Hello Glenn,

It is a bit unclear, but I'm assuming from your question that you performed FA as a data reduction method in order to generate composite variables from your scale data. You identified 5 named factors and now want to generate 5 variables consisting of factor scores for each the 5 variables (i.e., each of your 302 observations will have a single score for each of the 5 variables). If this is not correct, please clarify. Depending on the software you are using, the factor scores for these new variables can be generated automatically by the analysis software; so you do not have to calculated them. For example, in SAS, factor scores are generated by specifying the number of factors to be retained and by including an OUT statement, naming an output dataset in which the factor scores will be generated. Keep in mind that the correct method for this type of data reduction analysis is Principal Components Analysis (PCA) rather than FA, per se, although both are often referred to under the umbrella of factor analysis. 

There have been so many questions here about how to create scales for Likert-scored items that I have created a resource to address this topic:

https://www.researchgate.net/post/How_can_one_help_people_create_simple_scales_from_Likert-scored_items

Dear Glenn

If you mean that you want the score for each factor to reflect its weight, then you can go to the eigenvalue for each factor which reflects the variance explained by each the factor

Thanks kindly for the responses all. 

James, you have described what I'm doing.  However, I'm still not sure what I can do with my factor scores now that I have them. 

This is a quote from the paper of the method that I'm trying to emulate: "values were calculated using factor score weights from single sub-scale models that were then applied to individual raw scores and averaged to give a mean for each population sub-scale."

I think that this implies that they multiplied the weights of each factor by the score (and then took the mean)? I have a table of scores for each observation and a table of weights for each variable, but I'm still unsure how to 'apply' the weights to the scores. 

I am using R do to my analysis. 

Hi Glenn,

when they say "single sub-scale models" it rather implies, that they first conducted an overall FA finding the factors and loading items (but this step could have also been ommited), then again ran five FAs for each factor/scale, , i.e. the sub-scales. This in fact would be appropriate when using PCA instead of PFA., because PCA always overestimates communalities in a large item set with multiple factors. (However, without knowing the origninal study I can only speculate, but FA is widely misunderstood...). Then they gathered the factor score for each individual according to those five "independent" scale analyses. They then used those scores as weights (yes multiplication, or derived automatically as James points out) for each observation of the sub-scale items (the likert responses). Then, you are right, they took the mean for each sub-scale. And it is unclear, but it would make sense to sum up the weighted sub-scale scores for each participant, before reporting the overall mean, i.e. one sub-scale score for each participant goes into the overall mean.

(Edit: when I am reading my interpretation and the quote again, it looks incredibly dubious to me :))

Just for understanding...

There is a distinction you may consider regarding simple sum scores (adding items scores to a sum) or weighted sum scores. By definition a Factor Analysis of this kind you report is exploratory, meaning that factor scores (and loadings) depend on the characteristics of the sample. Thus, when you apply the factor scores, the reason is that you want to describe the sample, but not for making a generalized claim about the construction of the overall scale or factors. For the latter, simple sum scores would be more appropriate.

I don  not know how this weighting is done in R, but you surely find this with a little research on "factor scores in R"

Best, René

Glenn. If your data points relate to numerical  values such as income, age, test scores,  etc. then factor analysis is very appropriate.  Or you may very well have addressed the issue I am going to raise.  If so please just ignore this question. 

Are you certain that your Likert scales have provided scalar, or interval, data rather than ordinal data?  In other words, is the interval between values consistent across the range of values?  Is the interval between 1 & 2 the same as the interval between 2 & 3? Just because you have assigned numerical labels to the scales, does not mean that the values measured share the properties of numbers. 

If you can't affirm that the data is scalar, your analysis may be flawed.  For example, means and analysis of variance both assume interval data. 

I have been told there may be ways to generate interval data from Likert scales (Krosnick personal communication).  However, I am afraid I can't remember the research.  I believe it was to only label the endpoints of the scale but i could be wrong.   I also don't see how it fits other work reported by Krosnick. 

For example,  Krosnick, JA 1999, 'Survey Research', Annual Review of Psychology, vol. 50, pp. 537-67  shows that using numbers as labels on scales reduces reliability and validity compared to using words as labels.  If one uses words, one really can't assume the intervals between points on the scale are equivalent.

See also Saris, W, E., Revilla, M, Krosnick, JA & Shaeffer, EM 2010, 'Comparing Questions with Agree/Disagree Options to Questions with Item-Specific Options', Survey Research Methods, vol. 4, no. 1, pp. 61-79.  This article shows that item-specific scales yield much better data than generic scales such as Agree/Disagree.  However, if one uses item-specific scales, even if one uses numerical labels and one could be assured they were interval data, one is inherently measuring different intervals in the  different scales.

Perhaps more importantly, even if your scale does reflect numerical values such as number of hours watching TV,  your responses probably reflect the range used in the scale rather than actual data.  See 

Schwarz, N & Hippler, HJ 1987, 'What response effects may tell your respondents: informative functions of response alternatives', in H. J. Hippler, N. Schwarz and S. Sudman (eds), Social Information Processing and Survey Methodology, Springer-Verlag, New York.

Thanks for the responses Rene and David. 

Your ideas are very helpful Rene, I'll give that a shot, I'm pretty confident with doing FA in R now so it shouldn't be too tricky to do an FA for each sub-scale. 

Thanks for the information David. I understand that there is much disagreement on how to treat and analyse Likert data, hence why I decided to replicate the method in the original paper (something I'm starting to regret). 

When you say you followed the original authors, I assume this applies to your earlier quotation of them: "values were calculated using factor score weights from single sub-scale models that were then applied to individual raw scores and averaged to give a mean for each population sub-scale."

If so, it sounds like Rene and I have similar interpretations. First, the program (whichever one you use) produces weights for each item, and these are multiplied by the raw scores. The next step would be to sum up these weighted scores within each sub scale; averaging them simply makes the results more interpretable, because all it does is divide by a constant (i.e., the number of items in each sub scale).

This is a standard approach to such things, although I myself often use the simpler approach of just adding the scores without using factor weights. In terms of basic algebra, this assigns each item the same weight (implicitly, 1.0). This amounts to an assumption that there was little variation among the factor weights. For example, if all that items you include in a factor have loadings that range from .4 to .6, then using factor scores as weights will produce almost the same results as unit weighting.

FYI this unit weighting approach is equivalent to the reliability estimate you get from Cronbach's alpha, which assumes that you simply add all the items together. This is why alpha is often referred to as a "lower bound" for estimating reliability -- because using actual weights rather than equal weighting could give you a more reliable measure. Of course, if the alpha for each sub scale is already .8 or higher, then there is little to be gained by using a more complex weighting system.

Thanks kindly for your response David, I'm still working on this but I think I've worked out what I need to do now with your (and others here!) help. 

Kind regards,