Formula to use instead of model (average of 5 test subjects)

[bth]

Scenario: An average of 5 test subjects are brought into a study in subsequent months and are administered 1 monthly test for as long as they remain part of the study. The time they remain in the study is 3 months.

Could someone please explain a formula or technique to sum the number of expected tests per month and total if possible in lieu of having to model it out like below to capture the dynamic of subjects rolling on and off over time? My real example has many more subjects, so a formula would save a lot of time at scale and help with scenario analysis of shifting variables.

The answer may just be to pop it in excel and model, but thought I'd double check with some experts.

 

[Dr.Peterson]

Scenario: An average of 5 test subjects are brought into a study in subsequent months and are administered 1 monthly test for as long as they remain part of the study. The time they remain in the study is 3 months.

My first impression here, from the word "subsequent", was that this was part 2 of a multipart problem. But it sounds like this is a real situation, and all you mean is that each month, starting at some particular time, 5 people join the study, and they stay for 3 months each.

You talk about "an average of 5", but your model supposes that it is always exactly 5. I'll suppose you're okay with the latter, and don't need a deep statistical analysis of how the numbers might vary. (I wouldn't be able to do that for you.)

Could someone please explain a formula or technique to sum the number of expected tests per month and total if possible in lieu of having to model it out like below to capture the dynamic of subjects rolling on and off over time? My real example has many more subjects, so a formula would save a lot of time at scale and help with scenario analysis of shifting variables.

The answer may just be to pop it in excel and model, but thought I'd double check with some experts.

View attachment 9008

In the example, if you were to continue, you would find that it continues at 15 each month until you stop replacing people who leave. You are just ramping up to that steady state until you reach what might be called saturation (when there is a group present that started in each of the past three months.

If N people join each month and stay M months, then the totals in the study will be N, 2N, ... until you reach MN (because there are M "classes" of N people each). It will stay at MN until you stop taking in new people, at which time it will ramp back down, in reverse.

This doesn't seem hard, and it shouldn't be much different if the numbers aren't exactly steady.