Asymptotic distribution of Pearson Chi-Squared test statistic comparing conditional versus unconditional sampling

Question

I was trying to understand two different sampling schemes in terms of independence of 2 categorical variables.

Scheme 1: I have an infinite population where each member poses with two attributes. Those attributes are basically realizations from two independent binary variables (with equal success probabilities). Now I draw repeated independent samples with varying sample size (sample size is assumed to have poisson distribution). And for each sample, I construct a 2 way contingency table and estimate sample Chi-Squared. And now I calculate what is the distribution of the Chi-Squared.

Scheme 2: Same population. However here the sample size is fixed. Also I calculate the distribution of Chi-Squared distribution.

In Categorical data analysis, 1st one is Poisson sampling and 2nd one is Multinomial sampling. In the 2nd case, I know that the distribution of chi-squared will be Ch-Square 1. My question is in the 1st case what will be the distribution? Will it also be Ch-Square 1?

Thanks for your clarification.

Answer 1

In both cases, the statistic will have the same asymptotic distribution since it's the tabular proportions which determine the level of association in the 2 by 2 tables. Fixing the number of samples drawn in one of the margins of the table is often done to improve the power of statistical tests, such as case-control sampling where the distribution of disease in the population is rare and the exposure is expensive to determine. You would conceivably test 100,000s of cancer negative individuals before obtaining a cancer positive individual using simple random sampling.