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I'd like to know what PCA tells me about how the variables affect each other.

For example, let's say I've three variables Cholesterol, Exercise, Calorie Intake and Sleep. I want to know how Exercise, Sleep and Calorie Intake affect Cholesterol. Will the Cholesterol be lower (or higher) if I eat more calories or if I exercise more? After PCA I get the following:

Standard deviations:
[1] 2.0562689 0.4926162 0.2796596 0.1543862

Rotation:
                     PC1         PC2         PC3        PC4
Cholesterol  0.36138659 -0.65658877  0.58202985  0.3154872
Exercise    -0.08452251 -0.73016143 -0.59791083 -0.3197231
CalorieI     0.85667061  0.17337266 -0.07623608 -0.4798390
Sleep        0.35828920  0.07548102 -0.54583143  0.7536574

                          PC1     PC2    PC3     PC4
Standard deviation     2.0563 0.49262 0.2797 0.15439
Proportion of Variance 0.9246 0.05307 0.0171 0.00521
Cumulative Proportion  0.9246 0.97769 0.9948 1.00000

What I understand from this is how each individual component account for the variance in the data. The only thing I can say here is that PC1 and PC2 has a cumulative variance which account for ~98% which I implicitly interpret as PC3 and PC4 having no affect on the data at all. I understand that PC1 and PC2 are enough to explain all the data from the four given variables but does it say anything about how the variables affect each other?

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    $\begingroup$ I think Cholesterol be higher if you eat more calories through Low density lipoproteins. But PCA is not a tool to investigate how variables effect or affect each other. Regression techniques would be more appropriate for that. $\endgroup$ – ttnphns Aug 12 '15 at 15:44
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    $\begingroup$ PCA says absolutely nothing about how variables affect each other. It can only describe how they are mutually related. See stats.stackexchange.com/questions/534. $\endgroup$ – whuber Aug 12 '15 at 15:57
  • $\begingroup$ Yes, you're probably right, these are just values I made up it so that I could make my question more precise. Thanks for your answer $\endgroup$ – Lennart Aug 12 '15 at 15:57
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    $\begingroup$ My comment is not based on any analysis of your data. It doesn't matter what values your data have--you cannot use PCA (by itself) to use data like this to conclude anything about how changing calories or exercise will affect cholesterol. $\endgroup$ – whuber Aug 12 '15 at 15:59
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    $\begingroup$ @Antoni Use the language of association, relationship, variation, and correlation rather than of causes, effects, dependencies, responses, or explanations. The former describe patterns in the data--which it is legitimate to interpret--while the latter rely on information that is not in the data. $\endgroup$ – whuber Aug 12 '15 at 16:05
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If you want to see how cholesterol is affected by the different variables (sleep, exercise, and calorie intake), I would run a multiple regression with cholesterol as your dependent variable. After running your regression, you will be able to interpret the independent variables. These interpretations will tell you how each of the independent variables affects cholesterol.

PCA is generally used as a variable reducing technique. It is used when you find multicollinearity between variables in your regression. Thus, it will reduce the collinearity and make your regression coefficients more reliable. Since you will only have 3 variables in your regression, collinearity should not be a problem (at least I would hope not as the variables are all somewhat different).

I hope this helps!

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  • $\begingroup$ So to sum it up PCA should be used to reduce variables and how these variables affect each other should be looked in to with other methods. I'll read up on multiple regression. Thanks alot! $\endgroup$ – Lennart Aug 12 '15 at 15:55

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