Why it is said that large sample size can cause Type I error in Chi square test?
In adequate sample size may lead to chi square distribution; in this case, the distribution teds towards normality by increasing sample size. However, there are cases where the "nature" of the data is chi square and will not normalize when the sample size increases. Consider the general case of continuous data:
X2 = [(n - 1) / S2] / sigma2
As the sample size increases, where the difference between the observed variance and inferential variance are closed, the value of chi square tends to be large. This large value of chi square tends to provide the basis to reject the null hypothesis. The statistical background for this rationale is that small sample is said to be chi squared distributed, as the degree of freedom increases, the chi square will approach normalization. In the general case, this may be true. However, in a case where the nature of the data is truly chi square independent of sample size or degree of freed (df = n -1), the data distribution may remain chi squared. This insisting that as the sample size increases, chi square will become normalize---- is the source of Type I error.
TYPE I error: insisting that the alternative hypothesis (Ha) is correct or wrongly reject the null hypothesis (Ho). The chi square equation above would tend to lead rejection of Ho.
HOW TO AVOID THIS PROBLEM ? It is better to test for data distribution at various points, for instance at n = 30, if the data is normal, normal distribution could be seen. If normality is not seen, then repeat the test from the data set without increasing sample size. If through repeated test, the data remains non-normal or chi square distributed, it may mean that by its nature the data is indeed chi square.
SPECIFIC CASE IN DISCRETE DATA: Where the data is normal obtained from 2 x 2 table, for instance, the nature of the distribution is chi square. In this case, the distribution may not be normalized just by increasing sample size, i.e. normal data (yes | no) or coded (1,0).
..............................................
YES NO
..............................................
YES a b
..............................................
NO c d
..............................................
X2 = A / B
A = (n -1) (ad - bc)2
B = (a + b) (a + C) (b + d) (c + d)
see attached. p. 86.
LINK:
https://en.wikipedia.org/wiki/Chi-squared_distribution
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