As an earlier blog, I wondered about the equivalencies of the Chi-Square test and the two-proportion test while I was updating our Lean Six Sigma Green Belt material.
Since both are appropriate tests for comparing two proportions, they should provide an equivalent result. Lets see.
Chi-Square test and two-proportion test data
This data is provided in a summary format so that it can be tested using both methods.
I chose to have a lot of samples to make sure that there is no issue with low cell counts in the Chi-Square test for association (two-way table test). This test requires at least a count of 5 in each cell. Larger counts make it more accurate.
Chi-Square test
This test shows a significant difference in the two proportions.
Chi-Square Test for Association: Period, Worksheet columns
Rows: Period Columns: Worksheet columns
Defects Passed All
Before Change 487 30572 31059
439.7 30619.3
After Change 573 43235 43808
620.3 43187.7
All 1060 73807 74867
Cell Contents: Count
Expected count
Pearson Chi-Square = 8.802, DF = 1, P-Value = 0.003 Likelihood Ratio Chi-Square = 8.724, DF = 1, P-Value = 0.003
Now lets try the two-proportion test
Two-Proportion test
We will use the same summary data for the test.
Test and CI for Two Proportions
Sample X N Sample p 1 487 30572 0.015930 2 573 43235 0.013253
Difference = p (1) - p (2) Estimate for difference: 0.00267646 95% CI for difference: (0.000906807, 0.00444611) Test for difference = 0 (vs ? 0): Z = 2.96 P-Value = 0.003
Again the test shows that the two proportions are significantly different at a 95% confidence.
Summary of these results.
This is only a single test, but the results showed an equivalent p-value on both methods (at least within the rounding provided in a Minitab output). This is a comforting result, because both methods are acceptable to make this comparison. Since both null hypothesis statements are equivalent, the two proportions are equal, we should expect every hypothesis test with the same null and alternate hypothesis to provide the same p-value.