Suppose that I have two binary variables x and y, to test whether they are correlated, I have obtained the following samples.
(x, y)
1 1
1 1
1 1
1 0
1 1
0 1
1 1
1 1
1 1
1 1
Different correlation coefficients give the following results
Spearman: rho = -0.1111, p-value = 0.7599
Kendall: tau = -0.1111, p-value = 1
Pearson: rho = -0.1111, p-value = 0.7599
But telling from the data, we can be sure that x and y are some how correlated. What type of correlation coefficients can deal with this situation? Or what is the method to confirm the correlation, if there is no such correlation coefficient?
By looking at the data, you can definitely not infer that x and y are correlated!! That's the whole reason why you'r p-value is 76%: if there was no correlation, then in over three-quarters of all situations, you would end up with data like these.
You have ten pairs of observations. Eight of them are identical: (1,1). Only two deviate, both in a different direction: (0,1) and (1,0). No correlation coefficient in the world would call this 'significant'.
By looking at the data, you can definitely not infer that x and y are correlated!! That's the whole reason why you'r p-value is 76%: if there was no correlation, then in over three-quarters of all situations, you would end up with data like these.
You have ten pairs of observations. Eight of them are identical: (1,1). Only two deviate, both in a different direction: (0,1) and (1,0). No correlation coefficient in the world would call this 'significant'.
We record absence or presence of a dominant genetic marker in binary form (like your case) and instead of calculating correlation, we calculate resemblance (similarity) coefficients (simple matching (SM), Dice, Jaccard etc.).
In your case I would calculate SM= (a+d)/(a+b+c+d) = 8/(8+1+1+0)=0.8. Here, a=(1,1), d=(0,0), b=(1,0), c=(0,1). Hope this approach would help you.
Dear Yan
I agree with Casper.
But in problems like this it would be wise to consider besides the Tetrachoric correlation (a special case of the polychoric correlation) between the latents normal variables that may be supposed behind X and Y (http://en.wikipedia.org/wiki/Polychoric_correlation). For example SAS computes it with the PROC FREQ. In your example the lack of a value 0,0 do not lets it to do the computation.
I agree with Seyit: your data are binary, not quantitative. A correlation is not well suited for your data, but binary similarities like Jaccard, Dice, Sorensen, etc. are. There exist over 20 of these indices, depending on the origin of the data and the goals of your research. I suggest you to take a look at the thesis of Matthijs J. Warrens
(2008) called "Similarity Coefficients for Binary Data", from Leiden University, https://openaccess.leidenuniv.nl/bitstream/handle/1887/12987/Full%20thesis.pdf?sequence=2.
what did you do finally? my data are like yours, but I have more than 2 variables. could u help me please
Dear Nayereh
I concur with Casper, your data base is very small to find a significant correlation, but I think that in a matter like this it is best to use a tetrachoric correlation between the binary variables. See for example: https://en.wikipedia.org/wiki/Polychoric_correlation
really thanks for the quick answer. I have 58 isolates which are resistant (1) or sensitive (0) to 13 antibiotics. Is a tetrachoric correlation proper for my data?
Dear Nayereh
I do not know what is and isolate but if is x is the response to one antibiotic and y the response to another then I think that the tetrachoric correlation may be proper. If your response where X1, .. X13 from each antibiotic you may use the tetrachoric correlation too.
Dear Yan-Fu
I am sorry because I confounded my first interlocutor. What I wrote was for you both.
Thank you. One more question. How could I calculate this correlation? I searched but did not find an useful way. these are my data:
bacteria: antibiotic1 antibiotic2.....13
1 1
2 0
3 1
.
.
59
Dear Nayereh
I do not understand your question. Can you send to me a complete data table?
OK Nayereh, I understand it now. Your data may be analized as I said before. Between each pair of antibiotics you may have a tetrachoric correlation. Then you may obtain a 13x13 matrix of correlations and study its structure with a principal component analysis. May be that with many less than 13 dimensions you may reproduce most of the variability. What this may means is that most of the action of the antibiotics over the isolates of bacterias would be explained by similar characteristics of your 13 antibiotics. This is similar to the irt (item response theory) techniques that I used before for studing data from education studies.
How to calculate the correlation coefficient from mutual information from 000 to 111. Please indicate equations if any one know. I request you to give me some quick response.
The data 000 to 111 could be the outcome of a random vector (X1,X2,X3) which would follow trivariate bernoulli distribution. Any subset (Xi,Xj) would follow bi variate bernoulli. Now you can find the correlation of any two variables.
https://www.wiley.com/en-au/Discrete+Multivariate+Distributions-p-9780471128441
Dear Manzoor khan, Thank you for your answer but this is not helpful. Suggesting a book is not too helpful. I also can not download the book. Please help me to the point. Thank you.
Use inter reliability measure such as Cohen Kappa and weighted Cohen Kappa. Computation can be ease using "irr" R packages. Thank You
Direct measurement of correlation coefficient from two binary variables would not give you satisfactory result.
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