Dear Michael,

I would not try to make this analogy to ANOVA.
Unfortunately it is common that people get hooked
on an ANOVA way of thinking, which for FMRI is
really not that helpful as the GLM is often far more
natural in terms of what you want.

So if you just think about it in the context of the GLM
it is much simpler. The contrast is an exact weighting
of the betas (parameter estimates or PEs) so that
if an element is zero in the contrast, then that beta
is *not* included. With three levels the contrast
[-1 0 1] means both "is there a linear slope that is
greater than zero" and "is the response to level 3
greater than that to level 1". Both are exactly
equivalent but only for the 3 level case. In the 4 level
case the linear question would be [-3 -1 1 3] while
in the 5 level case it would be [-2 -1 0 1 2], which
are very different from [-1 0 0 1] or [-1 0 0 0 1].

I'm not sure if I can really explain it better than this.
I would just again suggest that you not get so hooked
up on the ANOVA ideas, as the mapping of ANOVA
quantities and GLM ones is complicated, as you can
see in the FEAT webpages. So it is *very* difficult to
develop a correct and intuitive insight that way.

All the best,
Mark


On 17 Dec 2010, at 20:36, Michael Harms wrote:

> Hi Mark,
>
> Sorry if I'm being dense, but I still don't see how [-1 0 1] tests for a
> "linear trend" in the context of which it is used on the FSL web page.
>
> My understanding of the viewpoint for treating [-1 0 1] as testing for a
> "linear trend" is that that comes out of the use of orthogonal
> polynomial coefficients in the ANOVA literature, and the justification
> for that terminology is that if you apply those contrast coefficients to
> the cell means of a quantitative factor (X) at equally spaced levels
> they will deliver the Type I (i.e., sequential) sums of squares for the
> increasing powers of X that you would get by regression.
>
> In that ANOVA framework, if you have 3 levels, and the mean (i.e.,
> "beta") of the 2nd level is changed dramatically, you are simultaneously
> changing the overall mean, which changes the relative relationships of
> the sums of squares. So that even though the weight in the contrast
> vector for the 2nd level is 0, the mean of that second level is still
> influencing the statistics of whether the "linear" contrast vector is
> significant through its impact on the total sums of squares.
>
> But as you noted, the case with FMRI is different. In the case of
> estimating say three distinct betas corresponding to three experimental
> manipulations, those betas are themselves estimated against the FMRI
> baseline, which in the context of the ANOVA analogy would actually
> represent a 4th level (right?). That is, the three betas in the FMRI
> case seem to have an independence relative to the overall
> "mean" (baseline) which isn't the case when using orthogonal polynomial
> coefficients to assess the "linear" effect in a 3 level ANOVA.
>
> Given that line of perhaps faulty reasoning, it still isn't clear to me
> why the [-1 0 1] contrast can be considered in the FMRI case to be
> testing a "linear trend" rather than solely testing the straightforward
> question of whether lev 3 is greater than lev 1.
>
> Please set me straight where I'm in error if I'm still not thinking
> about this correctly. (Am I thinking of the issue of the "ambiguous fMRI
> baseline" in the wrong manner?)
>
> cheers,
> -MH
>
> On Fri, 2010-12-17 at 08:12 +0000, Mark Jenkinson wrote:
>> Hi Michael,
>>
>> You are right that [-1 0 1] can be looked at either
>> as just comparing EV3 with Ev1 (i.e. levels 3 and 1)
>> or as looking for a linear trend. This is because the
>> slope of the linear trend is unaffected by the middle
>> point, as you understand from the example. You
>> are also right that in this example the quality of the fit
>> (or the residuals, or the statistics) is dependent on
>> this middle point. In a typical FMRI experiment this
>> is a bit different from this example though, in that you
>> normally specify a design matrix that has a separate
>> EV (regressor) for each level and then only use [-1 0 1]
>> as a contrast. This means that the parameter
>> estimate (PE) associated with the middle level's
>> regressor (EV) will not enter into the calculation at all.
>> The residuals are then not affected by this value and
>> also solely determined by the residuals from the
>> time series model.
>>
>> Put another way, setting up the design matrix and
>> contrasts is all about specifying a *model* of what
>> you think is happening, and if you are interested in
>> whether the slope of this line is zero or not, then you
>> need to extract a measurement of the slope of the line
>> and this is done with a [-1 0 1] contrast. The statistics
>> will depend on how big a slope was detected and how
>> big the residuals (in the timeseries fits) were, but not
>> on how strong the response is to the middle level. If
>> you were testing something other than the slope, or
>> had an even number of levels, then all the levels
>> would typically contribute, but this is the unusual case
>> where the slope is completely unaffected by the
>> response of the middle level, and hence the test for
>> a linear effect (is the slope different from zero or not)
>> is purely determined by the 1st and 3rd level responses.
>>
>> That is, only for the 3 level case (where there are only
>> 3 numbers to play with) does the mathematical formulation
>> of the linear trend question and the "is one level
>> bigger than another" question end up being the same thing.
>> For more levels they end up as different mathematical
>> formulations which is more intuitive.
>>
>> I hope this helps to clarify things.
>>
>> All the best,
>> Mark
>>
>>
>>
>> On 17 Dec 2010, at 00:20, Michael Harms wrote:
>>
>>> Hello,
>>> I've read several posts from the archives related to the statement on
>>> the FSL website that in a design with three levels of stimulation, the
>>> contrast "[-1 1 0] shows where the response to level 2 is greater than
>>> that for level 1" and "[-1 0 1] shows the general linear increase across
>>> all three levels."
>>>
>>> i.e.,
>>> https://www.jiscmail.ac.uk/cgi-bin/webadmin?
>>> A2=ind0901&L=FSL&D=0&1=FSL&9=A&I=-3&J=on&d=No+Match%3BMatch%
>>> 3BMatches&z=4&P=214931
>>>
>>> and
>>> https://www.jiscmail.ac.uk/cgi-bin/webadmin?A2=ind06&L=FSL&D=0&P=1810746
>>>
>>> Perhaps it is a matter of what is meant by a "linear" trend, but after
>>> reading these posts, I'm still confused about the interpretation of
>>> these contrasts in the case of 3 levels. (A related question came up on
>>> the SPM list recently, so I'm trying to understand the previous FSL
>>> posts on this matter).
>>>
>>> First, how is it that the contrast [-1 1 0] is simply checking for lev 2
>>> greater than lev 1 (and not saying anything about lev 3), whereas [-1 0
>>> 1] somehow checks for a "linear increase across all three levels"? That
>>> seems to be an inconsistency in interpretation that doesn't make sense,
>>> given that all you are doing is moving the position of the "0" in the
>>> contrast.
>>>
>>> Second, in the example that was provided, it was stated that if you plot
>>> x=[1 2 3] vs. y1 = [1 4 3] and y2 = [1 -4 3] that "you would still draw
>>> exactly the same regression line". While that statement is true in the
>>> sense that the regression line of x vs. y1 and x vs. y2 will both have
>>> positive slope, the r-squared of the best (least squares) fit is
>>> certainly dependent on the value of all elements of the y vector, and
>>> thus, at least in the sense that I think of a "linear trend" it is not
>>> the case that the "second point carries no information about the
>>> presence or absence of a linear trend".
>>>
>>> Any elaboration on these previous posts (or this issue in general) would
>>> be very helpful!
>>>
>>> thanks,
>>> -Mike H.
>>>
>>> --
>>> Michael Harms, Ph.D.
>>> --------------------------------------------------------------------
>>> Conte Center for the Neuroscience of Mental Disorders
>>> Washington University School of Medicine
>>>
>