I have a data-set about cities features like population, poverty, economic position, income etc. population, poverty and economic position are continues but poverty is based on a scoring. We have 100,70,50 and 30 scores for cities (only these values). cities with 100 score has highest poverty and cities with 30 score has lowest poverty. Now i want use PCA for this data-set.
* Is PCA suitable for this mixed data because one of the assumptions of PCA is continues inputs. If not, what are suitable techniques to do that? Finally i need rotated component matrix based on varimax rotation and need a software or package to do the rotation on loadings for me.
* Should i standardize data when using correlation matrix (not covariance) in PCA? (currently because of some theoretical aspects i select a city as base city and it's feature values as denominator of other cities values + removing outliers using x-mean(x)>=2S.D rule)
* Should i use correlation matrix here? we use it when we have different scales for features but here my data is scaleless.
* Normal distribution of inputs is an assumption of PCA? because my inputs don't show normal distribution, should i transform data using log()?
Working from the middle out:
(Package for getting rotated loadings reported) SPSS will run a PCA under the command Data Reduction, and one option is to ask for the rotated matrix, using varimax or other rotation approaches. I have to assume SAS does as well, but you will need a SAS guru for affirmation - I only do SPSS, SEM, and HLM.
(standardizing variables for correlation) ??? You *do* know that the bivariate correlation is based on standardized values, right? Or, to put it another way, yes, certainly, standardize your variables and run a correlation. Then compare it to the correlation on the original variables. **Exactly the Same**
(normal distribution assumption for PCA) It is an assumption, but it is a weak requirement. Since there is no probability estimate, you can use non-normal measures. However, you would learn additional information if you try normalizing your really skewed measures and running those in a PCA. If the loadings shift dramatically, you will know the extent to which that assumption is relevant to your situation. This book will help, if you want something to cite:
http://www.amazon.com/Principal-Comp.../dp/0387954422
(scaleless data) ?? All but one of your measures have a scale and that one at least has an order. Scaleless data are measures in which the order of the categories is not relevant, such as race (1=white, 2=black 3= asian etc or 1=black 2=asian 3=white etc)
(poverty measure) This is a strange measure, since it is really just 4 categories that are ordered. With them identified as 100, 70, 50, and 30, you are forcing the analysis to treat the categories as intervals that are differently spaced (20 between 1 and 2, 20 between 2 and 3, 30 between 3 and 4). You could just recode them as 1-4 if you want the categories to be treated as strict ordered intervals. You could recode 100 to 90 to make the interval coding even. Or you could try all kinds of recodes and see if they make any difference in the loadings - I love crass empiricism. :-)
Thank you for answer Julia. So based on assumption of PCA that inputs should be intervals, this is not a problem to have 1-4 values for an input?
Not if you are putting the measure into a scale with other items, as a means of data reduction. If you are testing psychometrics on your scale, I would try it a couple different ways to see which do and do not make a difference.
Thank you for answer. So as i mentioned i have different inputs with different scales and i select a city as base city and it's feature values as denominator of other cities values so new it is not like max min normalization that all values are between 0 and 1 [or -1 and 1] after normalization. Is this a true approach to use a city as base (for example capital)? Here how can i select a good boundary for mentioned inputs (for example you mentioned 1-4 scaling for poverty). There isn't different between 0-4 and 1-5 in this structure and my proposed normalization based on capital?
+ for more information i have two inputs with this behavior. in first one (100, 70, 50, and 30) and in second one (100, 70, 50, and 30), in first one higher values is better but in second input lower values is better.
I think Rotated Matrix under factor analysis will do the magic. delete variables which are less than your specified value and cross loadings then repeat the process again until a desire result is achieved. You can go ahead and name your new factors from the rotated component matrix.
Software: IBM SPSS, STATA, SAS
IBM SPSS is easy to use.
Rotation: IBM SPSS has two types of rotations:
Orthogonal: Varimax, Quartimax and Equamax: Factors are allowed not to correlate.
Oblique: Direct Oblimin, Promax: Factors are allowed to correlate.
Varimax: It maximize the variance of the squared loadings of a factor.
Quartimax: It minimizes the number of factors. It is not use for research studies.
Equamax: It is a compromise between Varimax and Quartimax
Direct Oblimin: It gives higher eigenvalues but reduced interpretability
Promax: It is computationally faster than the direct oblimin.
You can start with Varimax and conduct different alternatives i.e. eliminating few variables or adding to see which suits theory and your need.
You can use PCA or Factor analysis by using SPSS and SAS also. There are so many options of rotation like Varimax, Promax, Quatrimax etc. and in both the software it is easy to use. In SPSS its very easy.
If in values there is a big differences then log transformation you can use otherwise no need. you can make data normal but there is not any compulsion
I recommend you use correlation Matrix
PCA does not require normality
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