multidimensional scaling of variables, and rank correlations

correlation heatmap

A common way to visualize correlation matrices is the heatmap, so i’ll use that technique first (see below). While the rankings by academic and employer reputations are highly concordant, those by international faculty and student ratios are less so; and, the faculty–student ratio and faculty citation rankings have the weakest concordance of all, but are nevertheless positively correlated.

c <- cor(select(qswurus20, starts_with("rk_")), method = "kendall")
corrplot::corrplot(c)

This visualization is useful, but it’s very busy: To compare any pair of rankings, i have to find the cell in the grid corresponding to that pair and refer back to the color scale to assess its meaning. I can’t rely on the nearby cells for context, because they may be stronger or weaker than average and skew my interpretation. For example, the visibly weak associations between the faculty–student ratio and other rankings (the third row or column) happen to be arranged so that the slightly stronger among them, with the two reputational variables, are sandwiched between the even stronger associations between the two reputational rankings and between them and the faculty citations ranking; whereas its weaker associations are sandwiched between more typical, but still comparatively stronger, associations. A different ordering of the variables might “obscure” this pattern and “reveal” others.

The plot is also strictly pairwise: Every correlation between two rankings occupies its own cell—two, in fact, making almost half of the plot duplicative. This means that a subset analysis of, say, three rankings requires focusing on three cells at the corners of a right triangle while ignoring all the surrounding cells. This is not an easy visual task. It would be straightforward to create a new plot for any subset, but then the larger context of the remaining rankings would be lost.

correlation biplot

MDS of variables offers a natural alternative visualization: the biplot. As with MDS of cases, the point isn’t to overlay the case scores and variable loadings from a singular value decomposition, but to use the scores or loadings alone to endow the cases or variables with a Euclidean geometry they didn’t yet have. To that end, i’ll plot the variables as vectors with tails at the origin and heads at their fabricated coordinates \(Y=V\Lambda^{1/2}\):

c.eigen <- eigen(c)
c.coord <- c.eigen$vectors %*% diag(sqrt(c.eigen$values))
plot(c.coord, pch = NA, asp = 1, xlab = "", ylab = "")
arrows(0, 0, c.coord[, 1], c.coord[, 2])
text(c.coord, labels = rownames(c))

A more elegant ggplot2-style graphic can be rendered with ordr, with a unit circle included for reference:

library(ordr)
eigen_ord(c) %>%
  as_tbl_ord() %>%
  augment() %>%
  mutate_u(metric = stringr::str_remove(.name, "rk_")) %>%
  confer_inertia(1) %>%
  negate_to_nonneg_orthant("u") ->
  c_eigen
c_eigen %>%
  ggbiplot() +
  theme_minimal() +
  geom_unit_circle() +
  geom_u_vector() +
  geom_u_text_radiate(aes(label = metric)) +
  scale_x_continuous(expand = expand_scale(add = .4)) +
  scale_y_continuous(expand = expand_scale(add = .2)) +
  ggtitle("MDS of Kendall correlations between university rankings")

With respect to the pairwise correlations, the biplot is significantly less precise: Though the vectors all have unit length in \(\mathbb{R}^r\) (\(r\leq m=6\)), their projections onto the first two principal coordinates are much shorter, indicating that much of the geometric configuration requires additional dimensions to represent. Indeed, these coordinates capture only \(48.2\%+14.3\%=62.5\%\) of the inertia in the full representation. This means that the angles between the vectors must be interpreted with caution: For example, it looks like the academic and employer reputation rankings are extremely correlated, but the apparent alignment of the vectors could be an artifact of the projection, when in fact they “rise” and “fall” in opposite directions along the remaining dimensions. The correlation heatmap leaves no such ambiguity.

However, the biplot far surpasses the heatmap at parsimony: Each variable is represented by a single vector, and the angle cosines between the variable vectors roughly approximate their correlations. For instance, the rankings based on international student and faculty ratios have correlation around \(\cos(\frac{\pi}{4})=\frac{1}{\sqrt{2}}\), corresponding to either explaining half the “variance” in the other—not technically meaningful in the ranking context but a useful conceptual anchor. Meanwhile, the faculty–student ratio ranking is nearly independent of the faculty citation ranking, contrary to my intuition that these rankings would reflect a reverse association between research- and teaching-oriented institutions. The convenience of recognizing correlations as cosines may be worth the significant risk of error, especially since that error (the residual \(37.5\%\) of inertia) can be exactly quantified.

Moreover, the principal coordinates of the variable vectors indicate their loadings onto the first and second principal moments of inertia—the two dimensions that capture the most variation in the data. For example, the first principal coordinate is most aligned with the two reputational rankings, suggesting that a general prestige ranking is the strongest overall component of the several specific rankings. In contrast, the faculty–student ratio and faculty citation rankings load most strongly onto the second principal coordinate, suggesting that the divide between research- and teaching-focused institutions may yet be important to understanding how universities compare along these different metrics. These observations, provisional though they are, would be difficult to discern from the heatmap. More importantly, unlike the secondary patterns visible in the heatmap, these are in no sense artifacts of the layout but arise directly from the (correlational) data.

This last point means that observations made from a biplot can be validated from the MDS coordinates. In particular, we can examine the variables’ loadings onto the third principal coordinate, and we can check whether the reputational rankings are aligned or misaligned along it.

c_eigen %>%
  tidy(.matrix = "u") %>%
  select(-.name, -.matrix)

## # A tibble: 6 x 7
##     EV1     EV2     EV3     EV4     EV5      EV6 metric       
##   <dbl>   <dbl>   <dbl>   <dbl>   <dbl>    <dbl> <chr>        
## 1 0.834 -0.0907 -0.412   0.0430  0.0206 -0.351   academic     
## 2 0.795 -0.0964 -0.477  -0.0416  0.181   0.311   employer     
## 3 0.517  0.771   0.0480  0.331  -0.158   0.0372  ratio        
## 4 0.731 -0.352   0.239  -0.0278 -0.528   0.0685  citations    
## 5 0.631 -0.233   0.521   0.392   0.352  -0.00783 intl_faculty 
## 6 0.603  0.262   0.324  -0.665   0.140  -0.0312  intl_students

c_eigen %>%
  tidy(.matrix = "coord")

## # A tibble: 6 x 4
##   .name .values .inertia .prop_var
##   <fct>   <dbl>    <dbl>     <dbl>
## 1 EV1     2.89     2.89     0.482 
## 2 EV2     0.858    0.858    0.143 
## 3 EV3     0.833    0.833    0.139 
## 4 EV4     0.709    0.709    0.118 
## 5 EV5     0.480    0.480    0.0799
## 6 EV6     0.227    0.227    0.0379

Based on the third principal coordinates, the reputational rankings are aligned, as we knew already from the correlation matrix and heatmap. What’s a bit more interesting is that this component seems to separate these two rankings from those having to do with faculty citation rates and the international compositions of the faculty and student body. Based on the decomposition of inertia, this third principal coordinate is nearly as important as the second! It therefore makes sense to plot the two together:

c_eigen %>%
  ggbiplot(aes(x = 2, y = 3)) +
  theme_minimal() +
  geom_unit_circle() +
  geom_u_vector() +
  geom_u_text_radiate(aes(label = metric)) +
  scale_x_continuous(expand = expand_scale(add = .4)) +
  scale_y_continuous(expand = expand_scale(add = .4)) +
  ggtitle("MDS of Kendall correlations between university rankings",
          "Second and third principal coordinates")

The primary antitheses of the reputational rankings, after removing the first principal coordinate, are the two rankings based on international composition—and this axis is largely independent of the axis apparently distinguishing research- from teaching-oriented institutions. From my own limited knowledge, i’d hazard a guess that this reflects two tiers of international representation among students and faculty, one expressed by the most prestigious institutions that recruit highly qualified applicants from all over the world, and the other expressed by institutions that are not especially prestigious but are located in communities or regions with high percentages of international residents.

This is of course no more than idle speculation on my part! But a visualization scheme that encourages hypothesis generation is worth having on hand.