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In my line of work, we often hypothesize that one continuous predictor will have a stronger relationship with some outcome than another closely related (i.e., collinear) continuous predictor. We fit a multiple regression $y = a + b_{1}X_{1} + b_{2}X_{2}$ and start by checking to see if $b_{1} > b_{2}$. If it is, then yay, the relationship between those betas was in the predicted direction!

Even better is when $b_{1}$ is statistically significant while $b_{2}$ is not, however we can't stop here lest we commit the "difference in significance = significantly different" fallacy. And in my field, you can't publish without p-values, meaning we need some sort of null hypothesis test that is consistent with the verbal hypothesis that "predictor 1 is more strongly related to y than predictor 2", beyond merely showing that $b_{1} > b_{2}$ and/or $b_{1}$ is "statistically significant" while $b_{2}$ is not.

After scouring stack, I have identified at least four methods for doing this, all of which give me slightly different "stories" in my actual dataset. This leaves me with some paralysis about how to proceed given that I don't have a good sense of which method is generally considered to be the most persuasive. These are the methods I've found:

  1. A Wald test between the betas (as in this answer). In other words, compare the two partialized betas.
  2. Pivot the dataset longer so that continuous predictors 1 and 2 are stacked into a single column C alongside a grouping column G, and fit $y = a + b_{1}C + b_{2}G + b_{3}(C \times G)$, such as mentioned in this question. As far as I can tell, the interaction term in this model is equivalent to the difference between two unpartialized betas from one univariate regression with predictor 1 as the predictor and another with predictor 2 as the predictor.
  3. Use a linear hypothesis test (such as from the car package in R and outlined in this answer), which as I understand it does a model comparison of a model where $b_{1} = b_{2}$ vs the observed model.
  4. Use dominance analysis to determine whether/to what extent predictor 1 dominates predictor 2 (as nicely explained in this answer). It seems like significance tests of difference in dominance (if that's what we would call it) are less common, but there is an R package that seems to give you bootstrapped standard errors if you need them.

My questions are:

1. In your opinion, which of these methods conceptually most closely aligns with the verbal hypothesis that "predictor 1 relates more strongly to y than predictor 2"?

2. What exactly is the null hypothesis of each test, in a sentence?

3. In your opinion, which of these methods is most flexible to real-world issues one is likely to face, such as multicollinearity?

4. Is there another method not mentioned here that is worth mentioning? I would be open to a bayesian perspective as well.

P.S. I'm not super proficient at the math, so I've tried to explain in words what I can't express mathematically. Verbal answers are preferred over heavily mathematical answers, but both are appreciated!

Edit: For the sake of practicality, let's assume that a) both predictors are on the same scale and b) the sample size is large enough to reasonably address estimation issues (such as instability) caused by collinearity. The reason I mentioned the collinearity is that some methods may prefer to account for covariance between predictors.

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    $\begingroup$ Your meaning of "stronger" is arbitrary because it changes when the units of measurement of the explanatory variables are changed. But in situations where it is meaningful, your model is equivalent to $E[y]=a+b_1(X_1+X_2)+(b_2-b_1)X_2,$ which means you are asking about the purely routine test of whether the coefficient of $X_2$ is negative. $\endgroup$
    – whuber
    Nov 22, 2023 at 19:12
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    $\begingroup$ You mention that the variables are colinear. One big issue with colinearity is that the parameter estimates can change radically with very small changes in the data. Belsley gave an invented example where changes in the third significant digit of the data flipped the signs of the parameter estimates. So, I would say that no method that looks at the full equation is going to be stable or useful $\endgroup$
    – Peter Flom
    Nov 22, 2023 at 22:10
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    $\begingroup$ What I would do is look at two equations, one with X_1 and the other with X_2, and compare the standardized betas (because the unstandardized ones are arbitrary, as @whuber points out). But even that is a bit iffy, because the standardization is done on your particular data set and could be different in another set, so the interpretation is iffy. $\endgroup$
    – Peter Flom
    Nov 22, 2023 at 22:13
  • $\begingroup$ Those are both great points – I've added a caveat to the question. Thank you for pointing those out! $\endgroup$
    – mvanaman
    Nov 22, 2023 at 22:48
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    $\begingroup$ Will $b_1$ and $b_2$ always be both positive? To the extent that the "strength" of relationship is expressed by these coefficients, the strength difference would be expressed by $|b_1|-|b_2|$ rather than $b_1-b_2$, which is the same only if both are positive. (Only the dominance analysis will do its job regardless if I understand things correctly.) $\endgroup$ Nov 22, 2023 at 23:40

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Acknowledging that I provided the linked answer to #4 prior to providing a response.

  1. In your opinion, which of these methods conceptually most closely aligns with the verbal hypothesis that "predictor 1 relates more strongly to y than predictor 2"?

To an extent, it depends on what you are trying to convey to the audience. You note being interested in whether one predictor '...will have a stronger relationship...' with the outcome. What that means can affect the recommendation.

For simplicity, I'll assume $X_1$ and $X_2$ are binary variables (i.e., coded as $0$ and $1$) and that they have more or less the same mean (and are effectively standardized relative to one another).

Approach #1 and #3 both evaluate whether $b_1 - b_2 = 0$. That is, does changing from $0$ to $1$ for both $X_1$ and $X_2$ produce the same mean difference on $y$. If more interested in 'by how much' the predictor increases $y$/it's mean value (irrespective of impact on total predicted variation in $y$), this is likely preferred. For example in the evaluation of a policy or intervention, the variable that increases the mean the most might be the better way to evaluate the relative importance of variables as it suggests one 'works better' and might be better to use.

Approach #4, assuming you using general dominance statistics, evaluates whether the amount of variance explained in $y$ (as dominance analyzed; which accounts for an extensive set of increments to the $R^2$) by $X_1$ and $X_2$ is the same. The difference in dominance statistics gets more at relative explanatory utility and balances mean differences as reflected by the coefficients, with the relative variation in the predictors. This then speaks more to, in the context of the observed values, how much does the mean difference result in predicted differences on $y$. This is a more 'data dependent' measure than the mean difference given it's translated through variance at least in part.

I'm not sure #2 gives a useful test in this case and one would have to do a lot of work to get it to look like #1 and #3 (which #2 would be mimicking anyway). The linked example focuses on mean differences between groups (i.e., qualitatively different levels of some factor among observations) and does not, to my understanding, assume a data pivot/reshape to evaluate the effect of different variables as groups.

  1. What exactly is the null hypothesis of each test, in a sentence?

This was more or less stated in the set-up of each of the approaches above.

  1. In your opinion, which of these methods is most flexible to real-world issues one is likely to face, such as multicollinearity?

As was noted in the comments, regression coefficients, as such, are directly affected by multicollinearity and that's just a side effect of including correlated predictors.

Dominance analysis is 'order independent' (i.e., the $R^2$ component averages across all ways one could 'include' a predictor in the sequence of ways one could include all the predictors). People usually interpret that controlling for multicollinearity. That's probably not, strictly speaking, true. Dominance does, at a minimum, include $R^2$ values across a number of different sub-models and produces an additive decomposition of the $R^2$.

  1. Is there another method not mentioned here that is worth mentioning? I would be open to a bayesian perspective as well.

Some interesting work using Bayes factors that can allow for specific orders of coefficient magnitudes/explanatory power that could be of definite value as well (e.g., Xu, 2023). My sense is that such methods are most valuable in the cases that you have a specific ordering that you wish to test (and not an exploratory situation; which it seems like might be more like the case here).

References

Gu, X. (2023). Evaluating predictors’ relative importance using Bayes factors in regression models. Psychological Methods, 28(4), 825.

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  • $\begingroup$ thank you for that thorough answer! These are some great points. Your explanation of dominance was enlightening to me even over and above your answer to the linked question. $\endgroup$
    – mvanaman
    Dec 6, 2023 at 20:30

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