Bioinformatics Zen

Back from Madrid. I spent three weeks there on an excellent data analysis course, which I would recommend. Not only did I learn valuable techniques, I also got the chance to spend my evenings by the pool or in Sol eating tapas - which explains the lack of posts this July. I offer this brief tutorial in recompense, continuing the theme of data analysis.

Animesh made an excellent point on my previous post - principal components analysis is great and all that, but the results have very little biological meaning. Here I’m going to illustrate how you can relate PCA results to the original data. If you want to reproduce what I’m doing, the R code is here.

I’m using the same dataset as before, 200 crabs where five morphological characteristics have been measured for each. These are

• FL frontal lobe size (mm)
• RW rear width (mm)
• CL carapace length (mm)
• CW carapace width (mm)
• BD body depth (mm)

So jumping right in, carrying out PCA on the data, we get five components. The number of components you get will always be less than or equal to the number of columns in your data frame. Here’s the first three of these components from the crab data

FL	0.28898	0.32325	-0.50717
RW	0.19728	0.86472	0.41414
CL	0.5994	-0.19823	-0.17533
CW	0.66165	-0.28798	0.49138
BD	0.28373	0.15984	-0.54688

The first column is the first component, and so forth. Each row shows how much each characteristic of the original data adds to the given component, e.g. the weight for “frontal lobe” in the first component is 0.28898.

When you use PCA, what you want is for components to be able to discriminate points in the data. When I say discriminating, the components can be used to spread and separate, so you can get more of an idea of what’s happening. In terms of PCA, a discriminating component has both positive and negative values in the column. So we can see that the first component has no discriminating effect as all the values in the column are positive.

Looking to the second component, we can see two negative values, and three positive values, therefore the second component can be used to discriminate the data. To visualise this effect we’ll use the two most opposing characteristics - the most negative and the most positive values. In this case it’s rear width (RW) 0.87, and carapace width (CW) -0.29. Plotting these two characteristics we get the following graph.

Here you can see that data forms a nosey V shape. Imagine that you drew a line through each arm of the V, you would get two sets of data. Therefore, we might assume that there are two different distributions of carapace to rear width ratio.

We can test this by plotting the density of the crabs according to this ratio

There are two peaks, with some overlap. So the biological meaning of the second component is largely picking up the difference in ratios of rear and carapace widths. What it means, we’ll look at later on.

Looking at the third component, again this is discriminating, the most extreme values are carapace width and body depth. Plotting these, we get this figure.

This time we have two vaguely parallel lines, so again there appears to be different ratio of the two characteristics. A density plot illustrates the distribution of this ratio.

Two more overlapping but obvious distributions, so it would appear that the third component highlights a discriminating effect based on the ratio of body depth to carapace width.

We can plot the distribution of crabs based on these two ratios, which produces this figure.

I coloured the crabs based on their species and sex, I also used different point types to further discriminate sex. Looking at this plot, we can begin to derive biological meaning based on what we’ve learnt from principal components analysis.

The x-axis, which we derived from the second component, appears to be related to crab gender, as sex appears roughly separated based on this ratio. The boy crabs seem to have a larger carapace compared to rear width - their shells are relatively larger than their bums, compared with that of the girl crabs.

On the y-axis, the ratio derived from the third component, appears to separate crab species. The orange crabs have a larger body depth to their carapace width, compared with their blue counterparts.

So far, I haven’t so far plotted the component plots - which is often the first thing people do after performing PCA. So here they are, the first/second components.

And the second and third components plot

You can see that the first and second component plot doesn’t discriminate the crabs very well. This is why I chose to ignore the first component. The second/third component plot discriminates the data similarly to the morphology ratios plot, however the axis that would split gender and species have been rotated.

So here you have it, I hope this has illustrated how the data produced from PCA relates to your original data, and how you can begin to interpret it. Here’s the messages I’ve tried to convey.

• The first two components are not always the most useful.
• Look at the components (columns) with the positive and negative weights.
• Look at you original data in terms of the most positive and negative values for these components.