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

Yes - the Poisson distr. measures "arrivals", meaning the count of a variable occurring over a given interval. The recurrence of suicide is the event in this study. The "hazard ratio" refers to the rate of arrival, or the lambda parameter of the Poisson distribution.

Rather than just measuring group deviance from mean, as a chi-sq distr. would, modelling the data as Poisson directly measures the cross-group suicide recurrence rate itself.

• 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
• Let me correct myself: I believe from the abstract the researchers refer to two statistical analyses: (1) a survival analysis, which would use a contingency table and measure the suicide rate as group variable, and was found to be insignificant, and (2) a test of incidence rate, which would measure the suicide rate in time units, and thus follow a Poisson (rate of arrival) distribution, and which was found to be significant. I can't see the full article, so I can't explain any more beyond the abstract. – Big Dogs Clothing Company Jun 29 '18 at 1:52
• What seems invalid to me, is to set a theoretical fixed death rate of 3/83 based on the Placebo Group, thereby halving the variance. The death rate in the Placebo group could be randomly too high and the one in the Lithium Group randomly too low. – Constantin Volkmann Jun 29 '18 at 7:05
• I agree to @ConstantinVolkmann . Doing that seems as invalid as -for example- trying to compare two groups using a one sample t-test by using the mean of one group as the null hypothesis for the other group - instead of performing a two sample t-test. – Pere Jun 29 '18 at 10:07
• I'm not all that familiar with incidence rate testing, but it appears to be a two-sample test on Poisson parameters suited for a placebo/treatment design (statsdirect.com/help/rates/compare_crude_incidence_rates.htm). I don't understand the bit about "halving variance" - could you please elaborate? – Big Dogs Clothing Company Jun 29 '18 at 17:17

Indeed, the fractions of suicide $$\frac{3}{83}$$ and $$\frac{0}{84}$$ are different. That is correct. But for a good research report, we also want to express how much this observation may have been due to chance.

There are all sorts of ways to compare these numbers. To get these differences intuitively. Maybe it is more intuitive/straightforward to do some modeling/computing...

Probability of getting 0 cases

Let's consider the probability to get 0 cases out of 84 patients when the assumed suicide probability is $$p$$.

This is a binomial distribution, and the Poisson distribution is only an approximation of the binomial distribution.

Indeed the probability to get 0 or fewer suicides out of 84 when $$p = 3/83$$ is below 0.05

$$\begin{array}{rcl} P(K \leq k | n,p) &=& \sum_{i=0}^k {n\choose i} p^{i}(1-p)^{n-i} \\ P(K \leq 0 | n=84, p = \frac{3}{83}) &\approx& 0.04539407 \end{array}$$

The curve below shows how this probability depends on $$p$$

### plot curve
p <- seq(0.01,0.06, 10^-4)
y <- dbinom(x = 0, size = 84, prob = p)
plot(p, y,main="the probability to get 0 suicides out of 84 patients \n depending on suicide probability p",
type = "l", xlab = "p", ylab = "probability")

### plot lines and intercept
lines(c(1,1)*3/83,c(-10,10),lty=2)
text(0.036,0.38,"p = 3/83", col = 1, pos = 4, srt = -90, cex = 0.7)
lines(c(-1,1)*100,c(1,1)*0.05,lty=2)
text(0.05,0.05,"probabilty = 5%", col = 1, pos = 3, srt = 0, cex = 0.7)
points(3/83,dbinom(0,84,3/83), pch = 21, col = 1, bg = 0, cex =0.7)


Probability of opportunistic post-hoc comparison

You would assume that the null hypothesis is that the probability of suicide in both groups is the same: $$p = p_1 = p_2$$. The true suicide probability $$p$$ is unknown and this is what makes the analysis difficult.

In the study, the authors use one group as the reference and see whether, based on that one group, the other group is significantly different. But is that really significant?

Let's try it out with a simulation of how often we would infer that the bigger group is significantly larger than the smaller group.

the simulation shows clearly that with this approach you do not really get a true p-value. Given the hypothesis $$p = p_1 = p_2$$ the probability to reject this hypothesis is not constant and equal to the p-value. What the researchers compute is a p-value based on the assumption that the result from the other group is correct. But this is not the probability that they find a discrepancy.

### function to perform simulation
### the value returned, nneg/N_trials
### is how often the null get's rejected
###
rate <- function(p, n1 =84, n2 = 83, N_trials = 10^5) {
nneg <- 0 ### counters for how often the test fails

for (i in 1:N_trials) {

###  simulate N_trials experiments
x1 <- rbinom(1, size = n1, prob = p)
x2 <- rbinom(1, size = n2, prob = p)
p1 <- (x1/n1)
p2 <- (x2/n2)

### perform test
if (p1>p2) {
test <- pbinom(x2,n2,p1)
} else {
test <- pbinom(x1,n1,p2)
}

### add tp counter when negative test
if (test <= 0.05) {
nneg = nneg + 1
}
}
return (nneg/N_trials)
}

### Vectrize the function
rate <- Vectorize(rate,vectorize.args = "p")

### perform computations
set.seed(1)
p <- p <- 0.001 * 2^seq(0,9,0.25)
pr <- rate(p)

### plot results
plot(p, pr ,
main="probability of failing 5% test \n as function of p",
xlab = "true p", ylab = "5% fail probability",
cex = 0.7 , pch = 21, col = 1, bg = 1, log = "xy",
xlim = c(0.001,1), ylim = c(0.0001,0.2), xaxt = "n", yaxt = "n"
)

### logarithmic axis labels
axis(1, at = 0.001*c(c(1:10),c(2:10)*10,c(2:10)*100), labels = rep("",28))
axis(1, at = 0.001*c(1,10,100,1000))
axis(2, at = 0.0001*c(c(1:10),c(2:10)*10,c(2:10)*100), labels = rep("",28))
axis(2, at = 0.0001*c(1,10,100,1000), las = 2)

### this is what we expected 0.05 probability
lines(c(10^-6,1),c(1,1)*0.05, lty =3)

### this is a computation of the simulated probability
lines(p, 2*((1-pbinom(2, 84, prob = p))*dbinom(0, 83, prob = p)) , col =2, lty = 2 )

lines(p, 2*((1-pbinom(2, 84, prob = p))*dbinom(0, 83, prob = p) +
(1-pbinom(4, 84, prob = p))*dbinom(1, 83, prob = p)) , col =2, lty = 2 )

lines(p, 2*((1-pbinom(2, 84, prob = p))*dbinom(0, 83, prob = p) +
(1-pbinom(4, 84, prob = p))*dbinom(1, 83, prob = p) +
(1-pbinom(6, 84, prob = p))*dbinom(2, 83, prob = p)) , col =2, lty = 2 )


Intuitively we can understand why the the first part in the graph is much lower than 5%. The probability to reject the null hypothesis rises very fast. It is the probability to get the 0 cases in one group and 3 or more cases in the other group (this probability is much smaller than 5%). We have plotted this in the graph with a red curve (we have plotted 3 red curves, the others are including the cases '1 suicide in the one group and 5 or more in the other' plus ' 2 suicides in the one group and 7 or more in the other').

The reason why the second part in the graph is much higher than 5% is in part because the the situation is two sided. If the two groups are in reality equal then you get half the time the one group larger and the other time the other group larger. This increases the probability to find a significant difference by two. The other part of the reason is because the test performed (how does one group compare to the other) is not reflecting the hypothesis test (the two groups are the same). The computation of the p-value assumes that the one group is the reference but it does not take into account that the difference between the two groups may be due to the one group being occasionally higher while the is occasionally lower.

Sample distribution of results and $$\chi^2$$

Below we plot the distribution of the results when the true suicide probability is 0.03 and the group sizes are 1000. For large group sizes, the distribution becomes more smooth/continuous and it can be approximated with a normal distribution.

You can see that for a given $$p$$ the distribution has some sort of spherical distribution. Getting further away from the line 'group1 = group2' is less likely. This is what the $$\chi^2$$ test measures. It looks how far we get from the line 'group1 = group2' and determines whether this is unlikely or not (we do not know what the true $$p$$ is and so we can not determine the p-value exactly, but we can use the estimated $$p$$ to determine this, or use the highest p-value out of all possible parameters $$p$$). The $$\chi^2$$ test does this by approximvting the distribution by a normal distribution.

(this works well with large numbers, and this is why I plotted the situation with large numbers, for your case with group size 84 and number of cases 3 the $$\chi^2$$ test is not accurate)

In the same plot I have plotted the rejection boundaries that relate to the article that you referenced. These types, of boundaries are not accurate because they only consider one of the two groups as variable and the other group is considered as the reference. This underestimates the variability in the difference between the two groups.

prob <- 0.03
n <- 1000

### generate joint distribution
x <- 1:61
y <- 1:61
z <- matrix(rep(0,61*61),61)
for (i in 1:61) {
for(j in 1:61) {
p <- dbinom(i,n,prob=prob)*dbinom(j,n,prob=prob)
z[i,j] <- log(p)
}
}

range(z)

# contour plot
filled.contour(x,y,z,
xlab="group 1 cases",ylab="group 2 cases",
#levels=c(-500,-400,-300,-200,-100,-10:-1),
color.palette=function(n) {hsv(c(seq(0.15,0.7,length.out=n),0),
c(seq(0.7,0.2,length.out=n),0),
c(seq(1,0.7,length.out=n),0.9))},
levels=seq(-60,0,2),
plot.axes= c({
points(log(0.1+dat$$action_count),log(2/3+dat$$document_entropy))
points(log(0.1+dat$$action_count)[which(dat$$is_service == 1)],log(2/3+dat$$document_entropy)[which(dat$$is_service == 1)],pch=21,bg="red")
title("joint distribution \n for p = 0.03, n1 = n2 = 1000 ")
axis(1)
axis(2)
lines(c(0,10^4),c(0,10^4), lty = 2)
text(15,17, "group1 = group2" ,srt = 45, cex = 0.7)
px <- 1:60
lines(px,qbinom(0.05,1000,px/1000))
lines(qbinom(0.05,1000,px/1000),px)
text(30,19, "rejection boundary" ,srt = 40, cex = 0.7)
},""),
xlim=range(x)+c(-0.05,0.05),
ylim=range(y)+c(-0.05,0.05)
)


The bottom-line

The group sizes and the number of cases are too small to make reliable estimates.

The estimate they make is interresting but also slighlty optimistic. The above computations show how it is not a realistic p-value.

But, in addition, what we have not discusses above, the issue goes beyond the mathematics. The distribution of the test results is not a clean and nice binomial distribution (which is approximately Poisson distributed) and independent. The results might be biased which happens for instance when the experiment is not randomized or when the treatments are not blind. The test results migh be correlated, due to test procedures that are repeated within multiple subjects and not completely randomized. In this case the distributional assumptions are wrong (e.g. instead of i.i.d binomial distributed cases it might be including a random effect, or one might need to use a beta-binomial distribution instead).