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I have two datasets that are both organised as follows: an algorithm calculated a quality score (Q) of a specific image (image) that was processed using a certain algorithm (method). One has 315 observations, the other 2709. I first tried applying a simple linear regression model, only looking at method as a possible predictor, and looked at the QQ-plot.

fd <- lm(q ~ method, data=data)

QQ-plot for lm

I decided that the residuals did not follow a normal distribution. I then tried a Box-Cox transformation to fix the issue. The new model showed this QQ-plot:

bc <- boxcox(data_drim$q ~ data_drim$method)
(lambda <- bc$x[which.max(bc$y)])
powerTransform <- function(y, lambda1, lambda2 = NULL, method = "boxcox") {

  boxcoxTrans <- function(x, lam1, lam2 = NULL) {

    # if we set lambda2 to zero, it becomes the one parameter transformation
    lam2 <- ifelse(is.null(lam2), 0, lam2)

    if (lam1 == 0L) {
      log(y + lam2)
    } else {
      (((y + lam2)^lam1) - 1) / lam1
    }
  }

  switch(method
         , boxcox = boxcoxTrans(y, lambda1, lambda2)
         , tukey = y^lambda1
  )
}

# re-run with transformation
fd_bc <- lm(powerTransform(data_drim$q, lambda) ~ data_drim$method)

QQ-plot for new lm

I wasn't convinced yet, so I went with a non-parametric test instead: the Kruskal-Wallis test. I followed this up with paired comparison tests (Wilcoxon with BH correction).

My questions:

  1. Is this a good approach, given the results with the linear models? I read that you can ignore the normality assumption safely due to the central limit theorem once the number of observations surpasses 200, but I figured that was a 'per condition' number (or is it?), so I didn't go that way.

  2. Suppose I now want to investigate the interaction between different images and the method used. That's not something I can do with Kruskal-Wallis. I could have done that with a (g)lm, but I'm not sure how to go about that using non-parametric methods (if that is indeed what I should be doing).

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First, I'm not a fan of transforming variables to meet statistical criteria. This used to be necessary, but the advent of fast computers and good software has made it unnecessary. So, I like going non-parametric.

Second, there are a variety of methods of non-parametric regression. Depending on exactly what you are trying to find out, you might consider:

  • Quantile regression
  • Robust regression of various sorts
  • Trees and their offspring like forests
  • Spline regression or MARS
  • Generalized additive models

or something I'm not thinking of at the moment.

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  • 1
    $\begingroup$ Hi Peter, thanks a lot for your answer. I'm currently looking into quantile regression on the median. I get results that are along the lines of my earlier Kruskal-Wallis test (and follow-up comparisons), so that looks alright. Thanks! $\endgroup$ – Inkidu616 Nov 6 '18 at 12:48

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