# Can a dichotomous variable (yes/no) be merged with a Likert measure (1,2,3,4) using z scores?

I would like to create a composite scale using items that measure parent communication extracted from a UK panel survey (n=941). However although all 6 variables in question measure similar constructs - 2 of the items are dichotomous (yes/no) the other 4 are likert (1,2,3,4). My intention is to transform the 6 variables to z scores and then sum them to form the composite score which will be used in a path analysis. I need to know if it is correct to do this, i.e., merge dichotomous variable (yes/No) and Likert scale (1,2,3,4) after they have been standardized to z scores.

Pending this being ok, would it be necessary to do an exploratory factor analysis to ensure the the composite scale measures one underlying construct or it is enough to do a reliability analysis?

This analysis is forming part of my PhD so I need to defend what I am doing and I would appreciate any guidance anyone can give me.

• The following paper offers the solution to your question. A Mathematical Theory of Ability Measure based on partial credit item response. published in J. of App. Measurement, 2015 (Vol. 1). Apr 13, 2017 at 20:12

I'm not sure about the first part of your question. But regarding the second bit: a reliability analysis of itself does not tell you if you have one underlying construct or several. You can have a high cronbach-alpha (for reliability) in the presence of two or more factors. Definitely, do the factor analysis as well as reliability. You might also want to check out the latent variable and item response literature. Some of these models are set up to handle dichotomous and polytomous outcomes - which might deal with the z-score problem as well.

– Cody
Oct 23, 2012 at 9:21

I just checked the textbook Introduction to Meta-analysis (Chapter 7: Converting Among Effect Sizes) and it is possible to standardize the outcome measures to combine them using the z scores (as you already know). Whether or not you do this will depend on the origin of the data and how similar the questions are to each other. If you believe that the 6 questions are really asking the same thing but presented using two different effect estimates then it should be fine to combine them.

Hope this helps.

AMAS

– Cody
Oct 23, 2012 at 9:20

The simple, "classical test theory" approach that weighs every item equally would probably have you just standardize all the items and average the $z$-scores, but I don't think that's a good idea, because four-point Likert scales may not approximate a continuous dimension well enough (and a binary item definitely won't; it might not even make sense), nor can you be sure that the average of four polytomous items will be approximately continuous enough. I've seen it suggested that each item's Likert scale should have at least five options to approximate a normal distribution, and at least five Likert scale items should measure the same construct if their simple sum / average is to approximate a continuous dimension. (I've seen Berry, 1993 cited as a source for an argument that five or fewer options is unacceptable, but I haven't read it myself.)