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I'd like to test the hypothesis that there is a monotonic relationship between two variables, without assuming a specific model. What is the most robust (i.e. lowest probability of type-II error) way to do this?

I can think of a few options:

  • use a linear model of untransformed data. It'll be robust enough, even if I don't think the true relationship is linear.

  • look at rank-transformed data, e.g. with Spearman's rank correlation coefficient

  • use some kind of resampling approach, in which the order of the dependent variable is randomly shuffled. I'm not sure what statistic to compare in this approach.

Is there a fairly standard approach to this problem?

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  • $\begingroup$ Based on your suggestions, I'm guessing your data points are independent, and that the null hypothesis you want to test is of no association, i.e. a completely flat relationship. Linear regression with "robust" (i.e. heteroskedasticity-consistent) standard errors provides a Wald test for what you want, that is valid and efficient in large samples. It requires no assumption of true linearity. $\endgroup$ – guest Nov 28 '12 at 18:15
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Spearman's or Kendall's correlations are the standard way to do this.

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Could you use some sort of generalize additive model, where the dependent variables are relatedto the predictors as a smooth function, like done in the gam() function in R

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