# Why is odds ratio overlapping 1 while Chi-square P value is well below 0.05?

I have following values from an experiment:

   A   B
X 64  20
Y 62  11


I subject this to Chi-square test using following code:

from scipy.stats import chisquare
pval = chisquare([a,b], [c,d])[1]
print(pval)


Output is:

0.006421123271652286


This seems clearly significant (<0.05).

I now calculate odds ratio and its confidence intervals with above data using following formulae:

OR = (a*d) / (b*c)
se = math.sqrt((1/a)+(1/b)+(1/c)+(1/d))
lower  = np.exp(math.log(OR) - 1.96*se)
upper  = np.exp(math.log(OR) + 1.96*se)
print(OR, lower, upper)


Output is:

0.5677  0.2514   1.2819


( The confidence intervals agree with online calculator at https://select-statistics.co.uk/calculators/confidence-interval-calculator-odds-ratio/ )

So, confidence interval is very much overlapping 1, while I expected it to be on one side of 1 since P value was clearly significant.

I have following questions:

1. Where is the error and how can I correct it?

2. Would you call these data as statistically significant?

3. What test can I use so that P value and confidence intervals match?

Thanks for your help.

• Related (even possible duplicate): Why do my p-values differ between logistic regression output, chi-squared test, and the confidence interval for the OR? Nov 5, 2020 at 18:01
• There difference is very borderline, here it seems to be marked. Would you call these data as statistically significant?
– rnso
Nov 5, 2020 at 18:05
• Are you sure you are applying the software correctly? The p-value should be around 24%. Isn't chisquare supposed to be a "one-way chi square test"?
– whuber
Nov 5, 2020 at 18:24
• You were right about underlying incorrect use of software and about real P value (as shown in accepted answer below).
– rnso
Nov 6, 2020 at 7:32

## 1 Answer

The chisquare function tests given counts against expected counts. That's not what you intend. You're testing a contingency table. Use the chi2_contingency function with takes a table (nested array) as input and returns:

chi2: float
The test statistic.

p: float
The p-value of the test

dof: int
Degrees of freedom

expected: ndarray, same shape as observed
The expected frequencies, based on the marginal sums of the table.


The correct analysis gives a p-value of 0.24:

>>> from scipy.stats import chi2_contingency
>>> chi2_contingency([[64,20],[62,11]])
(1.3719790003937939, 0.24147215490328422, 1, array([[ 67.41401274,  16.58598726],
[ 58.58598726,  14.41401274]]))
>>>

• Yes, that was the error. Thanks.
– rnso
Nov 6, 2020 at 1:33