I have performed a VMR (variance-to-mean ratio) analysis on my data. I was wondering if there is a way to highlight that a particular difference between VMRs is significant?

Like in this plot, where I took a mean of VMR for each condition for two different groups I would like to compare. I see that in case of vmr_temp and vmr_ph they are different but how can I say that the difference is significant?

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#### Best Answer

Variance-mean ratio is most obviously pertinent for counted data where the easiest and simplest reference is the Poisson for which variance and mean are equal and the ratio is thus 1. Distributions which have ratios more than (less than) 1 are variously called over- (under-) dispersed and other distributions may be more appropriate reference distributions, although the form of any marginal distribution alone may not be decisive in determining appropriate models.

That does not seem implied here at all. The first signal that you are dealing with something different is that you report ratios $ll$ 1.

For measured (rather than counted) variables the ratio is more problematic as having dimensions the same as the measured variable. For example, a variable with length dimensions $L$ has variance with dimensions $L^2$, mean with dimensions $L$ and ratio with dimensions also $L$. A similar analysis applies to units of measurement.

In this case names like NaCl (sodium chloride), ph (pH?) and temp (temperature?) imply to me that you are dealing with measurements.

So, walk very, very carefully.

Comparing ratios across variables is utterly meaningless unless they are measured in the same units.

Comparing ratios for the same measured variable makes more sense, but comparing SD with mean would be immensely better. The ratio SD/mean, often called the coefficient of variation, is dimensionless and free of units and makes sense whenever measurements are necessarily all positive. (So, that rules out scales such as Fahrenheit or Celsius temperature for which the origin is arbitrary and negative values are possible.)

See threads here on the coefficient of variation such as How to interpret the coefficient of variation?

Independently of those details, likely to be crucial in this case, ratios of positive values are often best analysed on logarithmic scale, as large ratios typically vary more than small ratios. In this case, the variance-mean ratio is, on the whole, not a useful statistic for measured variables, so that advice remains separate.

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