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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?

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  • $\begingroup$ 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 $\frac{O-E}{\sqrt{E\times(1-\text{row.mag}/n)\times(1-\text{col.marg}/n)}}$. $\endgroup$
    – Antoni Parellada
    Commented Apr 10, 2017 at 20:07
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    $\begingroup$ 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. $\endgroup$
    – whuber ♦
    Commented Apr 10, 2017 at 20:07
  • $\begingroup$ 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. $\endgroup$
    – Antoni Parellada
    Commented Apr 10, 2017 at 20:45

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There is a way yes. Just take a step back and consider that by significance you probably mean a low p-value which is the probability of getting at least as "extreme" as this exact observed count (in your example 120) given the null hypothesis.

If you trace back how the expected count in that particular cell in your table is calculated then you can calculate the probability you want.

I don't know the context behind your data but I'll give an example.

Example: the count is the number of heads after 200 coin tosses. If the coin is unbiased the expected count is 100. The observed count is 120 and thus

P(at least 120 heads|coin is unbiased) = 0.0018

Which is a significant at significance level of 1%

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