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Sports Medicine

Research Article Volume 2 Issue 4

Survival ability of Indian and overseas batsmen on the cricket pitch in Indian premier league

Hemanta Saikia,1 Dibyojyoti Bhattacharjee2

1Assistant Professor, College of Sericulture, Assam Agricultural University, India
2Professor, Department of Statistics, Assam University, India

Correspondence: Hemanta Saikia, Assistant Professor, College of Sericulture, Assam Agricultural University, India

Received: June 28, 2018 | Published: July 2, 2018

Citation: Saikia H, Bhattacharjee D. Survival ability of Indian and overseas batsmen on the cricket pitch in Indian premier league. MOJ Sports Med. 2018;2(3):113-116. DOI: 10.15406/mojsm.2018.02.00057

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Abstract

Twenty20 format of cricket is a fast track ball game compared to the other formats of cricket viz. Test and One-day International (50-over a side). The Indian Premier League (IPL) is a national franchise based Twenty20 cricket tournament initiated by Board of Control for Cricket in India (BCCI). In this format, each batsman tries to score maximum runs in minimum balls. This fact increases the probability of dismissal of a batsman. As fall of wickets leads to the loss of resources of the batting side, thus it has an impact on the result of the game. This study tries to examine the survival ability of Indian and overseas batsmen in IPL 2012 season using a probabilistic model. The proposed model can be used to forecast the survival rate of the batsmen on the pitch in other format of cricket also, while the game is in progress. The findings of the study can be used to arrange the batting order of a team in Twenty20 cricket based on the match situation.

Keywords: cricket, probability, survival analysis, sport

Introduction

Cricket is an outdoor game played between two teams of eleven (11) players each in a circular ground. It is administered by certain rules and regulations, where the interaction between bat and ball takes place on a 22-yard hard surface in the middle of a circular ground called the cricket pitch. Unlike other sports, there are different versions of cricket. The different versions of cricket can be broadly classified as unlimited overs cricket (Test matches) and limited overs cricket (One-day and Twenty20). Indian Premier League (IPL) is a national franchise based Twenty20 format of cricket league initiated by Board of Control for Cricket in India (BCCI) in 2008. In IPL, each team faces only twenty (20) overs in a match, therefore, within these limited overs every batsman tries to score maximum runs in minimum balls. In the process of scoring runs quickly, the batsmen are exposed to the risk of losing their wicket. This fact increases the probability of dismissal of a batsman. As fall of wickets leads to the loss of resources of the batting side, so it has an impact on the result of the game. However, it does not mean that stability of a batsman on the pitch would help a team to win the match. Evidently, he should have scored runs as quickly as possible. Thus, in Twenty20 cricket, one can atleast measure how much time a batsman can survive or how many balls a batsman can face on the cricket pitch while batting. Therefore, the study makes an attempt to measure as well as compare the standing capability of Indian and overseas batsmen on the cricket pitch in IPL 2012 using survival analysis.

Survival analysis is defined as a set of methods for analyzing data where the outcome variable is the time until the occurrence of a particular event of interest.1 The event could be death due to cancer, occurrence of a disease, relief from a severe back pain, etc. Let us take an example to explain mathematical definition of survival function. Suppose the actual survival time of an individual (say) t which can be regarded as the value of a variable T (i.e. associated with the survival time). It can take any non-negative value. The different values that T can take have a probability distribution, so the variable T can be considered as a random variable. Now for the random variable T, the probability distribution function of T can be defined as F(t) and it is given by

F(t)=P(T<t)= 0 t f(x)dx MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb GaaiikaiaadshacaGGPaGaeyypa0JaamiuaiaacIcacaWGubGaeyip aWJaamiDaiaacMcacqGH9aqpjuaGdaWdXbGcbaqcLbsacaWGMbGaai ikaiaadIhacaGGPaGaaGPaVlaadsgacaWG4baaleaajugWaiaaicda aSqaaKqzadGaamiDaaqcLbsacqGHRiI8aaaa@4F91@      (1)

Which represents the probability that the survival time is less than some value t. Now the survival function is defined as the probability that the survival time is greater than or equal to t. Usually, it is denoted by S(t) and given by

S(t)=P(Tt)1P(T<t)1F(t)1 0 t f(x)dx MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGtb GaaiikaiaadshacaGGPaGaeyypa0JaamiuaiaacIcacaWGubGaeyyz ImRaamiDaiaacMcacqGHshI3caaIXaGaeyOeI0IaamiuaiaacIcaca WGubGaeyipaWJaamiDaiaacMcacqGHshI3caaIXaGaeyOeI0IaamOr aiaacIcacaWG0bGaaiykaiabgkDiElaaigdacqGHsisljuaGdaWdXb GcbaqcLbsacaWGMbGaaiikaiaadIhacaGGPaGaaGPaVlaadsgacaWG 4baaleaajugWaiaaicdaaSqaaKqzadGaamiDaaqcLbsacqGHRiI8aa aa@638A@      (2)

Therefore, the survival function can be used to represent the probability that an individual survives from the time origin to some time beyond t. The survival time or time to an event of interest can be measured in days, weeks, years, etc. in which the objects or subjects are followed over a specified period of time to pinpoint the event of interest occurs. Though its uses in medical, clinical trial, actuarial science, etc. are hefty, but still the application of survival analysis in sport (especially in cricket) is limited. A few studies have found in this regard are explicitly mentioned here. Danaher2 applied the survival analysis to find an estimate of a cricketer’s unknown batting average3 based on the product limit estimator. Similar product-limit estimation technique was adopted by Kimber and Hansford3 for assessing the batting performance of cricketers based on runs scored. The product limit estimator (PLE) is a non-parametric estimator originally proposed by Kaplan and Meier4 and it is defined as -

PLE= t i t ( 1 d i n i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadcfacaWGmbGaamyraiabg2da9Kqba+aadaqeqbGcbaqc fa4aaeWaaOqaaKqzGeGaaGymaiabgkHiTKqbaoaalaaakeaajugibi aadsgajuaGdaWgaaWcbaqcLbmacaWGPbaaleqaaaGcbaqcLbsacaWG Ubqcfa4aaSbaaSqaaKqzadGaamyAaaWcbeaaaaaakiaawIcacaGLPa aaaSqaaKqzadGaamiDaSWaaSbaaWqaaKqzadGaamyAaaadbeaajugW aiabgMYiHlaaykW7caWG0baaleqajugibiabg+Givdaaaa@5523@ , t 1   t 2   t 3    t n         MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadshajuaGpaWaaSbaaSqaaKqzadWdbiaaigdaaSWdaeqa aKqzGeWdbiaacckacaWG0bqcfa4damaaBaaaleaajugWa8qacaaIYa aal8aabeaajugib8qacaGGGcGaamiDaKqba+aadaWgaaWcbaqcLbma peGaaG4maaWcpaqabaqcLbsapeGaaiiOaiabgAci8kaacckacaWG0b qcfa4damaaBaaaleaajugWa8qacaWGUbaal8aabeaajugib8qacaGG GcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcaaaa@56CA@      (3)

Where ti be the observed times of n samples until the event of interest occurred from a given population. However, sometimes lack of information arises when observations have some information available for the event of interest but the information is not complete. This incomplete information is termed as censoring. If there is no censoring, ni is the number of survivors prior to time ti. But if there is a censoring, ni is the number of survivors minus the number of censored cases in the sample. To measure the survival capability of cricketers on the cricket pitch, censoring can be perceived in those situations when a batsman remains not-out in limited overs cricket. It can be termed as so-called right censoring data.

Following the work of Kimber and Hansford,4 product limit estimator was used by Das6 to estimate the adjusted batting average of some selected cricketers. He argues, it has been revealed from the past information that batsmen have a variable risk of getting dismissed based on their current score in the innings. Thus, he proposed to model the batsmen’s scores using generalized geometric distribution. A similar problem was also addressed by van Staden7 developing a new batting criterion named as ‘survival rate’. It is defined as the number of balls faced in all innings divided by the number of completed innings. Symbolically,

SV= Numberofballsfaced Numberofcompletedinnings = 1 n ( i=1 n b i + i=n+1 n+m b i * ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGtb GaamOvaiabg2da9Kqbaoaalaaakeaajugibiaad6eacaWG1bGaamyB aiaadkgacaWGLbGaamOCaiaaysW7caWGVbGaamOzaiaaysW7caWGIb GaamyyaiaadYgacaWGSbGaam4CaiaaysW7caWGMbGaamyyaiaadoga caWGLbGaamizaaGcbaqcLbsacaWGobGaamyDaiaad2gacaWGIbGaam yzaiaadkhacaaMe8Uaam4BaiaadAgacaaMe8Uaam4yaiaad+gacaWG TbGaamiCaiaadYgacaWGLbGaamiDaiaadwgacaWGKbGaaGjbVlaadM gacaWGUbGaamOBaiaadMgacaWGUbGaam4zaiaadohaaaGaeyypa0tc fa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacaWGUbaaaKqbaoaabm aakeaajuaGdaaeWbGcbaqcLbsacaWGIbqcfa4aaSbaaSqaaKqzadGa amyAaaWcbeaaaeaajugWaiaadMgacqGH9aqpcaaIXaaaleaajugWai aad6gaaKqzGeGaeyyeIuoacqGHRaWkjuaGdaaeWbGcbaqcLbsacaWG Ibqcfa4aa0baaSqaaKqzadGaamyAaaWcbaqcLbmacaGGQaaaaaWcba qcLbmacaWGPbGaeyypa0JaamOBaiabgUcaRiaaigdaaSqaaKqzadGa amOBaiabgUcaRiaad2gaaKqzGeGaeyyeIuoaaOGaayjkaiaawMcaaa aa@93E3@      (4)

where bi (i = 1, 2, … , n) represents the number of balls faced by the batsman in n completed innings and (i = n+1, n+2, …, n+m) represents the number of balls faced by the batsman in (n+m) not-out innings. Now on the basis of equation (2) and the batting average developed by Maini and Narayanan (2007) which is based on average exposure-to-risk (AVexposure), van Staden et al (2010) proposed a new meaningful batting average. It is based upon exposure using survival rate and it is defined as

A V survival = SV A V exposure MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadgeacaWGwbqcfa4damaaBaaaleaajugWa8qacaWGZbGa amyDaiaadkhacaWG2bGaamyAaiaadAhacaWGHbGaamiBaaWcpaqaba qcLbsapeGaeyypa0tcfa4damaalaaakeaajugibiaadofacaWGwbaa keaajugibiaadgeacaWGwbqcfa4aaSbaaSqaaKqzadGaciyzaiaacI hacaGGWbGaam4BaiaadohacaWG1bGaamOCaiaadwgaaSqabaaaaaaa @5278@

Where A V exposure = i=1 n x i + i=n+1 n+m x i * i=1 n r i + i=n+1 n+m r i * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGbb GaamOvaKqbaoaaBaaaleaajugWaiGacwgacaGG4bGaaiiCaiaad+ga caWGZbGaamyDaiaadkhacaWGLbaaleqaaKqzGeGaeyypa0tcfa4aaS aaaOqaaKqbaoaaqahakeaajugibiaadIhajuaGdaWgaaWcbaqcLbma caWGPbaaleqaaaqaaKqzadGaamyAaiabg2da9iaaigdaaSqaaKqzad GaamOBaaqcLbsacqGHris5aiabgUcaRKqbaoaaqahakeaajugibiaa dIhajuaGdaqhaaWcbaqcLbmacaWGPbaaleaajugWaiac0bUGQaaaaa WcbaqcLbmacaWGPbGaeyypa0JaamOBaiabgUcaRiaaigdaaSqaaKqz adGaamOBaiabgUcaRiaad2gaaKqzGeGaeyyeIuoaaOqaaKqbaoaaqa hakeaajugibiaadkhajuaGdaWgaaWcbaqcLbmacaWGPbaaleqaaaqa aKqzadGaamyAaiabg2da9iaaigdaaSqaaKqzadGaamOBaaqcLbsacq GHris5aiabgUcaRKqbaoaaqahakeaajugibiaadkhajuaGdaqhaaWc baqcLbmacaWGPbaaleaajugWaiac0rTGQaaaaaWcbaqcLbmacaWGPb Gaeyypa0JaamOBaiabgUcaRiaaigdaaSqaaKqzadGaamOBaiabgUca Riaad2gaaKqzGeGaeyyeIuoaaaaaaa@8AA5@      (5)

Where r 1 , r 2 ,..., r n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajugibiaacYcacaaMe8Ua amOCaKqbaoaaBaaaleaajugWaiaaikdaaSqabaqcLbsacaGGSaGaaG jbVlaac6cacaGGUaGaaiOlaiaacYcacaaMe8UaamOCaKqbaoaaBaaa leaajugWaiaad6gaaSqabaaaaa@4B99@  and r n+1 * , r n+2 * ,..., r n+m * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aa0baaSqaaKqzadGaamOBaiabgUcaRiaaigdaaSqaaKqzadGa iai1cQcaaaqcLbsacaGGSaGaaGjbVlaadkhajuaGdaqhaaWcbaqcLb macaWGUbGaey4kaSIaaGOmaaWcbaqcLbmacGaGilOkaaaajugibiaa cYcacaaMe8UaaiOlaiaac6cacaGGUaGaaiilaiaaysW7caWGYbqcfa 4aa0baaSqaaKqzadGaamOBaiabgUcaRiaad2gaaSqaaKqzadGaiaiY cQcaaaaaaa@5AAA@  denote the batsman’s exposure in n completed innings and m not-out innings respectively. Here, r i =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaamyAaaWcbeaajugibiabg2da9iaaigda aaa@3CAE@ if the score in ith innings is a completed score, which means that, the exposure is one for all completed innings.

Otherwise,  r i * ={ b i * / b ¯ ,ifthescoreisanotoutscoreand b i * < b ¯ 1,else MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aa0baaSqaaKqzadGaamyAaaWcbaqcLbmacGaGWkOkaaaajugi biabg2da9KqbaoaaceaajugibqaabeGcbaqcfa4aaSGbaOqaaKqzGe GaamOyaKqbaoaaDaaaleaajugWaiaadMgaaSqaaKqzadGaiaiPcQca aaaakeaajugibiqadkgagaqeaaaacaGGSaGaaGzbVlaadMgacaWGMb GaaGjbVlaadshacaWGObGaamyzaiaaysW7caWGZbGaam4yaiaad+ga caWGYbGaamyzaiaaysW7caWGPbGaam4CaiaaysW7caWGHbGaaGjbVl aad6gacaWGVbGaamiDaiabgkHiTiaad+gacaWG1bGaamiDaiaaysW7 caWGZbGaam4yaiaad+gacaWGYbGaamyzaiaaysW7caWGHbGaamOBai aadsgacaaMe8UaamOyaKqbaoaaDaaaleaajugWaiaadMgaaSqaaKqz adGaiaiMcQcaaaqcLbsacqGH8aapceWGIbGbaebaaOqaaKqzGeGaaG ymaiaacYcacaaMf8UaaGzbVlaaywW7caWGLbGaamiBaiaadohacaWG LbaaaOGaay5Eaaaaaa@87BC@

Where b ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGIb Gbaebaaaa@3785@ is the average number of balls faced by a batsman in his (m+n) innings and it is defined as,

b ¯ = Numberofballsfaced Numberofinnings = 1 n+m ( i=1 n b i + i=n+1 n+m b i * ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGIb GbaebacqGH9aqpjuaGdaWcaaGcbaqcLbsacaWGobGaamyDaiaad2ga caWGIbGaamyzaiaadkhacaaMe8Uaam4BaiaadAgacaaMe8UaamOyai aadggacaWGSbGaamiBaiaadohacaaMe8UaamOzaiaadggacaWGJbGa amyzaiaadsgaaOqaaKqzGeGaamOtaiaadwhacaWGTbGaamOyaiaadw gacaWGYbGaaGjbVlaad+gacaWGMbGaaGjbVlaadMgacaWGUbGaamOB aiaadMgacaWGUbGaam4zaiaadohaaaGaeyypa0tcfa4aaSaaaOqaaK qzGeGaaGymaaGcbaqcLbsacaWGUbGaey4kaSIaamyBaaaajuaGdaqa daGcbaqcfa4aaabCaOqaaKqzGeGaamOyaKqbaoaaBaaaleaajugWai aadMgaaSqabaaabaqcLbmacaWGPbGaeyypa0JaaGymaaWcbaqcLbma caWGUbaajugibiabggHiLdGaey4kaSscfa4aaabCaOqaaKqzGeGaam OyaKqbaoaaDaaaleaajugWaiaadMgaaSqaaKqzadGaiaiPcQcaaaaa leaajugWaiaadMgacqGH9aqpcaWGUbGaey4kaSIaaGymaaWcbaqcLb macaWGUbGaey4kaSIaamyBaaqcLbsacqGHris5aaGccaGLOaGaayzk aaaaaa@8C38@

At this point, it is very crucial to clarify that the word ‘survival rate’ stands here to exemplify the capability of a batsman to remain present on the cricket pitch during a match. So that there will be no confusion with the measure defined by van Staden7 (c.f. equation (4)). When we mentioned the word “survival” in terms of batsman in cricket, it could be the end of the time spent by a batsman on the pitch before dismissal. As mentioned earlier, time to an event of interest can be measured in days, weeks, etc. Similarly, a batsman’s ability to stand on the pitch in the match(es) can be measured in terms of number of balls faced by him. It can be considered as an outcome variable in so-called survival analysis. Thus, if a batsman has been facing consistently colossal number of balls in the match(es) then he would have the higher probability of survival on the pitch.

Methodology

The approach of this study is different than earlier studies, as it does not focus on any single performance statistics of batsman like batting average, strike rate, etc. Instead, a non-parametric estimator established by Kaplan and Meier (1958) a long back is used to calculate survival probability of batsman on the pitch based on the information of number of balls faced as well as whether the player was out or not-out in the match(es). The methodology applied here is discussed below in details. Let Oi = 1, if a batsman out in the ith ball of a match and 0 otherwise, ni be the number of batsmen survives prior to bi balls of a match, where bi is the observed number of balls faced by the batsmen in a match. However, as mentioned earlier, to measure the survival capability of batsmen, censoring (more specifically right censoring) can be perceived in those situation of a match when a batsman remains not-out in limited overs cricket. Therefore, if any batsman remains not-out in a limited overs cricket then ni be the number of batsmen survives minus the number of batsmen not-out in prior to bi balls of a match. Now the Kaplan-Meier (KM) estimator in terms of the game of cricket is defined as

S(b)= b i < b i+1 ( 1 O i n i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGtb GaaiikaiaadkgacaGGPaGaeyypa0tcfa4aaebCaOqaaKqbaoaabmaa keaajugibiaaigdacqGHsisljuaGdaWcaaGcbaqcLbsacaWGpbqcfa 4aaSbaaSqaaKqzadGaamyAaaWcbeaaaOqaaKqzGeGaamOBaKqbaoaa BaaaleaajugWaiaadMgaaSqabaaaaaGccaGLOaGaayzkaaaaleaaju gibiaadkgajuaGdaWgaaadbaqcLbmacaWGPbaameqaaKqzGeGaeyip aWJaamOyaKqbaoaaBaaameaajugWaiaadMgacqGHRaWkcaaIXaaame qaaaWcbaaajugibiabg+Givdaaaa@5708@  Where b1  b2  … and i = 1, 2, …, 120 (4)

Thus, the survival probabilities of the batsmen on the cricket pitch are computed using the above defined estimator. As we know that the precision of any estimate is reflected in the standard error of the estimate. Therefore, the standard error of KM estimate is computed as an essential aid to the interpretation of estimate. In this regard, Peto et. al. (1977) proposed a formula to compute standard error ofand it is defined as

se{ S ^ (b) }= S ^ (b) { 1 S ^ (b) } n i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb GaamyzaKqbaoaacmaakeaajugibiqadofagaqcaiaacIcacaWGIbGa aiykaaGccaGL7bGaayzFaaqcLbsacqGH9aqpjuaGdaWcaaGcbaqcLb saceWGtbGbaKaacaGGOaGaamOyaiaacMcacaaMc8Ecfa4aaOaaaOqa aKqbaoaacmaakeaajugibiaaykW7caaIXaGaeyOeI0Iabm4uayaaja GaaiikaiaadkgacaGGPaaakiaawUhacaGL9baaaSqabaaakeaajuaG daGcaaGcbaqcLbsacaWGUbqcfa4aaSbaaSqaaKqzadGaamyAaaWcbe aaaeqaaaaaaaa@55F6@      (5)

The expression (5) is conservative for standard error of because the standard errors will tend to be larger than they actually ought to be. Thus, the formula proposed by Greenwood’s (1926) for standard error of is usually recommended. The Greenwood’s (1926) standard error formula can be defined as

se{ S ^ (b) }= S ^ (b) { O i n i ( n i O i ) } 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb GaamyzaKqbaoaacmaakeaajugibiqadofagaqcaiaacIcacaWGIbGa aiykaaGccaGL7bGaayzFaaqcLbsacqGH9aqpceWGtbGbaKaacaGGOa GaamOyaiaacMcajuaGdaGadaGcbaqcfa4aaabCaOqaaKqbaoaalaaa keaajugibiaad+eajuaGdaWgaaWcbaqcLbmacaWGPbaaleqaaaGcba qcLbsacaWGUbqcfa4aaSbaaSqaaKqzadGaamyAaaWcbeaajuaGdaqa daGcbaqcLbsacaWGUbqcfa4aaSbaaSqaaKqzadGaamyAaaWcbeaaju gibiabgkHiTiaad+eajuaGdaWgaaWcbaqcLbmacaWGPbaaleqaaaGc caGLOaGaayzkaaaaaaWcbaaabaaajugibiabggHiLdaakiaawUhaca GL9baajuaGdaahaaWcbeqaaKqbaoaalaaaleaajugWaiaaigdaaSqa aKqzadGaaGOmaaaaaaaaaa@6438@  For bkbbk+1          (6)

Now confidence interval for the corresponding value of the survival function can be obtained based on the above standard error of KM estimate. Finally, based on the KM estimate, a survival plot is being depicted to identify the survival ability of Indian and overseas batsmen on the cricket pitch.

Hypothesis testing

Hypothesis testing is the modest way of examining survival ability of overseas and Indian batsmen on the cricket pitch. It allows assessing the extent to which whether an observed set of data are consistent with a particular hypothesis or not. Here the working hypothesis is that there is no difference between the survival ability of overseas and Indian batsmen on the cricket pitch. The well-known log-rank test is used to test the working hypothesis. This is the appropriate non-parametric test to use when the right censored data are non-informative, as the case is comparable here in case of not-out batsmen. In order to apply log-rank test, the survival ability of overseas and Indian batsmen computed separately. It compares observed and expected number of event of interest from both the groups. The groups in log-rank test are labeled as overseas (coded as “1”) and Indian (coded as “2”) players. Now suppose there are k different distinct balls, b1 < b2 < … < bk, where batsmen are out across the two groups. Let O1i be the individual batsman out in overseas group and O2i be batsman in Indian group. Again, suppose that n1i be the total number batsmen out in overseas group and n2i be the total number batsmen out in Indian group. Consequently, there are Oi = O1i+O2i batsmen out in a tournament from total of ni = n1i + n2i batsmen, at bi ball. Now, under the null hypothesis, it is defined as

W L = U L 2 V L ~ χ 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGxb qcfa4aaSbaaSqaaKqzadGaamitaaWcbeaajugibiabg2da9Kqbaoaa laaakeaajugibiaadwfajuaGdaqhaaWcbaqcLbmacaWGmbaaleaaju gWaiaaikdaaaaakeaajugibiaadAfajuaGdaWgaaWcbaqcLbmacaWG mbaaleqaaaaajugibiaac6hacqaHhpWyjuaGdaqhaaWcbaqcLbmaca aIXaaaleaajugWaiaaikdaaaaaaa@4E94@      (7)

Where U L = i=1 k ( O 1i E 1i ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGvb qcfa4aaSbaaSqaaKqzadGaamitaaWcbeaajugibiabg2da9Kqbaoaa qahakeaajuaGdaqadaGcbaqcLbsacaWGpbqcfa4aaSbaaSqaaKqzad GaaGymaiaadMgaaSqabaqcLbsacqGHsislcaWGfbqcfa4aaSbaaSqa aKqzadGaaGymaiaadMgaaSqabaaakiaawIcacaGLPaaaaSqaaKqzad GaamyAaiabg2da9iaaigdaaSqaaKqzadGaam4AaaqcLbsacqGHris5 aKqbaoaaCaaaleqabaqcLbmacaaIYaaaaaaa@54CC@ and V L =Var( U L )= i=1 k v 1i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb qcfa4aaSbaaSqaaKqzadGaamitaaWcbeaajugibiabg2da9iaadAfa caWGHbGaamOCaKqbaoaabmaakeaajugibiaadwfajuaGdaWgaaWcba qcLbmacaWGmbaaleqaaaGccaGLOaGaayzkaaqcLbsacqGH9aqpjuaG daaeWbGcbaqcLbsacaWG2bqcfa4aaSbaaSqaaKqzadGaaGymaiaadM gaaSqabaaabaqcLbmacaWGPbGaeyypa0JaaGymaaWcbaqcLbmacaWG RbaajugibiabggHiLdaaaa@54DC@

Since the different balls are independent of one another, the term V L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb qcfa4aaSbaaSqaaKqzadGaamitaaWcbeaaaaa@3A24@ (i.e. variance of U L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGvb qcfa4aaSbaaSqaaKqzadGaamitaaWcbeaaaaa@3A23@ ) is sum of the variances of O 1i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGpb qcfa4aaSbaaSqaaKqzadGaaGymaiaadMgaaSqabaaaaa@3AF6@ and it is given by

v 1i = n 1i n 2i O i ( n i O i ) n i 2 ( n i 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG2b qcfa4aaSbaaSqaaKqzadGaaGymaiaadMgaaSqabaqcLbsacqGH9aqp juaGdaWcaaGcbaqcLbsacaWGUbqcfa4aaSbaaSqaaKqzadGaaGymai aadMgaaSqabaqcLbsacaWGUbqcfa4aaSbaaSqaaKqzadGaaGOmaiaa dMgaaSqabaqcLbsacaWGpbqcfa4aaSbaaSqaaKqzadGaamyAaaWcbe aajuaGdaqadaGcbaqcLbsacaWGUbqcfa4aaSbaaSqaaKqzadGaamyA aaWcbeaajugibiabgkHiTiaad+eajuaGdaWgaaWcbaqcLbmacaWGPb aaleqaaaGccaGLOaGaayzkaaaabaqcLbsacaWGUbqcfa4aa0baaSqa aKqzadGaamyAaaWcbaqcLbmacaaIYaaaaKqbaoaabmaakeaajugibi aad6gajuaGdaWgaaWcbaqcLbmacaWGPbaaleqaaKqzGeGaeyOeI0Ia aGymaaGccaGLOaGaayzkaaaaaaaa@66AD@      (8)

The statistics WL summarizes the extent to which the observed number of survival balls in the two groups of batsmen deviate from those expected number of survival balls in the cricket pitch, under the null hypothesis of no group differences. The larger the value of log-rank test statistics (i.e. WL), greater the evidence against the null hypothesis.

Data consideration and analysis

In IPL 2012, each franchisee has maximum of eight overseas players per squad. However, only four of them can be played in the playing XI for each match. There were 14 league matches usually played by each of the IPL team prior to knock-out stage of the tournament. Therefore, a lot of Indian as well as overseas players' performance (i.e. number of balls faced per match and out or not-out) can be collected from the scorecard of the matches. All this scorecard information is collected from the website www.espncricinfo.com from 2012 season of the IPL. A total of 1123 players are considered for the study, out of which 459 are overseas players and 664 are Indian players. The results of the analyses based on the methodology discussed above are provided below. From the above table, it has been observed that there are 103 and 162 cases are censored (i.e. not-out cases) corresponding to overseas and Indian players respectively. Overall 23.6% not-out case is being found for a total of 1123 cases (Table 1) (Table 2) (Table 3) (Table 4). 

Player Type

Total N

N of Events

Censored

N

Percent

Overseas

459

356

103

22.40%

Indian

664

502

162

24.40%

Total

1123

858

265

23.60%

Table 1 Descriptive information of players in IPL 2012

Player Type

Mean

Estimate

Std. Error

95% Confidence Interval

Lower Bound

Upper Bound

Overseas

20.581

0.853

18.908

22.253

Indian

18.444

0.673

17.124

19.763

Overall

19.356

0.534

18.31

20.403

Table 2 Group means for survival time in terms of number of balls faced

Player Type

Median

Estimate

Std. Error

95% Confidence Interval

Lower Bound

Upper Bound

Overseas

15

1.058

12.927

17.073

Indian

15

0.728

13.573

16.427

Overall

15

0.626

13.774

16.226

Table 3 Group medians for survival time in terms of number of balls faced

Overall Comparisons

Log Rank (Mantel-Cox)

Chi-Square

Df

Sig.

4.289

1

0.038

Table 4 Hypothesis testing based on log-rank test

From the above survival plot, one can easily be observed that there is a very little difference between standing ability of overseas and Indian batsmen on the pitch in IPL 2012. Up to 18 balls there is no difference between overseas and Indian batsmen. However, after 20 balls and up to 57 balls, it has been observed that overseas batsmen have moderately more standing ability on the cricket pitch than Indian players in IPL 2012. The following log-rank test has also confirmed that there is a significant difference in terms of standing ability on the cricket pitch between overseas and Indian batsmen (as p-value is 0.038 < 0.05) in IPL 2012. Earlier researcher such as Stefani and Clarke,8 Harville and Smith,9 Clarke and Norman,10 Clarke and Allsopp11 and Allsopp and Clarke12 have acknowledged that there is a significant home advantage for home team in the game of cricket. Now, both survival plot as well as log-rank test, have confirmed that there is a advantage for overseas batsmen than Indian batsmen. Thus, the advantage for Indian batsmen playing in their home country or home ground is become a question mark in IPL 2012. This finding can be considered as future scope of this research. May be there was no significant home advantage in IPL 2012 for the Indian batsmen (Figure 1).

Figure 1 Survival functions of Indian and overseas batsmen in IPL 2014.

Conclusion

This study tries to examine the survival ability of Indian and overseas batsmen on the cricket pitch using survival analysis in Indian Premier League. The study has identified that overseas batsmen have moderately more standing ability on the cricket pitch than Indian batsmen after they faced 20 number of balls in IPL 2012. However, up to 20 number of balls, there is no difference between overseas and Indian batsmen in terms of standing ability on the cricket pitch. Since overseas batsmen have moderate advantage than Indian batsmen in IPL 2012; therefore, the home advantage for Indian players in IPL 2012 can be considered as future scope of the study. The proposed survival model can be used to forecast the survival rate of the batsmen in other format of cricket also, while the game is in progress. It can also be used to arrange the batting order of a team in the game of cricket based on the match situation.13‒15

Acknowledgments

None.

Conflicts of interest

Author declares that there is no conflict of interest.

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