# Help interpreting "poisson process" calculations from a paper

I'm reviewing a paper. It records the number of deaths in a population of a country due to drug related causes. The authors have calculated 5 year moving averages to better examine trends.

They then also calculated a confidence interval they call "Likely range of values around 5-year average".

So one such 5 year average is 260 deaths (average of deaths from 1996 to 2000). The "likely range..." around this value is (228, 292) rounded. This appears to be 260 +- 1.96 sqrt(260).

Is this a legitimate calculation? Is a moving average of a poisson process with high lambda normally distributed?

Calculations are listed on page 43, an explanation is found on page 12.

Yes, a moving average with large $\lambda$ will be close to Normally distributed. No, this calculation is not legitimate.

Because the cumulant generating function (cgf) of a Poisson distribution of intensity $\lambda$ is $$\psi_\lambda(t) = \lambda(e^{it}-1) = i(t\lambda) + \frac{1}{2!}(it\sqrt{\lambda})^2+\lambda^{-1/2}\frac{1}{3!}(it\sqrt{\lambda})^3 + \cdots$$

and the cgf of a Normal distribution with mean $\lambda$ and variance $\lambda$ is $$\phi_\lambda(t) = i(t\lambda) + \frac{1}{2!}(it\sqrt{\lambda})^2,$$

the terms

$$\lambda^{-1/2}\frac{1}{3!}(it\sqrt{\lambda})^3 + \cdots$$

express their difference. The factor of $\lambda^{-1/2}$ multiplying them indicates how rapidly the Poisson approaches the Normal as $\lambda$ increases. Look at the two distributions for $\lambda=260$:

The heights of the bars are the Poisson probabilities while the red curve gives the corresponding Normal densities (per unit interval). They are almost indistinguishable.

The calculation in question is described thus:

the broken grey lines show the likely range of random statistical variation around the 5-year moving average ... if the number of deaths can be represented as the result of a Poisson process, for which the underlying rate at which the events (deaths) occur is given by the 5-year moving average, then random year to year variation would result in only about one year in 20 having a figure outwith [sic] this range (which is a ‘95 per cent confidence interval’, calculated thus: the underlying rate of occurrence plus or minus 1.96 times its standard deviation;...

This is erroneous because it fails to account for variation in the average itself. To analyze that, let $X_1, X_2, \ldots, X_5$ be the counts in the five (non-overlapping) years and suppose they are independent (as they would be in a Poisson process). Then the "random year to year variation ... around the 5-year moving average" is the variance of $X_i - \bar X$ where $\bar X=(X_1+\cdots+X_5)/5$ is the five-year average. Due to independence, all five of these variances are the same, equal to the first, which can be computed as

$$\operatorname{Var}(X_1 - \bar X) = \operatorname{Var}\left(\frac{4}{5}X_1 - \frac{1}{5}X_2 - \cdots - \frac{1}{5}X_5\right) \\= \left(\frac{4}{5}\right)^2\lambda + \left(-\frac{1}{5}\right)^2\lambda + \cdots + \left(-\frac{1}{5}\right)^2\lambda = \frac{4}{5}\lambda.$$

Consequently the correct interval (it's not a confidence interval) for 95% of these variations would have a half-width of $$1.96 \sqrt{4\lambda/5}\approx 0.89(1.96\sqrt{\lambda}),$$ over 10% shorter than claimed. The discrepancy can be appreciated in this simulation of 100,000 five-year averages (giving 500,000 such deviations):

The gray bars are a histogram of annual deviations from the five-year means. The red curve uses the paper's formula. The blue curve uses the adjusted variance (multiplied by $4/5$). It is clear which is correct.

Note that neither of these calculations is truly appropriate in the application: the interval does not account for the fact that $\lambda$ is estimated (uncertainly) from the data. Accounting for this ought to widen the interval. As a result, the values computed in the paper will come out close to correct--but only because of this accidental near-cancellation of two separate errors!

So that you can check, here is the code (in R) used for the simulation.

lambda <- 260 # Annual rate
n <- 5        # Years in window
n.sim <- 1e5  # Windows to simulate

x <- matrix(rpois(n.sim*n, lambda), n.sim) # Annual values
x <- x - rowMeans(x)                       # Deviations from means

hist(x, breaks=seq(floor(min(x))-1/2, ceiling(max(x))+1/2), freq=FALSE,
col="White", border="#c0c0c0",
main="Simulated five-year deviations")
curve(dnorm(x, 0, sqrt(260)), col="Red", lwd=2, add=TRUE)       # As in the paper
curve(dnorm(x, 0, sqrt(260 * 4/5)), col="Blue", lwd=2, add=TRUE)# Corrected


By focusing on fluctuations in this short time series, the paper makes much of very little. Clearly there is a trend. A simple reasonable way to explore a trend is to fit a growth model, such as regressing the log counts against time. Here is a summary.

Residuals:
Min        1Q    Median        3Q       Max
-0.153579 -0.060968  0.008444  0.046664  0.213764

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.071e+02  7.593e+00  -14.10 3.60e-11 ***
x            5.638e-02  3.786e-03   14.89 1.45e-11 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.09763 on 18 degrees of freedom


This regression estimates a 5.6% annual trend ($p \lt 10^{-10}$: it's significant). All data fluctuate around the fit but remain within 25% of it. The typical amount of fluctuation is 10%.

Specialized models are not required. The large counts indicate a general linear model is unnecessary. There is little evidence of serial correlation, suggesting a time series model would add nothing to the analysis. To identify outliers and temporary fluctuations, one may supplement this simple regression with a robust smooth, such as Loess (shown here), IRLS, or possibly a GAM. Loess is used below; there are no outliers among its residuals. The models that allow for detailed temporary fluctuations tend to fit the data more closely and (accordingly) be more optimistic about the random fluctuations: that is, they probably overfit the data and thereby underestimate how much of the fluctuation should be viewed as random. Thus, the simpler regression model should be preferred provided it fits the data adequately.

The hollow dots show the data. The dashed black line is a Loess smooth (using the R loess function with a span of 1/2). The solid red line is the regression of log count against the year. The solid gray lines above and below it are a symmetric 95% prediction interval. Indeed, 95% of the data points lie within those lines and the remaining one is right on one of the lines. This implies none of the data points should be viewed as unusual in the context of this fit.

The usual diagnostic plots of residuals (not illustrated) show nothing is amiss; in particular, there are no outliers.

Consequently, a good description of these data is that

Drug-related deaths have been growing at 5.6% per year between 1996 and 2015, with no exceptional years to note.

This sentence replaces--and improves upon--more than a page of dense text in the report, spanning sections 3.1.2 and 3.1.3.

• Thanks so much, will digest over the next week and come back to you if I've any clarifications. Commented Nov 30, 2016 at 17:55
• This is a small masterclass in statistical thinking. Commented Dec 1, 2016 at 6:19
• Thanks so much for that whuber, had a proper look at it today and I'm fairly sold on why it's not a legitimate calculation. I knew the authors didn't quite understand what they were doing with their usage of the wording "confidence interval" but wanted to make sure they hadn't accidentally stumbled on something correct. So, the authors are attempting to identify outliers in each 5-year band, what would be a more suitable calculation? Also you say at the end that lambda is estimated, you mean the 5-year moving average lambda here? Commented Dec 1, 2016 at 18:14
• I added a quick analysis at the end to address your request for "a more suitable calculation." Much more could be said. For instance, about 5% of the variability is attributable to unavoidable variation in counts--but that doesn't account for all the variation occurring. Other factors are contributing to year-over-year variation. To understand them, one would have to bring in more data that might be related to the causes of that variation.
– whuber
Commented Dec 1, 2016 at 19:06
• Thanks whuber, will have a look over the weekend, thanks so much for all your help. Commented Dec 2, 2016 at 13:46