# Estimating yearly diagnosis rate from multiple studies

I want to estimate yearly diagnosis rates from some 8 studies. Each of the 8 studies contains the following data:

• $$n$$ = total number of patients (e.g., 100)
• $$t_{start}$$ = start timestamp of the diagnosis experiment (e.g., 2014)
• $$t_{end}$$ = end timestamp of diagnosis experiment (e.g., 2016)
• $$d$$ = number of patients diagnosed at end of experiment (e.g., 10)

I can calculate the yearly diagnosis rate for each study separately using the formula

\begin{align} r = exp\left(\frac{log\left(1+\frac{d}{n}\right)}{t_{end}-t_{start}}\right) - 1. \end{align}

(This is just from solving $$(1 + r)^{t_{end}-t_{start}} = 1+\frac{d}{n}$$ for the yearly diagnosis rate $$r$$. In the above example, it gives a yearly diagnosis rate of 4.9%.)

Sadly, the studies do not report the exact dates of enrollment and diagnosis for each patient, just aggregate start and end dates per study.

Now for my problem. The studies contain wildly varying patient numbers between 6 and 150 patients. It turns out that the smaller studies apparently report much higher yearly diagnosis rates than the larger studies. Ideally I want to create an aggregate statistic of the yearly diagnosis rate. However, what I was doing so far was simply taking the mean or median over the yearly diagnosis $$r_i$$ across all studies $$i=1..8$$, which skews upwards because the smaller, less reliable studies report higher diagnosis rates.

By taking mean or median across all studies, I'm basically putting as much trust in the smaller studies as in the larger studies, which feels wrong. I could just take a weighted average across studies by number of patients in the study, but is there any justification for doing that?

Could anyone help me out here? How can I aggregate the yearly diagnoses rates across my 8 studies in a justifiable manner?

Thanks a lot!

• Define "yearly diagnosis rates" first, then try to estimate it. It seems you mixed up 2 different things together. – user158565 Jun 20 at 2:37