Principal component analysis (PCA) has been gaining popularity as a tool to bring out strong patterns from complex biological datasets. We have answered the question “What is a PCA?” in this jargon-free blog post — check it out for a simple explanation of how PCA works. In a nutshell, PCA capture the essence of the data in a few principal components, which convey the most variation in the dataset.

# 1. A PCA plot shows clusters of samples based on their similarity.

**Figure 1. PCA plot**. For how to read it, see this blog post

PCA does not discard any samples or characteristics (variables). Instead, it reduces the overwhelming number of dimensions by constructing principal components (PCs). PCs describe variation and account for the varied influences of the original characteristics. Such influences, or loadings, can be traced back from the PCA plot to find out what produces the differences among clusters.

# 2. A loading plot shows how strongly each characteristic influences a principal component.

**Figure 2. Loading plot**

See how these vectors are pinned at the origin of PCs (PC1 = 0 and PC2 = 0)? Their project values on each PC show how much weight they have on that PC. In this example, NPC2 and CHIT1 strongly influence PC1, while GBA and LCAT have more say in PC2.

Another nice thing about loading plots: the angles between the vectors tell us how characteristics correlate with one another. Let’s look at Figure 2.

- When two vectors are close, forming a small angle, the two variables they represent are positively correlated. Example: APOD and PSAP
- If they meet each other at 90°, they are not likely to be correlated. Example: NPC2 and GBA.
- When they diverge and form a large angle (close to 180°), they are negative correlated. Example: NPC2 and MAG.

Now that you know all that, reading a PCA biplot is a piece of cake.

# 3. PCA biplot = PCA score plot + loading plot

**Figure 3. PCA biplot**

You probably notice that a PCA biplot simply merge an usual PCA plot with a plot of loadings. The arrangement is like this:

- Bottom axis: PC1 score.
- Left axis: PC2 score.
- Top axis: loadings on PC1.
- Right axis: loadings on PC2.

In other words, the left and bottom axes are of the PCA plot — use them to read PCA scores of the samples (dots). The top and right axes belong to the loading plot — use them to read how strongly each characteristic (vector) influence the principal components.

# 4. A scree plot displays how much variation each principal component captures from the data

A scree plot, on the other hand, is a diagnostic tool to check whether PCA works well on your data or not. Principal components are created in order of the amount of variation they cover: PC1 captures the most variation, PC2 — the second most, and so on. Each of them contributes some information of the data, and in a PCA, there are as many principal components as there are characteristics. Leaving out PCs and we lose information.

**Figure 4. PCA scree plot**

The good news is, if the first two or three PCs have capture most of the information, then we can ignore the rest without losing anything important. A scree plot shows how much variation each PC captures from the data. The y axis is eigenvalues, which essentially stand for the amount of variation. Use a scree plot to select the principal components to keep. An ideal curve should be steep, then bends at an “elbow” — this is your cutting-off point — and after that flattens out. In Figure 4, just PC 1,2, and 3 are enough to describe the data.

To deal with a not-so-ideal scree plot curve, there are a couple ways:

- Kaiser rule: pick PCs with eigenvalues of at least 1.
- Proportion of variance plot: the selected PCs should be able to describe at least 80% of the variance.

If you end up with too many principal components (more than 3), PCA might not be the best way to visualize your data. Instead, consider other dimension reduction techniques, such as t-SNE and MDS.

**In summary:** A PCA biplot shows both PC scores of samples (dots) and loadings of variables (vectors). The further away these vectors are from a PC origin, the more influence they have on that PC. Loading plots also hint at how variables correlate with one another: a small angle implies positive correlation, a large one suggests negative correlation, and a 90° angle indicates no correlation between two characteristics. A scree plot displays how much variation each principal component captures from the data. If the first two or three PCs are sufficient to describe the essence of the data, the scree plot is a steep curve that bends quickly and flattens out.

Looking for a way to create PCA biplots and scree plots easily? Try BioVinci, a drag and drop software that can run PCA and plot everything like nobody’s business in just a few clicks.

See a quick video of how to quickly run a PCA with BioVinci:

## Andrew Meyenn

Very good discussion.

I don’t understand why GBA and LCAT variables influence PC2, I would have through the influence of GBA was on PC1?

Could you explain please.

Thank You

## Tien

Hi Andrew, the vectors are not associated with the PC they point toward. The weight is presented by the value projected to the 0 axis of that PC. For example, GBA on PC1 is close to -1 but GBA on PC2 is close to -3. Therefore, GBA has more influence over PC2 than PC1.

## mahmood

Hi, In Figure 2, GBA is not closed to -1 for PC1. Isn’t that? it’s about -0.1.

## Greg

I spent a lot of time to find something similar to this

## Jeannie

Thanks, it is very informative

## Tracey

Principal component analysis (PCA) is a statistical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables (entities each of which takes on various numerical values) into a set of values of linearly uncorrelated variables called principal components.

## Aniko

Indeed. “Tracey” was too polite to point out that this sentence “A PCA plot shows clusters of samples based on their similarity” is just plain wrong. PCA has nothing to do with data grouping or similarity _per_se_.

Of course if you do a PCA on data that have been assigned to groups, and scatter-plot the transformed data in the coordinate system of the first two PCs then you can visualise the groups which you would not have been able to do in N>>2 dimensions.

## Levi Akundabweni

Would it be in order then to change the semantic “clusters” to ‘expression profiles’?

## Tien

Thank you for the comment. We will improve the accuracy of the post!

## Michael

Hi, I want to see if I am understanding this correctly:

Figure 1: Group 2 separates from the others due to PC1, but within group 2, there is a lot of variability captured by PC2?

Figure 3: Are the vectors that overlay a particular group, associated with that group? eg MAG, LCAT2 are mostly associated with Group 2? If so, they are associated with the subgroup on the bottom of Group 2? or perhaps only Group 4? Am I oversimplifying? More examples like these would help one understand. Apologies for being “clueless” 🙂

## Tien

Hi Michael. In figure 1, PC1 captures the most variation which happens to help separate the groups for this example dataset and PC2 captures 2nd most variation. For many other cases, 1 PC cannot separate a group this clearly. In figure 3, the vectors are not associated with the group they overlay. The weight is presented by the value projected to the 0 axis of that PC. For example, GBA on PC1 is close to -1 but GBA on PC2 is close to -3. Therefore, GBA has more influence over PC2 than PC1.

## Joren Wouters

Thanks a lot! Great explanation

## Md Mahmudul Hasan

Thanks a lot! Nice explanation!

## Yvonne

This was really helpful, thank you!

## Tien

Thank you for the kind words. We updated a post on 3D PCA as well, check it out here: https://blog.bioturing.com/2020/10/12/the-why-when-and-how-of-3d-pca/. If you have questions in other topics, please share with us, we’d love to help!

## Sadrakh Naibaho

how can i plot eigenvector on those coordinates (PC1 and PC2 axis)

## Tien

Hi Naibaho, you can try using BioVinci: https://vinci.bioturing.com/download to plot the eigenvector. Please follow the instruction in this video: https://www.youtube.com/watch?v=d2tILFSZMqQ&feature=emb_title

## SALOUA

Thank you very much. That was very helpful for me!

I understand the relationship between PCs and vectors, but what about “group 1, group 2 ..? how we can interpret their distribution?

Thank you

## SALOUA

I mean the dispersion of individuals

## Mextli

This is an amazing explication! Love it! thanks so much!