Last Monday some biostatisticians/epidemiologists from Australia published a paper about a “visualization tool which may allow greater understanding of medical and epidemiological data”:

H. Wand et al., “Quilt Plots: A Simple Tool for the Visualisation of Large Epidemiological Data“, PLoS ONE 9(1): e85047.

A brief look at the “paper” reveals that the quilt plot they propose is a special case of what is commonly known as a heat map, a point the authors acknowledge, with the caveat that they claim that

” ‘heat maps’ require the specification of 21 arguments including hierarchical clustering, weights for reordering the row and columns dendrogram, which are not always easily understood unless one has an extensive programming knowledge and skills. One of the aims of our paper is to present ‘‘quilt plots’’ as a useful tool with simply formulated R-functions that can be easily understood by researchers from different scientific backgrounds without high-level programming skills.”

In other words, the quilt plot is a simplified heat map and the authors think it should be used because specifying parameters for a heat map (in R) would require a terrifying skill known as programming. This is of course all completely ridiculous. Not only does usage of R not require programming skill, there are also simplified heat map functions in many programming languages/computation environments that are as simple as the quilt plot function.

The fact that a paper like this was published in a journal is preposterous, and indeed the authors and editor of the paper have been ridiculed on social media, blogs and in comments to their paper on the PLoS One website.

BUT…

Wand et al. do have one point… those 21 parameters are not an entirely trivial matter. In fact, the majority of computational biologists (including many who have been ridiculing Wand) appear not to understand heat maps themselves, despite repeatedly (ab)using them in their own work.

What are heat maps?

In the simplest case, heat maps are just the conversion of a table of numbers into a grid with colored squares, where the colors represent the magnitude of the numbers. In the quilt plot paper that is the type of heat map considered. However in applications such as gene expression analysis, heat maps are used to visualize distances between experiments.

Heat maps have been popular for visualizing multiple gene expression datasets since the publication of the “Eisengram” (or the guilt plot?). So when my student Lorian Schaeffer and I recently needed to create a heat map from RNA-Seq abundance estimates in multiple samples we are analyzing with Ryan Forster and Dirk Hockemeyer, we assumed there would be a standard method (and software) we could use. However when starting to look at the literature we quickly found 3 papers with 4 different opinions about which similarity measure to use:

There are also the folks who don’t worry too much and just try anything and everything (for example using the heatmap.2 function in R) hoping that some distance measure produces the figure they need for their paper. There are certainly a plethora of distance measures for them to try out. And even if none of the distance measures provide the needed figure, there is always the opportunity to play with the colors and shading to “highlight” the desired result. In other words, heat maps are great for cheating with what appears to be statistics.

We wondered…  what is the “right” way to make a heat map?

Consider first the obvious choice for measuring similarity: Euclidean distance. Suppose that we are estimating the distance between abundance estimates from two RNA-Seq experiments, where for simplicity we assume that there are only three transcripts (A,B,C). The two abundance estimates can be represented by 3-tuples $(p_A,p_B,p_C)$ and $(q_A,q_B,q_C)$such that both $p_A+p_B+p_C=1$ and $q_A+q_B+q_C=1$. If  $p_A=1$ and $q_A=0$, then the Euclidean distance is given by $\sqrt{1+q_B^2+q_C^2}$. This obviously depends on $q_B$ and $q_C$, a dependence  that is problematic. What has changed between the two RNA-Seq experiments is that transcript $A$ has gone from being the only one transcribed, to not being transcribed at all. It is difficult to justify a distance metric that depends on the relative changes in $q_B$ and $q_C$. Why, for example, should $(1,0,0)$ be closer to $(1,\frac{1}{2},\frac{1}{2})$ than to $(1,1,0)$?

The Jensen-Shannon divergence, defined for two distributions $P$ and $Q$ by

$JSD(P,Q) = \frac{1}{2}D(P\|M) + \frac{1}{2}D(Q\|M)$

where $M = \frac{1}{2}(P+Q)$ and $D(A\|B)$ is the Kullback-Leibler divergence, is an example of a distance measure that does not have this problem. For the example above the JSD is always $log2$ (regardless of $q_B$ and $q_C$). However the JSD is not a metric (hence the term divergence in its name). In particular, it does not satisfy the triangle inequality (which the Euclidean distance does). Interestingly, this defect can be rectified by replacing JSD with the square root of JSD (the JSD metric). Formal proofs that the square root of JSD is a metric were provided in “A new Metric for Probability Distributions” by Dominik Endres and Johannes Schindelin (2003), and separately (and independently) in “A new class of metric divergences on probability spaces and its applicability in statistics” by Ferdinand Österreicher and Igor Vajda (2003). The paper “Jensen-Shannon Divergence and Hilbert space embedding” by Bent Fuglede and Flemming Topsøe (2004) makes clear the mathematical origins for this result by showing that the square root of JSD can be isometrically embedded into Hilbert space (as a logarithmic spiral)

The 2-simplex with contour lines showing points equidistant
from the probability distribution (1/3, 1/3, 1/3) for the JSD metric.

The meaning of the JSD metric is not immediately apparent based on its definition, but a number of results provide some insight. First, the JSD metric can be approximated by Pearson’s $\chi^2$ distance  (Equation (7) in Endres and Schindelin). This relationship is confirmed in the numerical experiments of Sung-Hyuk Cha (see Figure 3 in “Comprehensive survey on distance/similarity measures between probability distance functions“, in particular the close relationship between JSD and the probabilistic symmetric $\chi^2$). There are also information theoretic and physical interpretations for the JSD metric stemming from the definition of JSD in terms of Kullback-Leibler divergence.

In “Transcript assembly and quantification by RNA-Seq reveals unannotated transcripts and isoform switching during cell differentiation“, Trapnell et al., Nature Biotechnology 28 (2010), we used the JSD metric to examine changes to relative isoform abundances in genes (see, for example, the Minard plot in Figure 2c). This application of the JSD metric makes sense, however the JSD metric  is not a panacea. Consider Figure 1 in the Merkin et al. paper mentioned above. It displays a heat map generated from 7713 genes (genes with singleton orthologs in the five species studied). Some of these genes will have higher expression, and therefore higher variance, than others. The nature of the JSD metric is such that those genes will dominate the distance function, so that the heat map is effectively generated only from the highly abundant genes. Since there is typically an (approximately) exponential distribution of transcript abundance this means that, in effect, very few genes dominate the analysis.

I started thinking about this issue with my student Nicolas Bray and we began by looking at the first obvious candidate for addressing the issue of domination by high variance genes: the Mahalanobis distance. Mahalanobis distance is an option in many heat map packages (e.g. in R), but has been used only rarely in publications (although there is some history of its use in the analyses of microarray data). Intuitively, Mahalanobis distance seeks to remedy the problem of genes with high variance among the samples dominating the distance calculation by appropriate normalization. This appears to have been the aim of the method in the Anders and Huber paper cited above, where the expression values are first normalized to obtain equal variance for each gene (the variance stabilization procedure). Mahalanobis distance goes a step further and better, by normalizing using the entire covariance matrix (as opposed to just its diagonal).

Intuitively, given a set of points in some dimension, the Mahalanobis distance is the Euclidean distance between the points after they have been transformed via a linear transformation that maps an ellipsoid fitted to the points to a sphere.  Formally, I think it is best understood in the following slightly more general terms:

Given an $n \times m$ expression matrix $X$ (rows=transcripts, columns=experiments), let $P=PCA(X)$ be the matrix consisting of projections of $X$ onto its principal components, and denote by $s^2_k(ij)$ the distance between projection of points i and j onto the kth component, i.e. $s^2_k(ij) = (P_{ik}-P_{jk})^2$. Let $\lambda_1,\ldots,\lambda_n$ be the singular values. For some $1 \leq p \leq n$, define the distance

$D_{ij} = \frac{s^2_1(ij)}{\lambda_1} + \cdots + \frac{s^2_p(ij)}{\lambda_p}$

When $n \leq m$ and $p=n$ then the distance D defined above is the Mahalanobis distance.

The Mahalanobis ellipses. In this figure the distance shown is from every point to the center (mean of the point) rather than between pairs of points. Mahalanobis distance is sometimes defined in this way. The figure is reproduced from this website. Note that the Anders-Huber heat map produces distances looking only at the variance in each direction (in this case horizontal and vertical) which assumes that the gene expression values are independent, or equivalently that the ellipse is not rotated.

It is interesting to note that D is defined even when $n > m$, providing a generalization of Mahalanobis distance for high-dimensional data.

The cutoff p involves ignoring the last few principal components. The reason one might want to do this is that the last few principal components presumably correspond to noise in the data. Amplifying this noise and treating it the same as the signal is not desirable. This is because as p increases the denominators $\lambda_p$ get smaller, and therefore have an increasing effect on the distance. So even though it makes sense to normalize by variance thereby allowing all genes to count the same, it is important to keep in mind that the last few principal components may be desirable to toss out. One way one could choose the appropriate threshold is by examination of a scree plot.

We’re still not completely happy with Mahalanobis distance. For example, unlike the Jensen-Shannon metric, it does not provide a metric over probability distributions. In functional genomics, almost all *Seq assays produce an output which is a (discrete) probability distribution (for example in RNA-Seq the output after quantification is a probability distribution on the set of transcripts). So making heat maps for such data seems to not be entirely trivial…

Does any of this matter?

The landmark Michael Eisen et al. paper “Cluster analysis and the display of genome-wide expression patterns“, PNAS 95 (1998), 14863–14868 describing the “Eisengram” was based on correlation as the distance measure between expression vectors. This has a similar problem to the issues we discussed above, namely that  abundant genes are weighted more heavily in the distance measure, and therefore they define the characteristics of the heat map. Yet the Eisengram and its variants have proven to be extremely popular and useful. It is fair to ask whether any of the issues I’ve raised matter in practice.

Depends. In many papers the heat map is a visualization tool intended for a qualitative exploration of the data. The issues discussed here touch on quantitative aspects, and in some applications changing distance measures may not change the qualitative results. Its difficult to say without reanalyzing data sets and (re)creating the heat maps with different parameters. Regardless, as expression technology continues to transition from microarrays to RNA-Seq, the demand for quantitative results is increasing. So I think it does matters how heat maps are made. Of course its easy to ridicule Handan Wand for her quilt plots, but I think those guilty of pasting ad-hoc heat maps based on arbitrary distance measures in their papers are really the ones that deserve a public spanking.

P.S. If you’re going to make your own heat map, after adhering to sound statistics, please use a colorblind-friendly palette.

P.P.S. In this post I have ignored the issue of clustering, namely how to order the rows and columns of heat maps so that similar expression profiles cluster together. This goes along with the problem of constructing meaningful dendograms, a visualization that has been a major factor in the popularization of the Eisengram. The choice of clustering algorithm is just as important as the choice of similarity measure, but I leave this for a future post.