I've been playing around with some data to make a p-chart recently, where I also have varying subgroup sizes. As I was reading up again on p-charts, I was surprised to see Donald Wheeler is actually a fan of using IR-charts where most textbooks would advise using a p-chart (or np-chart or c-chart or u-chart). His basic argument is that if the p-chart assumption of a binomial distribution is right, the IR-chart will give you about the same limits, but if the assumption of binomial distribution is wrong, the IR-chart will give you better empirical limits rather than using the p-chart's model-based limits. See e.g. https://www.qualitydigest.com/inside/quality-insider-article/what-about-p-charts-093011.html and https://www.qualitydigest.com/static/magazine/jul/spctool.html
Since Wheeler's books were my introduction to SPC, I tend to like his ideas. So for my case with data like yours, I'm thinking I may just make an IR-chart of the p-values. Another side benefit of me is that this will also give each lot equal weight, independent of sample size. A typical p chart or mean chart with varying sample size will give more weight to the lots have more data. That makes sense if your process is under control, and the sample size for each lot is not meaningful information. But in some cases with two-stage sampling designs, when there is a lot with poor performance characteristics they sample more data. So you can end up giving more weight to poor-performing lots.
One thing I love about SPC is that it's a very practical / applied field. In the end, you could make a p-chart and an IR-chart, and you could try making different p-charts with different values for LIMITN. With SPC, your goal is not to calculate the "correct" process limits in the same way that you might want to calculate the "correct" estimate of the variance of a parameter in some fancy statistical model. Instead, your goal (in my experience), is to calculate "useful" limits, where "useful" means do the process limits help you understand your process, and are they useful in helping bring the process under control and identifying lots with special cause variation.
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