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Mahmod Ohner

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Aug 4, 2024, 9:48:47 PM8/4/24

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Overview

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The same kinds of problems with these distributional assumptions can also arise when working in a regression (i.e. non-normal residuals, non-constant variance). Furthermore, we might run into additional problems if there is some kind of non-linearity in the relationship between the response and predictor (numeric) variables.

This chapter is about fixing models when the assumptions are not satisfied. What assumptions do we need to check? The test we are most likely to want to use with these data is an ANOVA, so the following assumptions must be evaluated:

Are the variances significantly different? Look at the box plots above. The data from the three samples seem to have rather different scatter. The sample from the rowan has less variation than that from the sycamore, and the sycamore has less variation than the oak. Does the scale-location plot tell the same story?

Based on these results, it looks like there is a highly significant difference in food collection rates across the three tree species. We know the data are problematic though, so the question is, does this result stand up when we deal with these problems?

The scale location-plot indicates that the constant variance assumption is now OK, i.e. the variance no longer increases with the fitted values. It looks like the log transformation seems to to have improved things quite a lot, but the diagnostics are still not perfect.

We could back-transform the means of the log-transformed data by taking the antilogs: \(10^x\) (for logs to the base 10) and \(e^x\) (for natural logs)22. When we back-transform data, however, we need to be aware of two things: (1) The back-transformed mean will not be the same as a mean calculated from the original data; (2) We have to be careful when we back-transform standard errors. If we want to display the back-transformed means on a bar plot, with some indication of the variability of the data, we must calculate the standard errors and then back transform the upper and lower limits, which will not then be symmetrical about the mean.

Clearly, in the case study above, a log-transformation alters the outcome of statistical tests applied to the data. It is not always the case that transforming the data will make the difference between a result being significant and not significant, or that the transformed data will give a less significant result.

Never use p-values to judge the success of a transformation! We use diagnostic plots to make that assessment. What we hope is that we can transform the response variable so that it conforms, at least approximately, to the assumptions of the statistical model we want to use, making the result from associated tests as reliable as possible.

Taking logarithms is only one of many possible transformations. Each type of transformation is appropriate to solving different problems in the data. The following is a summary of the three most commonly used transformations and the sort of situations they are useful for.

A log-transformation stretches out the left hand side (smaller values) of the distribution and squashes in the right hand side (larger values). This is obviously useful where the data set has a long tail to the right as in the example above.

Taking the square root of the data is often appropriate where the data are whole number counts (though the log transformation may also work here). This typically occurs where your data are counts of organisms (e.g. algal cells in fields of view under a microscope). The corresponding back-transformation is obviously \(x^2\).

In R the square root of a set of data can be taken using the sqrt function. However, note that there is no square function in the list. Taking squares is done using the ^ operator with the number 2 on the right (e.g. if the variable is called x, use x^2).

This transformation is generally used where the data are in the form of percentages or proportions. It can be shown in theory (even if not from the data you actually have) that such data are unlikely to be normally distributed. A correction for this effect is to take the inverse sine of the square roots of the original data, i.e. \(\arcsin \sqrtx\).

In fact we often use data transformations, perhaps without realising it, in many situations other than doing statistical tests. For example, when we look at a set of pH readings we are already working with data on a log scale because pH units (0-14) are actually the negative logarithms of the hydrogen ion concentration in the water. Similarly measurements of noise in decibels (dB), and seismic disturbance on the Richter scale, are actually logarithmic scales.

We have focussed on ANOVA here for the simple reason that the assumptions are a bit easier to evaluate compared to regression. However, exactly the same ideas apply when working with other kinds of models that lm can fit. The workflow is the same in every case: