I have two independents groups of two nominal variables (2x2 table).
To calculate the effect size should I use Odds Ratio or Phi Coefficient? Or both?
Can you you show us the 2x2 table and say a bit more about the context? E.g., what are the variables? And are you asking about what to report in a manuscript being submitted to a journal, or in some other context? What is conventionally reported in your field of research? Etc.! Thanks for clarifying.
PS- You may find this discussion interesting:
https://stats.stackexchange.com/questions/79848/phi-coefficient-and-odds-ratios
The odds ratio has the advantage that it has an explanation : it is the change in odds for a unit increase in the predictor. In the case of a 2x2 table, this corresponds to the effect of the binary variable on the odds.
On the other hand, no-one can explain what phi is. It's just Chi-squared rescaled between zero and one. People have attempted to divide it into tee-shirt sizes of small, medium and large, but why bother calculating a statistic to n decimal places if all you're going to do with it is replace it with vague words?
Will anyone speak up for phi? Has anyone found a way of explaining it, or an actual use?
There are lots of measures for the 2x2 association. As for Ronán Michael Conroy 's challenge, I'll skip, but Article Kappa Coefficients in Medical Research
speaks out for an alternative to both of these. I'd add the log of the odds ratio as also being useful. Depending on the context, separately reporting the sensitivity and specificity along with a measure of association is often useful for readers.
Yule's Y (aka., the coefficient of colligation) is a transformation of the odds ratio that ranges from -1 to +1. You might consider reporting it.
- https://en.wikipedia.org/wiki/Coefficient_of_colligation
Unfortunately, some of my old friends at McMaster reinvented Yule's Y and named it phi when using it as a substitute for kappa in 2x2 tables where the marginal frequencies are extreme. See pp. 452-453 of the following, for example:
- https://studentnurse.yolasite.com/resources/NCM104/Evidence_As_A_Nurse/Measuring%20Agreement%20Beyond%20Chance.PDF
Yule's Q, another transformation of the OR, is another option.
- https://en.wikipedia.org/wiki/Goodman_and_Kruskal%27s_gamma#Yule's_Q
Thinking about Yule's Y and Yule's Q again inspired me to plot one against the other for ln(OR) ranging from -5 to +5. See the attached image. My Stata code is below--feel free to check that I did not louse anything up.
clear
set obs 101
generate logor = (_n-51)/10
generate or = exp(logor)
generate Y = (sqrt(or)-1)/(sqrt(or)+1)
generate Q = (or-1)/(or+1)
*list, clean
scatter Y Q, ms(oh) ///
xtitle("Yule's Q") ytitle("Yule's Y") ///
title("Yule's Y vs Yule's Q") ///
subtitle("For ln(OR) ranging from -5 to 5") ///
xlab(-1(.2)1,grid) ylab(-1(.2)1,grid angle(h)) ///
xscale(range(-1 1)) yscale(range(-1 1))
Here's another pair of graphs plotting Y and Q against ln(OR). Notice that the sigmoid shape is much less pronounced for Y than for Q.
There is no single correct measure of effect size for any effect. Even for a 2x2 table there is a wide range of options. I personally would prefer the OR to phi in this case, but in other contexts other measures make more sense. So I don't think this can be answered in the abstract without more information about the data context and what you want to do.
Phi isn't usually great because it is just the Pearson correlation between the two binary variables coded 0/1. That can be misleading (especially if there is an underlying continuous latent variable measured in a very coarse way).
If there's an obvious outcome variable it may make more sense to use absolute or relative risks rather than OR. See https://en.wikipedia.org/wiki/Risk_difference These measures are base rate sensitive, though so its important to consider whether you want a portable measure than you can say compare across different data sets (with different base rates) or a measure of risk within a particular population (that incorporates the base rates).
Ahmad Al-shallawi – you say Use phi coefficient only.
This kind of advice is not very useful because you don't say why. Is phi more interpretable to the general reader? Is it more robust? Is it generalisable to tables with different marginals?
As you can see from the discussion, there is no single best answer, and the person who asked the question will have to make a decision based on the hypothesis and the audience they are communicating to.
So why phi? Have you an argument you would like to bring to the discussion?
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