# Chi-square & fisher's exact test output interpretation

I want to investigate the correlation between two nominal variables. I have executed chi-square and fisher exact test using SPSS.

I have also executed Cramer’s V (V=0.444, p=0.000) and phi coefficient (value=0.768, p=0.000) . My interpretation of the results is:

• The two variables are dependent
• The relationship is strong

Am I right?. Thanks in advance.

• The low p-value doesn't tell you that the effect is strong; with large sample sizes, even trivial effects will be statistically significant. – Glen_b -Reinstate Monica May 14 '17 at 1:03
• @Glen_b, I assumed that the relation is strong based on Cramer's V and Phi coefficient results and not on the p-value. – Pila Luffy May 14 '17 at 11:47
• Ah. That makes sense, sorry. – Glen_b -Reinstate Monica May 14 '17 at 11:48

Yes, you're correct on both accounts. These statistics strongly support the conclusion that the effect is both statistically significant the effect size is large compared to typical effect sizes for typical variables included as explanatory or control variables in a typical scientific or business analysis context.

We can make this concrete by looking at an example of a contingency table with similar summary statistics:

$$\begin{bmatrix} 10 & 3 & 2 & 1 \\ 2 & 10 & 2 & 3 \\ 1 & 2 & 11 & 3 \\ 0 & 3 & 2 & 12 \end{bmatrix}$$

Here, the accuracy is 64%. That is to say, out of the four possible class labels, each of which is approximately likely, the class label is in fact exactly the same 64% of the time, compared to the ~25% predicted by under the independent hypothesis. So it should be intuitively clear that is an example of two variables that exhibits strong dependence.

Yet the summary statistics for this example are similar to what you report: $\chi^2 = 58.279$, $df=9$, $p = 2.87\mathrm{e}{-09}$ for the chi-squared test with continuity correction. For effect sizes, $\phi = 0.746$, $v = 0.53$. (Because you say the variables are nominal, Cramer's v is slightly preferred to Pearson's $\phi$.) All of this is in the same ballpark as what you report, and suggests that your data exhibits a similarly obvious and strong relationship.

When we look for guidance on interpreting effect sizes, we'll see different opinions, but in general Cramer's V over 0.5 is considered extremely strong. However, such broad guidelines are not very useful without context. For example, one place where I use Cramer's V scores in my work is to perform data quality checks between databases. Cramer's V is used to compare database columns containing categorical data, and I expect to see v-scores over 0.9, otherwise the data quality is suspect. So in that context a 0.53 would be considered much too low, not "extremely strong." The point being that it depends very much on context. But in the context of analyzing a curated data set where variables have different semantic meanings, (say Highest Education Achieved and Income Level,) then v=0.53 would indicate an extremely strong relationship.

As you have opted to not provide details on the context I assume that it's sensitive and/or proprietary, but I hope these guidelines and examples will give you a point of reference while you are drawing your own conclusions.

• Excellent answer. I have a question, tho. What is phi when the table is larger than 2 x 2 ? – Sal Mangiafico Jul 16 '18 at 1:25

The first one is OK.

The second one has problem.

In jury trials, jury decides guilty or not guilty, judge makes the decision on punishment. In statistics, test is similar to jury on court. Based on the test you can get the conclusion to accept or reject null hypothesis. But you can not get the conclusion about the extent.

For the extent of relationship, need to check the percentages.

• Thank you for your response @a_statistician. The second interpretation is based on Cramer's V and Phi coefficient results. Should I check the percentages even if Cramer's V and Phi coefficient results shows that the relationship is strong? – Pila Luffy May 14 '17 at 11:54
• For second conclusion can be derived from Cramer's V and Phi coefficient. Maybe you can get confidence intervals of Cramer's V and Phi coefficient. The lower bounds of the CI maybe is the best measurement of the strength of the association. – user158565 May 14 '17 at 17:44