# Statistical significance of the residual in a particular cell in a Contingency table. Extrapolation from a Sample to the Population

The statistical significance of the overall residuals (expected - observed counts) in a contingency table can be tested through chi-square.

However, is a way to test whether a residual in a particular cell in a contingency table is significant and possibly estimate a confidence range?

E.g. if in a particular cell the expected counts are 100 and the observed counts are 120, we have a residual of 20 which is 20% of the expected value. Can we tell whether this difference is statistically significant? Assuming that the data represented a sample could we extrapolate the residual of the particular cell to the Population?

• After a chi square test, analyzing raw and standardized residuals is a way to get more information out of the contingency table. Alternatively the adjusted standardized residual is $\frac{O-E}{\sqrt{E\times(1-\text{row.mag}/n)\times(1-\text{col.marg}/n)}}$. Commented Apr 10, 2017 at 20:07
• It's a good question. Please note, though, that the correct way to assess a $\chi^2$ residual is to divide the difference (between the count and expected count) by the square root of the expectation. Here you would get $(120-100)/\sqrt{100}=2$ which is not large. (The size would typically be around $1$.) You might want to look into loglinear models.
– whuber
Commented Apr 10, 2017 at 20:07
• I don't have anything else to add to this article. Regarding @whuber suggestion, if you use R, I think R (and S-PLUS) Manual to Accompany Agresti’s Categorical Data Analysis (2002) by Laura Thompson is free online, and has examples on page 142. Commented Apr 10, 2017 at 20:45