# Covariate vs. factors

I have a survey problem where the dependent variable (ordinal) is in Likert-type scale (i.e. 1 to 5 from most satisfied to most dissatisfied) and two sets of independent variables. One set has 7 IVs (almost the same scale but 1-5 scale) and a set of 5 IVs with a scale of 1-6, both ordinal. See Which is applicable, ordinal or multinomial regression model?

Which is the best way to analyze this kind of problem ? Do I need to treat the IVs as factors or covariates?

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How is this question different from <stats.stackexchange.com/questions/61834/…; ? – Maarten Buis Jun 17 '13 at 8:05
Are your sets of (independent) variables each supposed to measure one construct? Are those standard scales from previous research? – Gala Jun 17 '13 at 8:06
@ Gael Laurans The sets measure the same DV. – Saroni Jun 17 '13 at 8:13
@ Maarten, I have tried deciphering results out of both ordinal and multinomial but there is no head-way..Maybe an error is on how I treat the IVs either factors or covariates. to my understanding covariates can be input or leave them out..factors are ones that the study is mainly based on..true? – Saroni Jun 17 '13 at 8:22
@Saroni I edited the question based on your comment. I think these changes make it worthwhile and distinct from the previous one but is it still interesting for you? – Gala Jun 17 '13 at 8:52

The distinction between a “factor” and a “covariate” is related to the nature of the predictor/independent variable.

A factor is a nominal variable that can take a number of values or levels and each level is associated with a different mean response on the dependent variable. Even if the factor is coded using numbers, these numbers have no particular meaning. For example, it's perfectly possible for group ‘2’ to have a lower mean value on the dependent variable than group ‘1’ and ‘3’. Behind the scenes, in a regular ANOVA/linear model, the groups can be represented by a set of “dummy variables” with a different coefficient for each group.

Ideally, a covariate should be a continuous and interval-level measure but in any case the values have to be meaningful because the relationship between covariates and outcome/dependent variable is quantitative. A simple linear model will have a single coefficient to capture this relationship. Other models (models with interactions, polynomial regression, splines, etc.) add some complications but it should be meaningful to think about the magnitude of the covariate.

The notion that “factors” are essential and “covariates” can be left out stems from common study designs in psychology and some other fields. Typically, the main variable of interest will be manipulated by experimentally setting it to a handful of levels whereas demographic variables (age, personality, etc.) are simply measured on a more-or-less continuous scale. Consequently, the “factor” must definitely figure in the analysis but the “covariates” could possibly be ignored. The experimental design can also ensure that different factors are not correlated and the groups are balanced, which is not necessarily the case if you are merely observing/measuring variables.

Mathematically, however it does not make any difference whether you look at it as an ANCOVA, in which the continuous variables are called “covariates”, or a multiple linear regression, in which continuous variables are simply predictors (see When should one use multiple regression with dummy coding vs. ANCOVA?).

You can also very much design a study where the main manipulation is quantitative (imagine something like manipulating the temperature of a room) but ancillary measures are binary (say gender). You would probably not call the temperature a “covariate” but it certainly should not be used as a “factor” in an ANOVA or left out of the model. Whether a variable is “essential” or was experimentally manipulated will change the interpretation but not necessarily the way it figures in the model.

In your case, whether it is reasonable to treat multi-item Likert scales as interval measures could be debated and will also depend on the specifics of the data but it is certainly pretty standard. They are definitely not nominal.

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Thanks Gael but 1 more thing. Can I send you the questionaire and the sample data set and advice where am going wrong..I feel pretty stupid but have learnt much from your contribution. email? – Saroni Jun 17 '13 at 10:02
Well, I am happy to spend time answering questions here because they give me some interesting things to think about and I learn a lot myself doing it, double-checking things or clarifying my thoughts but if you want someone to do the analysis for you, you should hire a proper statistical consultant (or, if relevant, see if your employer/university already has one on staff). Many people offer this kind of service – Gala Jun 17 '13 at 10:20
Thanks Gael. Have learnt something. – Saroni Jun 17 '13 at 10:32
One issue missing here is that when there are multiple factors the interactions among them are often analyzed but once one asks for interactions with covariates it's not strictly ANCOVA anymore. – John Sep 23 '13 at 19:27
@john That's because neither the question nor the answer are about ANCOVA and rather arbitrary notions of what should or should be called that. In fact, my point is that it doesn't matter at all (but I provided a link to a question in which you touch upon the issue). – Gala Sep 23 '13 at 19:35