# Rescaling a 5-point Likert scale to 6 points?

Two groups (A and B) were sent matching surveys with a 6-point Likert scale. However, due to an administrative error, Group A's survey had a 5-point Likert scale instead of a 6-point Likert scale. That is to say, Group A's survey has the mid-point 'No opinion' and Group B's survey has no mid-point but the two central points are 'Somewhat disagree' and 'Somewhat agree'.

I would like to compare how the two groups differ on the individual items of the survey. Is it possible to rescale the results so that they are comparable?

• @Nick I interpreted the question differently: I understand "two groups" to mean two non-identical groups. Thus, we can conceive of the data for a given survey item as being a sequence of 5 counts for group A and 6 counts for group B. – whuber Sep 10 at 17:24
• @whuber your interpretation is correct; there are two separate groups (patients and doctors) who are rating their experience of their consultation. So we have patient-doctor pairs who each filled out the questionnaire but the patients have a 5-point Likert while the doctors have a 6-point Likert. – spacegrasshopper Sep 10 at 17:47
• Well, that sounds more like Nick's interpretation! For each survey item and each pair you increment a count in one cell of his $5\times 6$ table corresponding to the doctor's answer and the patient's answer, respectively, thereby creating a $5\times 6$ table of counts for each item. It isn't apparent that you actually need to equate the two scales. – whuber Sep 10 at 18:54
• Do not hurry with simple linear rescaling from a 5 point range onto 6 point range. Inspect the two distributions first. – ttnphns Sep 10 at 20:55
• Not what you want to hear, but there isn't and can't be a single, simple fix here. – Nick Cox Sep 11 at 16:12

My personal experience with Likert scores is that it makes a difference in subjects' behavior whether or not there is a 'neutral' category. One possibility is that subjects who really have no opinion will try to be 'nice' by picking the slightly favorable response when no neutral response is available. So it is not obvious that any method of conversion from Likert-five to Likert-six is going to be without difficulties.

Otherwise, you might try multiplying scores 0 through 4 by $$5/4$$ to get five scores evenly spaced between 0 and 5.

Let's try a couple of examples with simulated data to see how this might work in practice. In the first simulation, available items are chosen at random, so there should be no difference between the converted Likert-5 items and the native Likert-6 ones, and a 2-sample Wilcoxon (rank sum) test finds no difference (P-value > 0.05).

set.seed(2020)
x = sample(0:4, 200, rep=T)
table(x)
x
0  1  2  3  4
38 37 46 39 40

x6 = 5*x/4
table(x6)
x6
0 1.25  2.5 3.75    5
38   37   46   39   40

y = sample(0:5, 200, rep=T)
table(y)
y
0  1  2  3  4  5
41 27 32 32 33 35

wilcox.test(x6,y)

Wilcoxon rank sum test with continuity correction

data:  x6 and y
W = 20343, p-value = 0.7648
alternative hypothesis:
true location shift is not equal to 0


Now we look at a simulation in which respondents tend to choose higher items (with no difference in 'behavior' with or without a neutral category). Again, no significant difference.

So with a couple of straightforward cases where the conversion 'ought to' work smoothly, it does work smoothly. You might explore other (possibly worrisome) scenarios with simulations of your own.

set.seed(910)
x = sample(0:4, 200, rep=T, p=(1:5)/15)
x6 = 5*x/4
table(x6)
x6
0 1.25  2.5 3.75    5
17   19   42   50   72
y = sample(0:5, 200, rep=T, p=(1:6)/21)
table(y)
y
0  1  2  3  4  5
8 20 29 38 59 46
wilcox.test(x6,y)

Wilcoxon rank sum test with continuity correction

data:  x6 and y
W = 20488, p-value = 0.668
alternative hypothesis:
true location shift is not equal to 0

• I might add, that adding here the consent bias (of which many respondents are not free), the neutral point on a scale is not the middle of it but rather shifted towards the biggest or right-hand end point. Thus, with a 7 point Likert rating scale (1 through 7) the neutral benchmark is 5, rather than 4, for a considerable percent of respondents. – ttnphns Sep 10 at 20:49
• @ttnphns: Thanks for fixing whatever typo you may have found in my Answer. Also, agree with your Comment. – BruceET Sep 10 at 21:30