The use of between-subjects covariates in within-subjects analyses is common and often useful. Your analysis is a split-plot (aka WSxBS) ANOVA with number size as WS and group as BS factor. The most interesting term, however, will be the WSxBS interaction with the null hypothesis "the effect of number size is the same for hemiplegic children and controls". If this H0 can be rejected, then simple effects analyses of the effect of number size per group (or vice versa) can be conducted. It is possible to enter basic processing speed as covariate into this model. The WSxBS interaction term will then be evaluated for subjects with covariate = 0 (i.e. the covariate will be held constant). As a consequence, if the group difference on the number size effect can be completely explained by baseline processing speed, this term will be zero. So the question is whether there is some specific effect of hemiplegia on number processing which is independent from basic processing speed. If this is the case, you will find it by entering the covariate. If this is not the case, you will see no effect because the covariate explains it all. In either case, running a model with and without covariate will be informative. A helpful (and concise) paper might be van Breukelen & van Dijk, J Int Neuropsych Soc 2007, 13, 903-904. All the best.

Hi Catherine,

I think it doesn't make sense to use processing speed as a covariate when comparing normal and hemiplegic children period (even without the within factor!). If I understood your research design correctly, then the covariate is confounded with the between-subjects factor (normal vs hemiplegic). So both variables arguably explain the same variance in the dependent variable. That's probably also why the  difference disappears when you enter the covariate (none of the variables has a unique contribution to explaining variance in the fitted statistical model).

See also: http://homepages.gold.ac.uk/aphome/ancova.pdf

regards,

Mario de Jonge

Try to use "path analysis"... Regression models handle well with this problem!

In a conventional ANOVA type model, no. The covariate varies between subjects so its effects are completely absorbed by the subjects term in the within subjects ANOVA.

However, it can make sense if you have a time-varying covariate (but you need to use a multilevel model or similar to incorporate that effect). It can also make sense if you are interested in covariate by factor interactions.

So for some of you, it makes sense to enter a covariate to see if the interaction is still significant after processing speed is considered. I think I understand the statistical rationale for that. However, I still do not understand it conceptually :

My results show that hemiplegic children are slower for number comparison than control children (main effect) but that it is true only for large numbers (interaction). However, when I enter processing speed as the covariate, the interaction disappears. What does it mean ? I do not understand how the difference in processing speed between my 2 populations can be responsible of the interaction. Hemiplegic children are very slow for large number comparisons and as fast as control children for when small numbers are concerned. On the other hand, control children do not experience difficulties in comparison whatever the size of numbers. The difficulties observed in hemiplegic children for large numbers are revealed because we compare their own reaction times in 2 conditions and this (within difference) is compared to the lack of difference in control children. Again, how processing speed per see can explain this pattern ?

I thought that, precisely, when interactions are obtained on 2 different populations, general differences between these populations cannot explain the results.

Rather: "The two populations differ on large scale, hence, the difference is real; but on smaller sets, there is something confounding your results... possibly time or in smaller population sizes, the "normal" children exhibit such large variation that any perceived difference with any other group vanishes." 

Covariates that differ between groups cannot be "corrected for" with ANCOVA. See Miller and Chapman, J Abnorm Psychol 2001, 110: 40-48.

Entering a covariate will eat up a degree of freedom and reduce power unless is also reduces error (in which case it might offset that loss of df).

Otherwise it will only matter if it is also entered as an interaction term. As a main effect it should just be taking variance from the within subjects error term.

BTW Miller and Chapman is, I believe, a deeply misunderstood paper - they are correct on this point (narrowly interpreted), but one should be careful on interpreting that paper.

Thom : What do you mean by time-varying covariate ?

Chiang-shan : My covariates always differ between group and this is precisely the reason why I enter them as covariates. I want to determine whether the effect is "due" to the difference that i have already identified (i.e., processing speed in my 2 populations). What do you mean exactly ?

Can you explain a bit more what you mean by 'processing speed' as your measure of this as a covariate may be an issue. For example, a measure of processing speed could be the time to compare numbers  - that would create a problem here?

I measured processing speed by a simple reaction task in which children had to press a key as soon as a target appears. And then I entered this variable as a covariate in the analysis of the time they take to compare numbers.

Not sure that is really measuring processing speed, more simple RT, motor/visual skills etc. unless the target requires some cognitive judgment. Am I clear, what you are trying to find is a large difference between 'big number' comparison for hemiplegic children vs. children withoput the problem, that is not apparent for small numbers, once processing speed is taken out of the comparison. If so you need to consider what you think causes this big number effect and how the covariate relates to this. In other words forget the stats for a moment and consider a way to take out processing speed in a way that is not confounded with the dimension that is causing the big number effect in hemiplegic children.

The use of between-subjects covariates in within-subjects analyses is common and often useful. Your analysis is a split-plot (aka WSxBS) ANOVA with number size as WS and group as BS factor. The most interesting term, however, will be the WSxBS interaction with the null hypothesis "the effect of number size is the same for hemiplegic children and controls". If this H0 can be rejected, then simple effects analyses of the effect of number size per group (or vice versa) can be conducted. It is possible to enter basic processing speed as covariate into this model. The WSxBS interaction term will then be evaluated for subjects with covariate = 0 (i.e. the covariate will be held constant). As a consequence, if the group difference on the number size effect can be completely explained by baseline processing speed, this term will be zero. So the question is whether there is some specific effect of hemiplegia on number processing which is independent from basic processing speed. If this is the case, you will find it by entering the covariate. If this is not the case, you will see no effect because the covariate explains it all. In either case, running a model with and without covariate will be informative. A helpful (and concise) paper might be van Breukelen & van Dijk, J Int Neuropsych Soc 2007, 13, 903-904. All the best.

Thank you Wolfgang. Your explanation is clear and I'll definitely read the paper you are talking about.

Maybe this link is also interesting for you: http://www.psyc.bbk.ac.uk/research/DNL/stats/Repeated_Measures_ANCOVA.html

This is precisely and exactly what I needed Ernest. Thank you so much.

Catherine, Given that you have pre-existing groups (hemiplegic and control), it appears to me that you may have to confront Lord's paradox (see Psych Bulletin- sorry I don't have the reference). By using analysis of covariance, you would in effect be saying, after you adjust the means, "What would the data be if the data are not what they are?"  Interpreting the results may be problematic. Anyway, it looks like an interesting study!