Admissions per year before and after antibiotics?

Question

I need to compare hospital admissions per year before and after the introduction of antibiotics in 100 patients (antibiotics introduced at various time points in the past).

Eg.

Before antibiotics: admissions: 2 start: Jan-09 end: Dec-13 years: 4.92 admissions/year: 0.4

After antibiotics: admissions:0 start: Dec-13 end: June 19 years: 5.5 admissions/year: 0

If I use a t test then 0 admissions in 1 month equals to 0 admissions in 5 years, which is not correct.

Would you recommend any specific software?

Do you have 100 patients before antibiotics and another 100 after? If only 100 altogether, how many before antibiotics and how many after? — BruceET, Nov 07 '19 at 23:15
These are 100 altogether and they are the same patients before and after antibiotics- just different time intervals for each: eg patient A was not on antibiotics for 2 years and then took them for 5 years while patient B was not on antibiotics for 7 years and then took them for 1 year. — Alex, Nov 08 '19 at 13:02
Duplicate of [your previous question](https://stats.stackexchange.com/questions/434992/comparing-variables-that-involve-time) — Lio Elbammalf, Nov 08 '19 at 15:42

kjetil b halvorsen · Answer 1 · 2019-11-08T19:45:24.287

I would start with Poisson regression, maybe with a random effect for patient. First, present the data in the following long format (you will need some transformations):

ID     Antibiotics   Admissions_total   Time_length ...
1      without         10                  3
1      with            8                   3
2      without         11                  4
2      with            3                   1
 ...

so you have to calculate total admissions, not per year, and length of period with/without in years/months. The length will then enter as an offset, that is, a variable with a known coefficient of 1. See Scaling vs Offsetting in Quasi-Poisson GLM. The model can then be written as $$ \text{Admissions-total}_i \sim \mathcal{Poisson}(e^{\lambda_i}) $$ where $\lambda_i= \mu + \tau_i + \text{Antibiotics}_i +\text{offset}(\log{\text{Time_length}})$. Here $\tau_i$ (if included in the model) is a patient random effect.

A simpler model without the random effect can be implemented in R with

mod_glm <- glm(Admissions_total ~ Antibiotics + offset(log(Time_length)), family=poisson, data=your_data_frame)

but as there might be large differences between patients, it is probably better to include random effects, which in R could be

library(lme4)
mod_lme4  <- glmer(Admissions_total ~ Antibiotics + offset(log(Time_length)) +(1 | ID), data=your_data_frame, family=poisson)

This could at least be a starting point. The model is called a Poisson rate regression.

While I used R in my example, this could be done with most statistical software.

Admissions per year before and after antibiotics?

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