Interpretability for chi-squared test?

Question

Sex	DiabetesNOTPresent	DiabetesYESPresent
Female	50	55
Male	124	70

So I have this chi-squared test that I performed. My result was statistically significant. The data relates to heart failure in patients at a certain hospital. How would I interpret this? Please correct me if I'm wrong but...

My finding is that the proportion of women that have diabetes is expected to be higher than the proportion in that of men, correct? Is this true for all patients coming into that hospital? Is it true for only the amount of patients similar to that of those found in my dataset, and this just repeats for every new "group" of patients? Would saying that "the proportion of men having diabetes is smaller than the proportion of women with diabetes" be just as valid?

Also, just because it is significant, does that mean that it's actually helpful? Or is the proportion difference just too small? Wouldn't the proportion difference be just as large whether I have 100 samples vs 10,000 samples... increasing as I have more patients?

Please nitpick everything I've spoken. This is probably one of the first chi-squared tests I've performed.

Answer 1

The chi-square test of a 2x2 contingency table such as this basically tests the following null hypothesis: gender should produce no difference in diabetes rates. Essentially, your chi-square test poses the following question: "Is the difference in diabetes rates by gender more than we would expect?".

In this case, you have a lot of males that do not have diabetes. Because this number is disproportionate, your chi square is consequently 6.78, giving a significant value. However, you don't know the strength of this association yet, so it may also help to also obtain Yule's Q coefficient. You can get this by using the psych package in R, using the Yule function. I demonstrate with your data below:

#### Construct Contingency Table ####
diabetes <- matrix(c(50,55,124,70),
                   ncol=2,
                   byrow=TRUE)
rownames(diabetes) <- c("Female","Male")
colnames(diabetes) <- c("No Diabetes","Diabetes")
diabetes

#### Test Table ####
chisq.test(diabetes) # chi square
psych::Yule(diabetes) # Yule's Q coefficient

The association is moderate, as shown by the result:

[1] -0.3217054

So to summarize, your test is significant, indicating that you can say with some level of certainty that we cannot support the null hypothesis: gender seems to be associated with diabetes rates. Your Yule coefficient explains that this association is moderate.

To answer some of your additional questions, this test only relates to the sample size you have, so it is not generalizable to all hospitals. This should make sense intuitively, as 1) your sample size isn't extremely large and 2) hospitals can vary a lot, such as how doctors are trained, access to resources, etc. Is this helpful? Certainly. While we would want more people to test and more sophisticated ways of tackling this question, this at least informs us that at the very minimum there is in fact a trend at this hospital, and it may (with caveats) indicate that this could be the case elsewhere. To your question about proportionality, there is no way of knowing whether or not this would hold for larger samples, but theoretically if the effect is consistent, the underlying assertion of the test would still hold. Only more testing in more settings can answer how generalizable your findings actually are.

To see what is going on under the hood, this video shows how to calculate both chi-square and Yule's coefficient by hand. This video is also an accessible summary of what chi-square tests do.

Answer 2

I'm responding to this post for future reference and to expand on a potentially interesting aspect that has not been fully covered in previous answers (i.e., the selection of an appropriate coefficient to measure this association).

A look at the proportions in the contingency table is essential to understand the "unilateral" nature of the association between gender and diabetes. The lack of "bilateral" association is evident when we analyse the cell proportions.

Your data displays:

• Female: 50 without diabetes, 55 with diabetes
• Male: 124 without diabetes, 70 with diabetes

From these counts, we calculate:

• Proportion of females without diabetes relative to all without diabetes = 50 / (50 + 124) = 0.287
• Proportion of males with diabetes relative to all with diabetes = 70 / (55 + 70) = 0.560

In a scenario of bilateral association, we would expect the proportions of cell 'a' (females without diabetes) and cell 'd' (males with diabetes) to be similar. The discrepancy between these proportions suggests an asymmetric association, primarily driven by the lower likelihood of males having diabetes.

In fact, we observe:

• A near-even split in the diabetic condition among females, with the odds being roughly 1.1
• A sizeable skew towards non-diabetes among males, with considerably lower odds of having diabetes (0.565)

This discrepancy highlights that the association is unidirectional; it is mainly driven by the lower likelihood of males having diabetes (OR 0.513) rather than an equally strong association for both genders.

The use of Yule's Q, suggested in the previous answer, is thus very appropriate here as it quantifies the strength of the asymmetry in the association, unlike symmetric measures such as the phi coefficient, which might misrepresent the true nature of the relationship by suggesting a mutual influence that is not really there.

By emphasising these proportions and their implications, one can provide a clearer and more precise explanation of why specific statistical measures are preferred and how they contribute to one's understanding of the data.

The following references might be helpful:

Mueller, J. H., & Schuessler, K. F. (1961). Statistical Reasoning in Sociology. Oxford & IBH Publishing Co. Google Books

Alberti, G. (2024). From Data to Insights: A Beginner’s Guide to Cross-Tabulation Analysis. Routledge - CRC Press. Google Books