Hi Sanjeev,

Good questions. Please see my responses below.

> 1) Do we skip a response variable from the dataset we

> input to principal

**In princomp, you wold just leave a variable out of the forumula. In principal, you enter a list of variables or a correlation matrix. Don't include the variable in the list.**
> 2) Princomp allows entry of a formula ? how do we

> achieve something similar using principal

**I'm not sure you can. The principal function expects a list of variables or a correlation matrix.**
> 2) I am unable to interpret the PC1, PC2 as well as

> the RC1, RC2 values. I understand PC1 and PC2

> contribute to 84% of the variance. So what variables

> are significant?

>

> Here is the output I obtained for mtcats? Any

> interpretation of results will be appreciated.

>

>

> >

> carss=principal(mtcars[,1:11],nfactors=2,score=TRUE,ro

> tate="none"

> > carss

> Principal Components Analysis

> Call: principal(r = mtcars[, 1:11], nfactors = 2,

> rotate = "none",

> scores = TRUE)

> dardized loadings based upon correlation matrix

> PC1 PC2 h2 u2

> .93 0.03 0.87 0.131

> cyl 0.96 0.07 0.93 0.071

> disp 0.95 -0.08 0.90 0.098

> hp 0.85 0.41 0.88 0.116

> drat -0.76 0.45 0.77 0.228

> wt 0.89 -0.23 0.85 0.154

> qsec -0.52 -0.75 0.83 0.165

> vs -0.79 -0.38 0.76 0.237

> am -0.60 0.70 0.85 0.146

> gear -0.53 0.75 0.85 0.150

> carb 0.55 0.67 0.76 0.244

>

> PC1 PC2

> .61 2.65

> Proportion Var 0.60 0.24

> Cumulative Var 0.60 0.84

>

> Test of the hypothesis that 2 factors are

> sufficient.

>

> The degrees of freedom for the null model are 55

> and the objective function was 15.4

> he degrees of freedom for the model are 34 and the

> objective function was 2.95

> The number of observations was 32 with Chi Square =

> 74.21 with prob < 8.1e-05

>

> Fit based upon off diagonal values = 0.99>

**It is vary rare to be able to interpret an unrotated solution that has more than on component. That is why we use rotation.**
]> >

> carssrc=principal(mtcars[,1:11],nfactors=2,score=TRUE,

> rotate="varimax"

> > carssrc

> Principal Components Analysis

> Call: principal(r = mtcars[, 1:11], nfactors = 2,

> rotate = "varimax",

> scores = TRUE)

> dardized loadings based upon correlation matrix

> RC1 RC2 h2 u2

> .68 -0.63 0.87 0.131

> cyl -0.64 0.72 0.93 0.071

> disp -0.73 0.60 0.90 0.098

> hp -0.32 0.88 0.88 0.116

> drat 0.85 -0.21 0.77 0.228

> wt -0.80 0.46 0.85 0.154

> qsec -0.16 -0.90 0.83 0.165

> vs 0.30 -0.82 0.76 0.237

> am 0.92 0.08 0.85 0.146

> gear 0.91 0.17 0.85 0.150

> carb 0.08 0.87 0.76 0.244

>

> RC1 RC2

> .67 4.59

> Proportion Var 0.42 0.42

> Cumulative Var 0.42 0.84

>

> Test of the hypothesis that 2 factors are

> sufficient.

>

> The degrees of freedom for the null model are 55

> and the objective function was 15.4

> he degrees of freedom for the model are 34 and the

> objective function was 2.95

> The number of observations was 32 with Chi Square =

> 74.21 with prob < 8.1e-05

>

> Fit based upon off diagonal values = 0.99>

>

> > carss$scores

> PC1 PC2

> 982 1.04919357

> Mazda RX4 Wag -0.2409801807 0.93709938

> Datsun 710 -1.0641633049 -0.08854287

> Hornet 4 Drive -0.1193693965 -1.42860381

> Hornet Sportabout 0.7559836345 -0.45608682

> Valiant -0.0214936918 -1.68432399

> Duster 360 1.1496507138 0.20246195

> Merc 240D -0.7869352163 -0.88580026

> Merc 230 -0.8757928531 -1.19917506

> Merc 280 -0.2015385197 -0.09794749

> Merc 280C -0.1949623577 -0.19581597

> Merc 450SE 0.8606317650 -0.41320595

> Merc 450SL 0.7840613648 -0.41305282

> Merc 450SLC 0.8226243650 -0.48470540

> Cadillac Fleetwood 1.4931345709 -0.50055025

> Lincoln Continental 1.5139372504 -0.44337838

> Chrysler Imperial 1.3756612988 -0.25460433

> Fiat 128 -1.4764769496 -0.17940646

> Honda Civic -1.6287652388 0.41619252

> Toyota Corolla -1.6211797995 -0.16884804

> Toyota Corona -0.7290594069 -1.28158472

> Dodge Challenger 0.8365260349 -0.61316242

> AMC Javelin 0.7134440470 -0.54801873

> Camaro Z28 1.1061255250 0.41160510

> Pontiac Firebird 0.8599075525 -0.52827815

> Fiat X1-9 -1.3683852489 -0.07327586

> Porsche 914-2 -1.0151008644 1.23716872

> Lotus Europa -1.2963042050 0.83345590

> Ford Pantera L 0.5256718947 2.11598496

> Ferrari Dino -0.0003790165 1.94614547

> Maserati Bora 1.0219431622 2.64780919

> Volvo 142E -0.9267860308 0.14125103

> >

**The interpretability of an exporatory principal components or factor analysis is only as good as the variables entered. If there is no reason to think that the variables will cluster to form meaningful composites, you probably will not find any.
**

In the rotated mtcars example

1. The higher a car's score on the RC1 component, the higher rear axle ratio (drat), and more forward gears, higher mpg, lower weight, fewer cylinders, and lower displacement, and probably a manual transmission. The lower the score, the opposite.

The same approach is used to interpret RC2. Here, higher scores on the component indicate greater horsepower, more carborators, more cylinders, lower quarter mile time, lower V/S, and lower mpg.

I don't know if these are useful components. It would depend on your knowledge of cars.

Hope this helps.