# Inference with one-sample Likert-type data

In case, one wants to do some inference with one-sample Likert-type data, what tests could one use? Signed Rank Test?

• What is the hypothesis you want to test? Sep 8, 2011 at 19:36
• Thanks, Aniko. I want to test whether the population is more biased towards one of the poles of the Likert scale, i.e., whether one of the halves of the scale attracts a larger share of the population. Sep 8, 2011 at 19:45
• @Paul if we assume 1 to 5 likert scale, do you mean "whether there are a higher percentage of 1s than 5s?" or do you mean "whether the mean of the scale is lower or higher than 3?" or something else? Sep 9, 2011 at 5:34
• Thanks, Jeromy. I mean "whether there are a higher percentage of 1s than 5s". Sep 9, 2011 at 20:47

For specificity, I will assume a 5-point scale in my answer. The usual approach is indeed the Signed Rank test with the middle value (3) serving as a location parameter. Note however that it does not test whether the median answer is 3. The null hypothesis is rather that the median is 3 and the distribution is symmetric around the median. So (with sufficient data) this null hypothesis will be rejected if either of those conditions is false.

So for example, if the true probabilities are (0%, 40%, 20%, 20%, 20%), then the signed rank test will be significant (with enough data):

> pvec <- c(0, 0.4, 0.2, 0.2, 0.2)
> x <- rmultinom(100, size=50, p=pvec) #generate 100 samples with n=50 each
> pvals <- apply(x, 2, function(x) wilcox.test(rep(1:5, x), mu0=3)\$p.value)
> mean(pvals < 0.05) #find the power
[1] 1


This is usually desirable, you just have to be careful wording the conclusions. If you truly care about the median, then a sign test should be used.

• Thanks, Aniko. True, I am using a 5-point scale. Your way works if the null hypothesis is not rejected; however, I am trying to establish that the median is larger than 3. Is there a way to deal with this? Sep 9, 2011 at 20:52
• @Paul I am not sure what do you mean. In the cautionary example the point is that the null hypothesis is always rejected even though the median is 3. But you could certainly argue that the rejection is appropriate because the values on the right tend to be higher. Sep 9, 2011 at 20:59
• @Aniko The point that I do not understand is that "you could certainly argue that the rejection is appropriate because the values on the right tend to be higher" when the median is 3. Could you please elaborate more upon that? Sep 10, 2011 at 12:25
• @Paul, I cannot make up your null hypothesis for you. If you only want to know whether the median is 3 regardless of how asymmetric the answers are otherwise, then you have to use a sign test. If, on the other hand, you think that "strongly agree" outweighs "disagree", then use the signed rank test, but don't call it a test for the median. Sep 12, 2011 at 14:11