Statistics of statisticians: Critical mass of statistics and operational research groups in the UK

Statistics of statisticians : Critical mass of statistics and operational resear c h groups in the U K Ralph K enna Applie d M athemat ics Research Centre, Cov entr y University , Cov entr y , CV1 5F B, Eng- land. Ber trand Berche Statistical Phys ics Group, In stitut Jea n L amour † , CNRS – Na ncy Universit ´ e – UPVM, B.P . 70239, F – 54506 V and œuvre l ` es Nancy Cedex, F rance Summary . Using a recently dev eloped model, inspired by mean field theory in statistical physics , an d data fro m the UK’ s Research Assessment E x ercise, we a nalyse the rela- tionship between the quality of statistics and operationa l research groups and the quan- tity researche rs in them. Similar to other academic disciplines, w e provide ev idence for a linea r depen dency of quality on quant ity up to a n up per cri tical ma ss, whi ch is interp reted as the av erage maxim um number of colle agues with whom a resea rcher can communi- cate meaning fully within a research grou p . T he model also predicts a low er cri tical mass, which research groups should striv e to achiev e to a v oid e xtinction. For statistics and op- erational resea rch, the lower cr itical mass is estimat ed to be 9 ± 3 . The upp er critical mass, beyond wh ich research qua lity does no t signi ficantly d epend on group size, is about twice this v alue. 1. Introduction The notion of cri tic a l mass in r esearch has been ar ound for a long time without prop er definition. As gov ernments, funding councils and universities s e ek indica tors to mea sure r esearch quality and to pursue grea ter efficiencies in the research sector, critical mass is bec oming a n increasingly imp ortant concept at manageria l and p olicy - making level. Ho wev er, until very rece ntly there hav e b een no successful a ttempts to quantify this notion (Harrison, 2009). It has b een descr ibe d by E vidence (2010) as “some minim um size thr e shold for effectiv e performanc e ” and, as suc h, has be en link ed to the idea that b e ne fit ac c rues thr ough incr ease of scale of r esearch groups. How ev er, although Evidence (2010) demons tr ated “a r elationship of some kind b etw een larger units and r elatively high citation impact” , indications of such a threshold have b een lacking. W e recently presented a mo del for the rela tionship b etw een quality of rese arch groups and their quantit y (Kenna a nd Ber che, 2 010a ). This mo del was inspire d by mean-field theories o f statistica l physics and allowed for a quantitativ e definition of critical mass . In fact there ar e tw o cr itical masses in resear ch a nd their v alues are discipline dep endent. Instead of a thres hold group size ab ov e which research quality improv es, w e have s hown that ther e is a break po in t or upp er critic al mass b eyond which the linear dep endency of rese a rch quality on gr oup quantit y r e duces. Denoting † Lab oratoire associ´ e au CNRS U MR 7198 2 K e nna and Berche this v alue by N c , we show ed that the strength of the ov er all resear ch sector in a given discipline is impr ov ed by suppo rting gr o ups whose size ar e less than N c , provided they are bigger than a s e c ond critical mass, which we denote by N k . Groups whose s ize are smaller than N k are vunerable and should seek to achiev e the low er critical mass for long-term viability . The t wo critical masses ar e r elated by a s caling relation, N c = 2 N k . (1) W e classify research gr oups of size N within a given discipline a s sma ll, medium and large acco rding to whether N < N k , N k ≤ N < N c or N ≥ N c , r esp ectively . W e recently determined the critical masses of a multitude of aca demic disciplines by applying statistical analyses to the results of the UK’s most recent R ese ar ch Assess- ment Exer cise (RAE) in which the quality of resea rch groups w ere meas ured (Kenna and Berche, 2010b). Notably absent fro m our analaysis, how e ver, w ere the statistics and op eratio nal resea rch gr oups, a s these were less str aightforw a rd to analyse than other sub ject ar eas. Here we r ectify this o mission by a car eful analysis o f these disci- plines. O ur main res ult is that the low er critical mass, which statistics and op eratio nal resear ch gro ups s hould a ttain to b e viable in the long ter m, is N k = 9 ± 3 . (2) In Sectio n 2 we summarize our mo del and how w e derive critical masses from it. W e a lso discuss the r esearch assessment exercise. In Sectio n 3 we apply the mo del and statistical analysis to the r esults of the RAE for statistics and op erationa l resea r ch groups. W e co nclude in Section 4, wher e implica tions for p o licy and manag ement are briefly discuss e d. 2. Quality and quantity in resear ch Our mo del is bas e d on the idea that resear ch g r oups are c omplex systems , for which the pro p er ties of the whole ar e not simple sums of the corr esp onding prop er ties of the individual parts. Instead, interactions b etw een individuals within resea rch gro ups hav e to a lso b e taken into account. The s tr ength o f an individual within a res earch group is a function of many factor s: their intrinsic ca libr e and training , their teaching and a dministrative lo a ds, librar y facilities, jour na l a c c ess, extra mural collab or a tion, the quality o f management, and even co nfidence g ained by previous successes a s well as the pr e stige of the institution and o ther factors. W e denote the av erag e individual resear ch strength within the g th resear ch group in a given aca demic discipline, resulting from all of thes e (and any other) factor s by a . The overall calibre of a resea rch gro up comprising N individuals is als o depe nden t on the exten t o f, and streng th of, the communication links b etw een them. W e denote the av era ge strength of the N ( N − 1) / 2 int eractions b etw een the N individuals in the g th group by b . The ov er all streng th of the gr oup is ther e fore given by S = N a + 1 2 N ( N − 1) b . (3) How ever, once the size of a res earch g roup b e comes to o la rge (say ab ov e a cut- off v alue N c ), meaning ful communication be t ween al l pa ir s of individuals b ecomes Criti cal mass of research groups 3 impo ssible. In this case, the group may fragment into N subg r oups, of av erage size M = N / N , say . If the average strength of interaction be tw een the subgro ups is c , the ov era ll strength of the group b ecomes S = N a + 1 2 N ( M − 1) b + 1 2 N ( N − 1) c . (4) W e denote by h S i the exp e cte d streng th of a gr oup o f size N a nd we define the quality of suc h a research gro up to b e the av er a ge str ength p er head: s = S N . (5) Gathering ter ms of the same order in N , we a rrive at a form for the exp e cte d dep en- dency of research-group qua lity on re search-group quantit y , h s i =  a 1 + b 1 N if N ≤ N c a 2 + b 2 N if N ≥ N c . (6) W e cons ide r ed the effect o n the ov erall strength of a discipline by a dding new resear chers (Kenna and Berche, 201 0a). Asking the question whether it is b etter, on av era ge, to allo ca te new resea rchers to a group with N > N c or N < N c mem b e r s, we found that the latter is prefer able provided N > N k , where N k is given by Eq.(1). This is eq uiv a le n t to maximising the gra dien t o f the str ength function h S ( N ) i . W e also considered the consequences of transfer ring resear chers from la rge to sma ll/medium groups and fo und that such a movemen t is expec ted to b e b eneficial to so ciety as a whole, provided the r ecipient gro up is not to o sma ll (i.e., provided, ag ain, tha t it has ov er N k mem b e r s). Thu s there ar e tw o critical masses in resear ch, whic h we name lower ( N k ) and upp er ( N c ). Of these, the former co rresp onds mor e closely to the traditional, in tuitiv e notion of critical mass, although there is no threshold v alue beyond which r esearch qua lit y suddenly improv es (Evidence , 2010). T o implemen t the mo del (6), we require a set of e mpir ical data o n the q uality a nd quantit y o f res earch gro ups . The RAE is a n ev aluation pro ce s s undertaken approxi- mately every 5 years on beha lf of the funding b o dies for universities in the UK. The results of the RAE ar e used to a llo cate funding to such highe r education institutes for the subse q uent years. The last RAE was carried out in 20 08. Research groups were examined to de ter mine the prop ortion of resear ch submitted catego rized as follows: • 4*: Qua lity that is world-leading in terms of or iginality , sig nificance and rigour • 3*: Quality that is internationally excellent in terms of origina lity , significance and rigour but which nonetheless falls sho rt of the highest standards of excellence • 2*: Q ua lit y that is recognise d internationally in terms o f or iginality , significa nce and r igour • 1*: Quality that is r ecognised nationally in terms of orig inality , sig nificance and rigour • Unclassified: Quality that falls b elow the standar d of nationally recognised work. 4 K e nna and Berche A formula is then used to deter mine how funding is distributed to resear ch groups. The 2 009 formula used by the Higher Education F unding Council for Engla nd weighs each ra nk in such a w ay that 4* and 3* res earch resp ectively receive seven and three times the amount of funding allo cated to 2* res e a rch, and 1* and unclassified resear ch attract no funding. This funding formula may therefo re b e co ns idered to repr esent a measurement of quality of e a ch resea r ch group. (In 2 0 10, after lobbying by the larg er, resear ch intensiv e universities the English funding formula was changed s o that 4* resear ch r eceives nine times the funding allo cated to 2* resea rch. W e hav e chec ked that the 2 010 formula pro duces no significant change to the results presented here.) F rom the outset, we ackno wledg e that there ar e obvious a ssumptions underlying our analysis and limits to what can b e achiev ed. Fir stly , we use the term “ group” in the sens e of RAE. This means the co llection o f staff included in a submissio n to one o f the 67 Units of Assessment (UOA’s). RAE groups are no t always identical to administrative departments within universities, but we a ssume that they repres ent a coherent gro up for r e search purp oses . Individuals submitted to RAE are drawn fro m academic staff who were in p os t and on the payroll of the submitting higher educatio n institution on the census date (31 Octob er 2007 ). W e assume that the RAE pro cess is fair and unbiased a nd that the scor e s a r e reasona bly reliable a nd robust. Deviations from these a s sumptions contribute to noise in the s ystem. Statistical analyses and a list of the critical masses for a v ariety of academic disciplines (not including statistics a nd op erational research) are given in (Kenna a nd Berche, 201 0b). In the next se ction, we per form a similar a na lysis for the sta tistics and op erationa l research groups submitted to RAE 2008. 3. Statistical analysis of statistics and operational re sear ch groups The Statistics a nd Op e rational Research UOA at RAE 20 08 included theor etical, applied and metho dolo gical appro a ches to statistics, pro bability and op era tio nal re- search. The r e were 30 submissio ns comprising 388.8 individuals (with fractions cor re- sp onding to part-time staff ) a nd group sizes rang ed from N = 2 to N = 30, with mean group size 13. W e find it use ful to compare to the Applied Ma thematics UOA b eca use of the high degree of ov erla p b etw ee n the tw o disciplines. There w ere 45 s ubmissions in applied mathematics entailing 850 .05 individuals in groups o f size N = 1 to N = 80 . 3 with mean group size 1 8 . 9. The 30 submissions for statistics and o per ational res e arch are lis ted in T able 1 . Also listed a re the num b er s o f staff submitted and the re sultant quality sco re. In Fig. 1(a), w e plo t RAE-mea sured quality scores against group quantit y for the Applied Mathematics UOA. As exp ected from (6), r esearch qua lity indeed tends to increase linearly with gro up size N up to a breakp oint, estimated a t N c = 12 . 5 ± 1 . 8 and which splits the 4 5 r esearch teams into 16 small/medium g roups and 29 lar ge ones. The co efficient of determina tion is meas ur ed to b e R 2 = 0 . 74 and the data passes the Kolmog orov-Smirnov normality test. The P v alue for the null hypothesis that there is no underlying c o rrelatio n b e t ween qualit y and quantit y is less than 0 . 001 , indicating that this ca n b e rejected. The presence of the breakp oint is evidenced by the P v alue for the hypo thes is that the slop es to the left and r ight coincide. This is also le s s tha n 0 . 001 , so the h y po thesis can be rejected. The dep endency of qua lit y o n quantit y co n tin ues at a reduced level to the right o f the break po in t as the P v alue for Criti cal mass of research groups 5 T able 1. Universities whi ch submi tted to the Stati stics and Operation al Re- search UOA at RAE 2008 , listed al phabe tically tog ether w ith th e numbers of staff submitted N and quali ty measurement s s . Index Universit y N s 1 Bath 15.00 42. 14 2 Bristol 23.00 48. 57 3 Brunel 10.00 35.71 4 Cam bridge 16.00 52. 86 5 Durham 11.60 30. 71 6 Glasg o w 13. 00 35.71 7 Green wich 2.00 22.8 6 8 Imperial 13.90 50. 00 9 Join t submission: Edinburgh & Heriot-W att 30.00 31. 43 10 Kent 12.00 43 .57 11 Lancaster 21.65 39. 29 12 Leeds 11.00 46. 43 13 Live rp ool 5.00 22.14 14 London Metrop olitan 4.00 19.2 9 15 London School of Economics & Pol itical Science 13.00 37.1 4 16 Manc h ester 10.90 39. 29 17 New castle 13.00 35. 00 18 Nottingham 9. 00 45.71 19 Op en Universit y 7.00 33.5 7 20 Oxford 24.50 62. 86 21 Plymouth 4.00 19.2 9 22 Queen Mary 8.20 29.2 9 23 Reading 7.7 0 25.71 24 Salford 9.80 22.8 6 25 Sheffield 10. 70 35.71 26 Southampton 28.90 40. 71 27 St And rews 7.00 36.4 3 28 Strathclyde 10 .33 29.29 29 Universit y College London 10.50 32. 86 30 W arwick 24.00 48. 57 Mean: 36.50 12. 96 6 K e nna and Berche 0 25 50 75 100 0 10 20 30 40 50 60 70 80 90 100 s N (a) 0 25 50 75 100 0 5 10 15 20 25 30 35 s N (b) Fig. 1. P anel (a) depicts quali ty of research versus quantit y of researchers for the Appl ied Mathemati cs UOA at RAE 200 8 toge ther wi th the b est fit to mod el (6) and 95% confidence inter val. Panel (b) is the equivalent plot f o r all statistics and operatio nal research groups. v anishing slo p e to the right is 0 . 001. In Fig. 1(b), the e q uiv alent full data set for the Statistics and Op era tional Re- search UOA is plotted, and the differ ence b etw een this data set and that for Applied Mathematics is immediately appa r ent. A co rrelation b etw een qualit y a nd quantit y is visible up to ab out N = 24 , beyond which there a re only tw o data p oints. How ever, the relatively hig h v alue o f the break p o int compa r ed to that of applied mathematics (exp e c ted to be a closely related discipline) gives ca us e for c o ncern, as do es the neg- ative slop e on the r ight. No other discipline ana lysed in (Kenna and Berche, 2010 b) exhibited such a phenomenon and this concern is the reaso n for the o mission of a n analysis of s tatistics a nd op erationa l r esearch there. How ever, close r insp ection of the data r eveals that the submissio n with the lar gest N v alue, and that corr esp onding to the r ightmost p oint in Fig. 1(b) is in fact a joint submission b etw een Edin burgh and Heriot-W att universities. This w as the only joint submission in this s ub ject ar ea. Arguing that this submission do es not r epresent a single co hesive “resear ch g r oup” in the same spirit as the other s in the discipline, we may consider the co rresp onding data p oint to b e an o utlier and omit it from the analysis. The r e maining data are depicted by cr osses (in red online) in the qua lit y versus quantit y plot of Fig . 2(a), in which the E din burgh/Herio t-W att datum is represented by a black cir c le. The solid line is a piecewise linear regress ion to the data for which the dashed cur ves represent the 95% confidence in terv al. One finds a breakp oint at N c = 17 . 4 ± 5 . 6. The co efficient of determinatio n is R 2 = 0 . 6 0 a nd the data pass e s the Kolmog orov-Smirnov no rmality test. As for applied mathematics, the P v alue for the a bsence of a corr elation b etw een quality a nd quantit y is less than 0 . 00 1 . How ever, unlike applied ma thematics, the P v alue for the a bsence of corr elation b etw een s and N for la rge g roups is 0 . 9, so this hypothesis cannot b e dismiss ed. This observ atio n is consistent with the results fo r other disciplines presented in (Kenna and Berche, 2010b), where w e found tha t research quality tends to s aturate in la rge gr oups provided N k > 7 . Also unlike in applied mathematics, the P v alue for the co incide nc e of slo pe s on either side o f the transition N c is 0 . 2 and the cor resp onding hypothesis cannot b e safely disgar ded. W e nonetheless arrive at the es timate for the low er critical mass for Criti cal mass of research groups 7 0 25 50 75 100 0 5 10 15 20 25 30 35 s N (a) 0 25 50 75 100 0 10 20 30 40 50 60 70 80 90 100 s N (b) Fig. 2. (a ) The same data as in Fig. 1(b ), but o mitting that correspondin g to the jo int submission of Edinburgh and Heriot-Watt universities (which correspond s to the black disc) from the fitting procedur e. (b ) A comparison between statistics & operati onal research (“ + ” symbols and solid line (red online) ) and appl ied mathemati cs (“ × ” symbols and dashed line (blue onli ne)). statistics and op er ational re search given in Eq.(2). This result app ears r easonable as it is close to that of applied mathematics, whic h is N k = 6 ± 1 . Of course it is poss ible to fit to other a ns¨ atze, such as polynomia ls, log-linear cur ves and p ower-la ws. The re s ults o f such fits are given in T able 2. Unlike o ur mo del (6) how ever, these ans¨ atze are not based o n micro scopic considera tio ns and interpretation of, a nd co mparisons b etw een the co rresp onding r esults ar e mor e difficult. Indeed, we know o f no wa y to extr act critical ma sses from these pro cedur es. Edin burg h and Heriot-W a tt Universities also submitted jointly to the Applied Ma thema tics UOA at RAE 200 8. W e find that the results of the fit to (6) are not a ppr eciably a ffected by removing the datum corresp onding to this joint s ubmission. Notwithstanding this, the statistics rep orted in T able 2 for applied mathematics co rresp ond to the data set with E din burgh/Herio t-W att remov ed. Thes e r esults are almost identical to those presented in (K enna a nd Berche, 2010a ;2010 b) for the full data set. T o further compa re statistics and op er ational res earch to a pplied mathematics, we plot the sets of data co rresp onding to b oth UOA ’s in Fig. 2(b) together with the fits coming fro m the mo del (6). The similarities in their critical masses are evident, a s are the s imilarities b etw een slop es of the piece wis e linear fits, although that for sta tis- tics and op erationa l r esearch is s hifted slightly ab ov e that for a pplied mathematics, indicating a cons istent ly better av er age per formance for comparably s ized g roups o r problems with the RAE due to the absence of a systematic approach to no r malize scores b et ween disciplines. W e be lie ve the latter is the more likely scenario . In any case, it is clea r tha t in comparis o n to applied mathematics, there ar e r elatively few statistics and op erational research teams in the UK and, of those, there are even few er which a re sup ercr itica l (and there fore op era ting with sufficient resources) in s ize. This suggests that gr eater inv estment in this sub ject a r ea is requir ed to a chiev e optimal resear ch efficiency . T o illustr ate the sup erior it y o f the mo del over the alterna tive idea that there is no rela tio nship b etw een quality and q uantit y in re search, we plot in Fig. 3 the devia- tions of the data fro m the predictions coming from b oth scena rios. In each case the data are plo tted a gainst the index v alues listed in T a ble 1, which co rresp ond to an 8 K e nna and Berche T able 2. Results for th e mode l (6) and f or alter nati v e fitting an s ¨ atze. The Edinburgh /Heri ot-Wa tt jo int submission s have b een re mov ed from an alyses of both disciplin es. Ansatz for h s ( N ) i P arameter Applied Statistics & and mathematics operational R 2 -v alue researc h a 1 + b 1 N if N ≤ N c a 1 5 ± 4 15 ± 5 a 2 + b 2 N if N ≥ N c b 1 2 . 5 ± 0 . 6 1 . 9 ± 0 . 5 a 2 32 ± 13 51 ± 35 b 2 0 . 4 ± 0 . 1 0 ± 2 N c 12 ± 2 18 ± 6 R 2 74 . 2 60.3% A 0 + A 1 N + A 2 N 2 A 0 13 ± 3 12 ± 6 A 1 1 . 50 . 3 2 . 9 ± 0 . 9 A 2 − 0 . 012 ± 0 . 003 − 0 . 059 ± 0 . 027 R 2 67 . 2% 59.9% B 0 + B 1 N + B 2 N 2 + B 3 N 3 B 0 8 ± 4 17 ± 9 B 1 2 . 4 ± 0 . 5 1 ± 3 B 2 − 0 . 05 ± 0 . 02 0 . 10 ± 0 . 2 B 2 0 . 0003 ± 0 . 0002 − 0 . 004 ± 0 . 005 R 2 70.6% 61.1% C 0 + C 1 N C 2 C 0 − 112 ± 231 − 15 ± 75 C 1 115 ± 227 27 ± 66 C 2 0 . 1 ± 0 . 2 0 . 3 ± 0 . 5 R 2 72.5% 57.5% D 0 + D 1 ln ( N + D 2 ) D 0 − 4 ± 10 − 16 ± 44 D 1 14 ± 3 20 ± 13 D 2 0 . 9 ± 1 . 5 − 4 ± 8 R 2 72.8% 57.9% Criti cal mass of research groups 9 -50 -25 0 25 50 0 32 s - s index | (a) -50 -25 0 25 50 0 32 s - index (b) Fig. 3. (a) Quali ty m easuremen ts normali sed to the ov erall mean f or statisti cs and operational research and (b) r enor malise d to t he expectation v al ues h s i given in Eq.( 6). The ti ghter distri - bution of the data about the line in (b) demonstrates the v a lidity of the model. In both plots, the abscissae index the universities listed alphabetical ly in T able 1. alphab etical ordering of the institutes which submitted to the Statistics and Op era - tional Research UOA. In Fig. 3(a), the differ ences be tw een the quality scor es and the mean quality v a lue of the 3 0 r esearch gr oups are plotted. The r ange and standa rd deviation co r resp onding to this plot are 43.6 and 10.5 resp ectively (43.6 and 10 .7 if Edinburgh/Heriot-W att is ex cluded). In Fig. 3(b), the deviations from the exp ecta- tion v alues coming from the mo del (6) a re plotted. The range and standard devia tion asso ciated with this plot (excluding Edinburgh/Heriot-W a tt) are 26.1 and 6 .7 , resp ec- tively . The tighter distribution of the data in Fig. 3(b) ov er Fig. 3(a) illustrates the v alidity of the mo del. Plots o f the type g iven in Fig. 3(a) form the bas is on which resear ch gr oups are ranked p ost RAE, with tea ms a bove and below the line deemed to be p erfor ming ab ov e and b elow av e r age, resp ectively . How e ver, such ra nkings do not compa r e like with like as they fail to take size, and hence resources , into a c c ount. W e sug gest that Fig. 3(b) for ms the bas is of a b etter system as in this plo t, p er formances are compar ed to the av er ages for teams of given sizes. Fig. 3(b) takes size into account and g ives a better indication of which g roups are punc hing ab ov e a nd b elow their weigh ts. 4. Conclusions T o summaris e, we have applied a mean- field inspired mo del to examine the relatio n- ship betw een the quality of resea rch teams in statistics and op era tio nal resea rch and the quantit y of re searchers in those teams. Our empiric al data is taken fro m the most rece nt Resea rch Assess men t E x ercise in the UK. W e find that, when an outly- ing amalg amated group is omitted the dep endency of quality upo n qua ntit y for this sub ject area is s imilar to, and co ns istent with, a multitude of o ther disc iplines which were rep o r ted o n in (Kenna and Ber che, 2010b). The mo del allows the definition of t wo critica l mass es for the discipline. the research quality of s mall ( N < N k ) and medium ( N k ≤ N < N c ) teams is strongly dep endent on the num b er of resea rchers in the gr oup. Beyond N c , larg e tea ms tend to fr agment and resear ch quality is no longer cor r elated with group size. The low er cr itical mass for s tatistics a nd op era - 10 K enn a and Berche tional resear ch is determined to b e N k = 9 ± 3, and the upp er v alue is ab out twice that. These v alues co mpa re satisfactorily to the equiv alent fo r applied ma thematics which has N k = 6 ± 1. T o further contextualize these v alues, we q uote from Kenna and Ber che (2 010b) the res ults N k ≤ 2 for pur e mathematics (a rela tively solitary resear ch discipline) and N k = 2 0 ± 4 for medical sciences (a highly collab or ative one). Notwit hstanding the fac t that some statisticia ns a nd op erationa l resea rchers were submitted to RAE 2008 as part of teams in other disciplines such as business , eco- nomics, engineering and epidemiolo g y , ab out a qua rter of statistics/op er a tional re- search gr oups submitted to RAE a re sub-cr itical, with N < N k = 9 , and therefore vulnerable. These tea ms need to strive to attain critical mass. Of the 29 tea ms ex- cluding the Edinburgh/Heriot-W a tt combination, only five (17%) hav e size a bove the upper c r itical mass of N c = 1 8. Therefor e the ma jority of statistics and op erationa l resear ch teams within the UK are under- r esourced in terms of sta ff num b ers. W e sug - gest that to increa se r esearch efficiency for this discipline in vestmen t is needed. This conclusion para llels that o f Smith and Staets k y (200 7 ) for the teaching o f statistics in the UK. References Harrison, M. (200 9) Do es high quality resea rch r equire cr itical mass ? In The Ques- tion of R& D S p e cialisation: Persp e ct ives and Policy I mplic ations (eds D. Pon tik akis, D. Kr iakou and R. v an B av al), pp 57 -59. Europ ea n Commission: JRC T ec hnical and Scient ific Rep orts. Evidence, a division of T ho mpson Reuters (Scientific) Ltd. (201 0) The futu r e of t he UK University r ese ar ch b ase (rep ort for Universities UK). Kenna, R. and Ber che, B . (2010 a ) The extensive na ture of group quality . EPL , 90 , 58002 . Kenna, R. and Berche, B. (2010 b) Cr itical mass and the dep endency of resea rch quality on gro up siz e . Scientometrics DO I: 10 .1007/ s1119 2-010-0282-9. Smith, T.M.F. and Staetsk y , L. (20 07) The teaching of statistics in UK universities. J. R. Statist. So c. A , 170 , Part 3, pp. 1-42. Ac kno wledgem en ts W e are gr ateful to Houshang Mashhoudy and Neville Hunt for inspiring discus sions.