Correct way to express a mean rate between two groups

Question

I am a novice in statistics so pardon my rather simple question.

I have a retrospective case-control study, I would like to show the rate of admissions (ie number of admissions per year) of my cases and controls. I have individual level data on a = total number of admissions of a given patient during follow up period y = total number of years of follow up of a given patient

and I have n1 = number of cases and n2 = number of controls.

If I would like to show the mean number of admissions per year of the cases, and the mean number of admissions of controls per year (with SDs), should I be doing

mean= sum(a/y)/n for both groups, and standard deviation? in R I can easily do this by calculating a new column r (admission rate of an individual) = a/y, then do mean(r) and sd(r)

or should it be

mean= sum(a)/sum(y) for both groups (sum of n1 admissions, divided by sum of y1 years of follow up for cases; and sum of n2 admissions, divided by sum of n2 years of follow up for controls) - however, how do I derive standard deviation from this?

Answer 1

Edited:

total cases: n1
total control: n2

each case/control has $a$ admission numbers in $y$ period of time.

If $a$ and $y$ are varying/different for each patient $i$

Mean number of admissions per year of the cases:

The mean, $\mu$ is $\frac{\sum_{i}^{n_1}(a_i)}{\sum_{i}^{n_1}(y_i)} = \frac{a_1 + a_2 + ... + a_{n1}}{y_1 + y_2 + ... + y_{n1}}$

basically, add all the individual patient admissions together and divide by adding all the individual patient's periods.

special case: If $a$ and $y$ is the same for all patients then mean number of admissions per year of the cases:= is total number of cases $(n1 \cdot a)$ divided by total time $(n1 \cdot y)$

mean, $\mu$ = $\frac{(n_1 \cdot a)}{(n_1 \cdot y)}$

replace n1 by n2 above and you can get the mean number of admissions of controls per year.

for standard deviation you just need to apply the formula.

$\frac{1}{\sum_{i}^{n_1}(y_i)} \cdot \sum_{i}^{n_1}(a_i -\mu)^2 = \frac{(a_1 - \mu)^2 + (a_2 - \mu)^2 + .... + (a_{n_1} - \mu)^2} {y_1 + y_2 + ... + y_{n1}}$

change n1 to n2 for standard deviation in controls.