Using Rates in a Poisson Distribution OSHA keeps records of fatal injuries in deaths per 100,000 full time equivalent workers.  I'd like to see if improvement over time is significant from 2006 (3.7) to now (2.8).
My instinct is to model this as a Poisson distribution for a population of a million full time workers, set lambda to 28 and the number of events to 37, but I am concerned that this would be invalid since a Poisson distribution is supposed to describe discrete events rather than rates.  Am I setting myself up for invalid results?
 A: You can model it as a rate if you have the number of fatal injuries (diagram #1 in the link provided by whuber) and the total number of workers each year. Because the numbers of workers are not shown, they can be approximated by dividing number of injuries by the rate (diagram #2) per 100000 and then multiply by 100000.
In Poisson regression, rates are modeled using offset variables. In short, you're taking the population size for each year (number of workers each year) into account without estimating regression coefficients for the population size. This is brilliantly explained by the user ocram in the first answer to another question: When to use an offset in a Poisson regression? In that example, time is used as the offset variable to model rate of events per time, but the idea is the same as in this situation.
Now, all we have to do is enter the data from the link, calculate the approximate total number of workers for each year, and then run a Poisson regression model using the log total numbers as the offset variable. Using R notation and output:  
# enter number of fatal work injuries 2006-2015 from diagram #1
events <- c(4808, 4613, 4183, 3488, 3651, 3642, 3571, 3635, 3728, 3751)

# enter rates from diagram #2
rates <- c(3.7, 3.5, 3.2, 2.8, 3.0, 2.9, 2.8, 2.8, 2.8, 2.8)

# years 2006-2015 where 2006 = 1
year <- seq(1:10)

# calculate approximate population size each year
population <- 100000 * events/rates

summary(glm(events ~ offset(log(population)) + year, family=poisson))

Giving the results:
Coefficients:
              Estimate Std. Error z value Pr(>|z|)    
(Intercept) -10.240924   0.010503 -975.08   <2e-16 ***
year         -0.030207   0.001747  -17.29   <2e-16 ***

As you can see, there is a highly significant effect of time on the rates of fatal injuries. The rate of fatal injuries for a particular year is estimated as exp(-0.030207) = 97% of the rate in the past year. Now, just to check our results, we'll calculate the fatal injury rate in 2015 using data from 2006: 3.7 * exp(-0.030207) ^ 9 = 2.82 which is what we expected.
