Question

Suppose an experiment will randomly divide 40 cases between two possible treatments, \(A\) and \(B,\) and will then record two possible outcomes, Successful or Not successful. The outline of a two-way table is shown in Table 2.14. In each case below, fill in the table with possible values to show: (a) A clear association between treatment and outcome. (b) No association at all between treatment and outcome. Table 2.14 Fill in the blanks to show (a) Association or (b) No association $$\begin{array}{|l|c|c|c|}\hline & \text { Successful } & \text { Not successful } & \text { Total } \\\\\hline \text { Treatment A } & & & 20 \\\\\hline \text { Treatment B } & & & 20 \\\\\hline \text { Total } & & & 40 \\\\\hline\end{array}$$

Short answer

For a clear association between treatment and outcome, Treatment A resulted in 17 successful and 3 not successful outcomes while Treatment B resulted in 3 successful and 17 not successful outcomes. For no association between treatment and outcome, both Treatment A and B each resulted in 10 successful and 10 not successful outcomes.

Step-by-step solution

Association between Treatment and Outcome

In case (a), we have to show a clear connection between the treatment type and outcome. We can do this by assigning majority of successful outcomes to one treatment and most of the not successful outcomes to the other. Let's say for instance, Treatment A results in 17 successful outcomes and 3 not successful outcomes. Conversely, Treatment B results in 3 successful outcomes and 17 not successful ones. This clearly shows an association as we see that Treatment A generally leads to successful outcomes and Treatment B to unsuccessful ones.

Two-way Table with Association between Treatment and Outcome

\(\begin{array}{|l|c|c|c|}\hline & \text{Successful} & \text{Not Successful} & \text{Total} \\\hline \text{Treatment A } &17&3 &20 \\\hline \text{Treatment B } &3&17& 20 \\\hline \text{Total } &20&20& 40 \\\hline\end{array}\)

No Association between Treatment and Outcome

In case (b), we're required to show no association at all between the treatment type and the outcome. This means the success and failure rates should be the same for both treatments. Let's evenly distribute the outcomes, for example, with each treatment having 10 successful outcomes and 10 not successful outcomes.

Two-way Table with No Association between Treatment and Outcome

\(\begin{array}{|l|c|c|c|}\hline & \text{Successful} & \text{Not Successful} & \text{Total} \\\hline \text{Treatment A } &10&10 &20 \\\hline \text{Treatment B } &10&10& 20 \\\hline \text{Total } &20&20& 40 \\\hline\end{array}\)