# Sampling a changing distribution over time

I'm interested in learning more about sampling a changing population over time. For example, these statistics from the University of Washington coronavirus testing show a 7% positive rate day after day. Some news outlets are interpreting this as a positive: "One encouraging sign is that in Washington State, which had an early outbreak, the number of positive tests appears stable."

Can someone suggest a branch of statistics or papers for further study that would help me assess the veracity of such claims? My intuitive reasoning, as described below, shows why I am questioning the optimistic interpretation, but I would like to learn a more formalized framework.

For instance, let's say that the population proportion that tests positive for an attribute is 7%, and that proportion is stable. If I took a random sample each day, I should be getting 7% consistently, subject to some variance.

However, let's say the population proportion is subject to change over time. Let's also add the assumption that I am trying to prioritize sampling people (without replacement) that I believe are most likely to be positive. Eventually I will sample the whole population. Under these new assumptions, if the population proportion were stable, you would expect my initial sample proportions to overshoot, and my final sample proportions to under shoot as I ran out of likely positives. That would imply that if my sample proportion was constant and did NOT drift down over time, then my population proportion MUST be increasing over time. And if my sample proportion was increasing over time, then my population proportion must be increasing by a heck of a lot over time.

These are just qualitative conclusions I can come to. Is there a branch of statistics I can learn about to help me make concrete conclusions about a changing population proportion given sample proportions over time? I am sure that as you vary the sample size across time, the conclusions also change. Any framework to help understand the tweaks and their effects would be great.

Thank you.

• You are making a lot of assumptions that are not stated. Is the total population stable? What exactly does "I believe are most likely to be positive" imply? Does that change with time? For Covid-19, it is suspected that there are many more people who have the virus than test positive for it. That reduces the mortality.
– Carl
Mar 21 '20 at 6:32
• Hi Carl, I was looking at the University of Washington test results where they've seen a 7% positive rate and it's been fairly stable over consecutive days. People are interpreting it as a good sign that the virus isn't spreading but the intuition I laid out above would suggest otherwise. That was just the spark, and I'm more looking for general suggestions on statistical techniques and sub-branches that address problems like this so I can learn more about the first principles. Though, any thoughts you have on the specific problem would also be appreciated! Mar 21 '20 at 14:30
• If there was a peer reviewed study you have looked at, then including the reference within your question and structuring the question in that context would be at least put the question into a context that can be responded to. Please edit the question accordingly.
– Carl
Mar 21 '20 at 16:05
• Thanks, I have edited the question. Hopefully that makes it easier to address. Mar 21 '20 at 16:33