Performing Under Difficulty: The Magical Pressure Index

Abstract

This chapter introduces an interesting concept of measuring performance of cricketers considering the match situation under which the player performed. The difficulty level, under which the team is performing, is quantified introducing a measure called as the pressure index in the chapter. The pressure index is an inclusive measure that can be utilized to measure batting performance both individually and between partners, bowling performance, determining turning point of the match, comparing run chases and predicting the outcome of the match. Considering the huge area of applicability of the index, we call it magical. The pressure index also opens up several another area of research for people interested in sports analytics. The concept of pressure index can be extended to other sports as well. We believe that the pressure index has in its store several other hidden dimensions which are left for the readers to explore.

Appendices

Appendix 7.1: Wicket Weights of Different Batting Positions as in Lemmer (2005)

Batting position (i)	Wicket weight wi
1	1.30
2	1.35
3	1.40
4	1.45
5	1.38
6	1.18
7	0.98
8	0.79
9	0.59
10	0.39
11	0.19

Appendix 7.2: Binary Logistic Regression

In some cases, it so happens that subjects under consideration can be classified into one of the two groups based on their performance on a set of explanatory variables (X, say). In such a case, we are rather interested to compute the probability that a subject belongs to one of the two categories given the values of the explanatory variables. Let us call the proportion as p. Estimating the value of p based on a set of explanatory variables (X) or building models for estimating the value of p given values of the explanatory variables X is called logistic regression.

Binary type data, i.e., data of the form Yes and No comes from a random sample that has a binomial distribution with probability of success p. Often the value of the probability of success is unknown and is to be estimated using a model given the data. Since the binary type data cannot follow a normal distribution, an essential condition for using ordinary regression, so one needs a new type of regression model to do this job called the logistic regression.

A logistic regression model ultimately gives you an estimate of p, the probability that a particular outcome will occur in a binary set up. The estimate is based on information from one or more explanatory variables X = (X₁, X₂, …, X_k).

Logistic Regression can be used whenever an individual is to be classified into one of two populations. However, if the population into which an individual to be classified is more than two then we reach to the multinomial logistic regression.

• Assumptions of Binary Logistic Regression

• Since it assumes that P(Y = 1) is the probability of the event occurring, so it is essential that the dependent variable (also referred to as the response variable) is being coded accordingly. The factor level 1 (i.e. Y = 1) of the dependent variable should represent the desired outcome.

• The independent variables (also referred to as the explanatory variables) should be independent of each other.

• It requires quite a large number of samples because the coefficients are estimated by MLE method which is less powerful than the OLS method.

• Derivation of the Regression Equation

Here, we try to understand the binomial case:

Let Y be a binary variable having two outcomes 0 and 1, such that,

$$P\left( {Y = 1} \right) + P\left( {Y = 0} \right) = 1$$

We are in search of a relation of the form

$$P\left( {Y = 1} \right) = a + b^{\prime} {\mathbf{X}}$$

Let us now introduce the example concept of odds in favor. The odds in favor of an event Y = 1 is the ratio between P(Y = 1): P(Y ≠ 1). Thus,

$${\text{Odds}}\left( {Y = 1} \right) = \frac{P(Y = 1)}{P(Y \ne 1)} = \frac{P(Y = 1)}{P(Y = 0)} = \frac{P(Y = 1)}{1 - P(Y = 1)}$$

Unlike P(Y = 1), odds (Y = 1) does not have an upper limit, however like probability odds ratio cannot be negative. However, with log transformation the value of the variable ranges from −∞ to ∞. Thus,

log_e [Odds (Y = 1)] = log_e$\left[ {\frac{P(Y = 1)}{1 - P(Y = 1)}} \right]$ is called as logit of Y.

So, logit (Y) = log_e$\left[ {\frac{P(Y = 1)}{1 - P(Y = 1)}} \right].$

Let us assume that the logit is a function of some independent variables X₁, X₂, …, X_k etc. which may be discrete or continuous, i.e.,

logit (Y) = $a + b_{1} X_{1} + b_{2} X_{2} + \cdots + b_{k} X_{k}$

⇒ log [odds(Y = 1)] = $a + b_{1} X_{1} + b_{2} X_{2} + \cdots + b_{k} X_{k}$

⇒ odds(Y = 1) = $e^{{a + b_{1} X_{1} + b_{2} X_{2} + \cdots + b_{k} X_{k} }}$

⇒ $\frac{P(Y = 1)}{1 - P(Y = 1)} = e^{{a + b_{1} X_{1} + b_{2} X_{2} + \cdots + b_{k} X_{k} }}$

⇒ $P(Y = 1) = {e}^{{a + b_{1} X_{1} + b_{2} X_{2} + \cdots + b_{k} X_{k} }} - P(Y = 1){e}^{{a + b_{1} X_{1} + b_{2} X_{2} + \cdots + b_{k} X_{k} }}$

⇒ $(1 + e^{{a + b_{1} X_{1} + b_{2} X_{2} + \cdots + b_{k} X_{k} }} )P(Y = 1) = {e}^{{a + b_{1} X_{1} + b_{2} X_{2} + \cdots + b_{k} X_{k} }}$

⇒ $P(Y = 1) = \frac{{{e}^{{a + b_{1} X_{1} + b_{2} X_{2} + \cdots + b_{k} X_{k} }} }}{{1 + {e}^{{a + b_{1} X_{1} + b_{2} X_{2} + \cdots + b_{k} X_{k} }} }}$

Or more precisely we can write P(Y = 1) as P(Y = 1|X₁, X₂, …, X_k) writing, $a + b_{1} X_{1} + b_{2} X_{2} + \cdots + b_{k} X_{k}$ = z, we have,

$$P\left( Y \right) = \frac{{{e}^{z} }}{{1 + {e}^{z} }} = \frac{1}{{\frac{{1 + {e}^{z} }}{{{e}^{z} }}}} = \frac{1}{{1 + {e}^{ - z} }} = \frac{1}{{1 + \exp \{ - (a + b_{1} X_{1} + b_{2} X_{2} + \cdots + b_{k} X_{k} )\} }}$$

Thus, the logit model is a form of the logistic curve and hence the name.

• Understanding the coefficients of the model

We have,

$${\text{logit}}\left( Y \right) = a + b_{1} X_{1} + b_{2} X_{2} + \cdots + b_{k} X_{k}$$

Now,

$$\begin{aligned} a & = {\text{log odds for individual for all}}\;\;X_{i} = 0 \\ & = {\text{the odds that would have resulted for a logistic model}} \\ & {\text{without any }}X{\text{ at all}}. \\ \end{aligned}$$

Now, in order to understand the b’s let us consider the change in logit (Y) when one of the X’s varies when all others are kept fixed.

Say: Fix X₂, X₃, …, X_k to 0 and vary X₁ from 0 to 1.

Let, logit P₁(Y) when X₁ = 1 and

logit P₀(Y) when X₁ = 0, while all others are fixed at 0 as mentioned earlier. Thus,

$${\text{logit }}P_{1} \left( Y \right) - {\text{logit }}P_{0} \left( Y \right) = a + b_{1} X_{1} - a = b_{1} X_{1} = b_{1} \;\;{\text{as}}\;\;X_{1} = 1$$

Thus, b₁ represents the change in the log odds i.e. logit that would result from a one unit change in the variable X₁ when other variables are fixed.