Adjusting the factor \(\frac{1}{400}\) for Test cricket

Finding a suitable estimate for \(\frac{1}{400}\)

Additional factors: Impacts of home ground and toss

Training period: June 17, 2017 to June 17, 2021

    Pearson's Chi-squared test with Yates' continuity correction

data:  home_impact
X-squared = 26.016, df = 1, p-value = 3.386e-07

    Pearson's Chi-squared test with Yates' continuity correction

data:  toss_impact
X-squared = 9.216, df = 1, p-value = 0.002399

Additional factors: Impacts of home ground and toss

In a hypothetical match between \(i\) (home team) and \(j\) (away team), the expected scores of a team \(i\) can be formulated as a combination of the partial expected scores of team \(i\), each of which is an attempt to introduce the two additional impacts: home (\(h\)) and toss (\(t\)). \[\begin{equation} E_{i,home} = \frac{1}{1 + 10 ^ {- \frac{1}{20} {(R_i - R_j + h_{i, j}) g(RD_j)}}} \label{7} \end{equation}\] \[\begin{equation} E_{i,toss} = \frac{1}{1 + 10 ^ {- \frac{1}{20} {(R_i - R_j + t_{i, i}) g(RD_j)}}} \label{8} \end{equation}\] Similarly, the expected scores of team \(j\) can be formulated as a combination of the partial expected scores of team \(j\) given by \[\begin{equation} E_{j,away} = \frac{1}{1 + 10 ^ {- \frac{1}{20} {(R_j - R_i + a_{j, i}) g(RD_j)}}} \label{9} \end{equation}\] \[\begin{equation} E_{j,toss} = \frac{1}{1 + 10 ^ {- \frac{1}{20} {(R_j - R_i + t_{j, i}) g(RD_j)}}} \label{10} \end{equation}\]

\(h_{i,j}\) and \(a_{j,i}\)’s constitute the home impact (h) whereas \(t_{i, i}\), \(t_{j, i}\), \(t_{j, j}\), and \(t_{i, j}\),’s constitute the toss impact (t).

Formulation: Impacts of home ground and toss

\[ \begin{aligned} h_{i,j} &= \frac{(matches\ won\ by\ i\ vs\ j + 0.5 \times matches\ drawn) - (matches\ lost\ by\ i\ vs\ j + 0.5 \times matches\ drawn)}{matches\ played\ between\ i\ and\ j} \\ &= \frac{matches\ won\ by\ i\ vs\ j - matches\ lost\ by\ i\ vs\ j}{matches\ played\ between\ i\ and\ j} \end{aligned} \]

\[ \begin{aligned} a_{j,i} &= \frac{(matches\ won\ by\ j\ vs\ i + 0.5 \times matches\ drawn) - (matches\ lost\ by\ j\ vs\ i + 0.5 \times matches\ drawn)}{matches\ played\ between\ j\ and\ i} \\ &= \frac{matches\ won\ by\ j\ vs\ i - matches\ lost\ by\ j\ vs\ i}{matches\ played\ between\ i\ and\ j} \end{aligned} \]

Clearly, \(h_{i, j} = - a_{j, i} \ \ \ \ \forall \ i, j\)

\[ toss_{i,i} = \begin{cases} toss\ win\ impact\ in\ host\ country\ i\ & \text{if team}\ i \text{ wins the toss} \\ toss\ lose\ impact\ in\ host\ country\ i\ & \text{if team}\ i \text{ wins the toss} \\ \end{cases} \]

\[ toss_{j,i} = \begin{cases} toss\ win\ impact\ in\ host\ country\ i\ & \text{if team}\ j \text{ wins the toss} \\ toss\ lose\ impact\ in\ host\ country\ i\ & \text{if team}\ j \text{ loses the toss} \\ \end{cases} \]

Expected scores in the form of standard distributions

Observe that by replacing \(10\) by \(e\), the partial expected scores resemble distribution functions of logistic distributions with respect to random variables \(U = \tan (\frac{\pi}{2}H)\) and \(V = \tan (\frac{\pi}{2}T)\). The monotone increasing, bijective transformation \(x \mapsto \tan (\frac{\pi}{2}x)\) can be taken to remove the theoretical ambiguity of \(H\) and \(T\) taking only values in the range \([-1, 1]\) but it would create complexities in exponential terms taking values \(- \infty\) and \(\infty\) which would lead to expected scores turning out to be \(0\) and \(1\) more often than not.

\[\begin{equation} F(h) = \frac{1}{1 + e ^ {- \frac{1}{20} {(R_i - R_j + h) g(RD_j)}}} = \frac{1}{1 + e ^ {- \frac{g(RD_j)}{20} (h - (R_j - R_i))}} \label{11} \end{equation}\] \[\begin{equation} F(t) = \frac{1}{1 + e ^ {- \frac{1}{20} {(R_i - R_j + t) g(RD_j)}}} = \frac{1}{1 + e ^ {- \frac{g(RD_j)}{20} (t - (R_j - R_i))}} \label{12} \end{equation}\]

Similarity between impacts and logistic distributions

    Asymptotic one-sample Kolmogorov-Smirnov test

data:  h_values
D = 0.09386, p-value = 0.1696
alternative hypothesis: two-sided

    Asymptotic one-sample Kolmogorov-Smirnov test

data:  t_values
D = 0.035952, p-value = 0.9936
alternative hypothesis: two-sided

Kolmogorov-Smirnov tests suggest the acceptance of null hypotheses in both cases which imply that both \(H\) and \(T\) are derived from logistic distributions.

Farlie-Gumbel-Morgernstern family of copula

Gumbel proposed a model with an association parameter \(\omega \in [1, 1]\), a distribution of the Farlie-Gumbel-Morgernstern type called the Gumbel type 2 distribution, \[ (x, y) \mapsto F(x)G(y) \left[ 1 + \omega (1 - F(x))(1 - G(y)) \right], \quad x, y \in \mathbb{R} \] The association parameter \(\omega\) satisfies the relation \(\rho = \frac{\omega}{3}\), but only over a limited range \(|\rho| \leq \frac{1}{3}\), where \(\rho\) denotes Spearman’s correlation coefficient. It can be shown that, \[ \begin{aligned} \rho(X,Y) &= 12 \int_0^1 \int_0^1 ( C_{\omega}(x, y) - xy ) \, dx dy \\ &= 12 \int_0^1 \int_0^1 ( xy [1 + \omega (1-x)(1-y)] - xy) \, dx dy \\ &= \frac{\omega}{3} \end{aligned} \]

For the training dataset used in this study, \(\rho\) turns out to be \(-0.1812159\) so the above choice of association parameter (\(\omega = -0.5436477\)) can be used.

Formulation of Expected scores using FGM family

Data based on training period

Exploratory Data Analysis: Density plots

Exploratory Data Analysis: Trend Analysis

Applying the model on WTC 2021-23

Comparison of rankings after ICC WTC 2021-23

The expected scores in most matches turn out to be fairly decent from a predictive perspective as the model correctly predicts the winner in 44 of the 56 non drawn matches. The changes in ratings and rating deviations are also justified in accordance with the result of the matches.

The above table is as per ICC’s ratings immediately after the completion of WTC 2021-23 cycle, on July 31, 2023.

Confidence Intervals of Expected Scores

\(E_A\) can be written as \(p = p_1 p_2 [1 + \omega (1 - p_1) (1 - p_2)]\) where,

\[ p_1 = \frac{1}{1 + e^{-L_1}} \ \ ; \hspace{1cm} L_1 = \frac{g(RD_B)}{20} (h - (R_B - R_A)) \] \[ p_2 = \frac{1}{1 + e^{-L_2}} \ \ ; \hspace{1cm} L_2 = \frac{g(RD_B)}{20} (t - (R_B - R_A)) \]

Expected scores are functions of \(h\) and \(t\) and hence, essentially functions of \(\boldsymbol{L} = (L_1, L_2)\).

\[\begin{equation} \frac{\partial p}{\partial L_1} = p_2[1 + (1 + 2p_1p_2 -2p_1 - p_2)\omega] \frac{e^{-L_1}}{(1 + e^{-L_1})^2} \label{14} \end{equation}\]

\[\begin{equation} \frac{\partial p}{\partial L_2} = p_1[1 + (1 + 2p_1p_2 -2p_2 - p_1)\omega] \frac{e^{-L_2}}{(1 + e^{-L_2})^2} \label{15} \end{equation}\]

\[\begin{equation} \operatorname{Var}(L_1) = \left(\frac{g(RD_B)}{20}\right)^2 \operatorname{Var}(h) \label{16} \end{equation}\]

\[\begin{equation} \operatorname{Var}(L_2) = \left(\frac{g(RD_B)}{20}\right)^2 \operatorname{Var}(t) \label{17} \end{equation}\]

\[\begin{equation} \operatorname{Cov}(L_1, L_2) = \left(\frac{g(RD_B)}{20}\right)^2 \operatorname{Cov}(h, t) \label{18} \end{equation}\]

Using the above equations, \(\operatorname{Var}(p)\) can be obtained by, \[\begin{equation} \operatorname{Var}(p) = \begin{pmatrix} \frac{\partial p}{\partial L_1} & \frac{\partial p}{\partial L_2} \end{pmatrix} \operatorname{Cov}(\boldsymbol{L}) \begin{pmatrix} \frac{\partial p}{\partial L_1} \\[6pt] \frac{\partial p}{\partial L_2} \end{pmatrix} \label{19} \end{equation}\]

Confidence Intervals for Expected Scores in WTC 2021-23

Prediction of drawn Test matches

Keeping the effect of exogenous variables such as rain-affected matches or other hazardous weather conditions, flat pitches, imposition of suspensions or draws due to political and/or various security concerns etc. in mind, two of the most obvious performance based factors to infer on draws are:

• Part of the probability that none of \(A\) and \(B\) wins : \((1 - E_A - E_B)\)
• Closeness of \(E_A\) and \(E_B\) : \(|E_A - E_B|\)

Handling both factors to predict draws

Define, for \(\alpha \in (0,1)\), a convex combinbation, \[ D_{\alpha, A, B} = \alpha (1 - E_A - E_B) + (1 - \alpha) |E_A - E_B| \]

A well predicted drawn Test match should produce a high value of \(D_{\alpha, A, B}\) for a certain choice of \(\alpha \in (0,1)\). Equivalently, we might be interested to find a choice of \(\alpha\) for which a significantly large proportion of matches having high \(D_{\alpha, A, B}\) values actually result in draws. A trade-off occurs between choices of \(\alpha \in (0,1)\) and choices of a significantly large proportion i.e, \(q\)-th quantile of the \(D_{\alpha, A, B}\) values of all the matches in a specific time period.

Trade-off between choice of \(\alpha\) and \(q\)-th quantile of \(D_{\alpha, A, B}\)

The following table shows how many of the 12 drawn Test matches during WTC 2021-23 could be predicted by a trade-off between the choices of \(\alpha\) and \(q\)-th cutoff quantile.

Choosing \(\alpha\) and cutoff \(q\)-th quantile

For the chosen time period, \(\alpha \in [0.55, 0.65]\) yields the highest proportion (\(9\) out of \(12\)) of accurately predicted drawn Test matches. A higher proportion is maintained for the particular choice of \(\alpha = 0.6\) for several choices of the quantile \(q\). Based on our findings from the test data, we can consider \((0.6, 0.67)\) to be a sensible choice of \((\alpha, q)\).

Margin of Victory (MOV)

Let us reformulate \(S_A\) as,

\[ S_A = \begin{cases} \frac{1 + MOV}{2} & \text{if team } A \text{ wins the match} \\ \frac{1 - MOV}{2} & \text{if team } A \text{ loses the match} \\ \frac{1}{2} & \text{otherwise} \end{cases} \]

where \(MOV\) for \(i\)-th match is defined to be,

\[ MOV_i = \left(\frac{R_i - R_{min}}{range_{R}}\right)^{\beta_i} \left(\frac{W_i - W_{min}}{range_{W}}\right)^{1 - \beta_i} + I_i \left(\frac{E4P_i + ERM_i}{TR_i}\right) \]

The scaling used in the reformulation of \(S_A\) is constructed in a way that the mean \((= \frac{1}{2})\) is conserved from the original definition of \(S_A\).

Notations used in formulation of MOV

\[ R_i = \text{margin of victory by runs in } i \text{-th match} \] \[ W_i = \text{margin of victory by wickets in } i \text{-th match} \] \[ R_{min} = \text{minimum margin of runs in which a match is won} \] \[ W_{min} = \text{minimum margin of wickets in which a match is won} \] \[ \beta_i = \begin{cases} 1 & \text{if } i \text{-th match is won by margin of runs} \\ 0 & \text{otherwise} \end{cases} \] \[ E4P_i = \frac{E4R_i}{E4R} = \frac{\text{Expected 4th innings score of winner in } i \text{-th match}}{\text{Overall expected 4th innings score}} \] \[ ERM_i = \text{margin of victory by excess runs over an innings in } i \text{-th match} \] \[ TR_i = \text{Total runs scored in } i \text{-th match} \] \[ I_i = \begin{cases} 1 & \text{if } i \text{-th match is won by an innings margin} \\ 0 & \text{otherwise} \end{cases} \]

Expected runs through Survival probability estimates

If a random variable \(X_A\) denotes the runs scored by team \(A\) in a Test innings while getting all-out and it follows a certain distributional assumption with density \(f_A\), then we can estimate the expected runs scored in a Test innings while team \(A\) have either declared (after scoring \(x_A\) runs) or not batted at all (\(x_A = 0\)), by calculating the Mean Residual Life (MRL) of team \(A\),

\[ MRL_{A} = \mathbb{E} [X_A | X_A > x_A] = \frac{\int_{x_A}^{\infty} t f_A(t) dt}{\int_{x_A}^{\infty} f_A(t) dt} \]

We can bound the upper limit to \(952\) (highest ever recorded team score in an innings in Test cricket) instead of \(\infty\) to avoid overestimation.

Fitted distributions of 4th innings team scores

The following well known distributions are observed to fit well for 4th innings of Test matches corresponding to each team based on the training period.

Using the corresponding distributional assumptions, the expected 4th innings scores of every team during ‘declarations’ and ‘did not bat’ have been calculated. The average expected 4th innings score (\(E4R\)) based on the training period has been found out to be \(233.6189\).

Expected scores in declared innings

We could similarly find the expected scores at the end of innings on the \(12\) occassions of declarations during ICC WTC 2021-23.

Note that considering only runs scored \((x)\) until declaration might draw inaccurate inferences, hence number of wickets fell until declaration has also been taken into account and runs per dismissal have been accordingly used to calculate the MRL values.

The final proposed model

We previously updated the ratings of teams using the formula,

\[ r_A' = r_A + (RD_A)^2 g(RD_B) (S_A - E_A) = r_A + \frac{g(RD_B) (S_A - E_A)}{\sqrt{\frac{1}{(RD_A)^2} + \frac{1}{d^2}}} \]

Using the Expected Scores (\(E_A\)) as explained earlier and the revised formula of \(S_A\), ratings of teams can now be updated as,

\[ r_A' = \begin{cases} r_A + \frac{g(RD_B) \left(\frac{1 + MOV_i}{2} - E_A \right)}{\sqrt{\frac{1}{(RD_A)^2} + \frac{1}{d^2}}} & \text{if team $A$ wins the } i \text{-th match} \\ \\ r_A + \frac{g(RD_B) \left(\frac{1 - MOV_i}{2} - E_A \right)}{\sqrt{\frac{1}{(RD_A)^2} + \frac{1}{d^2}}} & \text{if team $A$ loses the } i \text{-th match} \\ \\ r_A + \frac{g(RD_B) \left(\frac{1}{2} - E_A \right)}{\sqrt{\frac{1}{(RD_A)^2} + \frac{1}{d^2}}} & \text{if the } i \text{-th match results in a draw} \end{cases} \]

Applying the final proposed model on WTC 2021-23

Comparison of rankings at the end of WTC 2021-23

Although the ratings marginally differ, the rankings of all the teams remain exactly same when compared with respect to the improvised Glicko’s model and our final proposed model. The gradual changes of ratings of each team can be studied through trend curves.

Trend comparison of chronological ratings of teams

Conclusion

• By incorporating key contextual non performance based factors such as home advantage and toss impact and a crucial performance based factor Margin of Victory, the enhanced model provides a more accurate and fair assessment of team performance in Test cricket.

• The model demonstrated superior predictive accuracy, correctly forecasting the outcomes of 44 out of 56 non-drawn matches during the ICC World Test Championship 2021-23. Statistical tests showed that home ground and toss advantages are significant factors in deciding the outcome of a Test match.

• A suitable choice of \(\alpha\) and cutoff \(q\)-th quantile resulted in maximising the prediction of proportion (9 out of 12) of drawn Test matches during a given time period based on a certain training period.

• The inclusion of the Margin of Victory \((MOV)\) allowed for more dynamic rating adjustments, reflecting the degree of dominance in match outcomes instead of awarding equal proportions of rating points irrespective of the margins of victory.