Neutral umpires and leg before wicket decisions in test cricket

Transcription

1 J. R. Statist. Soc. A (2006) 169, Part 4, pp Neutral umpires and leg before wicket decisions in test cricket Trevor J. Ringrose Cranfield University, Shrivenham, UK [Received April Final revision February 2006] Summary. The proportions of dismissals which were leg before wicket in test cricket matches during are analysed by using generalized linear and mixed models to assess the effect of location, team and the presence or otherwise of neutral umpires. The location and the team batting are clearly significant. There is also clear evidence of an interaction showing that some teams players are out leg before wicket less often at home but there is no further interaction with neutral umpires, suggesting that this is not caused simply by home umpires favouring the home team. Keywords: Cricket; Generalized linear mixed model; Hierarchical generalized linear model 1. Introduction Until recently the two umpires in an international cricket test-match were always both provided by the home country, sometimes leading to accusations of bias by supporters of visiting teams. Hence cricket has come into line with other major international team sports and the umpires in all test-matches are now neutral, i.e. from countries other than the two that are involved. It is therefore of interest to see whether this change has had any measurable effects. Suspicions of bias usually concerned marginal decisions, often relating to run-outs, catches behind the wicket or leg before wicket (LBW). In most test-matches there are only a small number of such difficult marginal decisions, but they can be critical to the result. The ideal way to assess the prevalence, or otherwise, of bias would be to look at a very large sample of slow-motion replays of such marginal decisions, but this is clearly very labour intensive as well as carrying substantial difficulties of sampling design and so is largely impractical. Without such extensive use of replays it is difficult to see how to investigate run out or caught behind decisions, but LBWs are amenable to statistical analysis. The LBW decision is probably the most difficult and controversial that an umpire has to make. The ball must hit the batsman on the pad, without previously hitting the bat, and the umpire must be convinced that it would have gone on to hit the stumps. Furthermore, the ball must not have pitched outside the leg stump and normally must have hit the pad in front of the stumps. This last restriction is relaxed if, in the umpire s opinion, the batsman did not play a shot, in which case the ball can have hit the pad outside the off stump. There are few more complicated decisions for any sporting official, so it is hardly surprising that mistakes are sometimes made, as well as mistaken assessments of whether there were mistakes or not, or that LBW decisions lend themselves to conspiracy theories. Address for correspondence: Trevor J. Ringrose, Applied Mathematics and Operational Research Group, Cranfield University, Royal Military College of Science Shrivenham, Swindon, SN6 8LA, UK. T.J.Ringrose@cranfield.ac.uk 2006 Royal Statistical Society /06/169903

2 904 T. J. Ringrose Hence we consider the use of historical data, specifically all test-matches that were played between early 1978 and early It is a popular perception that biased umpiring may manifest itself as differential LBW rates between home and away teams, so we therefore investigate the proportion of dismissals which are LBW, subdivided by batting side, bowling side and where the game was played. Variation over some factors is expected; others may suggest bias. It is particularly interesting to do this now, to see whether the introduction of neutral umpires has had any effect. The LBW rate is clearly a crude measure, and the results that are obtained will be open to different interpretations, but nevertheless it seems worth assessing whether data covering 25 years and the switch to neutral umpires provide any evidence at all to substantiate the conspiracy theories. Section 2 describes previous analyses of test-match LBW data and Section 3 describes the data set that is used here. Section 4 then describes the fitting of various regression models, Section 5 mentions possibilities for further work and Section 6 discusses the results. 2. Previous work Sumner and Mobley (1981) considered all test-matches in , calculating the proportion of dismissals which were LBW for home and away teams in each country. Binomial two-sample tests for equality showed that away batsmen were out LBW more often in Australia (p<0:05), India and Pakistan (p<0:01). They suggested that in India and Pakistan the predominance of slower bowlers on slow pitches encourages batsmen to play forward more, so they are unlikely to be out LBW. However, these data cannot disentangle effects due to location, batting team and bowling team. Croucher (1982) used simple breakdowns of types of dismissal by team and location for England versus Australia in He concluded that English batsmen were out LBW equally often in each country, whereas Australian higher order batsmen (1 5) were out LBW more often in England than Australia, and the reverse was true for their lower order batsmen (6 11). Crowe and Middeldorp (1996) used generalized linear models (GLMs) to look at LBW rates for test-matches in Australia in They modelled the LBW rates as a function of batting team and found significantly higher rates for England, South Africa and Sri Lanka, but they suggested that this may be due to unfamiliarity with pitches or different batting and bowling styles. The present author used GLMs to analyse data, of the type that are used in this paper, for in a talk at the 1988 statistics research students meeting in Guildford and later analysed data now including neutral umpires for , talking at the Royal Statistical Society conference in Strathclyde in 1998 and the British Association for the Advancement of Science meeting in Sheffield in 1999 (Duckworth, 1999). We now update these data to , which greatly increases the number of tests with neutral umpires, and consider the use of mixed models to allow for the paired nature of the data (home team and away team in each series). 3. The data The data that are used cover all (men s) test-match series from New Zealand versus England in to New Zealand versus Pakistan in (inclusively). The data were collated by the author from the test-match score-cards in the Playfair Cricket Annual from 1979 to 2004 (Frindall, 2004). There are a total of 293 test series involving 10 countries. Of these, 125 had

3 Leg before Wicket Decisions in Test Cricket 905 two home umpires, 128 had one home and one neutral umpire and 40 had two neutral umpires. Until 1989 all test series had featured only home umpires, but in the early 1990s some series, and by 1995 all series, had one home and one neutral umpire. The exceptions to this were that a handful of series between Asian teams had two neutral umpires. Since early 2002 all test-match series have had two neutral umpires. Hence any effect of neutral umpires will be very difficult to distinguish from changes over time that were caused by other factors, since although there is some overlap it has essentially been a move from zero to one to two neutral umpires in each match. Test cricket has clearly changed greatly over this period, in particular with the appearance of reverse swing and the rise of match winning spinners in the 1990s and the recent increase in run rates, which may or may not impact on LBW rates. For each team in each series we record two counts and five factors, as follows: Y, the number of wickets which were LBW; N, the total number of wickets which fell to a bowler (run-outs and handled the ball omitted); B, the batting team, coded as 1 for England, 2 for Australia, 3 for the West Indies, 4 for India, 5 for Pakistan, 6 for New Zealand, 7 for Sri Lanka, 8 for South Africa, 9 for Zimbabwe and 10 for Bangladesh; F, the fielding team, with coding as for B (note that B F); A, the batting team at home (1) or away (2); U, for two home umpires (1), one neutral umpire (2) or two neutral umpires (3); P the place where the match was played (P = B if A = 1 and P = F if A = 2). An obvious feature of these data is that, since test-matches are (almost) never played at neutral grounds, any three of B, F, A and P immediately imply the other. In fact one match was played at a neutral venue, Pakistan versus Sri Lanka in the final of the Asian test championship in Bangladesh in March Test-matches are played in series of up to six matches, and in all cases the number of neutral umpires per match was constant throughout the series, though the identity and nationality of the neutrals usually changed. Data are summarized at the level of series rather than the individual test-match, since matches in the same series are played by substantially the same set of players, so we might expect correlation between them, and summing Y and N over the series is a simple way to obviate the need to model this correlation. All summaries and analyses are conditional on N, which is itself variable, tending to be lower for both teams if it rains or the pitches favour batsmen. In addition, the stronger team will often lose fewer wickets than their opponents, and when a team loses fewer than 20 wickets in a match those lost will be predominantly from the top of the batting order, i.e. the best batsmen. Hence N for a team winning a series will tend to contain a higher proportion of wickets of better batsmen than will N for their defeated opponents. If better batsmen are either more or less likely to be LBW than poorer batsmen, this will affect the LBW proportion. In addition, if bias manifests itself as away batsmen being more likely to be given out caught behind the wicket then this increases N without changing Y, decreasing the LBW proportion. Here we assume that both effects, if they exist, are small. Table 1 shows a crude summary of these data, with LBWs as a proportion of total dismissals (to a named bowler), cross-classified by batting and fielding team. The overall percentage of LBWs is 16.57%, with Australia, India, Pakistan and Sri Lanka all dismissing their opponents LBW more often (as a percentage) than they themselves are dismissed LBW, whereas for the others the reverse is true. The largest value in Table 1 is the astonishing figure of 30.75% of West Indian players being dismissed LBW against Pakistan. These can be subdivided by location and type of umpire, showing for example 15.96% LBWs with all-home umpires, 16.60% with one neutral umpire and 19.05% with two neutral umpires. Similarly we can pick out that in England, the West Indies and New Zealand the away team has a lower LBW rate than the home team, whereas in Australia the away team s rate is slightly higher

4 906 T. J. Ringrose Table 1. Sum of Y (first row for each team), sum of N (second row) and Y as a percentage of N (third row), by batting and fielding side, all matches Batting Results for the following fielding teams: Sum team England Australia West India Pakistan New Sri South Zimbab- Bangla- Indies Zealand Lanka Africa we desh England Australia West Indies India Pakistan New Zealand Sri Lanka South Africa Zimbabwe Bangladesh Sum and in India, Pakistan and Sri Lanka the away team s LBW rate is far higher than the home team s. However, we can also pick out that, when batting away from home with home umpires, Indian and Pakistani batsmen had the lowest LBW rates. However, simplistic comparisons like this are clearly vulnerable to the usual problems of attention being drawn to unusual features that could easily have occurred by chance, illustrating the need for a proper analysis. 4. Analysis A natural way to analyse these data is to use a GLM with a response of Y successes out of N and binomial errors, with the usual choice of link functions (McCullagh and Nelder, 1989). All series involving Bangladesh have been omitted from these analyses. This is partly because they have played far fewer games, all with at least one neutral umpire, and partly because they are so

5 Leg before Wicket Decisions in Test Cricket 907 much weaker than all other test-playing sides that it is not clear whether their results are really comparable. The dramatic weakening of Zimbabwe in early 2004 occurred after the cut-off point for the data that are used here. In addition, the one test-match that was played on a neutral ground is also omitted. This leaves 278 series between nine teams and so 556 observed Y, N pairs. Only four of the five explanatory variables (factors) can be used: U plus any three of B, F, A or P. Since A has only two levels rather than nine this should clearly be included, so U and A with any two of B, F or P are used. Inclusion of P as one of the other two factors would help interpretability as this represents something which in a given series is the same for both sides. All analyses are performed in GenStat (Payne, 2002), so the aliasing strategy is for all factor level effects to be relative to that for level 1. Hence the base-line for B, F and P is England, whereas for A and U it is playing at home with two home umpires. All of Zimbabwe s tests, and nearly all of South Africa s, have been since the introduction of neutral umpires, so some interaction terms are not estimable, affecting the degrees of freedom in Table 2 in Section 4.3. Some effects are expected. A batsman who plays primarily off the front foot is less likely to be LBW than one who plays more off the back foot. Hence if the balance of front foot and back foot play differs between countries then this will affect the LBW rate, making B significant. Fast short-pitched deliveries will result in fewer LBWs, so if the proportion of such deliveries bowled differs between countries then F may be significant. Pitches in each country have fairly consistent characteristics, so if the nature of the pitch affects the LBW rate then P will be significant. It is less clear what form this effect will take, since for example on hard pitches encouraging short-pitched bowling the ball is more likely to bounce higher, reducing the chance of LBW, but batsmen are more likely to play off the back foot, increasing the chance of LBW. The most interesting effects are A and the interactions. If A is significant then LBW rates differ between the home and away teams, whereas if A:P is significant then LBW rates differ between the home and away team more in some countries than others. If U is significant then this just means that LBW rates overall have changed since the introduction of neutral umpires, whereas if A:U is significant then the difference between LBW rates for home and away teams depends on the presence of neutral umpires, and if A:U:P is significant then this difference is greater in some countries than in others. Similar interpretations apply with B or F in place of P No series effect If no series effect is included, ignoring the paired nature of the data, then for example all cases of England batting away to Australia with two home umpires are grouped together. This was the approach that was used in the author s previous work noted in Section 2, and the 556 pairs now become 284 combinations. Hence an initial model is the GLM Y i Bi.n i, π i / g.π i / = x T i β, i = 1,...,284,.1/ where y i LBWs are observed out of n i dismissals, each combination of factor levels is coded into x i, g. / is the link function and β the vector of regression parameters. The link function made little difference, so only results for the logit link g.p/ = log{p=.1 p/} are reported. There is evidence of moderate overdispersion, i.e. that var.y i / = φπ i.1 π i / with φ > 1. For most models the ratio of deviance to degrees of freedom gives ˆφ 1:4 (McCullagh and Nelder (1989), page 128). This suggests that LBW rates for a given set of factor levels are not constant but vary between series Series effect Since there is a natural pairing in the data and overdispersion we consider a series effect (fixed or random). Now each team in each series is a separate observation and a new factor series,

6 908 T. J. Ringrose S, takes the same level for both the home and the away team s results in each series. This models some series having higher proportions of LBWs than others between the same teams due to factors which affect both teams roughly equally. For example, there is anecdotal evidence that some umpires are more willing to give LBWs than others. Similarly, both teams are playing on the same pitches in the same weather, though conditions can change during a game. However, S does not account for overdispersion that is caused by effects which do not affect both teams equally, such as changes in team composition. If S is a fixed effect then the model is overparameterized, causing problems of aliasing and collinearity. However, we are already implicitly regarding the series as a sample of those that could have been played, so S can be treated as a random effect. This seems the natural way to model the series effect as a nuisance variable, thus omitting it from further analysis and interpretation. For this reason interactions involving S are not considered. Now consider the generalized linear mixed model (GLMM) Y i u j Bi.n i, π i / g.π i / = x T i β + u j, i = 1,...,556,.2/ where u j N.0, σs 2 / is the random series effect with j =i=2 (assuming that the data are presented in pairs, halves rounded up) and g. / is the link function. See McCulloch and Searle (2001), sections 8.1, 10.3, 10.4, for more details of GLMMs. The model (2) can also be viewed as a binomial normal hierarchical generalized linear model (HGLM), as defined by Lee and Nelder (1996, 2001). In the standard GLMM approach the fixed effect parameters β are estimated by using the likelihood in which the u j must be integrated out. HGLMs are instead based on the h-likelihood h where the u j are regarded as (unobserved) data and hence no integration is required (Lee and Nelder (1996), pages ). However, because GLMM fitting usually involves numerical approximation of a difficult integral there is often little difference in practice between the two. This was the case here, where the results from the GLMM and the HGLM fitting approaches were very similar. This was also true for the binomial beta HGLM, where u j in model (2) is replaced by v j = log{u j =.1 u j /} and u j follows a beta distribution (Lee and Nelder (1996), pages ). Hence only the binomial normal HGLM results are reported Results The changes in deviance used in model selection for both the GLM (1) and the HGLM (2) are set out in Table 2, with P used as the base-line as it is clearly the most important main Table 2. Forward model selection, where the HGLM also includes random effect S Model Results for the GLM Difference in degrees Results for the HGLM of freedom Deviance Degrees of Difference in 2h Difference freedom deviance in 2h P P + B + A + U P + B + A + U + A:B P + B + A + U + A:B + U:B P + B + A + U + A:B + U:B A:U P + B + A + U + A:B + U:B A:U + A:B:U The parameters are defined in the text, Section 3.

7 Leg before Wicket Decisions in Test Cricket 909 effect. Model selection for nested GLMs is based on comparing differences in deviances with χ 2 critical values whereas that for nested fixed effects in HGLMs is similarly based on comparing the difference in 2h with χ 2 critical values (Lee and Nelder (1996), section 6.1, and Lee and Nelder (2002), page 201). Although reported in a forward selection style, all sensible subsets of the variables were considered. In both the GLM and the HGLM standard residual plots showed no problems with the models fitted. In both cases all main effects are needed, with P and B preferable to F. In the GLM, interactions involving A, B and U are significant if overdispersion is ignored, but if it is accounted for (Davison (2003), page 515) then they are not. In the HGLM, adding A:B to the main effects decreases 2h by 27.6 for only eight more parameters (significant at the 0.1% level on χ 2 8 ), but with most of the overdispersion now accounted for by S no other interactions are significant. The random effect S is clearly significant, with Table 3 showing that log. ˆσ s 2/= 2:884 and so ˆσ2 s =0:0559. Hence the scale of between-series Table 3. Parameter estimates, standard errors and t-ratios for the GLM and HGLM Description Term Results for GLM Results for HGLM P + U + A + B + A:B P + U + A + B + A:B + S Estimate Standard t Estimate Standard t error error Constant In Australia P In West Indies P In India P In Pakistan P In New Zealand P In Sri Lanka P In South Africa P In Zimbabwe P Australia B West Indies B India B Pakistan B New Zealand B Sri Lanka B South Africa B Zimbabwe B Away A neutral umpire U neutral umpires U Australia away B2.A West Indies away B3.A India away B4.A Pakistan away B5.A New Zealand away B6.A Sri Lanka away B7.A South Africa away B8.A Zimbabwe away B9.A Series log.σs 2 / Reference category England at home with two home umpires.

8 910 T. J. Ringrose variation is that approximately 95% of series effects will be within ±0:11 on the same scale as the fixed effect parameter estimates in Table 3. The scaled deviance (Lee and Nelder (2002), page 201) is on degrees of freedom, suggesting that there is still some overdispersion that is not accounted for by S, because of those factors affecting the two teams in a series differently. An important result is that nothing involving A:U is significant, so there is no evidence that neutral umpires have affected the difference between home and away LBW rates. The significance of U is due to the recent increase in LBW rates, though this could be caused by differences in either playing or umpiring styles. Parameter estimates from the HGLM with fixed effects P + U + A + B + A:B aregivenin Table 3, together with those from the equivalent GLM, purely for comparison. The two sets of results are clearly very similar; in all cases the printed GLM standard errors are slightly smaller because they do not include the overdispersion. Table 3 shows a clear difference in (the elements of) P between the three subcontinental countries India, Pakistan and Sri Lanka, whose pitches produce more LBWs than those in England (the reference category), versus the others, whose pitches may produce fewer. Indeed, a clear pattern emerges where Australia, India, Pakistan and Sri Lanka have P>0, B<0 and A:B > 0, whereas the West Indies, New Zealand, South Africa and Zimbabwe have the reverse. The only parameters which spoil this neat pattern are that Australia has P<0 and South Africa has (just) B<0. For all except South Africa B is larger in magnitude than A:B, so the batting team away effect does not quite cancel the batting team effect. Many of these parameter estimates have small t-ratios, but this only means that they do not differ significantly from the reference category, England at home. This suggests that England are much closer to the second group of countries (the West Indies, New Zealand, South Africa, Zimbabwe). Both P and S can be viewed as nuisance factors, so, because U also affects both teams equally, it is the elements of AÅB that are of most interest. The clearest conclusion is that if one of (Australia, India, Pakistan, Sri Lanka) plays at home to one of (England, the West Indies, New Zealand, South Africa, Zimbabwe) then the home team has a much lower LBW rate. Other comparisons are less conclusive but suggest that one of (Australia, India, Pakistan, Sri Lanka) playing away to one of (England, the West Indies, New Zealand, South Africa, Zimbabwe) will still tend to have a lower LBW rate than their opponents, whereas in a series between two of (Australia, India, Pakistan, Sri Lanka) the home team will have the lower LBW rate and in a series between two of (England, the West Indies, New Zealand, South Africa, Zimbabwe) the LBW rates will be roughly equal. 5. Open questions and possible further work The particular ground on which matches were played could be included as a factor, on the assumption that the nature of the pitch at a ground is consistent. For example, Sydney usually favours spin more than any other Australian pitch and Headingley favours seam bowling more than any other English (test) pitch. Domestic cricket and limited overs internationals could be investigated to assess the effects of B =F and P B, F respectively, but differences in the quality of pitches and styles and standards of play mean that they are unlikely to be meaningfully comparable with test-match data. Most football fans believe that some referees favour the home team, especially if it is a powerful club or country, but many would also agree that this is not conscious bias but simply weakness, of giving in to the pressure of large crowds. This may also apply to cricket, so a potentially interesting covariate would be the overall size of the crowd, either for a match or over a series, used as a proxy for crowd pressure on the umpires. This might help to assess

9 Leg before Wicket Decisions in Test Cricket 911 whether any perceived problem was due simply to poor umpiring, with bad decisions tending to favour the home team more often than not. An inexperienced neutral umpire would then be just as likely to favour the home team as an inexperienced home umpire. Dealing with the problem of winning teams losing a higher proportion of wickets of specialist batsmen (Section 3) would require information on which batsmen were dismissed. However, it would not be enough simply to code these as top order (1 5) or lower order (6 11) as in Croucher (1982), as this grossly oversimplifies things. The batting skill of numbers 6 9 in particular can vary dramatically between teams and over time, an extreme example being Adam Gilchrist, a world class batsman who currently usually bats at number 7 for Australia. Accounting for this feature properly would entail substantial work in both collecting and interpreting the data; simply recording the number in the order of the batsmen dismissed would be inadequate. 6. Conclusions A key conclusion is that, although LBW rates have increased slightly since the introduction of neutral umpires, there is no evidence that this effect differs between teams or locations. In particular, there is no evidence that the disparity between home and away LBW rates has been affected by neutral umpires. A two-sentence summary of the team and location effects could be as follows. Subcontinental pitches produce a higher proportion of LBWs than elsewhere. Compared with batsmen from other countries, subcontinental batsmen have a lower proportion of dismissals by LBW, especially when at home, and Australian batsmen have a lower proportion out LBW when at home. Clearly we cannot tell whether the effects of team and location are caused by umpiring or by styles of play. However, if they are caused by umpiring then neutral umpires have behaved similarly to home umpires, and there is no evidence that neutral umpires have remedied any bias in LBW rates. Finally, notwithstanding the above, we should welcome neutral umpires, not because they are from a third country, but because they are increasingly being selected on merit from a small group of the world s best umpires. Acknowledgement The author thanks Professor Youngjo Lee for his swift and helpful responses to questions about HGLMs. References Croucher, J. S. (1982) Anglo-Australian test cricket dismissals Bull. Appl. Statist., 9, Crowe, S. M. and Middeldorp, J. (1996) A comparison of leg before wicket rates between Australians and their visiting teams for test cricket series played in Australia, Statistician, 45, Davison, A. C. (2003) Statistical Models. Cambridge: Cambridge University Press. Duckworth, F. (1999) Sporting headlines. RSS News, 27, no. 3, 1 2. Frindall, W. (2004) Playfair Cricket Annual London: Headline. Lee, Y. and Nelder, J. A. (1996) Hierarchical generalized linear models (with discussion). J. R. Statist. Soc. B, 58, Lee, Y. and Nelder, J. A. (2001) Hierarchical generalised linear models: a synthesis of generalised linear models, random-effects models and structured dispersions. Biometrika, 88, Lee, Y. and Nelder, J. A. (2002) Analysis of ulcer data using hierarchical generalized linear models. Statist. Med., 21, McCullagh, P. and Nelder, J. A. (1989) Generalized Linear Models, 2nd edn. London: Chapman and Hall. McCulloch, C. E. and Searle, S. R. (2001) Generalized, Linear and Mixed Models. New York: Wiley. Payne, R. W. (ed.) (2002) The Guide to GenStat: Release 6.1. Harpenden: Lawes Agricultural Trust. Sumner, J. and Mobley, M. (1981) Are cricket umpires biased? New Scient., July 2nd,