Dear Steven,

At 23:59 05/06/2000 -0400, Grant, Steven (NIDA) wrote:
| After comparing analyzing some single pixel values with a standard
| GLM statistical package (SAS and JMP), I think I now understand the
| solution and problems associated obtaining a correlation between a
| difference in a covariate obtained across a baseline and active
| condition and a difference in the scans associated with each
| condition.  I would like to confirm that my understanding is
| correct.

OK.

Just for the spectators, let's confirm that the model is:

Y_iq = A_q + C * 0.5*s_iq + error

...where:
	Y_iq is the baseline (q=1) / active (q=2) scan on subject i
		i = (i=1,...,N) N = number of subjects
	A_q is the baseline / drug effect
	s_iq is the covariate
		(possibly after transformation including mean centering)
	C is the slope parameter for the covariate
	B_i is the subject effect

...so the design matrix has:
	2 columns indicating baseline / drug
	1 column of the covariate
	n columns indicating the subject

...and would be entered into SPM as:
	- multi-subject, conditions and covariates
	- two conditions, one scan per condition per subject
	- one covariate

...and the discussion is about what effect of different 
centering/transformation schemes for s_iq are:

Shoot...

| Given the caveat that there is only one scan per condition for each
| subject and using the Conditions and Covariate design in SPM99, no
| covariate interactions, no covariate centering, then as Andrew
| Holmes (and others) have stated, a t-contrast of 0 0 1 will map the
| positive correlations between the change in the covariate with the
| change in the scans.  This same answer is obtained whether the
| covariate is entered as raw data, or mean centered data, or whether
| the baseline covariate value is constant across all subject (e.g.,
| baseline covariate = 0).

Correct.

| The t-value given by SPM can be converted
| to a correlation coefficient by the standard r to t formula using N-
| 2 for degrees of freedom (not the d.f. given in the SPM table).

Not quite. The r to t formula is for a simple linear regression model, and 
is only applicable to such a model. If the d.f. given in the SPM table are 
not equal to N-2 then you do not have a simple linear regression. The 
models we've been considering can be shown to be simple linear regressions 
of the within subject scan difference on the difference in covariate. (In 
the same way that a blocked two group ANOVA with one scan per block per 
group is the same as a two-sided paired t-test.) Here N is the number of 
subjects in a multi-subject analysis, or the number of scans in a 
single-subject analysis. Note that the converse is not true, having d.f. of 
N-2 doesn't imply you have a simple linear regression.

If you have a multiple regression (as you will have if you use AnCova 
global normalisation for example), then an SPM{t} for a covariate is 
equivalent to a test of non-zero *partial* correlation of that covariate 
with the scan voxel values, having removed the effects of the other terms 
in the model. The partial correlation is related to the sums of squares 
associated with a contrast, and could be derived within SPM using the ESS 
image written out when examining F-contrasts.

The partial correlation is the square root of the proportion of the 
residual sum of squares from the reduced model that is explained by 
including the covariate. So, if ResSS is the residual sum of squares for 
your full model, and ESS is the extra sum of squares due to the covariate, 
the partial correlation r(C|X_0) = sqrt(ESS/(ResSS-ESS)). See Altman 
"Practical Statistics for Medical Research", S12.4.9.

                             ----------------

| The problem arises if you wish to derive the condition effects from
| the same analysis. In this case, the form of the covariate will
| make a great difference.

It's not really a problem, it's just that in centering the covariate you 
re-define what the condition effect is. As I said yesterday, mean-centering 
the covariate (so that it's mean is zero) defines the condition effect as 
being at the mean covariate value. Not mean centering defines the condition 
effects to be at a zero covariate value. ...and there are instances where 
the latter is of interest.

| If you use the raw covariate values, then
| the size of the condition effect will be overestimated (i.e., larger
| t-value, lower p-value) for the reasons explained by Karl Friston in
| recent postings.

Could go either way, and depends on the relationship between the covariate 
and the response, as well as on the mean of the covariate.

| One unfortunate result is that a conjunction
| analysis between the condition and covariate effects will be overly-
| liberal.

If the covariate isn't mean centred, then the contrasts for the condition 
and covariate effects will be non-orthogonal, and therefore dependent, 
precluding a simple conjunction analysis. There's more to it though: 
Non-orthogonal contrasts put forward for conjunction are reduced into an 
new orthogonal set of contrasts within SPM. (Similarly, in a non-orthogonal 
model, you can work out orthogonal contrasts. So even with "raw" 
covariates, you could calculate a contrast that essentially effected the 
mean correction. J-B has the details...) Further, more seriously, at low 
degrees of freedom the SPM{t}'s for each contrast won't be independent 
anyway, even if the contrasts themselves are orthogonal, because the 
SPM{t}'s all share the same variance estimator in the denominator.

                             ----------------

| If the covariate is mean-centered for each condition, then the
| condition effects will be properly estimated.

If you're defining your condition effects to be at the mean covariate 
value, then (circularly), yes.

| Alternatively, one could perform 2 separate analyses, one just to
| examine the covariate effects, and a second condition-only design
| without the covariate to get the condition effects.

That wouldn't work I'm afraid.

If there is a covariate effect then a condition-only design will not model 
it (obviously), leaving it in the errors, which will not then be 
independent and identically normally distributed. So the model (and ensuing 
inference) would be technically invalid. (You'd probably get an increased 
variance estimate pulling one way, and an extra degre of freedom pulling 
the other.)

If you mean center the covariate, the condition effects for a covariate & 
condition model will be the same as for a condition only design. ...but 
don't forget the variance and degrees of freedom!

                             ----------------

As Karl says, it's generally safer to mean correct covariates, unless you 
are not interested in the condition effect (when you can be lazy and not 
bother), or are interested in the condition effect at a particular adjusted 
value of the covariate (when you want the condition effects to be affected 
by the covariate).


Hope this helps,

-andrew



+ - Dr Andrew Holmes			mailto:[log in to unmask]
| Robertson Centre for Biostatistics 	(  ,) / _)(  ,)
| Boyd Orr Building, University Ave.,	 )  \( (_  ) ,\
| Glasgow. G12 8QQ  Scotland, UK.    	(_)\_)\__)(___/
+ -					http://www.rcb.gla.ac.uk/



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