# How to test if proportions are the same across categories?

I am working with collision data and want to run a few tests. I thought a chi-square was what I wanted, but it doesn't let me say what I want to. Then I thought a z-test of two proportions was what I wanted, but I realized I'm violating the major assumption of independence.

DV: Injury severity of collision (either fatal/severe or non-fatal/severe). IV: Road classification that collision occurred on (principal, primary, major, collector, secondary and local).

I want to be able to determine:

1. Is the distribution of crash severity the same for all road classifications? I would conduct six chi-square test of homogeneity, where each record would take on six new dummy variables for each road class—the collision was either on a primary road (=TRUE) or not on a primary road (=FALSE); on a major road (=TRUE) or not on a major road (=FALSE); etc. The results are below:

2. Is there a significant difference between the proportion of collisions that were fatal/severe and the proportion of collisions that were non-fatal/severe? This is where I messed up, forgetting that these two are not independent (the more fatal collisions, the fewer non-fatal collisions). Still my incorrect results are below. However, what I want to be able to say with statistical confidence is, "Collisions are between X and X% more likely to be fatal when they occur on Y-type of roads than other types of roads."

Does anyone have any advice on how to choose the right test to say what I want? Any other comments on what would be a better test? Or follow-up tests?

• The complement of Fatal / severe is anything else. So there should have been $4$ crashes on Principal road classes total, or there is a problem with your data. Regarding 1, are the degree of injury classifications ordered (ie Fatal / severe > Visible injury > Pain > Property damage)? I also don't understand your data setup in 1: 341 accidents are False for Fatal / severe for Principal roads, but that isn't the sum of the numbers of non-fatal / non-severe accidents for Principal roads, & it isn't the sum of the numbers of Fatal / severe for other road types. – gung - Reinstate Monica Oct 28 '15 at 22:37
• There were 4 crashes on Principal when you add fatal/severe (1) and non-fatal/severe (3). For the first question (chi-sq), the sum of True and False in each injury severity should be the same for each road class. E.g. Principal roads have 341 False and 1 True for Fatal/severe (=342). Major roads have 181 False and 161 True for Fatal/severe (=342). Yes, these are ordinal (fatal being worse, property damage being least severe). These tables have correct data, but it might be confusing to look at them together. Try looking at them completely separate. – plnnr Oct 28 '15 at 23:41

For 1, get rid of all the False rows. All of the information you need to work with is contained in the True rows. At that point, you could do a chi-squared test. However, you may prefer to use ordinal logistic regression as the levels of degree of injury are ordered, and the chi-squared test does not take that fact into account.

For 2, your percentages are computed column-wise. You want them to be row-wise. If you got rid of those percentages and just had a contingency table with two columns and 6 rows, you could do another chi-squared test (or a logistic regression).

• In addition to what gung proposes, you could do a correspondence analysis on the (modified) table to an aid in interpretation. – kjetil b halvorsen Oct 29 '15 at 9:52