# StatQuest: Linear Discriminant Analysis (LDA), clearly explained

Here it is, folks! By popular demand, a StatQuest on linear discriminant analysis (LDA)!

Also, because you asked for it, here’s some sample R code that shows you how to get LDA working in R.

If all went well, you should get a graph that looks like this: # Heatmaps – how to draw and interpret them.

We’ve all seen heatmaps a million times before, but there are a lot of arbitrary decisions made when drawing them. In this video, I cover the various things you do to create a heatmap using hierarchical clustering.

By request, here are the powerpoint slides. These are for your personal use. Instead of sharing these to your friends, please ask them watch the video because view and subscriber metrics are very important to me when it comes to reporting how many people use StatQuest to understand stats.

# RNA-seq: The Pitfalls of Technical Replicates

Science, like most things, isn’t perfect. Because individual mice (or other biological samples, like humans) are different and because the way we measure things isn’t perfect, every time we do an experiment, we get slightly different results.

We cope with this variation by doing the same experiment a bunch of times. Once we’ve collected a lot of samples, we typically calculate the average and the standard deviation. If we’re worried about the biological variation (the variation from each individual mouse or human) we do lots of biological replicates, and if we’re worried about technical variation, we do technical replicates (remeasure the same individual multiple times). Often we’ll do both.

For simple and inexpensive experiments, it’s easy to do lots of replicates, both biological and technical. However, RNA-seq is time consuming and expensive, and we want to squeeze as much as we can out of the samples that we have (which can be very limited!) The good news is that because of the way technical variation affects RNA-seq results, we only need to do biological replicates with this type of experiment.

Here’s an example of biological variation alone (imagine there is no technical variation). In this example, we’re measuring read counts for an imaginary gene “X”. Each mouse has either more or less than the average for all mice. If we do enough samples, our estimate of the average of all mice will be pretty good because the measurements that are above the mean will be canceled out by the measurements that are below the mean. This figure shows how biological variation gives us measurements that are always a little larger or smaller than the mean.

When we add technical variation to the mix, RNA-seq will result in random values that either add or subtract read counts from each mouse’s measurement. Here’s an example where I’ve colored the biological variation orange and the technical variation green. This figure shows how technical variation can increase or decrease the read counts we get for each sample. It also shows the equation for the average read count.

When we calculate the average measurement from our samples, we end up with a term for biological variation and a term for technical variation. Just like before, the term for biological variation will go to zero as we add samples, because the positive values will be canceled out by the negative values. However, because the technical variation is similarly distributed, it will cancel itself out for the same reason. Thus, without doing a single technical replicate (and only doing biological replicates) we can reduce the affects of biological and technical variation and get a good estimate of the mean. This last figure shows how the biological and technical variation can be separated into separate terms. Each of these terms goes to zero as we add more biological replicates.

One problem that is often overlooked is that technical replicates alone will only eliminate the term for technical variation. The term for biological variation will only be reinforced, and this can lead to misleading results. See the video for more details.

# PCA clearly explained

Update: A lot of people ask to see example code that shows how to actually do PCA. Here’s how to do it in R

[sourcecode language=”R”]
## In this example, the data is in a matrix called
## data.matrix
## columns are individual samples (i.e. cells)
## rows are measurements taken for all the samples (i.e. genes)
## Just for the sake of the example, here’s some made up data…
data.matrix <- matrix(nrow=100, ncol=10)
colnames(data.matrix) <- c(
paste("wt", 1:5, sep=""),
paste("ko", 1:5, sep=""))
rownames(data.matrix) <- paste("gene", 1:100, sep="")
for (i in 1:100) {
wt.values <- rpois(5, lambda=sample(x=10:1000, size=1))
ko.values <- rpois(5, lambda=sample(x=10:1000, size=1))

data.matrix[i,] <- c(wt.values, ko.values)
}

pca <- prcomp(t(data.matrix), scale=TRUE)

## plot pc1 and pc2
plot(pca\$x[,1], pca\$x[,2])

## get the name of the sample (cell) with the highest pc1 value
rownames(pca\$x)[order(pca\$x[ ,1], decreasing=TRUE)]

## get the name of the top 10 measurements (genes) that contribute
## most to pc1.
gene_score_ranked <- sort(gene_scores, decreasing=TRUE)
top_10_genes <- names(gene_score_ranked[1:10])

top_10_genes ## show the names of the top 10 genes

## get the scree information
pca.var <- pca\$sdev^2
scree <- pca.var/sum(pca.var)
plot((scree[1:10]*100), main="Scree Plot", xlab="Principal Component", ylab="Percent Variation")
[/sourcecode]

RNA-seq results often contain a PCA (Principal Component Analysis) or MDS plot. Usually we use these graphs to verify that the control samples cluster together. However, there’s a lot more going on, and if you are willing to dive in, you can extract a lot more information from these plots. The good news is that PCA only sounds complicated. Conceptually, it’s actually quite simple.

In this video I clearly explain how PCA graphs are generated, how to interpret them, and how to determine if the plot is informative or not. I then show how you can extract extra information out of the data used to draw the graph.

# RPKM, FPKM and TPM, clearly explained

Warning: This StatQuest is specifically for people who do high-throughput RNA sequencing (RNA-seq). If that’s what you’re interested in, quest on!

It used to be when you did RNA-seq, you reported your results in RPKM (Reads Per Kilobase Million) or FPKM (Fragments Per Kilobase Million). However, TPM (Transcripts Per Million) is now becoming quite popular. Since there seems to be a lot of confusion about these terms, I thought I’d use a StatQuest to clear everything up.

These three metrics attempt to normalize for sequencing depth and gene length. Here’s how you do it for RPKM:

1. Count up the total reads in a sample and divide that number by 1,000,000 – this is our “per million” scaling factor.
2. Divide the read counts by the “per million” scaling factor. This normalizes for sequencing depth, giving you reads per million (RPM)
3. Divide the RPM values by the length of the gene, in kilobases. This gives you RPKM.

FPKM is very similar to RPKM. RPKM was made for single-end RNA-seq, where every read corresponded to a single fragment that was sequenced. FPKM was made for paired-end RNA-seq. With paired-end RNA-seq, two reads can correspond to a single fragment, or, if one read in the pair did not map, one read can correspond to a single fragment. The only difference between RPKM and FPKM is that FPKM takes into account that two reads can map to one fragment (and so it doesn’t count this fragment twice).

TPM is very similar to RPKM and FPKM. The only difference is the order of operations. Here’s how you calculate TPM:

1. Divide the read counts by the length of each gene in kilobases. This gives you reads per kilobase (RPK).
2. Count up all the RPK values in a sample and divide this number by 1,000,000. This is your “per million” scaling factor.
3. Divide the RPK values by the “per million” scaling factor. This gives you TPM.

So you see, when calculating TPM, the only difference is that you normalize for gene length first, and then normalize for sequencing depth second. However, the effects of this difference are quite profound.

When you use TPM, the sum of all TPMs in each sample are the same. This makes it easier to compare the proportion of reads that mapped to a gene in each sample. In contrast, with RPKM and FPKM, the sum of the normalized reads in each sample may be different, and this makes it harder to compare samples directly.

Here’s an example. If the TPM for gene A in Sample 1 is 3.33 and the TPM in sample B is 3.33, then I know that the exact same proportion of total reads mapped to gene A in both samples. This is because the sum of the TPMs in both samples always add up to the same number (so the denominator required to calculate the proportions is the same, regardless of what sample you are looking at.)

With RPKM or FPKM, the sum of normalized reads in each sample can be different. Thus, if the RPKM for gene A in Sample 1 is 3.33 and the RPKM in Sample 2 is 3.33, I would not know if the same proportion of reads in Sample 1 mapped to gene A as in Sample 2. This is because the denominator required to calculate the proportion could be different for the two samples.