Variable importance assessed by MR vs. by Spearman's rank-order coefficient

Question

I am looking for an answer to the following questions:

• Consider a multiple linear regression model with 2 IVs ($X_1$ and $X_2$). Now, suppose that there is a strong monotonic relationship between $X_1$ and the DV (using the Spearman rank coefficient), and that a weak relationship exists between $X_2$ and DV. Does this mean that $X_1$ will have the highest beta weight ($\beta_{X_1} > \beta_{X_2}$)?
• For example, if we say that IV1 has more predictive power than IV2 and so on, how can we interpret this statement in term of monotonicity between each IV and the DV?

Answer 1

It is not necessarily true that the ordering of Spearman's rank coefficient will be the same as the ordering of the betas returned by a multiple regression model fit. There can be several reasons for this, here are a few that I can think of quickly (there may well be more):

Range restrictions:
A restricted range will not affect (on average--i.e., the expected value of) the estimated slope of a variable (although it will its significance), but it will strongly affect the correlation (either Spearman's or Pearson's). Thus, by adjusting the ranges of $X_1$ and $X_2$, it is possible to flip the orderings back and forth. Here's an example where $X_1$ ranges from 0 to 10, and $X_2$ ranges from 0 to 3:

x1   0         5       10         0        5       10        0         5        10
x2   0         0        0         1.50     1.5      1.5      3         3         3
y  -48.09667 -12.12629 17.93941 -57.59159 12.30414  6.521197 4.300887 58.36051 -55.91287

rho(x1,y) = 0.3689324;     rho(x2,y) = 0.05270463 
beta_x1   = 2.331171;      beta_x2   = 5.448008

Multicollinearity:
If your covariates are correlated with each other ($r(X_1, X_2)\ne0$) all kinds of counter-intuitive results can occur. Perhaps one of the most counter-intuitive is the suppressor effect (you can read about it at the bottom of this page--but read the whole page, it's worth your time). Here's an example where $X_1$ acts as a suppressor:

x1 -0.029557 0.606060 1.8436 -1.20882  0.46996  0.22534  0.23217 -1.35094 -0.787902
x2  0.165287 0.086891 2.2664 -0.69033  0.23273  0.18010 -0.32480 -1.36021 -0.556156
y   0.432622 0.223160 1.1213  0.42582 -0.94566 -1.00691  0.15683 -0.54717  0.056125

rho(x1,y) =  0.2333333;        rho(x2,y) = 0.1
beta_x1   = -0.593140024;      beta_x2   = 0.874374951

MR betas are linear relationships, Spearman's rho isn't necessarily:
The rectilinear line of best fit can have a very shallow slope (i.e., small beta weight) if the true function is curvilinear, whereas Spearman's correlation coefficient is completely unaffected by this. Here's an example of a situation where $Y$ is related to the log of $X_2$:

x1 3        6        9        3       6        9        3        6        9
x2 3        3        3        6       6        6        9        9        9
y  1.942158 4.222256 4.672984 4.85704 5.061267 5.381291 4.154654 5.875549 7.183006  

rho(x1,y) = 0.6324555;     rho(x2,y) = 0.5797509  
beta_x1   = 0.3490794;      beta_x2  = 0.3542117  

(I don't know if the latter two examples are 'fair', in that they rely on a misspecification of the multiple regression model--that is, the model that was fit was not the true data generating process--but we probably never fit the model that exactly mirrors the true data generating process in real life, and these are cases where the orderings flip.)

Thus, the answers to your two questions are: No, having a higher Spearman rank coefficient does not mean that it will have the highest beta weight; and we cannot interpret orderings of beta weights in terms of orderings of monotonicity between the IV and DV.