You cannot interpret PCs by concentrating the behavior of a single variables. Each PC corresponds to the collective behavior of the original variables. So, when you say contribution of protein content is +ve/-ve at a certain PC, you should also pay attention to the contribution of other variables to the same PC. So each PC represents how these variables with significant contributions move together. It is highly probable that PC1 represents the collective behavior of a set of variables positive correlated with protein content, while PC 3 represents the collective behavior of ANOTHER set of variables negatively correlated with protein content. Each sample of yours is the result of the superposition of these two effects (among other significant PCs, of course) weighted by their eigenvalues.

You cannot interpret PCs by concentrating the behavior of a single variables. Each PC corresponds to the collective behavior of the original variables. So, when you say contribution of protein content is +ve/-ve at a certain PC, you should also pay attention to the contribution of other variables to the same PC. So each PC represents how these variables with significant contributions move together. It is highly probable that PC1 represents the collective behavior of a set of variables positive correlated with protein content, while PC 3 represents the collective behavior of ANOTHER set of variables negatively correlated with protein content. Each sample of yours is the result of the superposition of these two effects (among other significant PCs, of course) weighted by their eigenvalues.

Hi Vanessa,

I agree with Burak that you cannot interpret a trait is related to two PCs simply looking at a PC plot. PCA analysis is generally performed to see the relation of a set of variable (such as spectra in your case) and where a sample stands in a PC plot. Whole lot of variabilty present in your sampling population might be explained by two or more than two PCs depending upon source of variation in your samples and samples are placed close or apart based on eigenvalues.

Hi Vanessa,

I agree with Burak and Umesh. Each PC is related to all your initial variables. Your assumption that PC1 and PC3 are related to the concentration of different proteins will only be possible if you have the differents proteins as initial variables. Maybe PC3 does not add more information to your analysis, you have too see the eigenvalues. PC1 represents the higher variability in your initial data.

I think that in order to interpret PCA axes, you need to look at the eigenvectors that scored highest and lowest at each axis. Doing so will allow you to interpret the "gradient" of a given axis.

Hi Vanessa, I meant to look at the eigenvectors of each axis. Not all of them, but the ones that scored the highest and the lowest in each axis. This is how we interpret the gradient of each axis in Ecology and Paleoecology. That is why it doesn't matter that a single variable scored high in Axis 1 and maybe Axis 3. If you have questions about how to do this, let me know. Best wishes.

Dear Vanessa,

I have always found it useful to look at the scoring plot together with the loading plot. The score plot, which you refer as PC1 shows you how your samples are related to each other. The loading plot, on the other hand, shows you which variables are important for each component and also how the variables are related to each other. It may very well be that the same variable are important in more than one principal component.

Hope this helps!

If you find it aesthetically unpleasing to have a both positive and negative values associated with “protein content”, feel free to multiply the offensive PCs by -1. The signs of the PCs are arbitrary. You could also apply a factor rotation, an orthogonal transformation of your PCA factors, to yield more intuitive scores. See:

Johnson, Richard Arnold, and Dean W. Wichern. Applied multivariate statistical analysis. Vol. 5. No. 8. Upper Saddle River, NJ: Prentice Hall, 2002.

Two possible reasons:

Simplest to explain is if two different proteins give similar fluorescence spectra that can be easily confused.

The second depends on the maths of PCA. PCA is based on co-variance and in complex data sets it is possible to have some correlation between constituents. Perfectly correlated constituents (R2 = 1) will always appear together and not be separable by PCA methods. Strongly correlated constituents will appear together in high ranked PCs but then the small portion of unrelated variation in the constituents will appear in a low ranked PC describing the divergence of the constituents that appeared concurrently previously. This means the high PC is describing one aspect of the proteins behaviour, the lower PC a different one and so the two need to be considered together to get a full picture.

PCA is not intended to explain your data in terms of pure constituents, it is designed to give you the most compact explanation of the major sources of variance in the dataset whether those sources are pure or mixed. Because it is based on co-variance then it is very powerful for exploring how constituents can interact with each other.

If specific and quantitative information on a particular protein is what you need then regression methods against knowns, curve resolution methods (many related to PCA) or classical least squares fitting against standard spectra are better suited to that job.

As Kevin says the sign of the PC is irrelevant - the central operation of PCA squares the data, so all signs become arbitrary.

Burak's answer basically says it all. But see the book by Brian Manly (1986). A pc is either an average of all contributing variables weighted by their loadings (if all loadings have the same sign), or a weighted contrast between variables with positive loadings and variables with negative loadings. Furthermore, you should consider whether your variable of interest is significantly contributing to both pcs. I suggest you read my article about estimating significances of PCA metrics.

Furthermore, the answer by Kevin Keane is also a very good one. This is a classical PCA feature: the original covariance (or correlation) matrix is the sum of all products of eigenvalues and respective eigenvectors. Therefore, you can multiply the eigenvector by -1 assuming you also multiply the respective eigenvalue by -1. This poses no problem to the weighted averages or weigthed contrasts the eigenvectors represent as the loadings preserve their magnitude i.e. the proportions between variables within each axis are preserved.

Important: do not mind Mauro de Toledo answers as they are rubish: each axis corresponds exactly to one eigenvector. Therefore, you can not have several eigenvectors on each axis. Mauro has definitely mistaken the eigenvectors by their loadings i.e. the score of each variable in each eigenvector. But one should restrain himself from answering to questions about subjects he/she does not master.

If N variables are highly correlated then they will all load out on the SAME Principal Component (Eigenvector), not different ones. Should you try to remove the bias by making single correlations for different wavelengths of Fluo versus concentration. Also quenching may interfere for self absorption at quite high concentration. PCA was created for reducing variables number to a few components but if variables are correlated, and mainly in social science context.

It is so false PCA is mainly for the social science context! The original work presenting PCA: Pearson K. 1901. On lines and planes of closest fit to systems of points in space. Philosophical Magazine 2: 559-572.. After it came thousands of works using PCA in all kinds of science. State-of-the-art science submits satelite data from ocean color and infra-red emitions to PCA algorithms to estimate CO2 in the oceans! PCA is used in all kinds of technologies and in factory production quality control! Furthermore, N highly correlated variables will not load out on the same eigenvector ... unless they all follow the same correlation pattern! Read my article on PCA, is all explained there. People, stop writing nonsense about subjects you do not master!