# interpreting chi square test of independence result

so I've conducted a chi square test of independence to determine whether there was a relationship between payment method (paper and no paper) vs churn (churn and no churn) and determined that there was indeed a relationship between the two as seen in the picture I attached. however, how can I know whether someone who uses no paper is more likely or less likely to churn as compared to those who uses paper? can I just look at the frequencies?

Compare observed and expected counts in the cell of the table for 'NoPaper' by 'Churn'. Alternatively, look at Pearson residuals.

Here is the chi-squared test in R for your data.

MAT = matrix(c(102, 195, 26, 172), byrow=T, nrow=2)
chisq.out = chisq.test(MAT, cor=F)
chisq.out

Pearson's Chi-squared test

data:  MAT
X-squared = 27.882, df = 1, p-value = 1.29e-07
chisq.out$$obs [,1] [,2] [1,] 102 195 [2,] 26 172 chisq.out$$exp
[,1]  [,2]
[1,] 76.8 220.2
[2,] 51.2 146.8
chisq.out\$resi
[,1]      [,2]
[1,]  2.875543 -1.698212
[2,] -3.521807  2.079876


The chi-squared statistic is $$Q = \sum \frac{(X_{ij} - E_{ij})^2}{E_{ij}},$$ where $$X_{ij}$$ are observed counts, $$E_{ij}$$ are expected counts (obtained from row and column totals), and the sum is taken over all four cells of the table. The null hypothesis that paperless payments and churn are independent is strongly rejected with a very small P-value.

The observed count in the cell for paperless and churn is only 26, whereas the expected count assuming Paper and Churn are independent factors is 51.2. So, according to your data, it seems that paperless payments are associated with reduced churn.

Pearson residuals are $$\pm\sqrt{\frac{(X_{ij} - E_{ij})^2}{E_{ij}}},$$ where the sign is the same as for the difference $$X_{ij} - E_{ij}.$$ When $$Q$$ is sufficiently large to reject the null hypothesis of independence, then it is customary to look at the Pearson residuals with the larger absolute values (especially absolute values exceeding 2 or 3) to interpret the practical consequences of the observed association. Here the key Pearson residual $$-3.52$$ is for the cell I mentioned earlier.

• thank u so much! this is amazingly helpful! Apr 5, 2020 at 1:36