# How wise is the use of ANCOVA when groups differ on the covariate?

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)?

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is this covariate associated with the dependent variable? – Macro Jan 24 '12 at 3:55
yes covariate is body length and dependent variable is head length. Can I work with HL/BL ratio or what should I do? – Noro Jan 24 '12 at 4:04
We need a little more information to help you. What defines the groups? Is this a randomized study? What do you mean by "needing" a p-value above 0.05? – guest Jan 24 '12 at 6:38
It's not clear what the problem is here. If there is a significant difference in the relationship between the covariate and your response variable in different groups, why is that a problem? – Peter Ellis Jan 24 '12 at 10:34
My groups are defined by sex. I want to find out, if they differ in head size. I wanted to run ANCOVA with body length as covariate, but my groups significantly differ on covariate. Isn´t this problem for running and interpreting ANCOVA? I assume, that other assumtions are not broken. – Noro Jan 24 '12 at 11:30

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.

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Many thanks for your help – Noro Jan 25 '12 at 1:33
An ANCOVA is about power, and shouldn't be used to "control for" differences. See dionysus.psych.wisc.edu/coursewebsites/PSY710/Readings/… and stats.stackexchange.com/questions/5808/…. Although an argument could be made that the differences are small enough not to worry about the issues presented. – Matt Albrecht Jan 25 '12 at 2:18
Terrific answer. – rolando2 Jan 25 '12 at 2:26
@MattAlbrecht thanks for the interesting reading. In practice and especially in observational studies, ANCOVA is often used to control for differences rather than as a way of increasing power, which is more typically a focus in experimental settings (though certainly in observational settings power is of great concern also). Should people wish to apply ANCOVA as a way to statistically control for differences, they need to understand conceptually what they are achieving with it. – Anne Z. Jan 25 '12 at 2:59
@MattAlbrecht M&C say that unknowing application of ANCOVA can give biased treatment effect estimates, that there's often no simple way to control for group differences. It's a good suggestion to treat the covariate as a substantive variable and develop a model in which it is carefully introduced. You still may have specification error and biased treatment effect estimates. But at least you don't just assume the simplistic ANCOVA model. I'm not sure that's relevant here--the relationship between body length and head size does seem like it could be modeled simply--but it's an important caution. – Anne Z. Jan 25 '12 at 13:52

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.

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1. different body size is not feature, I have unfortunately more younger females and older males. But my data for every group are still normally distributed. – Noro Jan 24 '12 at 12:08

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)=