My groups differ on covariate

Question

In this case I presume loss of ANCOVA power, so I don´t know what type of analysis should I run next. There was significant difference in covariate between groups (p=0,008). Is there some solution? Could you help me please? Can I run ANCOVA if my groups differ on covariate (I know, that I can, but is it right solution)?

Answer 1

Yes, you can run an ANCOVA when you have significant differences on the covariate, but you should be aware of the reason for the caution against it and should be prepared to make arguments for why it's acceptable in your case.

A significant difference on a covariate in an ANCOVA is problematic because it suggests that ANCOVA may not be an appropriate way to analyze the data, not because of a loss of power. ANCOVA is used to statistically control for small differences on the covariate. If the differences are too great, the analysis itself becomes suspect.

ANCOVA calculates adjusted means and then compares these means to each other. Group means are adjusted using a common linear regression slope that estimates the relationship between the covariate and the dependent variable. For example, if you are looking at males, the adjusted mean would be calculated as follows:

$$ \overline{Y}_{male(adj)} = \overline{Y}_{male} - b_1 (\overline{X}_{male} - \overline{X}_{T}) $$

This is telling you what the average head size for males might be if males had the same average body length as the entire sample (and if other ANCOVA assumptions hold, such as homogeneity of the regression). The $b_1$ term is the common slope from fitting a regression of head size on body length for the entire sample.

This is essentially a "what-if" analysis--exploring what would happen if individuals from your two groups did not differ on the covariate, in this case body length. You might want to ask yourself if it makes sense to consider a case where animals from the different groups actually had the same body size. If there is quite a bit of overlap in body lengths, it may be a reasonable hypothetical scenario. But if the animals are of wildly different sizes, it may not make sense to run ANCOVA and statistically correct for body length differences. For example, if you were running an analysis on mice and elephants, would it make sense to consider a hypothetical situation where their weights were set to the average of the entire sample? No mouse would ever be that big and no elephant would ever be that small.

It would be useful to make a scatterplot of head size vs. body length with different symbols for each group so you can demonstrate how much (or how little) the sizes overlap. You might want to do this to inspect homogeneity of the regression lines anyway. If there isn't much overlap, you could limit your analysis to a set of animals whose sizes do overlap, for example by matching. This is one benefit of matching--it prevents you from making unwarranted extrapolations.

Your question seems to be "do animals relative head sizes differ?" where head size is relative to body length. You could use the ratio of head size to body length as your outcome, but you may need to transform it to make it normally distributed. If head size is always smaller than body length, I might consider a logit transform as that often transforms proportions into normally distributed data. The arcsine-square root transformation is also sometimes used with such data.

Answer 2

For the problem of when the covariate is correlated with the independent variable of interest, you could either:

1. accept that what you're studying is a feature, not a bug, and simply not use a covariate
2. use a different covariate that is not correlated with independent variable if you have one
3. try and match the two groups on the covariate of interest from within your sample (or collect more from the needed population)

However, 3 may not be suitable in light of 1, e.g., schizophrenia is associated with cognitive deficits, so trying to match a schizophrenic group with a control group on IQ will not be representative of the control population.

Answer 3

I think you have to be careful when constructing an index as a proportion (Head length/total length) since you are assuming an isometric relationship between those variables, and in most case the biological relation between those variables are allometric but never isometric:

if isometry exist then: HeadL=b*TotalL, when dividing all terms in order to produce an index/ratio: HeadL/TotalL= b (in this case you have controled the influence of TotalL on the index)

However if non isometry exists (or you have an Y-intercept different from 0)=

HeadL= a + b*TotalL then

HeadL/TotalL= a/TotalL + b (so in this simple case you still have the influence of TotalL in the index, so if they differ in the covariate (TotalL) they still have influence on the proportion - so it is not a solution, especially if there exist a power relation between those variables)

There are many critiques regarding the uses of indexes/ratios in relationships that are not isometricals. I do not think that would be a proper solution for your problem. Another option could be testing a heterogeneous ANCOVA model.