This question extends Meg's question and Glen_b's comments/answer found here. I'm wondering about the appropriateness of two approaches to conducting post-hoc comparisons following the (omnibus) chi-square test of a larger than 2-by-2 contingency table. I searched and searched The Google and she was unable to provide a definitive answer.
Say we are asked whether the arrival mode of trauma patients differs between patients admitted to two medical facilities. Some example data is provided in the contingency table below (with row percentages given we are interested in between-facility differences).
Facility Ambulance Transfer Private Vehicle Total A 300 (60%) 150 (30%) 50 (10%) 500 (100%) B 400 (80%) 80 (16%) 20 ( 4%) 500 (100%)
The omnibus 2-by-3 chi-square test indicated that there is some association/difference between arrival mode and facility (p < .001). Under this scenario, the post-hoc comparisons would require breaking the 2-by-3 contingency table into a series of 2-by-2 tables. There are two approaches to do this.
The first approach I was taught back in the day would require omission of data within one column. For example, we would omit the Private Vehicle column and compare the rates of Ambulance vs. Transfer between the two facilities. Updated data are provided below.
Facility Ambulance Transfer Total A 300 (67%) 150 (33%) 450 (100%) B 400 (83%) 80 (17%) 480 (100%)
You will notice that the relative percentages increased (to different extents between facilities) because we reduced the overall denominator by excluding those arriving to the facility via private vehicle. Here, the chi-square test is statistically significant (p < .001) and indicated that the number of patients arriving via ambulance to Facility A was lower than expected, whereas the number of patients arriving via ambulance to Facility B was higher than expected (and vice versa for Transfer).
The second approach would be to maintain the total number of patients in the denominator by aggregating columns to allow direct comparison of the original within-arrival mode percentages between facilities. For example, comparing the percentages of patients arriving via ambulance by asking whether 60% in Facility A is statistically smaller than 80% in Facility B. Updated data are provided below.
Facility Ambulance Not Ambulance Total A 300 (60%) 200 (40%) 500 (100%) B 400 (80%) 100 (20%) 500 (100%)
Here, the relative percentages remained the same as the original 2-by-3 contingency table given we maintained the denominator. The chi-square test is also statistically significant (p < .001) and also indicated that the number of patients arriving via ambulance to Facility A was lower than expected, whereas the number of patients arriving via ambulance to Facility B was higher than expected (albeit with different rates compared to the first approach).
As a side note, the question of 60% vs. 80% (or, 67% vs. 83%) could be answered equivalently via the two-proportion z-test.
So, when considering these two approaches, my overall question is whether both approaches are appropriate statistically. I am uneasy about the first approach given that the updated percentages are contingent (not a pun) on how many patients were removed from columns not being considered in the comparison. Thoughts are appreciated and should provide an answer to a question that appears to be buried deep in the bowels of textbooks and the internet.