[R] scale or not to scale that is the question - prcomp

[Duncan Murdoch]

On 8/19/2009 10:14 AM, Petr PIKAL wrote:
> Duncan Murdoch <murdoch at stats.uwo.ca> napsal dne 19.08.2009 15:25:00:
> 
>> On 19/08/2009 9:02 AM, Petr PIKAL wrote:
>> > Thank you
>> > 
>> > Duncan Murdoch <murdoch at stats.uwo.ca> napsal dne 19.08.2009 14:49:52:
>> > 
>> >> On 19/08/2009 8:31 AM, Petr PIKAL wrote:
>> >>> Dear all
>> >>>
>> > 
>> > <snip>
>> > 
>> >> I would say the answer depends on the meaning of the variables.  In 
> the 
>> >> unusual case that they are measured in dimensionless units, it might 
>> >> make sense not to scale.  But if you are using arbitrary units of 
>> >> measurement, do you want your answer to depend on them?  For example, 
> if 
>> > 
>> >> you change from Kg to mg, the numbers will become much larger, the 
>> >> variable will contribute much more variance, and it will become a 
> more 
>> >> important part of the largest principal component.  Is that sensible?
>> > 
>> > Basically variables are in percentages (all between 0 and 6%) except 
> dus 
>> > which is present or not present (for the purpose of prcomp transformed 
> to 
>> > 0/1 by as.numeric:). The only variable which is not such is iep which 
> is 
>> > basically in range 5-8. So ranges of all variables are quite similar. 
>> > 
>> > What surprises me is that biplot without scaling I can interpret by 
> used 
>> > variables while biplot with scaling is totally different and those two 
> 
>> > pictures does not match at all. This is what surprised me as I would 
>> > expected just a small difference between results from those two 
> settings 
>> > as all numbers are quite comparable and does not differ much.
>> 
>> 
>> If you look at the standard deviations in the two cases, I think you can 
> 
>> see why this happens:
>> 
>> Scaled:
>> 
>> Standard deviations:
>> [1] 1.3335175 1.2311551 1.0583667 0.7258295 0.2429397
>> 
>> Not Scaled:
>> 
>> Standard deviations:
>> [1] 1.0030048 0.8400923 0.5679976 0.3845088 0.1531582
>> 
>> 
>> The first two sds are close, so small changes to the data will affect 
> 
> I see. But I would expect that changes to data made by scaling would not 
> change it in such a way that unscaled and scaled results are completely 
> different.
> 
>> their direction a lot.  Your biplots look at the 2nd and 3rd components.
> 
> Yes because grouping in 2nd and 3rd component biplot can be easily 
> explained by values of some variables (without scaling). 
> 
> I must admit that I do not use prcomp much often and usually scaling can 
> give me "explainable" result, especially if I use it to "variable 
> reduction". Therefore I am reluctant to use it in this case.
> 
> when I try "more standard" way
> 
>> fit<-lm(iep~sio2+al2o3+p2o5+as.numeric(dus), data=rglp)
>> summary(fit)
> 
> Call:
> lm(formula = iep ~ sio2 + al2o3 + p2o5 + as.numeric(dus), data = rglp)
> 
> Residuals:
>      Min       1Q   Median       3Q      Max 
> -0.41751 -0.15568 -0.03613  0.20124  0.43046 
> 
> Coefficients:
>                 Estimate Std. Error t value Pr(>|t|) 
> (Intercept)      7.12085    0.62257  11.438 8.24e-08 ***
> sio2            -0.67250    0.20953  -3.210 0.007498 ** 
> al2o3            0.40534    0.08641   4.691 0.000522 ***
> p2o5            -0.76909    0.11103  -6.927 1.59e-05 ***
> as.numeric(dus) -0.64020    0.18101  -3.537 0.004094 ** 
> 
> I get quite plausible result which can be interpreted without problems.
> 
> My data is a result of designed experiment (more or less :) and therefore 
> all variables are significant. Is that the reason why scaling may bye 
> inappropriate in this case?

No, I think it's just that the cloud of points is approximately 
spherical in the first 2 or 3 principal components, so the principal 
component directions are somewhat arbitrary.  You just got lucky that 
the 2nd and 3rd components are interpretable:  I wouldn't put too much 
faith in being able to repeat that if you went out and collected a new 
set of data using the same design.

Duncan Murdoch