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.

My made-up “toy” data:

> x module 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 Residuals: module 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 Residuals: student 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”.

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 DEA NA -10 -20 NA -40 B 1 NA -10 NA -30 C 1 1 NA NA -20D0 0 0 NA-10E 1 1 12NA

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(!is.na(diffs)) } } return(result) } 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$coefficients[is.na(result$coefficients)] <- 0 class(result) <- c("meddiff_fit", "list") return(result) }

© David Firth, April 2019

**To cite this entry:**

Firth, D (2019). Robust measurement from a 2-way table. Weblog entry at URL https://statgeek.net/2019/04/26/robust-measurement-from-a-2-way-table/

]]>

**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:

- Set the record straight, in relation to some incorrect reporting of Part 1 in the specialist media.
- 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.

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

- At https://www.researchprofessional.com/0/rr/he/agencies/ofs/2019/OfS-grade-inflation-analysis-not-fit-for-purpose–says-expert.html, an article entitled “OfS grade inflation analysis not fit for purpose, says expert”
- At https://www.researchresearch.com/news/article/?articleId=1379083, which seems to be a straight copy of the same article (I have not checked in detail).

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):

- 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. - 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. - 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.

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.)

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:

2010-11 University A University B Firsts Other Firsts Other h 1000 0 h 500 500 i 0 1000 i 500 500 2016-17 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

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- 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

.

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 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:

.

So it is superficially very similar. But the all-important term 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 https://statgeek.net/2019/01/07/part-2-further-comments-on-ofs-grade-inflation-report/

**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 https://wonkhe.com/blogs/policy-watch-ofs-report-on-grade-inflation/)

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.

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.

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 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 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).

Yes.

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.

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.

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!

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.

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 https://statgeek.net/2019/01/02/office-for-students-report-on-grade-inflation

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.)

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 and for, respectively, salaries between*T*and £55.55k, and salaries over £55.55k. (Currently is irrelevant, and 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**:

For **salary between T and £55.55k**:

For **salary over £55.55k**:

Solving those last two equations is simple, and results in

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:

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.

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.

Much depends on the value of .

The calculation above gives the value of 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 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 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 *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.

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.

## 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) $loss_percent [1] 4 $employee_contribution_percent [1] 11 $accrual_reciprocal [1] 75 $DC_employer_rate_below_55.55k [1] 14.33333 $DC_employer_rate_above_55.55k [1] 9 #------------------------------------------------------------ ### This time for a smaller (2%) loss, ### with specified employee contribution > flatcurve(2.0, E = 10.5) $loss_percent [1] 2 $employee_contribution_percent [1] 10.5 $accrual_reciprocal [1] 70.80745 $DC_employer_rate_below_55.55k [1] 16.33333 $DC_employer_rate_above_55.55k [1] 11 #------------------------------------------------------------ ### Finally, my personal favourite: ### --- status quo with just the "match" lost > flatcurve(1, A = 100/75) $loss_percent [1] 1 $employee_contribution_percent [1] 8 $accrual_reciprocal [1] 75 $DC_employer_rate_below_55.55k [1] 17.33333 $DC_employer_rate_above_55.55k [1] 12

© David Firth, March 2018

**To cite this entry:**

Firth, D (2018). Simple maths of a fairer USS deal. Weblog entry at URL https://statgeek.net/2018/03/16/simple-maths-of-a-fairer-uss-deal/

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.

The 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*?

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.

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.

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 https://statgeek.net/2018/03/15/uss-proposals-tail-wagging-the-dog/

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**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:

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) + 1/100) ## sacrifice arising from income between the thresholds s2 <- (s > new_threshold) * (min(s, old_threshold) - new_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") dev.off()

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

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

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 https://statgeek.net/2018/03/13/latest-uss-proposal-who-would-lose-most/.

## 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?!

** 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.

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 http://brianosmith.blogspot.co.uk/ (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.

]]>

This post is a little off-topic, as the exercise I am about to illustrate is not one that most corpus linguists will have to engage in. However, I think it is a good example of why a mathematical approach to statistics (instead of the usual rote-learning of tests) is…]]>

Following a recent Twitter exchange with Ian Dryden, I was thinking I’d write something about the risk of default for the USS pension scheme. But then I came across this other new post at corp.ling.stats — which has already done what I was intending!

This post is a little off-topic, as the exercise I am about to illustrate is not one that most corpus linguists will have to engage in.

However, I think it is a good example of why a mathematical approach to statistics (instead of the usual rote-learning of tests) is extremely valuable.

At the time of writing nearly two hundred thousand university staff in the UK are active members of a pension scheme called USS. This scheme draws in income from these members and pays out to pensioners. Every three years the pension is valued, which is not a simple process. The valuation consists of two aspects, both uncertain:

- to value the liabilities of the pension fund, which means the obligations to current pensioners and future pensioners (current active members), and
- to value the future asset value of the pension fund…

View original post 2,753 more words

- Latest USS proposal: Who would lose most?
- USS proposals: Tail wagging the dog?
- Simple maths of a fairer USS deal

In this post I’ll follow up on the previous one, USS pension scheme and fairness, in the light of this week’s UCU proposals (proposals which provide a basis for renewed talks with Universities UK about the future of USS).

My purpose here is to suggest that *a fairer solution* could also be *more affordable* — in that it would help fund the defined-benefit section of USS, but would also reduce (and might perhaps even eliminate?) the upward pressure on employer and employee contribution rates.

The following summary excerpt is taken from https://www.ucu.org.uk/article/9364/Further-talks-agreed-in-universities-pensions-dispute:

The University and College Union (UCU) deserves all of the many congratulations it is currently getting, for having produced such a proposal this week — in particular, a proposal that has at last persuaded the employers (Universities UK) to take part in further talks about the future of USS.

That said: I should admit to feeling disappointed by the 3rd and 4th bullet points above! USS members would all be getting a slightly worse pension, in return for an appreciable increase in the contribution rates (both employee and employer contribution rates).

**More crucially** — if I have understood it correctly — the implied loss of pension (as a percentage of contributions paid) would be *greatest for those USS members whose salary is £55k or lower*. Higher-paid members of USS would be required to make a smaller sacrifice — *substantially* smaller, in the case of those with the very highest salaries — relative to the overall size of their pensions.

Recognising this unfair distribution of the pain, and thinking about how best to fix it, leads naturally to an additional device that could make the proposed changes fairer *and* more easily affordable.

The current USS setup has a slightly *progressive* aspect, which is that the 18% employer contribution for salary over the £55k threshold pays only 12% into the member’s defined-contribution (DC) pension pot. The remaining 6% therefore helps to support the running costs of USS, and (mainly, it seems safe to presume) the defined-benefit part of USS.

The suggestion I want to make here is that future USS should become *more* progressive — that is, USS should move further in that same, progressive direction. (As mentioned in the previous post, the current USS proposal would increase that 12% figure to 13.25%, thereby making the scheme appreciably *less* progressive. And as I have argued in that previous post, that seems completely indefensible.)

The **specific suggestion**:

- For salary between the current £55k threshold and some specified higher threshold, pay
*x*% from the employer contribution into the employee’s DC pot. The value of*x*could be left at 12, for example. - For salary above the
*higher*of the two thresholds, pay*y*% from the employer contribution into the employee’s DC pot, where*y*is smaller than*x.*

Even if *x* remains at 12, such a device would allow (through suitable choice of the higher threshold and the value of *y) *substantial savings to be made from the employer contributions on higher salaries — savings which could then be used directly to support the threatened, defined-benefit part of USS.

The precise arithmetic on this becomes possible only with detailed knowledge (not available to me) of the distribution of USS-member salaries over £55k. Some general considerations on the choice of the higher threshold, and of *y*, are:

- The higher threshold clearly should not be so high as to make the resultant savings too small to be of much consequence.
- The value of
*y*probably needs to be at least as high as the employee contribution rate (currently 8%), otherwise it could become unattractive for those with the highest salaries to remain in USS.

**Illustrative example**:

Just to give a sense of what might be possible through this device. Let’s suppose:

- salary between £55k and £75k gets 12% into the DC pot, from the employer contribution — as now.
- salary over £75 gets 8% (i.e., the current employee contribution rate) instead of 12%.

Then the **amount saved**, which could then be used to support the defined-benefit part of USS, would be **4% of all member salaries over £75k**.

It is unclear to me, in absence of enough data to determine it, whether the ‘Robin Hood’ device just described would be **enough** to completely eliminate the need for a change in the accrual rate and/or increased contributions (as proposed by UCU in their points 3 and 4 above).

What *is* clear to me, though, is that such a device would help eliminate the unfairness described above. And it would, at least, *reduce* the need for any changes in accrual or contribution rates, even if such need is not completely eliminated.

© David Firth, March 2018

**To cite this entry:**

Firth, D (2018). Future USS: Robin Hood can help? Weblog entry at URL https://statgeek.net/2018/03/01/future-uss-robin-hood-can-help/.

The legend of Robin Hood, ‘feared by the bad, loved by the good’, will already be known to most people who have grown up in England. There are countless stories of Robin Hood ‘persuading’ the rich to part with their money, for the benefit of poorer folk.

For anyone interested, there’s a lot to read about it here: http://www.robinhoodlegend.com/.

One aspect that I particularly like is that my home city of *Wakefield* is one of the (many!) places that lay claim to Robin Hood as one if its own townspeople. But then there’s all that romantic Sherwood Forest nonsense…

- Future USS: Robin Hood can help?
- Latest USS proposal: Who would lose most?
- USS proposals: Tail wagging the dog?
- Simple maths of a fairer USS deal

The Universities Superannuation Scheme (USS) is among the largest pension schemes in the UK. It provides pensions for academic and other staff in most of the UK’s universities — and right now it is the subject of strike action in around 60 universities. The strike relates to substantial proposed changes to the pension scheme.

Most of the current arguments are about the long-term **affordability** of USS in its present form. I do not consider myself sufficiently expert to add much that would be useful on that (crucial) aspect of USS — so that’s not the subject of this post. Instead, here I will take a look at the current USS scheme in regard to aspects of **fairness**; and I’ll examine the new proposals and their implied direction of travel, in relation to fairness.

Why am I writing this? Well, I needed to find out enough about USS and the new proposals to be able to explain — to friends, family, colleagues in other countries, etc. — what all the current argument is about. One thing I found — quite separately from the arguments about affordability of a defined-benefit pension scheme — was that the new proposals seem to have the wrong direction of travel in relation to (at least *my* notions of) fairness.

At the end I’ll make a specific suggestion towards improving the currently proposed changes to USS.

Back in the mists of time, when I first joined USS, the university that appointed me was (rightly) very keen to stress how great the pension scheme was. I would need to pay 6.35% of salary, and the university would contribute an additional 18.55% of my salary, to provide a guaranteed pension based on whatever salary level I had achieved at retirement. This was regarded as an important part of the deal: our salaries were considerably lower than we could be paid elsewhere, but the promise of a decent pension would compensate to some extent.

I remember well, a few years later in 1997, the widespread feeling of outrage and dismay when the universities collectively decided to reduce their contributions from 18.55% to 14%. In 1997 the USS pension fund was substantially in surplus, and the universities succumbed to the natural temptation of a ‘contributions holiday’. (It turned out to be a very long holiday: the universities’ contribution rate remained at 14% until 2011.)

Right now, in 2018, employer contribution is 18%, and USS members themselves pay 8% of salary into the scheme (with an option to increase that to 9% in return for a matching 1% increase of the employer contribution to the USS member’s “defined contribution” account with USS~~).~~

In relation to this **history**, let me mention here two aspects relating to **fairness**:

- The ‘
**final salary**‘ basis for determining the amount of pension payable was (in my opinion!) blatantly unfair. As someone who stands to benefit from it, I think I am in a good position to say this. Members’ life-long contributions to USS were being used disproportionately to pay the pensions of those who were paid the highest salaries, and especially those whose salaries became high relatively late in their careers. - The long
**contributions holiday**taken by universities between 1997 and 2011 is clearly connected with whatever problems USS has today. The universities ought to be prepared to substantially*increase*their contributions to the USS pot when the historic surplus has been run down partly as a result of the earlier (substantial!) ‘holiday’. It would be completely unfair to expect future USS members to pay the cost of the ‘holiday’.

This excerpt from https://www.uss.co.uk/~/media/document-libraries/uss/member/member-guides/post-april-2016/a-new-employees-guide-to-joining-uss.pdf gives a very rough summary of the current USS setup:

In relation to **fairness** of the current setup, I want to note three things:

- The unfair
**final salary**aspect has now gone (following the changes to USS that were made in 2016). Instead, we now have ‘Career Revalued Benefits’ (CRB), based on contributions from annual salary up to the ‘threshold’ amount of around £55k. This results in lower pensions for many (perhaps most) USS members, especially those whose salaries increase substantially in mid-late career. But CRB does seem a fairer basis than ‘final salary’, for determining the relationship between life-long contributions and the level of pension ultimately provided. (The 1/75 multiplier is arguably not high enough; but that relates more to affordability than to fairness.) - The current USS setup has an appealing,
**progressive**aspect: the employer contribution of 18% supports only a 12% contribution to the ‘USS Investment Builder’ pension pot that relates to to annual salary in excess of the £55k threshold. The implication of this is that some (maybe most?) of the remaining 6% of employer contribution, for salaries over £55k, gets used in support of the defined-benefit CRB scheme that applies to the first £55k of every member’s salary. As a principle, this seems right and fair: a priority for USS should be to ensure that even the least well-paid members get a decent pension. Appropriateness of the precise details (12%*versus*perhaps some lower figure or a decreasing schedule for higher salaries; and the £55k level of the salary ‘threshold’) is of course open to debate, still. - The previous two points were about ways in which USS currently
*is*fair. For balance, let me mention one aspect of current USS that does seem unfair, still. The USS scheme has always, as far as I know, included specific pension provision for the surviving**spouse/partner and/or other dependants**of a member who dies. While this clearly is a benefit to those members who*have*got such dependants, it implies some level of subsidy from the contributions — whether made directly, or by the employer — of any member*without*such dependants. (Again I write as someone who*does*benefit from this; but I don’t regard it as fair!)

The following excerpt is from https://www.uss.co.uk/how-uss-is-run/valuation/2017-valuation-updates/proposed-changes-to-future-uss-benefits:

My thoughts on the specific numbers there, in terms of **fairness**, are as follows. (The **first item in the following list is by far the most important** of the three; the other two are quite minor by comparison.)

- For salary amounts over the current threshold of £55k, the proposal would actually
*increase*the employers’ pension-pot contributions (from the current 12%, to 13.25%). This runs counter to the ‘progressive’ aspect, mentioned above, of the current USS setup. Moreover, it implies that*less*of the employer contribution on salary amounts over £55k — that is to say, less than now — would actually be used to service the defined-benefit commitments of USS. This seems absurd. It represents**a direction of travel that’s completely unfair**: it would systematically shift the future cost of those defined-benefit commitments, away from the higher-paid members, onto those whose salary is mostly or entirely below the current £55k threshold. - The suggested flexibility to adjust member contributions downwards to 4%, while still getting 13.25% employer contributions, looks good on the face of it. But who would it benefit most? Most members with low to middling incomes will probably want or need to contribute 8%, still, in order to (aim to) ensure that they have a decent pension at retirement. Probably the
**main beneficiaries of this flexibility would be those members with the highest salaries**, who could use it to reduce or eliminate potential breaches of the HMRC allowances (either the Annual Allowance for pension contributions, or the Lifetime Allowance for total pension-pot size). - That last bullet-point in the above excerpt looks relatively minor. It suggests shifting some of the cost of USS death benefits away from employers, to members — if I have understood it correctly. The details of this are not yet clear, at least to me; but it could imply that the cost of pensions for widows/partners and other dependants — which I have argued above is already unfair — would be carried more directly than before by members, i.e., through their own direct contributions rather than their employers’.

My comments in this post really are quite separate from the on-going arguments about affordability of the defined-benefit part of USS. But…

If those arguments about affordability do result in substantial changes being made, then I really hope that considerations of fairness will play a major role. As things stand at present, the proposed USS changes would appear to be quite neutral or even beneficial to those members with the very highest salaries, but clearly detrimental to the majority.

A more positive take on the points made above would be that there is plenty of room for manoeuvre, towards the negotiation of a future USS that could work better for all. The universities plainly are *not* currently at the absolute limit of what is affordable. The proposal to contribute *more* to the pension pots of the highest paid, rather than less, makes this abundantly clear (as indeed does the USS history, as mentioned above). A future USS that’s designed to be more *progressive* than the present version — i.e., with the highest paid receiving gradually *lower* marginal rates of employer contribution to their pension-pots — could perhaps make the defined-benefit component of USS work out to be (even more) clearly affordable?

**Update** (4 March 2018): The follow-up post **Future USS: Robin Hood can help?** formulates a concrete suggestion along the lines just indicated above — a simple device that would make USS both fairer *and* more affordable.

© David Firth, February 2018

**To cite this entry:**

Firth, D (2018). USS pension scheme and fairness. Weblog entry at URL https://statgeek.net/2018/02/26/uss-pension-scheme-and-fairness/.

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.]