Variables aggregation

[Serena Menny]

Hello,

After aggregating the initial 6 variables (x1, x2, x3, x4, x5, and x6) into 3 larger variables based on the results of a principal component analysis (PCA), with a Cronbach's alpha of 0.7245, I further aggregated these 3 variables into one overall variable. However, the Cronbach's alpha of this final aggregation was found to be 0.4346. The question arises: can I proceed with aggregating the 3 variables into one, given that individually they had an alpha of 0.7245, or is it not advisable due to the lower alpha of the combined variables (0.4346)?

Thank you,

[Clyde Schechter]

This is actually not a statistical issue. The answer to your question depends on what you are trying to measure with these variables.

Cronbach's alpha is a measure of the internal consistency of the responses to items in a scale. It tells you, loosely speaking, the extent to which people who tend to respond in a given way to one of the items tend to respond the same way to the other items in the scale. If you are trying to measure beliefs and attitudes, usually high internal consistency is a desirable attribute of a scale. It enables you to conclude that the summed or averaged scale responses provide a stronger signal-to-noise ratio than the individual items do, and that the items are collectively gathering information about a common construct. Scales constructed by factor or principal components analysis are typically of this nature as the factor or PC analysis identifies item subsets that cohere in this same sense.

But sometimes internal consistency is not important, or even undesirable, for a scale to achieve it's purposes. If, say, a bank wants to rate its customers on their degree of "loyalty" to the bank they might have items like:

1. Does this customer have a checking account with us?
2. Does this customer have a savings account with us?
3. Does this customer have certificates of deposit with us?
4. Does this customer have an investment account with us?
5. Does this customer have a retirement account with us?
6. Does this customer have a mortage with us?
7. Does this customer have a car loan with us?
8. Does this customer have a credit card we issue?
9. Does this customer have an installment loan with us?

and maybe more such items--you get the idea. Since most people have some limits on their finances, these items are not likely to be correlated with each other in a typical bank customer population. If anything, most people might only be able to afford a few of these, so that having some precludes having others and the correlation among the items could even be negative. Such a scale would have a low Cronbach's alpha, perhaps even negative. But clearly the sum of these items would be a very good measure of customer engagement with the bank.

So, no, you can't make this decision by looking at numbers. You have to think about what each variable means and what you are trying to get at by combining them to decide whether Cronbach's alpha is relevant or not.

[Serena Menny]

This is actually not a statistical issue. The answer to your question depends on what you are trying to measure with these variables.

Cronbach's alpha is a measure of the internal consistency of the responses to items in a scale. It tells you, loosely speaking, the extent to which people who tend to respond in a given way to one of the items tend to respond the same way to the other items in the scale. If you are trying to measure beliefs and attitudes, usually high internal consistency is a desirable attribute of a scale. It enables you to conclude that the summed or averaged scale responses provide a stronger signal-to-noise ratio than the individual items do, and that the items are collectively gathering information about a common construct. Scales constructed by factor or principal components analysis are typically of this nature as the factor or PC analysis identifies item subsets that cohere in this same sense.

But sometimes internal consistency is not important, or even undesirable, for a scale to achieve it's purposes. If, say, a bank wants to rate its customers on their degree of "loyalty" to the bank they might have items like:

1. Does this customer have a checking account with us?
2. Does this customer have a savings account with us?
3. Does this customer have certificates of deposit with us?
4. Does this customer have an investment account with us?
5. Does this customer have a retirement account with us?
6. Does this customer have a mortage with us?
7. Does this customer have a car loan with us?
8. Does this customer have a credit card we issue?
9. Does this customer have an installment loan with us?

and maybe more such items--you get the idea. Since most people have some limits on their finances, these items are not likely to be correlated with each other in a typical bank customer population. If anything, most people might only be able to afford a few of these, so that having some precludes having others and the correlation among the items could even be negative. Such a scale would have a low Cronbach's alpha, perhaps even negative. But clearly the sum of these items would be a very good measure of customer engagement with the bank.

So, no, you can't make this decision by looking at numbers. You have to think about what each variable means and what you are trying to get at by combining them to decide whether Cronbach's alpha is relevant or not.

Thanks a lot! Your answer was really helpful!