# Chi Square versus Poisson distribution

In a study which analyses the effect of Lithium on suicide rates, the results were the following:

• Placebo group: 3 suicides in 83 patients
• Lithium group: 0 suicides in 84 patients

My first approach would be to apply the Chi Square test, which does not result in a significant difference (chi=3.0184).

The authors however, suggest the following, finding a significant difference between the groups (p=0.049):

As a post hoc analysis, differences between intervention groups with regard to completed suicides were examined based on determining the probability of zero events in the lithium group on the expectation of 3 ⁄ 83 events in the placebo group on grounds of a Poisson distribution

Is this a valid approach, what am I missing?

• Consider 'Fisher exact text'. While 3 suicides are tragic and regrettable, the count is not large enough to show statistical significance. // Chi-squared statistic does not have dist'n $\mathsf{Chisq}(1)$ because of low expected counts in some cells. And P-value 0.049 is just barely significant, even if it were accurate. // Admittedly, not exactly the same thing, but you couldn't declare a coin biased, based on three Heads out of three tosses. – BruceET Jul 19 '18 at 2:13

• Could you please explain how this justifies the implicit assertion that the difference is "significant" with p = 0.049? Given--as the abstract says--the 167 patients were randomized into those two groups, and such randomization will place all 3 suicides into the same group with a probability of $41/167\approx 25\%,$ it's difficult to see how this result could possibly be considered significant. – whuber Jun 29 '18 at 0:13