Afraim / Andreas -
Below are a number of points about the recent mailings on
efficiency estimation using the formula:
trace(c'*inv(xX.X'*xX.X)*c)^-1 (Friston et al, 99)
1. This formula is about maximising signal (not signal:noise
ratio, which is a more relevant measure). It assumes that
noise characteristics are independent of the design (which
may not hold).
2. This formula does not allow for band-pass filtering
(the extent of highpass filtering being the mean-correction
applied through the constant column of designs).
Thus the high efficiencies that can obtain for designs
with dominant power at low-frequencies would not be
realised in fMRI data, which is dominated by low-
frequency noise. One could allow for bandpass filtering
by incorporating the smoothing filter S in the formula:
trace(c'*pinv(S*X)*S*S'*pinv(S*X)'*c)^-1 (Friston et al, 00)
(assuming no intrinsic autocorrelation) where X=xX.X and
S=K in SPM99 (as Tom Nichols also pointed out). The
pratical problem with this is that the K filter matrix is not
determined until one has chosen the high and low-pass
cut-offs in the estimation stage of SPM99, which may
prove a hassle if one is using the stochastic design stage
to create multiple different designs.
3. Alternatively, one can calculate the Estimated Measurable
Power, as described in Josephs & Henson (1999), which
incorporates bandpass filtering. Note however that this
measure only applies to a single contrast; it does not
accommodate covariation between correlated columns of a
design matrix, which can reduce efficiency, and which is
accommodated by the "trace" formula above.
4. Note that the efficiency of finite designs may not correspond
to the asymptotic efficiency, which is more appropriate if one
wants to make generalisations about optimal SOA and ordering
of events. It was an initial transient effect for example that gave
rise to Afraim's unexpected results for a range of fixed SOAs.
The large initial rise in signal ('upstroke') produced high efficiency
at very short SOAs. If the contribution of this transient effect is
minimised (by using much longer sequences, or ignoring the
initial few scans), the plot of efficiency against (fixed) SOAs
is once again in line with asymptotic results: low efficiency at
short SOAs, peaking to highest efficiency at about 16-18s.
(Personally, I only make use of the asymptotic results - ie the
general lessons about optimal SOAs and ordering - in designing
my experiments, rather than iteratively optimising finite designs.)
5. Note that to calculate efficiency of blocked designs, one
still needs to convolve with the HRF. This explained the second
puzzling aspect of Afraim's results (see below). Furthermore,
if one wants to compare blocked with randomised ordering
of event-types, with a minimal SOA between events, one
should really model blocks as runs of events rather than boxcars.
6. In answer to Andreas's question, the magnitude of the contrast
weights will affect the the above measure of efficiency, given that
only takes into account the signal. The magnitude of the contrast
weights (length of the contrast vector) will not of course affect the
real sensitivity of the design, which depends on the ratio of signal
to noise. This can be seen by the effective normalisation of the
t-statistic by the contrast weights: c'*B / sqrt(s^2*c'*inv(X'*X)*c),
where B are the parameter estimates and s^2 the error variance estimate.
In other words, only compare designs using the same set of contrasts
(see my previous email). Either of Andreas's contrasts, [0.5 0.5 -1]
or [1 1 -2], are fine however for testing statistical effects (and will
give the same answer).
7. In answer to Andreas's more specific questions, the formula
does apply to epoch-designs (though see 5); including other basis
functions (such as derivatives) will not make any difference
because they are orthogonal; and you could include only
selected columns of the design matrix, though you will get different
answers if the ignored colums are correlated with your contrasts
of interest.
8. Finally, note of course that all these efficiency measures assume
a linear summation of responses, which is unlikely to hold at short
SOAs. Nonetheless, when Karl allowed for nonlinearities of the
type captured by second-order Volterra kernels (Friston et al, 1998),
which impacted with SOAs of <~8s, it appeared that the advantages
of short SOAs still outweighed these nonlinearities for SOAs down
to about 1s. This remains a theoretical prediction however....
Rik
Dr Afraim Haddadi wrote:
> Dear Rik,
>
> Thank you for your help. This is what I understood from our
> conversatioon the other day. Please feel free to comment
> and/or forward onto the helpline.
>
> 1) The efficiency estimation is not meaningfull when applied
> to unconvolved boxcars as it is the shape of the HRF that
> determines this. As expected, convolution with an HRF
> produces the more familiar pattern:
>
> SOA Length Efficiency
> 14 7 666.1003
> 20 10 860.8268
> 28 14 817.9594
> 42 21 649.6377
> 100 50 415.9499
>
> (1000 scans TR=1 fixed length, fixed SOA epochs)
>
> 2) With regards to events. the calculation can be misleading
> at short SOAs:
>
> SOA Efficiency
> 0.5 3348.2
> 1 793.9949
> 2 178.0398
> 4 54.3206
> 8 455.1452
> 16 1114.1
> 32 837.3594
> 64 466.4965
>
> (1000 scans TR=1 fixed SOA events/HRF)
>
> This being due to the fact that linear summation of the HRF
> is assumed in constructing the regressor such that the picture
> at short SOAs is dominated by the effects of a large
> initial upstroke. A more realistic estimation may be derived
> by ignoring the top segment of the design matrix in
> this case, leaving one with a more familiar inverted-U
> shape.
>
> Much obliged,
>
> Afraim
> -----------
> Dr A Salek-Haddadi
> Clinical Research Fellow
> Institute of Neurology
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