composite index with PCA derived weights

[bmkraju]

composite index with PCA derived weights

Seasons greetings !

I was trying to develop a composite index of vulnerability including 38 indicators. I am using weights obtained from Principal component analysis (PCA) in the lines of Nagar & Basu (2002).

I got negative weights to 7 variables in the first instance. I was redoing PCA by dropping one of the variables at a time that got highest negative weight. I am repeating this till I get positive weights to the variables carried. Now i am left with 18 variables with positive weights.

Is this procedure OK. Dear Seniors, Please suggest me the way if I am wrong.

With best regards,
BMK Raju
Scientist
INDIA

[Dason]

Hi,

Please only post a topic once.

Why are you doing this? Why do you require the variables to have positive weight?

[bmkraju]

Sorry for posting in two groups. I thought the groups are mutually exclusive.

Really Thanks for responding.

In fact, I want to assess vulnerability of state (spatial unit) to extreme climatic variability. My variables are like incidence of drought, dry spells, intense rainfall events etc. Now I want to combine these variables and develop a composite index of vulnerability. The hick up is these variables are not completely independent. (Example: drought and dry spells are correlated). So, I cannot linearly combine the variables which are not independent. Therefore I used PCA to generate independent (underlying) variables. These are linearly combined taking the variance explained by each PCA axis as weight.

Before PCA, all the variables are transformed using the following rescaling method.

X*=(X-min)/(max-min) if the variable is positively related to vulnerability
X*=(Max-X)/( max-min) if the variable is negatively related to vulnerability

As a result, higher the value of a transformed variable higher the vulnerability. All variables are brought to one direction.

Weights under went for each original variable through PCA were worked out. Say a variable’s loading on a PCA axis j is Zj and corresponding eigen value is Ej. Then the weight realized for a given variable in the index will be ∑_j▒〖(Ej*Zj)〗

I am getting this value for few variables as negative. When all of my transformed variables are positively related to vulnerability, why do I get negative weights to certain variables. How to proceed further. Please guide me.