# Normality scale variables consisting of several ordinal variables

I have five ordinal variables with a 7-point likert scale attached to them, where all 7 possible answers have a "meaning" (for example: 1 = totally disagree, 7 = totally agree). These five variables together are presumed to measure "Feelings of Autonomy", this has been confirmed by several analyses. Now I have constructed this "Feelings of Autonomy" scale by having SPSS calculate the mean of all five ordinal variables. Hence these 5 variables together form a scale, in which the values can contain decimals. Before conducting a regression analysis, I have to check normality of the scale variables. My scale variable is constructed by using five ORDINAL variables (it's not exactly the same as a scale variable representing seconds, height etc.), and therefore my question: Do I need to check for normality?

• Not so; regression doesn't depend on normality of the response. The bigger deal is whether averaging ordinal variables is substantively meaningful and gives you a variable suitable for regression, which on this evidence is hardly to judge. Jan 16 '17 at 21:00
• There may be some confusion here between the concepts of "ordinal" and "continuous." Discrete variables can be measured on an interval or ratio scale whereas ordinal variables can be continuous. Some purists would object to your taking the mean of ordinal scales although I believe that, in practice, this is very unlikely to lead you to a false conclusion. The assumption of normality is another thing altogether. Of course only continuous variables can be normal but likert scales (and means of likert scales) are sometimes close enough to normal for many practical purposes. Jan 16 '17 at 21:03
• It's the IV. The 5 ordinal variables & the scale they form, stem from an existing questionnaire. Regarding my data, analyses confirm the theory behind this dimension (Feelings of Autonomy). So assuming - for the sake of my thesis - that creating a "scale" variable by calculating a mean for each respondent is a justifiable choice: Does it pose the same kind of threat to my regression as non-normality in continuous variables that are not constructed out of ordinal variables, and why? By the way: I know there is always debate regarding the necessity of normality etc., but that discussion aside. Jan 16 '17 at 23:21
• Regarding the assumption of normality with ordinal variables: As Frederick Lord said "the numbers don't remember where they came from." Jan 17 '17 at 0:09

Before conducting a regression analysis, I have to check normality of the scale variables.

This notion is wrong. There is no assumption of normality of either the DV nor the IVs in regression. The usual hypothesis tests and confidence intervals and prediction intervals make use of an assumption of normality, of the error term in the regression model (equivalently, the conditional distribution of the DV is assumed to be normal) but you can't assess that by looking at the DV itself. The DV might be very far from normally distributed (e.g. it might be skewed or bimodal) without any problem for the assumptions of your inference. [It shouldn't matter what the distribution of the IVs is at all. There are things you might worry about but that isn't one of them.]

You know that the conditional distribution of the DV cannot actually be normal (so it's pointless to use hypotheses testing for it), but that's not really the relevant question, which is whether the extent to which it's non-normal will badly affect your properties of your inference. (i.e. its about how much effect your non-normality has, not whether it is normal; you already know it's not)

(In any case, there are alternative possible tests - and indeed estimators if that were felt necessary - one could use without assuming normality)

My scale variable is constructed by using five ORDINAL variables (it's not exactly the same as a scale variable representing seconds, height etc.),

Note that in order to add the components, you already assumed at the moment you added them that the components were interval -- as soon as you say a "5" plus a "2" is the same as a "3" plus a "4" or a "6" plus a "1" (and for that matter, a "5" on the second scale plus a "2" on the first scale) -- calling them all $7$. When you do that (along with all the other things that are treated as equal by the process of addition), you incorporate the assumption that "5" - "4" = "3" - "2" (and so forth), which is explicitly assuming you have interval data. You no longer have ordinal scales you already treated them all as interval.

So whether or not it was okay to add them (i.e. treat them as interval), the choice about whether they could be treated that way was already made back at the point you added the components. This is not something you can figure out from whether or not the sum looks normal (that's not relevant to any aspect of your question).

Do I need to check for normality?

If you need to assume something is normal it would be good to have some reason to think it's not so far from normal that it would badly affect your results -- or to avoid the assumption if you don't have a suitable reason to think so (but again, normality assumptions could be avoided in your inference, whether you use ordinary least squares linear regression or some other form of linear model). But if you are going to assess the assumption beware of looking at the wrong thing (per the earlier comments).

• Thank you for your answer. So if I understand you correctly: 1. I have to look at the normality of the errors of the regression model (Normal Q-Q Plot of the studentized residuals)? 2. By creating a mean of the 5 ordinal variables I have already (indirectly) assumed they're interval? 3. The previous makes my question about the difference between my IV and an interval variable consisting of metric data redundant? Jan 16 '17 at 23:42
• @Eline 1. I didn't suggest you have to do anything, but yes, if you were going to do an assessment of whether the errors were reasonably consistent with normality the most obvious first thing to do would be to examine a normal Q-Q plot of residuals). Even if they're fairly clearly not normal it doesn't necessarily imply there's a serious problem but it will let you get a sense of the kind of non-normality and how large it might be (though one should also keep in mind that residuals tend to be somewhat more normal-looking than the errors they approximate). ...ctd Jan 16 '17 at 23:53
• ctd... and from there it may be possible to get an idea of how much it might matter. 2. I'd say you assumed it quite directly, but I suppose that depends on how aware one is of what looking at "5" and "2" and recording 7 (and so on) really implies. One might add numerals without thinking about what it means and then argue that the consequence was implicit rather than explicit. 3. As far as I can see you don't ask a question about that in your post. You state something, but there doesn't seem to even be an implied question. Jan 16 '17 at 23:53
• Sorry, I forgot to mention I asked that additional question in my reaction on the comments of Nick and David. I meant "have to.." as "if I want to do it correctly I have to.." Thank you, everything is clear now. Jan 17 '17 at 0:07
• I don't know that "if you want to do it correctly, you have to ..." is a correct interpretation of what I said. There's more than one way to conclude assumptions are reasonable (you might have some prior reason, either from similar studies or by some theoretical reasoning to think such a scale will yield a reasonably normal error-distribution for example), or to figure out the kind of deviations you should see can make little difference or to avoid the need to make such assumptions at all. ...ctd Jan 17 '17 at 0:37