The questionnaires and the obtained data were not efficient.

Good Luck.

An eigenvalue less than 1 means that the PC explains less than a single original variable explained, i.e. it has no value, the original variable was better than the new variable PC2.

This would fit with factor rotation producing a second factor that is related to a single variable. That single variable contributed to PC1, so less than 100% of its variance remains available to subsequent PCs or factors- which is why low rank PCs dependent on a single variable must have an eigenvalue is less than 1.

There are many possible reasons for this, 3 of which are:

1) you are studying a dataset created by a single data generating process, so you would expect only 1 PC.

2) you have multiple data generating processes but a poor experimental design and have created a very noisy dataset so there is only enough 'signal' in your data to exceed the noise in the 1st PC

3) you have multiple data generating processes, good experimental design but poor data curation. Have you validated any data collection procedures, coding transformations, data processing? Are all your variables on the same scale and dynamic range? If not, did you centre and scale your data?

What software did you do with this analysis? SPSS?

Thank you James Renwick Beattie , I did performed both PCA and Factor analysis using R matrix. According to the R matrix, 5 of variable are relatively correlated each other between 0.62 and 0.85. but, when I assumed to use second factor which has an eigenvalue lower than 1 to reach more than 80% of variation, first factor explains 5 variables and second factor explains one variable when I subjected the varimax rotation. May I explain still use second component and second factor? In fact, I used the RSM of central composite design and trying to visualize the all runs using PCA. So I also want to see how the responses are correlated each other. Do you think using PCA and factor analysis can be logical for CCD runs?

Thank you @Abolfazl Ghoodjani I used the SAS for analyzing the data.

If all you want to know is which variables interact then PCA and factor analysis works well. The relative contribution of each variable to the factor/PC will give you a direct measure of how much the variable is involved in that interaction (note that PCs are unit vectors, so the sum of the squared weights is 1). This means that you can readily determine what % of a PC/factor's variance is explained by individual variables: it is simply the square of the weight x100.

Thank you James Renwick Beattie for your answer, it was very helpful for me to think about them. I will think and do more research about the functionality of PCA and factor analysis for my purpose. Basically, I know the general logic about them, but I am trying to understand deeply for the use of them. Thanks again.

Best,

Serap

No law against using these. It depends what you are using them for. If the second PC was great at differentiating the 18 groups you would want to tell your readers that (even if some of the commentators above think this should be hidden from the readers). Often a scale is constructed such that there is a large general factor and then other factors. Bi-factor and higher order (here second order) factor analysis could be used (or just having correlated factors, but if the general factor accounts for a lot these other methods often work better). These can be done in free software (e.g., the lavaan package, http://lavaan.ugent.be/). Once the general factor is included, a clearer pattern sometimes emerges. My bigger question is how you are going to compare the groups. 18 is a lot. There are methods around that are appropriate to compare groups by multiple variables.

Dear Seraphs,

you can but you have to keep in mind that in that case it is possibly better to include all the items in your regression. An Eigenvalue lower than one means that the factor does not "amplify" the effect of each component and thus it explains less than the components.

An additional check you can do is to verify whether you can obtain a good factor by dropping some of the components that have a very low contribution because of the low correlation. You can preliminarily check whether the Cronbach Alpha can be improved by dropping some components and then, re-run your factor analysis with a reduced number of components.

Hope this helps

Best regards

Andrea

Thank you Daniel Wright for your answer it was very helpful to me. I am going to check the significance of second PC, then I will decide to use or not to use. I am reading more about both analyses to fit with my research.

Thank you Andrea Moro for your response. Yes, you are right Maybe I should drop one of the variable since factor analysis has already merged two variables in the plot. I am planning to re-run factor analysis after it.

Thanks,

Serap