I perform the following experiment: each observer performs 3 tasks to evaluate his strength (S variables), additionally each of them perform 2 IQ tasks. So, I have a dataset with 18 rows for observers and 5 columns: 3 for S and 2 for IQ variables.

How can I interpret it?

Can I say that S and IQ are positively correlated (according to PC1)? Or negatively correlated (according to PC2)?

### UPDATE:

As I understand from comments and answers I have situation similar to this:

Is it correct interpretation?

### Technical details below

1) I perform PCA and study first 3 components for significance. To do it I sample all variables independently with replacement. Thus, I obtain a null-distribution of explained residual Variance for each component (for first component it was obviously explained total variance). I found that observed values for first and second components are very improbable for null-distribution (p-value<<0.01). Thus, I conclude, that 1 and 2 principle components are significant.

2) I check whether I have "noisy" variables in my dataset. To address this question I perform sampling of each variable separately and obtain the null-distribution for case that this variable is independent from others. I found that all variables has similar input in explained residual variance for both first and second components. Perfect.

Thus, I conclude, (A) that first and second component explain significant amount of residual variance, (B) first component loading are of the same size, second component loading dissect 5 variable to 2 groups, which are of a different sign. These groups are the same as biology under variables (physical or mental abilities).

Now comes a most hard part. I should explain these results to my grandmother (boss, student, reviewer... I am not sure what will be easier).

I may say that more muscles you have more clever You are, because all 5 variables projections on first component are cocorrelated (or something like this). But I have no ideas how to explain different sign for "muscles" and "brain" on the second component.

If I will have different signs for a first component loadings everything will be easier.

• If I will have different signs for a first component loadings everything will be easier - why? – amoeba Feb 2 '17 at 15:15
• Then i may say, the more strong You are - the smaller IQ You have. – zlon Feb 2 '17 at 15:15
• Why do you do a PCA? What is the question you are trying to answer? If it’s only about the sign of correlations between S and IQ variables, why don’t you simply look at the correlation matrix? – Elvis Feb 2 '17 at 15:47
• It is similar. But correlation matrix leads to a multiple comparisons problem. My question was to find anything in this data set. I could not use factor analysis or GLME because i have no predictors and predictable variables. All my variables were born equal. Correlation matrix - rather noisy approach, moreover in a similar experiment I have 28 variables and 100 observers all correlations will gone due to multiple comparisons. – zlon Feb 2 '17 at 16:21