# Hypothesis Testing using dichotomous and 7-point Likert scale

I am working on testing the following hypothesis:

H1: People who are worried about their health will consume yogurt H0: People who are worried about their health will not consume yogurt

I have interviewed 46 respondents using a 7-point Likert scale for the independent variable ('I am worried about my health' 1)Totally disagree 2) Disagree 3)slightly disagree 4)nor agree nor disagree, 5)slightly agree, 6) agree, 7) totally agree). The other question is dichotomous - Do you consume yogurt? 1)Yes 2)No.

I would like to test this hypothesis using the variables above even though one is in Likert scale and the other is dichotomous.

I recommend rewording them as follows:

• H0: Ratings of worry about health do not relate to yogurt consumption.
• HA: Ratings of worry about health [do] relate to yogurt consumption.

This avoids dichotomizing people into groups who do / don't worry about health, which is unnecessary because you have richer information than that. In addition to who does and does not worry about health, you know who slightly worries, who totally does not worry, who's neutral, etc. Don't waste that!

To correlate your ordinal worry variable with the binary yogurt variable, I recommend calculating Kendall's $\tau$. This effect size estimate is appropriate for ordinal (including binary) data and facilitates nonparametric hypothesis testing. In , the command is cor.test(x,y,method='kendall') where x and y are your two variables.

As an effect size estimate, I prefer to interpret $\tau$ on the scale of Pearson's r, as I am more familiar with that scale and expect most audiences to be as well. To convert $\tau$ to the scale of r, use $r = \sin\big(\tau\cdot\frac \pi 2 \big)$ as I mentioned in another answer. Full disclosure: I also borrowed much of the content of this answer from another I wrote earlier today.

• First of all thank you so much for answering! I was checking on how to calculate the Kendall T and I found this link youtube.com/watch?v=V4MgE43SrgM . This guy ranks the grades of teachers and correlates them with the grades of students and calculates the Concordance and Dicordance to reach the Kendall T. In my case, I have 7 to 1 Likerts scale data and 1 to 0 binary data so I am not sure how to calculate the kendall. Thank you. Aug 16 '14 at 16:47
• +1 ... but a nit-pick on some details. Compare the Pearson correlation and Kendall's tau on the following data: x <- c(17.91, 24.13, 8.58, 21.02, 5.47, 30.35, 14.80, 27.24, 33.46, 11.69) ... and ... y <- c(6.06, 8.53, 3.59, 11.00, 15.94, 20.88, 25.82, 18.41, 13.47, 23.35). Sep 2 '14 at 1:12
• ... I'm guessing the statement about the relation between $\tau$ and $r$ was intended under some restricted set of conditions (rather than always). Does it always apply when one variable is binary? Sep 2 '14 at 2:03
• @Glen_b: Oh, that was dumb of me. As per Bollen & Barb (1981), binning continuous variables will attenuate their Pearson correlation, so that's probably a general way to get a smaller r than $\tau$. I've rephrased that sentence and removed the parenthetical BS. Thanks for catching that! I don't think I can even try to defend my initial claim as true under any specific circumstances...not sure what I was thinking TBH. Sep 2 '14 at 17:39
• @Nick There are many, many situations where it's usually true; when I first read that I figured that maybe it was true -- but I couldn't help tugging at the thread. I think you might even be able to claim that under some reasonable set of conditions it's nearly always true, but I haven't really gone into that. I'd be very interested in what could be said about it. One might have more luck specifying conditions under which the relation holds between Spearman's rho and Kendall's tau. Sep 2 '14 at 22:40