Understanding exact tests for clinical trial data

Question

A clinical trial is done with 400 persons suffering from a particular disease, to find out whether a treatment is better than placebo. They are randomised to receive treatment or placebo (200 participants each). The outcome studied is how many get cured. The results are shown in the following 2x2 table:

\begin{array} {|r|r|} \hline \text{ } & \text{Treatment group} & \text{Placebo group}\\ \hline \text{Cured} & 172 & 151 \\ \hline \text{Not cured} & 28 & 49 \\ \hline \text{Total} & 200 & 200 \\ \hline \end{array}

The odds ratio calculated from this table is $1.99$. The objective now is to test the null hypothesis (odds ratio = 1) against the alternate hypothesis (odds ratio is not 1). Ludbrook's 2008 article describes an exact test for this scenario:

The formula for executing a two-sided randomization test, adapted to a 2x2 table with the constraint that the column totals are fixed (single conditioning), is:

P=(All tables for which the summary statistic is at least as extreme as that observed, in either direction)/All possible tables with the same column totals

I am a bit confused about what exactly it means. Does it mean I should form all possible tables with 200 treatment and 200 control participants, with each participant having a 50% chance of getting cured? Then there would be $2^{200} \times 2^{200}=2^{400}$ possible tables, each being equally likely. I would then calculate what fraction of these tables give an odds ratio equally or more extreme than the one I got experimentally, i.e. $1.99$. This would give me the p-value.

Is this the correct interpretation? If not, why?

If so, why the assumption of 50% cure rate? Why not 20%, 70%, 90%, or any other number?

(I would have contacted the author directly, but it turns out he is deceased. That is why I asked this question here.)

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Reference

John Ludbrook, Analysis of 2 × 2 tables of frequencies: matching test to experimental design, International Journal of Epidemiology, Volume 37, Issue 6, December 2008, Pages 1430–1435, https://doi.org/10.1093/ije/dyn162

Answer 1

Let's analyse this $2\times 2$ contigenty table.

• 172 of the 200 treated patients got cured, that means $\frac{172}{200}=86\%$

• 151 of the 200 untreated patients got cured, that means $\frac{151}{200}=75.5\%$

being 86>75.5 the treatment looks work.

Now the question is: 86 is really greater then 75.5 or the difference is due to the random variability of the phenomena?

To get an answer we can do the $\chi^2$ test

• the first table is your contingenty table

• the second one, is the expected table, under the hypothesis that there is no difference in treatment group or placebo group. (every expected value is calculated under independence hypothesis, i.e. $161.5=\frac{323\times 200}{400}$)

• the third table is the test. Every cell is calculated as $\frac{[\text{Observed}-\text{Expected}]^2}{\text{Expected}}$

• the total test is 7.09 that means a p-value of $0.8\%$, using a chi square distribution with $(2-1)\times (2-1)=1$ degree of freedom.

CONCLUDING: the test has a high significant statistical level. The data are enough to reject the hypotesis of OR=1 (the treatment is good to get cured)

Answer 2

Fisher's exact test is based on a hypergeometric distribution.

Fisher's Exact Test in R. As implemented in R statistical software, the results of the two-sided test are as follows:

TABL = rbind(c(172,151), c(28,49))
TABL
     [,1] [,2]
[1,]  172  151
[2,]   28   49

fisher.test(TABL)

        Fisher's Exact Test for Count Data

data:  TABL
p-value = 0.01088
alternative hypothesis: 
   true odds ratio is not equal to 1
95 percent confidence interval:
 1.160626 3.464351
sample estimates:
odds ratio 
  1.989975 

Based on Hypergeometric Distribution. Here is one way to explain the connection to a hypergeometric distribution. Suppose we draw 77 patients not cured from among the 400 patients (200 Treatment and 200 Placebo), what is the probability I'll see exactly 28 in the Treatment group? That's the following hypergeometric probability:

$$P(X=28)=\frac{{200\choose 28}{200\choose 49}}{{400\choose 77}}= 0.000295.$$

This is evaluated in R by computing the binomial coefficients or by using R's hypergeometric PDF function dhyper.

choose(200,28)*choose(200,49)/choose(400,77)
[1] 0.002917137
dhyper(28, 200,200, 77)
[1] 0.002917137

One-sided P-value: However, the P-value of a one-sided test would be $P(X\le 28) = 0.00544,$ which can be evaluated by summing 29 hypergeometric probabilities or by using R's hypergeometric CDF function phyper:

sum(dhyper(0:28, 200,200, 77))
[1] 0.005441333
phyper(28, 200,200, 77)
[1] 0.005441333

Two-Sided P-value: Finally, the P-value for a 2-sided test is the probability of a more extreme result in either direction: $P(X \le 28) + P(X \ge 49) = 0.01088,$ which is the P-value shown in the R printout from Fisher's Exact test above.

sum(dhyper(49:77, 200,200, 77))
[1] 0.005441333
2*phyper(28, 200,200, 77)
[1] 0.01088267
sum(dhyper(c(0:28, 49:77), 200,200, 77))
[1] 0.01088267

In the plot of the relevant hypergeometric PDF below, the two-sided P-value is the sum of the heights of the bars outside the vertical dotted lines. [The relevant hypergeometric distribution is precisely symmetrical because Treatment and Placebo groups are of exactly the same size. One might say that there are ${400 \choose 77} = 4.47 \times 10^{56}$ possible $2 \times 2$ tables matching the experimenal outcomes, but this hypergeometric distribution contains the information about them needed for a valid test.]

k = 0:77;  PDF = dhyper(k, 200,200, 77)
plot(k, PDF, type="h", col="blue", lwd=2, main="Hypergeometric PDF")
  abline(v=c(28.5, 48.5), col="red", lwd=2, lty="dotted")