a relatively high number for each module that was found relatively “easy”, and lower numbers for modules that were relatively “hard”.

A standard method for doing this robustly (i.e., in such a way that the analysis is not influenced too strongly by the performance of a small number of students) is the clever median polish method due to J W Tukey.  My university department has been using median polish now for several years, to identify any strong “module effects” that ought to be taken into account when assessing each student’s overall performance in their degree course.

Median polish works mostly OK, it seems: it gives answers that broadly make sense.  But there are some well known problems, including that it matters which way round the table is presented (i.e., “rows are students”, versus “rows are modules”) — the answer will depend on that.  So median polish is actually not just one method, but two.

When my university department asked me recently to implement its annual median-polish exercise in R, I could not resist thinking a bit about whether there might be something even better than median polish, for this specific purpose of identifying the column effects (module effects) robustly.  This led me to look at some simple “toy” examples, to help understand the principles.  I’ll just show one such example here, to illustrate how it’s possible to do better than median polish in this particular context.

Example: 5 modules, 3 students

My made-up “toy” data:

> x
student  A  B  C  D  E
      i NA NA NA 45 60
      j NA NA NA 55 60
      k 10 20 30 NA 50

There were five modules (labelled A,B,C,D,E).  Students i, j and k each took a selection of those modules.  It’s a small dataset, but that is deliberate: we can see easily what’s going on in a table this small.  Module E was easier than the others, for example; and student k looks to be the weakest student (since k was outperformed by the other two students in module E, the only one that they all took).

I will call the above table perfect, as far as the measurement of module effects is concerned.  If we assign module effects (−20, −10, 0, 10, 20) to the five modules A,B,C,D,E respectively, then for every pair of modules the observed within-student differences are centered upon the relevant difference in those module effects.  For example, look at modules D and E: student i scores 15 points more in E, while j scores 5 points more in E, and the median of those two differences is 10 — the same as the difference between the proposed “perfect” module effects for D and E.

When we perform median polish on this table, we get different answers depending on whether we apply the method to the table directly, or to its transpose:

> medpolish(x, na.rm = TRUE, maxiter = 20)
Median Polish Results (Dataset: "x")

Overall: 38.75

Row Effects:
    i     j     k 
 0.00  5.00 -8.75 

Column Effects:
     A      B      C      D      E 
-20.00 -10.00   0.00   8.75  20.00 

student  A  B  C    D     E
      i NA NA NA -2.5  1.25
      j NA NA NA  2.5 -3.75
      k  0  0  0   NA  0.00

> medpolish(t(x), na.rm = TRUE, maxiter = 20)
Median Polish Results (Dataset: "t(x)")

Overall: 36.25

Row Effects:
     A      B      C      D      E 
-20.00 -10.00   0.00  11.25  20.00 

Column Effects:
     i      j      k 
 0.625  5.625 -6.250 

module      i      j  k
     A     NA     NA  0
     B     NA     NA  0
     C     NA     NA  0
     D -3.125  1.875 NA
     E  3.125 -1.875  0

Neither of those answers is the same as the “perfect” module-effect measurement that was mentioned above.  The module effect for D as computed by median polish is either 8.75 or 11.25, depending on the orientation of the input table — but not the “perfect 10”.

A better method: Median difference analysis

I decided to implement, in place of median polish, a simple non-iterative method that targets directly the notion of “perfect” measurement that is mentioned above.

The method is in two stages.

Stage 1 computes within-student differences and takes the median of those, for each possible module pair.  For our toy example:

> md <- meddiff(x)
   A   B   C  D   E
A NA -10 -20 NA -40
B  1  NA -10 NA -30
C  1   1  NA NA -20
D  0   0   0 NA -10
E  1   1   1  2  NA

The result here has all of the available median-difference values above the diagonal.  Below the diagonal is the count of how many differences were used in computing each one of those medians.  So, for example, the median difference between modules  D and E is −10; and that was computed from 2 students’ exam scores.

Stage 2 then fits a linear model to the median-difference values, using weighted least squares.  The linear model finds the vector of module effects that most closely approximates the available median differences (i.e., best approximates the numbers above the diagonal).  The weights are simply the counts from the lower triangle of the above matrix.

In this “perfect” example, we achieve the desired perfect answer (which here is presented with E as the “reference” module):

> fit(md)$coefficients
  A   B   C   D   E 
-40 -30 -20 -10   0

My plan now is to make these simple R functions robust enough to use for our students’ actual exam marks, and to add also inference on the module-effect values (via a suitably designed bootstrap calculation).

For now, here are my prototype functions in case anyone else wants to play with them:

meddiff <- function(xmat) {
    ## rows are students, columns are modules
    S <- nrow(xmat)
    M <- ncol(xmat)
    result <- matrix(NA, M, M)
    rownames(result) <- colnames(result) <- colnames(xmat)
    for (m in 1:(M-1)) {
        for (mm in (m+1):M) {
            diffs <- xmat[, m] - xmat[, mm]
            ## upper triangle
            result[m, mm] <- median(diffs, na.rm = TRUE)
            ## lower triangle
            result[mm, m] <- sum(!

fit <- function(m) {
    ## matrix m needs to be fully connected above the diagonal
    upper <- upper.tri(m)
    diffs <- m[upper]
    weights <- t(m)[upper]
    rows <- factor(row(m)[upper])
    cols <- factor(col(m)[upper])
    X <- cbind(model.matrix(~ rows - 1), 0) - 
           cbind(0, model.matrix(~ cols - 1))
    colnames(X) <- colnames(m)
    rownames(X) <- paste0(colnames(m)[rows], "-", colnames(m)[cols])
    result <- lm.wfit(X, diffs, weights)
    result$coefficients[$coefficients)] <- 0
    class(result) <- c("meddiff_fit", "list")

© David Firth, April 2019

To cite this entry:
Firth, D (2019). Robust measurement from a 2-way table. Weblog entry at URL


Part 2, further comments on OfS grade-inflation report


Update, 2019-01-07: I am pleased to say that the online media article that I complained about in Sec 1 below has now been amended by its author(s), to correct the false attributions.  I am grateful to Chris Parr for helping to sort this out.

In my post a few days ago (which I’ll now call “Part 1”) I looked at aspects of the statistical methods used in a report by the UK government’s Office for Students, about “grade inflation” in English universities.  This second post continues on the same topic.

In this Part 2 I will do two things:

  1. Set the record straight, in relation to some incorrect reporting of Part 1 in the specialist media.
  2. Suggest a new statistical method that (in my opinion) is better than the one used in the OfS report.

The more substantial stuff will be the second bullet there (and of course I wish I didn’t need to do the first bullet at all).   In this post (at section 2 below) I will just outline a better method, by using the same artificial example that I gave in Part 1: hopefully that will be enough to give the general idea, to both specialist and non-specialist readers.  Later I will follow up (in my intended Part 3) with a more detailed description of the suggested better method; that Part 3 post will be suitable mainly for readers with more specialist background in Statistics.

1.  For the record

I am aware of two places where the analysis I gave in Part 1 has been reported:

The first link there is to a paywalled site, I think.  The second one appears to be in the public domain.  I do not recommend following either of those links, though!  If anyone reading this wants to know about what I wrote in Part 1, then my advice is just to read Part 1 directly.

Here I want to mention three specific ways in which that article misrepresents what I wrote in Part 1.  Points 2 and 3 here are the more important ones, I think (but #1 is also slightly troubling, to me):

  1. The article refers to my blog post as “a review commissioned by HE”.  The reality is that a journalist called Chris Parr had emailed me just before Christmas.  In the email Chris introduced himself as “I’m a journalist at Research Fortnight”, and the request he made in the email (in relation to the newly published OfS report) was “Would you or someone you know be interested in taking a look?”.  I had heard of Research Fortnight.  And I was indeed interested in taking a look at the methods used in the OfS report.  But until the above-mentioned article came to my attention, I had never even heard of a publication named HE.  Possibly I am mistaken in this, but to my mind the phrase “a review commissioned by HE” indicates some kind of formal arrangement between HE and me, with specified deliverables and perhaps even payment for the work.  There was in fact no such “commission” for the work that I did.  I merely spent some time during the Christmas break thinking about the methods used in the OfS report, and then I wrote a blog post (and told Chris Parr that I had done that).  And let me repeat: I had never even heard of HE (nor of the article’s apparent author, which was not Chris Parr).  No payment was offered or demanded.  I mention all this here only in case anyone who has read that article  got a wrong impression from it.
  2. The article contains this false statement: “The data is too complex for a reliable statistical method to be used, he said”.  The “he” there refers to me, David Firth.  I said no such thing, neither in my blog post nor in any email correspondence with Chris Parr.  Indeed, it is not something I ever would say: the phrase “data…too complex for a reliable statistical method” is a nonsense.
  3. The article contains this false statement: “He calls the OfS analysis an example of Simpson’s paradox”.  Again, the “he” in that statement refers to me.  But I did not call the OfS analysis an example of Simpson’s paradox, either in my blog post or anywhere else.  (And nor could I have, since I do not have access to the OfS dataset.)  What I actually wrote in my blog post was that my own artificial, specially-constructed example was an instance of Simpson’s paradox — which is not even close to the same thing!

The article mentioned above seems to have had an agenda that was very different from giving a faithful and informative account of my comments on the OfS report.  I suppose that’s journalistic license (although I would naively have expected better from a specialist publication to which my own university appears to subscribe).  The false attribution of misleading statements is not something I can accept, though, and that is why I have written specifically about that here.

To be completely clear:

  • The article mentioned above is misleading.  I do not recommend it to anyone.
  • All of my posts in this blog are my own work, not commissioned by anyone.  In particular, none of what I’ll continue to write below (and also in Part 3 of this extended blog post, when I get to that), about the OfS report, was requested by any journalist.

2.  Towards a better (statistical) measurement model

I have to admit that in Part 1 I ran out of steam at one point, specifically where — in response to my own question about what would be a better way than the method used in the OfS report — I wrote “I do not have an answer“.  I could have and should have done better than that.

Below I will outline a fairly simple approach that overcomes the specific pitfall I identified in Part 1, i.e., the fact that measurement at too high a level of aggregation can give misleading answers.  I will demonstrate my suggested new approach through the same, contrived example that I used in Part 1.  This should be enough to convey the basic idea, I hope.  [Full generality for the analysis of real data will demand a more detailed and more technical treatment of a hierarchical statistical model; I’ll do that later, when I come to write Part 3.]

On reflection, I think a lot of the criticism seen by the OfS report since its publication relates to the use of the word “explain” in that report.  And indeed, that was a factor also in my own (mentioned above) “I do not have an answer” comment.  It seems obvious — to me, anyway — that any serious attempt to explain apparent increases in the awarding of First Class degrees would need to take account of a lot more than just the attributes of students when they enter university.  With the data used in the OfS report I think the best that one can hope to do is to measure those apparent increases (or decreases), in such a way that the measurement is a “fair” one that appropriately takes account of incoming student attributes and their fluctuation over time.  If we take that attitude — i.e, that the aim is only to measure things well, not to explain them — then I do think it is possible to devise a better statistical analysis, for that purpose, than the one that was used in the OfS report.

(I fully recognise that this actually was the attitude taken in the OfS work!  It is just unfortunate that the OfS report’s use of the word “explain”, which I think was intended there mainly as a technical word with its meaning defined by a statistical regression model, inevitably leads readers of the report to think more broadly about substantive explanations for any apparent changes in degree-class distributions.)

2.1  Those “toy” data again, and a better statistical model

Recall the setup of the simple example from Part 1:  Two academic years, two types of university, two types of student.  The data are as follows:

  University A           University B
    Firsts  Other          Firsts  Other
  h   1000      0        h    500    500 
  i      0   1000        i    500    500
  University A          University B
    Firsts  Other          Firsts  Other 
  h   1800    200       h       0      0
  i      0      0       i     500   1500

Our measurement (of change) should reflect the fact that, for each type of student within each university, where information is available, the percentage awarded Firsts actually decreased (in this example).

Change in percent awarded firsts:
  University A, student type h:  100% --> 90%
  University A, student type i:   no data
  University B, student type h:   no data
  University B, student type i:   50% --> 25%

This provides the key to specification of a suitable (statistical) measurement model:

  • measure the changes at the lowest level of aggregation possible;
  • then, if aggregate conclusions are wanted, combine the separate measurements in some sensible way.

In our simple example, “lowest level of aggregation possible” means that we should measure the change separately for each type of student within each university.  (In the real OfS data, there’s a lower level of aggregation that will be more appropriate, since different degree courses within a university ought to be distinguished too — they have different student intakes, different teaching, different exam boards, etc.)

In Statistics this kind of analysis is often called a stratified analysis.  The quantity of interest (which here is the change in % awarded Firsts) is measured separately in several pre-specified strata, and those measurements are then combined if needed (either through a formal statistical model, or less formally by simple or weighted averaging).

In our simple example above, there are 4 strata (corresponding to 2 types of student within each of 2 universities).  In our specific dataset there is information about the change in just 2 of those strata, and we can summarize that information as follows:

  • in University A, student type i saw their percentage of Firsts reduced by 10%;
  • in University B, student type h saw their percentage of Firsts reduced by 50%.

That’s all the information in the data, about changes in the rate at which Firsts are awarded.  (It was a deliberately small dataset!)

If a combined, “sector-wide” measure of change is wanted, then the separate, stratum-specific measures need to be combined somehow.  To some extent this is arbitrary, and the choice of a combination method ought to depend on the purpose of such a sector-wide measure and (especially) on the interpretation desired for it.  I might find time to write more about this later in Part 3.

For now, let me just recall what was the “sector-wide” measurement that resulted from analysis (shown in Part 1) of the above dataset using the OfS report’s method.  The result obtained by that method was a sector-wide increase of 7.5% in the rate at which Firsts are awarded — which is plainly misleading in the face of data that shows substantial decreases in both universities.  Whilst I do not much like the OfS Report’s “compare with 2010” approach, it does have the benefit of transparency and in my “toy” example it is easy to apply to the stratified analysis:

2016-17          Expected Firsts       Actual
                 based on 2010-11
  University A         2000             1800
  University B         1000              500
  Total                3000             2300

— from which we could report a sector-wide decrease of 700/3000 = 23.3% in the awarding of Firsts, once student attributes are taken properly into account.  (This could be viewed as just a suitably weighted average of the 10% and 50% decreases seen in University A and University B respectively.)

As before, I have made the full R code available (as an update to my earlier R Markdown document).  For those who don’t use R, I attach here also a PDF copy of that: grade-inflation-example.pdf

2.2  Generalising the better model: More strata, more time-points

The essential idea of a better measurement model is presented above in the context of a small “toy” example, but the real data are of course much bigger and more complex.

The key to generalising the model will simply be to recognise that it can be expressed in the form of a logistic regression model (that’s the same kind of model that was used in the OfS report; but the “better” logistic regression model structure is different, in that it needs to include a term that defines the strata within which measurement takes place).

This will be developed further in Part 3, which will be more technical in flavour than Parts 1 and 2 of this blog-post thread have been.  Just by way of a taster, let me show here the mathematical form of the logistic-regression representation of the “toy” data analysis shown above.  With notation

  • u for providers (universities); u is either A or B in the toy example
  • t for type of student; t is either h or i in the toy example
  • y for years; y is either 2010-11 or 2016-17 in the toy example
  • \pi_{uty} for the probability of a First in year y, for students of type t in university u

the logistic regression model corresponding to the analysis above is

\log\left(\pi_{uty}\over 1-\pi_{uty}\right) = \alpha_{ut} + \beta_{uy}.

This is readily generalized to situations involving more strata (more universities u and student types t, and also degree-courses within universities).  There were just 4 stratum parameters \alpha_{Ah},\alpha_{Ai}, \alpha_{Bh}, \alpha_{Bi} in the above example, but more strata are easily accommodated.

The model is readily generalized also, in a similar way, to more than 2 years of data.

For comparison, the corresponding logistic regression model as used in the OfS report looks like this:

\log\left(\pi_{uty}\over 1-\pi_{uty}\right) = \alpha_{t} + \beta_{uy}.

So it is superficially very similar.  But the all-important term \alpha_{ut} that determines the necessary strata within universities is missing from the OfS model.

I will aim to flesh this out a bit in a new Part 3 post within the next few days, if time permits.  For now I suppose the model I’m suggesting here needs a name (i.e., a name that identifies it more clearly than just “my better model”!)  Naming things is not my strong point, unfortunately!  But, for now at least, I will term the analysis introduced above “stratified by available student attributes” — or “SASA model” for short.

(The key word there is “stratified”.)

© David Firth, January 2019

To cite this entry:
Firth, D (2019). Part 2, further comments on OfS grade-inflation report. Weblog entry at URL

Office for Students report on “grade inflation”


Update, 2019-01-07: There’s now also Part 2 of this blog post, for those who are keen to know more!

Chris Parr, a journalist for Research Professional, asked me to look at a recent report, Analysis of degree classifications over time: Changes in graduate attainment.  The report was published by the UK government’s Office for Students (OfS) on 19 December 2018, along with a headline-grabbing press release:


The report uses a statistical method — the widely used method of logistic regression — to devise a yardstick by which each English university (and indeed the English university sector as a whole) is to be measured, in terms of their tendency to award the top degree classes (First Class and Upper Second Class honours degrees).  The OfS report looks specifically at the extent to which apparent “grade inflation” in recent years can be explained by changes in student-attribute data available to OfS (which include grades in pre-university qualifications, and also some other characteristics such as gender and ethnicity).

I write here as an experienced academic, who has worked at the University of Warwick (in England) for the last 15 years.  At the end, below, I will briefly express some opinions based upon that general experience (and it should be noted that everything I write here is my own — definitely not an official view from the University of Warwick!)

My specific expertise, though, is in statistical methods, and this post will focus mainly on that aspect of the OfS report.  (For a more wide-ranging critique, see for example

Parts of what I say below will get a bit technical, but I will aim to write first in a non-technical way about the big issue here, which is just how difficult it is to devise a meaningful measurement of “grade inflation” from available data.  My impression is that, unfortunately, the OfS report has either not recognised the difficulty or has chosen to neglect it.  In my view the methods used in the report are not actually fit for their intended purpose.

1.  Analysis of an idealized dataset

In much the same way as when I give a lecture, I will aim here to expose the key issue through a relatively simple, concocted example.  The real data from all universities over several years are of course quite complex; but the essence can be captured in a much smaller set of idealized data, the advantage of which is that it allows a crucial difficulty to be seen quite directly.

An imagined setup: Two academic years, two types of university, two types of student

Suppose (purely for simplicity) that there are just two identifiable types of university (or, if you prefer, just two distinct universities) — let’s call them A and B.

Suppose also (purely for simplicity) that all of the measurable characteristics of students can be encapsulated in a single binary indicator: every student is known to be either of type h or of type i, say.  (Maybe h for hardworking and i for idle?)

Now let’s imagine the data from two academic years — say the years 2010-11 and 2016-17 as in the OfS report — on the numbers of First Class and Other graduates.

The 2010-11 data looks like this, say:

  University A           University B
    Firsts  Other          Firsts  Other
  h   1000      0        h    500    500 
  i      0   1000        i    500    500

The two universities have identical intakes in 2010-11 (equal numbers of type h and type i students).  Students of type h do a lot better at University A than do students of type i; whereas University B awards a First equally often to the two types of student.

Now let’s suppose that, in the years that follow 2010-11,

  • students (who all know which type they are) learn to target the “right” university for themselves
  • both universities A and B tighten their final degree criteria, so as to make it harder (for both student types h and i) to achieve a First.

As a result of those behavioural changes, the 2016-17 data might look like this:

  University A          University B
    Firsts  Other          Firsts Other 
  h   1800    200       h       0     0
  i      0      0       i     500  1500

Now we can combine the data from the two universities, so as to look at how degree classes across the whole university sector have changed over time:

  Combined data from both universities:
    2010-11                  2016-17
          Firsts  Other            Firsts  Other
        h   1500    500          h   1800    200
        i    500   1500          i    500   1500
          -------------            -------------
    Total   2000   2000      Total   2300   1700
        %     50     50              57.5   42.5

The conclusion (not!)

The last table shown above would be interpreted, according to the methodology of the OfS report, as showing an unexplained increase of 7.5 percentage points in the awarding of first-class degrees.

(It is 7.5 percentage points because that’s the difference between 50% Firsts in 2010-11 and 57.5% Firsts in 2016-17.  And it is unexplained — in the OfS report’s terminology — because the composition of the student body was unchanged, with 50% of each type h and i in both years.)

But such a conclusion would be completely misleading.  In this constructed example, both universities actually made it harder for every type of student to get a First in 2016-17 than in 2010-11.

The real conclusion

The constructed example used above should be enough to demonstrate that the method developed in the OfS report does not necessarily measure what it intends to.

The constructed example was deliberately made both simple and quite extreme, in order to make the point as clearly as possible.  The real data are of course more complex, and patterns such as shifts in the behaviour of students and/or institutions will usually be less severe (and will always be less obvious) than they were in my constructed example.  The point of the constructed example is merely to demonstrate that any conclusions drawn from this kind of combined analysis of all universities will be unreliable, and such conclusions will often be incorrect (sometimes severely so).

That false conclusion is just an instance of Simpson’s Paradox, right?


The phenomenon of analysing aggregate data to obtain (usually incorrect) conclusions about disaggregated behaviour is often (in Statistics) called ecological inference or the ecological fallacy.  In extreme cases, even the direction of effects can be apparently reversed (as in the example above) — and in such cases the word “paradox” does seem merited.

Logistic regression

The simple example above was (deliberately) easy enough to understand without any fancy statistical methods.  For more complex settings, especially when there are several “explanatory” variables to take into account, the method of logistic regression is a natural tool to choose (as indeed the authors of the OfS report did).

It might be thought that a relatively sophisticated tool such as logistic regression can solve the problem that was highlighted above.  But that is not the case.  The method of logistic regression, with its results aggregated as described in the OfS report, merely yields the same (incorrect) conclusions in the artificial example above.

For anyone reading this who wants to see the details: here is the full code in R, with some bits of commentary.

2.  So, what is a better way?

The above has shown how the application of a statistical method can result in potentially very misleading results.

Unfortunately, it is hard for me (and perhaps just as hard for anyone else?) to come up with a purely statistical remedy — i.e., a better statistical method.

The problem of measuring “grade inflation” is an intrinsically difficult one to solve.  Subject-specific Boards of Examiners — which is where the degree classification decisions are actually made within universities — work very hard (in my experience) to be fair to all students, including those students who have graduated with the same degree title in previous years or decades.  This last point demands attention to the maintenance of standards through time.  Undoubtedly, though, there are other pressures in play — pressures that might still result in “grade inflation” through a gradual lowering of standards, despite the efforts of exam boards to maintain those standards.  (Such pressures could include the publication of %Firsts and similar summaries, in league tables of university courses for example.)   And even if standards are successfully held constant, there could still be apparent grade-inflation wherever actual achievement of graduates is improving over time, due to such things as increased emphasis on high-quality teaching in universities, or improvements in the range of options and the information made available to students (who can then make better choices for their degree courses).

I should admit that I do not have an answer!

3.  A few (more technical) notes

a.  For the artificial example above, I focused on the difficulty caused by aggregating university-level data to draw a conclusion about the whole sector.  But the problem does not go away if instead we want to draw conclusions about individual universities, because each university comprises several subject-specific exam boards (which is where the degree classification decisions are actually made).  Any statistical model that aims to measure successfully an aspect of behaviour (such as grade inflation) would need to consider data at the right level of disaggregation — which in this instance would be the separate Boards of Examiners within each university.

b.  Many (perhaps all?) of the reported standard errors attached to estimates in the OfS report seem, to my eye, unrealistically small.  It is unclear how they were calculated, though, so I cannot judge this reliably.  (A more general point related to this: It would be good if the OfS report’s authors could publish their complete code for the analysis, so that others can check it and understand fully what was done.)

c.  In tables D2 and D3 of the OfS report, the model’s parameterization is not made clear enough to understand it fully.  Specifically, how should the Year estimates be interpreted — do they, for example, relate to one specific university?  (Again, giving access to the analysis code would help with understanding this in full detail.)

d.  In equations E2 and E3 of the OfS report, it seems that some independence assumptions (or, at least, uncorrelatedness)  have been made.  I missed the justification for those; and it is unclear to me whether all of them are indeed justifiable.

e.  The calculation of thresholds for “significance flags” as used in the OfS report is opaque.  It is unclear to me how to interpret such statistical significance, in the present context.

4.  Opinion

This topic seems to me to be a really important one for universities to be constantly aware of, both qualitatively and quantitatively.

Unfortunately I am unconvinced that the analysis presented in this OfS report contributes any reliable insights.  This is worrying (to me, and probably to many others in academia) because the Office for Students is an important government body for the university sector.

It is especially troubling that the OfS appears to base aspects of its regulation of universities upon such a flawed approach to measurement.  As someone who has served in many boards of examiners, at various different universities in the UK and abroad (including as an external examiner when called upon), I cannot help feeling that a lot of careful work by such exam boards is in danger of simply being dismissed as “unexplained”, on the basis of some well-intentioned but inadequate statistical analysis.  The written reports of exam boards, and especially of the external examiners who moderate standards across the sector, would surely be a much better guide than that?

© David Firth, January 2019

To cite this entry:
Firth, D (2019). Office for Students report on “grade inflation”. Weblog entry at URL

Simple maths of a fairer USS deal


In yesterday’s post I showed a graph, followed by some comments to suggest that future USS proposals with a flatter (or even increasing) “percent lost” curve would be fairer (and, as I argued earlier in my Robin Hood post, more affordable at the same time).

It’s now clear to me that my suggestion seemed a bit cryptic to many (maybe most!) who read it yesterday.  So here I will try to show more specifically how to achieve a flat curve.  (This is not because I think flat is optimal.  It’s mainly because it’s easy to explain.  As already mentioned, it might not be a bad idea if the curve was actually to increase a bit as salary levels increase; that would allow those with higher salaries to feel happy that they are doing their bit towards the sustainable future of USS.)

Flattening the curve

The graph below is the same as yesterday’s but with a flat (blue, dashed) line drawn at the level of 4% lost across all salary levels.


I drew the line at 4% here just as an example, to illustrate the calculation.  The actual level needed — i.e, the “affordable” level for universities —  would need to be determined by negotiation; but the maths is essentially the same, whatever the level (within reason).

Let’s suppose we want to adjust the USS contribution and benefits parameters to achieve just such a flat “percent lost” curve, at the 4% level.  How is that done?

I will assume here the same adjustable parameters that UUK and UCU appear to have in mind, namely:

  • employee contribution rate E (as percentage of salary — currently 8; was 8.7 in the 12 March proposal; was 8 in the January proposal)
  • threshold salary T, over which defined benefit (DB) pension entitlement ceases (which is currently £55.55k; was £42k in the 12 March proposal; and was £0 in the January proposal)
  • accrual rate A, in the DB pension.  Expressed here in percentage points (currently 100/75; was 100/85 in the 12 March proposal; and not relevant to the January proposal).
  • employer contribution rate (%) to the defined contribution (DC) part of USS pension.  Let’s allow different rates C_1 and C_2 for, respectively, salaries between T and £55.55k, and salaries over £55.55k. (Currently C_1 is irrelevant, and C_2 is 13 (max); these were both set at 12 in the 12th March proposal; and were both 13.25 in the January proposal.)

I will assume also, as all the recent proposals do, that the 1% USS match possibility is lost to all members.

Then, to get to 4% lost across the board, we need simply to solve the following linear equations.  (To see where these came from, please see this earlier post.)

For salary up to T:

 (E - 8) + 19(100/75 - A) + 1] = 4.

For salary between T and £55.55k:

  -8 + 19(100/75) - C_1 + 1 = 4.

For salary over £55.55k:

 13 - C_2 = 4.

Solving those last two equations is simple, and results in

 C_1 = 14.33, \qquad C_2 = 9.

The first equation above clearly allows more freedom: it’s just one equation, with two unknowns, so there are many solutions available.  Three example solutions, still based the illustrative 4% loss level across all salary levels, are:

 E=8, \qquad A = 1.175 = 100/85.1

 E = 8.7, \qquad A = 1.21 = 100/82.6

 E = 11, \qquad A = 100/75.

At the end here I’ll give code in R to do the above calculation quite generally, i.e., for any desired percentage loss level.  First let me just make a few remarks relating to all this.


Choice of threshold

Note that the value of T does not enter into the above calculation.  Clearly there will be (negotiable) interplay between T and the required percentage loss, though, for a given level of affordability.

Choice of C_2

Much depends on the value of C_2.

The calculation above gives the value of C_2 needed for a flat “percent lost” curve, at any given level for the percent lost (which was 4% in the example above).

To achieve an increasing “percent lost” curve, we could simply reduce the value of C_2 further than the answer given by the above calculation.  Alternatively, as suggested in my earlier Robin Hood post, USS could apply a lower value of C_2 only for salaries above some higher threshold — i.e., in much the same spirit as progressive taxation of income.

Just as with income tax, it would be important not to set C_2 too small, otherwise the highest-paid members would quite likely want to leave USS.  There is clearly a delicate balance to be struck, at the top end of the salary spectrum.

But it is clear that if the higher-paid were to sacrifice at least as much as everyone else, in proportion to their salary, then that would allow the overall level of “percent lost” to be appreciably reduced, which would benefit the vast majority of USS members.

Determination of the overall “percent lost”

Everything written here constitutes a methodology to help with finding a good solution.  As mentioned at the top here, the actual solution — and in particular, the actual level of USS member pain (if any) deemed to be necessary to keep USS afloat — will be a matter for negotiation.  The maths here can help inform that negotiation, though.

Code for solving the above equations

## Function to compute the USS parameters needed for a
## flat "percent lost" curve
## Function arguments are:
## loss: in percentage points, the constant loss desired
## E: employee contribution, in percentage points
## A: the DB accrual rate
## Exactly one of E and A must be specified (ie, not NULL).
## Example calls:
## flatcurve(4.0, A = 100/75)
## flatcurve(2.0, E = 10.5)
## flatcurve(1.0, A = 100/75)  # status quo, just 1% "match" lost

flatcurve <- function(loss, E = NULL, A = NULL){

    if (is.null(E) && is.null(A)) {
        stop("E and A can't both be NULL")}
    if (!is.null(E) && !is.null(A)) {
        stop("one of {E, A} must be NULL")}

    c1 <- 19 * (100/75) - (7 + loss)
    c2 <- 13 - loss

    if (is.null(E)) {
        E <- 7 + loss - (19 * (100/75 - A))

    if (is.null(A)) {
        A <- (E - 7 - loss + (19 * 100/75)) / 19

return(list(loss_percent = loss,
            employee_contribution_percent = E,
            accrual_reciprocal = 100/A,
            DC_employer_rate_below_55.55k = c1,
            DC_employer_rate_above_55.55k = c2))

The above function will run in base R.

Here are three examples of its use (copied from an interactive session in R):

###  Specify 4% loss level, 
###  still using the current USS DB accrual rate

> flatcurve(4.0, A = 100/75)
[1] 4

[1] 11

[1] 75

[1] 14.33333

[1] 9

###  This time for a smaller (2%) loss, 
###  with specified employee contribution

> flatcurve(2.0, E = 10.5)
[1] 2

[1] 10.5

[1] 70.80745

[1] 16.33333

[1] 11

### Finally, my personal favourite:
### --- status quo with just the "match" lost

> flatcurve(1, A = 100/75)
[1] 1

[1] 8

[1] 75

[1] 17.33333

[1] 12

© David Firth, March 2018

To cite this entry:
Firth, D (2018). Simple maths of a fairer USS deal. Weblog entry at URL

USS proposals: Tail wagging the dog?


Update on 16 March: There’s now a follow-up post to this one, which gives more detail on how (mathematically) to achieve a fairer sharing-out of whatever level of USS member pain might ultimately be deemed necessary.  See Simple maths of a fairer USS deal (but ideally only after reading the necessary background, below!).

In response to my previous post, “Latest USS proposal: Who would lose most?“, someone asked me about doing the same calculation for the USS JNC-supported proposals from January.  For a summary of those January proposals and my comments about their fairness, please see my earlier post “USS pension scheme and fairness“.

Anyway, the calculation is quite simple, and it led to the following graph.  The black curve is as in my previous post, and the red one is from the same calculation done for the January USS proposal.

lost-comparisonThe red curve shows just over 5% effective loss of salary for those below the current £55.55k USS threshold, and then a fairly sharp decline to less than 2% lost at the salaries of the very highest-paid professors, managers and administrators.  Under the January proposals, higher-paid staff would contribute proportionately less to the “rescue package” for USS — less, even, than under the March proposals.  (And if the salary axis were to be extended indefinitely, the red curve would actually cross the zero-line: that’s because in the January proposals the defined-contribution rate from employers would actually have increased from (max) 13% to 13.25%.)

In terms of unequal sharing of the “pain”, then, the January proposal was even worse than the March one.

At the bottom here I’ll give the R code and a few words of explanation for the calculation of the red curve above.

But the main topic of this post arises from a remarkable feature of the above graph! At the current USS threshold salary of £55.55k, the amount lost is the same — it’s 5.08% under both proposals.  Which led me to wonder: is that a coincidence, or was it actually a (pretty weird!) constraint used in the recent UUK-UCU negotiations?  And then to wonder: might the best solution (i.e., for the same cost) be to do something that gives a better graph than either of the two proposals seen so far?

Tail wagging the dog?

The fact that the loss under the March proposal tops out at 5.08%, exactly (to 2 decimals, anyway) the same as in the January proposal, seems unlikely to be a coincidence?

If it’s not a coincidence, then a plausible route to the March proposal, at the UUK-UCU negotiating table, could have been along the lines of:

How can we re-work the January proposal to

  • retain defined benefit, up to some (presumably reduced) threshold and with some (presumably reduced) accrual rate,

while at the same time

  • nobody loses more than the maximum 5.08% that’s in the January proposal
  • the employer contribution rate to the DC pots of high earners is not reduced below the current standard (i.e., without the “match”) level of 12%


Those constraints, coupled with total cost to employers, would lead naturally to a family of solutions indexed by just two adjustable constants, namely

  • the threshold salary up to which DB pension applies (previously £55.55k)
  • the DB accrual rate (previously 1/75)

— and it seems plausible that the suggested (12 March 2018) new threshold of £42k and accrual rate of 1/85 were simply selected as the preferred candidate (among many such potential solutions) to offer to UUK and UCU members.

But the curve ought to be flat, or even increasing!

The two constraints listed as second and third bullets in the above essentially fix the position of the part of the black curve that applies to salaries over £55.55k.  That’s what I mean by “tail wagging the dog”.  Those constraints inevitably result in a solution that implies substantial losses for those with low or moderate incomes.

Once this is recognised, it becomes natural to ask: what should the shape of that “percentage loss” curve be?

The answer is surely a matter of opinion.

Those wishing to preserve substantial pension contributions at high salary levels, at the expense of those at lower salary levels, would want a curve that decreases to the right — as seen in the above curves for the January and March proposals.

For myself, I would argue the opposite: The “percent lost” curve should either be roughly constant, or might reasonably even increase as salary increases.  (The obvious parallel being progressive rates of income tax: those who can afford to pay more, pay more.)

I had made a specific suggestion along these lines, in this earlier post:

The details of any solution that satisfies the “percent loss roughly constant, or even increasing” requirement clearly would need to depend on data that’s not so widely available (mainly, the distribution of all salaries for USS members).

But first the principle of fairness needs to be recognised.  And once that is accepted, the constraints underlying future UUK-UCU negotiations would need to change radically — i.e., definitely away from those last two bullets in the above display.

Calculation of the red curve

In the previous post I gave R code for the black curve.  Here is the corresponding calculation behind the red curve:

sacrifice.Jan <- function(salary) { # salary in thousands
    old_threshold <- 55.55
    s <- salary

## sacrifice arising from income up to old_threshold
    s2 <- min(s, old_threshold)
    r2 <- s2 * (19/75 + 1/100 - (13.25 + 8)/100)

## sacrifice (max) arising from income over the old threshold
## -- note that this is negative
    r3 <- (s > old_threshold) * (s - old_threshold) * 
                (13 - 13.25)/100

    return(r2 + r3)

## A vector of salary values up to £150k
salaries <- (1:1500) / 10

## Compute percent of salary that would be lost, 
## at each salary level
sacrifices <- 100 * sapply(salaries, sacrifice.Jan) / salaries

In essence:

  • salary under £55.55k would lose the defined benefit (that’s the 19/75 part) and the 1% “match”, and in its place would get 21.25% as defined contribution.  The sum of these parts is the computed loss r2.
  • salary over £55.55k would gain the difference between potential 13% employer contribution and the proposed new rate of 13.25% (that’s the negative value r3 in the code).

© David Firth, March 2018

To cite this entry:
Firth, D (2018). USS proposals: Tail wagging the dog?. Weblog entry at URL


Latest USS proposal: Who would lose most?


Update on 16 March: After reading this post, you might perhaps be interested in these follow-ups:

Update, 14 March: Some details in the original post yesterday were not quite right, and so the graph/numbers that appear in the now-corrected version below are different in detail from yesterday’s.  But the overall picture is unchanged.  (If you really want to know about those changes in the detail, please see my note in Appendix 2 at the bottom of the post about that.)

Yesterday (March 12th) the UUK/UCU negotiations at ACAS concluded with an agreement document.

In this post I’ll look at the numbers in those proposed interim changes to the Universities Superannuation Scheme, to work out how much money would effectively be lost by USS members at each salary level.

This is inevitably a fairly rough calculation, but its results don’t really demand more precision.  The picture is very clear: the cost of “saving” USS would be felt most by USS members with low or moderate incomes.

The effective marginal rates at which money is lost by members are (as calculated below):

  • 4.7% on salary up to £42k
  • 6.3% on salary between £42k and the current USS threshold salary of £55.55k
  • 1.0% (at most) on salary over £55.55k

This translates into the following relationship between salary and the percentage of total salary lost:


The two “kinks” in that graph reflect the discontinuities in marginal rates, at £42k and at £55.55k.

The vertical lines drawn in green are current full-time pay grades at a typical university (with no London allowance or other extras): grade 6 is the pay of many Research Associates and Teaching Fellows, for example; grade 7 is the pay of most Lecturers; grade 8 is the pay of Senior Lecturers and Readers; and grade 9 is the pay of Professors and other senior staff.  (I have mentioned only academic and research staff here, but the same grades apply also to administrative and technical staff in UK universities.)

The long decay to the right continues indefinitely, ultimately approaching an asymptote at 1% lost, i.e., for those with absolutely stratospheric salaries (if such people are actually members of USS, still, that is — though I would guess that many are not).

In the rest of this post I’ll give the details of the calculation that leads to the above numbers and graph.  (For people who prefer a list of numbers to a graphical display, I have also added the numbers as an Appendix at the bottom of this post.)

Just here, though, let me again comment on how unfair this “remedy” would be.  The unfairness should be obvious from the above graph: those who are paid most, and would stand to benefit most from being in USS, would contribute least, in percentage terms, in this proposed move towards the future sustainability of USS.  For a more general view on this unfairness, see also my previous two posts in this “USS” category:

The calculation

It suffices to consider salaries in three distinct bands.  In each salary band, we can calculate how much is lost, per unit of salary.

The following code in R reproduces the graph drawn above.  A brief explanation is then given, beneath the displayed code.

## This code runs in base R.

## Function to compute the amount that would be lost annually (£k)
## at any given salary level
sacrifice <- function(salary) { # salary in thousands
    old_threshold <- 55.55
    new_threshold <- 42
    s <- salary

## sacrifice arising from income up to the new threshold
    r1 <- min(s, new_threshold) * ((8.7 - 8)/100 +
                                    19 * (1/75 - 1/85) +

## sacrifice arising from income between the thresholds
    s2 <- (s > new_threshold) * (min(s, old_threshold) - 
    r2 <- s2 * ((8.7 - 8)/100 + (19/75 - (12 + 8.7)/100) + 1/100)

## sacrifice (max) arising from income over the old threshold
    r3 <- (s > old_threshold) * (s - old_threshold) * (1/100)

    return(r1 + r2 + r3)

## A vector of salary values up to £150k
salaries <- (1:1500) / 10

## Compute percent of salary that would be lost, 
## at each salary level
sacrifices <- 100 * sapply(salaries, sacrifice) / salaries

## Plot the result
svg(file = "lost.svg", width = 8, height = 4)
plot(salaries, sacrifices, type = "l",
 xlab = "salary (thousands)", ylab = "percent lost",
 main = "Percent of salary lost under UUK-UCU agreement 2018-03-12")
abline(v = c(29, 39, 48, 61), col = "green")
text(x = c(34, 44, 54, 75), y = 2.8,
 labels = c("6", "7", "8", "9"), col = "green")

Band 1: Salary up to £42k

Most contributions from this part of salary go to the “defined benefit” part of USS. The new proposal would see 8.7% of member’s salary up to £42k going in to this, as opposed to 8.0% at present. The return (i.e., the value of the defined-benefit pension) can readily be calculated using the standard HMRC formula, the one that is used for Annual Allowance purposes. Under current USS, the value of this part is 19 times (s/75), where s is either £42k or the member’s salary if the salary is less than £42k. Under yesterday’s proposals, the value of this part would fall to 19 times (s/85). Under yesterday’s proposals, USS members would also lose the possibility to add 1% “matching” employer contribution to an additional, defined-contribution pension pot. The amount lost to each member, relating to salary in this first band, is then the sum of the additional contribution made and the amount of pension value lost: that is r1 in the above code.

Band 2:  Salary between £42k and £55.55k

Now, for salaries greater than £42k, let s2 be the smaller of (salary minus £42k) and (£55.55k minus £42k). Then current USS has members contributing 8% of s2 in the defined-benefit part, for a return of 19 times s2/75. Yesterday’s proposal would change the contribution to 8.7% of s2, for a return of s2 times (12% + 8.7%). And again, the possibility of 1% matching employer contribution to the defined-contribution pot would be lost. The amount lost to each member, relating to salary in this second band, is again just the sum of the additional contribution made and the amount of pension value lost: that is r2 in the above code.

Band 3: Salary over £55.55k

Relating to salary above the current £55.55k threshold, the loss would be limited to loss of the 1% matching employer contribution.  This is computed as r3 in the above code. (In practice this will be an upper bound on what is lost.  Those USS members with the very highest salaries are likely also to face issues relating to the HMRC Annual Allowance and Lifetime Allowance limits, in which case the loss of the matching employer contribution could be worth substantially less than 1% to them.)


I have reproduced the full calculation here, with code, because I found the result of the calculation so shocking!  If anyone reading this thinks I have made a mistake in the calculation, please do let me know. If it is correct — and right now I have no reason to suspect otherwise — then I confess I’m alarmed that this is actually being proposed as a potential solution, even as an interim solution for the next 3 years, to the perceived problems with USS.  It shakes my faith in those who have been involved in negotiating it.  With seemingly intelligent people on both sides of the table, how could they possibly come up with something as bad as this?

© David Firth, March 2018

To cite this entry: Firth, D (2018). Latest USS proposal: Who would lose most?  Weblog entry at URL

Appendix 1: A tabular view of what’s in the graph

## Make a table for anyone who wants more detail than the graph
salary <- c(10:55, 55.55, 56:100, 150)
percent_lost <- round(100 * sapply(salary, sacrifice) / salary, 2)
salary <- 1000 * salary
my_table <- data.frame(salary, percent_lost)

That’s the code for making a little table, showing the same numbers as those in the above graph.

Here is the resulting table:

salary    %
 10000 4.68 -- I started the table at £10k for no good reason
 11000 4.68
 41000 4.68
 42000 4.68 -- the proposed new threshold
 43000 4.72
 44000 4.76
 45000 4.79
 46000 4.82
 47000 4.86
 48000 4.89
 49000 4.92
 50000 4.94
 51000 4.97
 52000 5.00
 53000 5.02
 54000 5.05
 55000 5.07
 55550 5.08 -- current USS threshold, highest % of salary lost
 56000 5.05
 57000 4.98
 58000 4.91
 59000 4.84
 60000 4.78
 61000 4.72
 62000 4.66
 63000 4.60
 64000 4.54
 65000 4.49
 66000 4.44
 67000 4.39
 68000 4.34
 69000 4.29
 70000 4.24
 71000 4.19
 72000 4.15
 73000 4.11
 74000 4.07
 75000 4.02
 76000 3.98
 77000 3.95
 78000 3.91
 79000 3.87
 80000 3.84
 81000 3.80
 82000 3.77
 83000 3.73
 84000 3.70
 85000 3.67
 86000 3.64
 87000 3.61
 88000 3.58
 89000 3.55
 90000 3.52
 91000 3.49
 92000 3.47
 93000 3.44
 94000 3.41
 95000 3.39
 96000 3.36
 97000 3.34
 98000 3.31
 99000 3.29
100000 3.27
150000 2.51 -- possibly there are even some salaries this high?!

Appendix 2: Details of the update made on 14 March

Many thanks to all who gave feedback on the original posting, yesterday (13 March).

In response to that feedback, I made two substantive changes to the calculation.  This Appendix gives details of those changes, for those who are interested (and for the record).

Neither change affects the story qualitatively: only the detailed numbers have changed a bit.

Change 1: Use of HMRC multiplier 19 rather than 23

The HMRC calculations for Annual Allowance and Lifetime Allowance purposes are different in detail: the former uses a multiplier of 19 times pension to value USS defined benefits, while the latter uses 23 (i.e., in place of 19).  In yesterday’s post I had used 23.  The updated figures calculated above use multiplier 19 instead.

Mainly I decided to use the smaller figure as it’s a bit more conservative, in relation to the value lost through the proposed reduction of defined benefits.  (I certainly don’t want to be accused of bias in the other direction, through having picked the larger multiplier.)

The effect on the calculated numbers is mainly to reduce the height of the “spike” that appears in the graph, around the £55k salary level.  The spike is still there; it’s just a bit smaller.

My friend Jon commented that the actual value of a defined-benefit pension is harder to quantify than the HMRC formula would suggest — and that it’s likely to be dependent on age and perhaps other factors.  This is undoubtedly true, and certainly I would not suggest that anyone should use the above numbers for their own financial planning!  Rather, the aim here was (only) to show through a simple, transparent calculation how the losses arising from current proposals would differ — in rough, average terms — between pay levels.

Since writing my post yesterday I found that I am not alone in having done a calculation like this: see also (and maybe there are others too?).

Change 2: Inclusion of the USS “Match” at all salary levels

Several people pointed out to me that the USS “Match” possibility is available at all salary levels.  So it’s a benefit that would be lost at all salary levels, under the 12 March agreement.  In yesterday’s post I had taken it into account only at salaries over £55.55k: that (relatively minor) error is now corrected, in the revised figures shown above.


Exit poll for June 2017 election (UK)



It has been a while since I posted anything here, but I can’t resist this one.

Let me just give three numbers.  The first two are:

  • 314, the number of seats predicted for the largest party (Conservatives) in the UK House of Commons, at 10pm in Thursday (i.e., before even a single vote had been counted) from the exit poll commissioned jointly by broadcasters BBC, ITV and Sky.
  • 318, the actual number of seats that were won by the Conservatives, now that all the votes have been counted.

That highly accurate prediction changed the whole story on election night: most of the pre-election voting intention polls had predicted a substantial Conservative majority.  (And certainly that’s what Theresa May had expected to achieve when she made the mistake of calling a snap election, 3 years early.)  But the exit poll prediction made it pretty clear that the Conservatives would either not achieve a majority (for which 326 seats would be needed), or at best would be returned with a very small majority such as the one they held before the election.  Media commentary turned quickly to how a government might be formed in the seemingly likely event of a hung Parliament, and what the future might be for Mrs May.  The financial markets moved quite substantially, too, in the moments after 10pm.

For more details on the exit poll, its history, and the methods used to achieve that kind of predictive accuracy, see Exit Polling Explained.

The third number I want to mention here is

  • 2.1.0

That’s the version of R that I had at the time of the 2005 General Election, when I completed the development of a fairly extensive set of R functions to use in connection with the exit poll (which at that time was done for BBC and ITV jointly).  Amazingly (to me!) the code that I wrote back in 2001–2005 still works fine.  My friend and former colleague Jouni Kuha, who stepped in as election-day statistician for the BBC when I gave it up after 2005, told me today that (with some tweaks, I presume!) it all works brilliantly still, as the basis for an extremely high-pressure data analysis on election day/night.  Very pleasing indeed; and strong testimony to the heroic efforts of the R Core Development Team, to keep everything stable with a view to the long term.

As suggested by that kind tweet reproduced above from the RSS President, David Spiegelhalter: Thursday’s performance was quite a triumph for the practical art and science of Statistics.  [And I think I am allowed to say this, since on this occasion I was not even there!  The credit for Thursday’s work goes to Jouni Kuha, along with John Curtice, Steve Fisher and the rest of the academic team of analysts who worked in the secret exit-poll “bunker” on 8 June.]


R and citations


We’re hosting the international useR! conference at Warwick this summer, and I thought it might be interesting to try to get some data on how the use of R is growing. I decided to look at scholarly citations to R, mainly because I know where to find the relevant information.

I have access to the ISI Web of Knowledge, as well as to Google Scholar. The data below comes from the ISI Web of Knowledge database, which counts (mainly?) citations found in academic journals.

Background: How R is cited
Since version 0.90.0 of R, which was released in November 1999, the distributed software has included a FAQ document containing (among many other things) information on how to cite R. Initially (in 1999) the instruction given in the FAQwas to cite

When R version 1.8.1 was released in November 2003 the advice on citing R changed: people using Rin published work were asked to cite

The “2003” part of the citation advice has changed with each passing year; for example when R 1.9.1 was released (in June 2004) it was updated to “2004”.

ISI Web of Knowledge: Getting the data
Finding the citation counts by searching the ISI database directlydoes not work, because:

  1. the ISI database does not index Journal of Computational and Graphical Statistics as far back as 1996; and
  2. the “R Core Development Team” citations are (rightly) not counted as citations to journal articles, so they also are not directly indexed.

So here is what I did: I looked up published papers in the ISI index which I knew would cite R correctly. [This was easy; for example my friend Achim Zeileis has published many papers of this kind, so a lot of the results were delivered through a search for his name as an author.] For each such paper, the citation of interest would appear in its references. I then asked the Web of Knowledge search engine for all other papers which cited the same source, with the resulting counts tabulated by year of publication.

It seems that the ISI database aims to associate a unique identifier with each cited item, including items that are not themselves indexed as journal articles in the database. This is what made the approach described above possible.

There’s a hitch, though! It seems that, for some cited items, more than one identifier gets used. Thus it is hard to be sure that the counts below include all of the citations to R: indeed, as I mention further below, I am pretty sure that my search will have missed some citations to R, where the identifier assigned by ISI was not their “normal” one. (This probably seems a bit cryptic, but should become clearer from the table below.)

Citation counts
As extracted from the ISI Web of Knowledge on 25 June 2011:

ISI identifier 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Total
5 15 18 43 131 290 472 528 435 419 449 378 396 3579
39 123 91 57 39 25 14 388
16 235 421 327 289 187 126 1601
42 397 531 511 445 366 2292
5 39 75 41 25 10 195
55 438 849 656 461 2459
92 714 962 733 2501
208 1402 1906 3516
7 21 44 72
172 1363 1535
205 205
1 12 14 25 36 81 93 262
Total 5 15 18 43 131 290 528 945 1452 1964 3143 4354 5717 18605

For the “R Development Core Team (year)” citations, the peak appears about 2 years after the year concerned. This presumably reflects journal review and backlog times.

There are almost certainly some ISI identifiers missing from the above table (and, as a result, almost certainly some citations not yet counted by me). For example, the number of citations found above to R Development Core Team (2009) is lower than might be expected given the general rate of growth that is evident in the table: there is probably at least one other identifier by which such citations are labelled in the ISI database (I just haven’t found it/them yet!). If anyone reading this can help with finding the “missing” identifiers and associated citation counts, I would be grateful.

The graph below shows the citations found within each year since 1998.

© David Firth, June 2011

To cite this entry:
Firth, D (2011). R and citations. Weblog entry at URL


The graph shows the citations found within each year since 1998.

[Click on the graph to view it at a larger size.]

Citations to Ihaka and Gentleman (1996) and to R Core Development Team (any year) are distinguished in the graph, and the total count of the two kinds of citation is also shown.