A treatment is suggested for a particular illness. The results of treating a number of patients chosen at random from those in a hospital suffering from the illness are shown in the following table, in which the entries $a, b, c, d$ are numbers of patients.

$\begin{array}{lcc} & \text { Recovery } & \text { Non-recovery } \\ \text { Untreated } & a & b \\ \text { Treated } & c & d\end{array}$

Describe the use of Pearson's $\chi^{2}$ statistic in testing whether the treatment affects recovery, and outline a justification derived from the generalised likelihood ratio statistic. Show that

$$\chi^{2}=\frac{(a d-b c)^{2}(a+b+c+d)}{(a+b)(c+d)(a+c)(b+d)}$$

[Hint: You may find it helpful to observe that $a(a+b+c+d)-(a+b)(a+c)=a d-b c .]$

Comment on the use of this statistical technique when

$$a=50, \quad b=10, \quad c=15, \quad d=5 .$$