Posted by B. Douglas Ward on October 31, 2000 at 10:51:40:
In Reply to: Re: 3dDeconvolve -- what is the noise? posted by Craig Stark on October 26, 2000 at 19:11:21:
Craig:
I stand by my previous statement:
"However, if the experiment is well designed, I don't believe that failure
to adequately model effect B would introduce a bias into the estimate of the
magnitude of the A effect. Obviously, if the experiment is poorly designed,
say if A always follows B, then it would be possible for the time-lagged
B effect to be falsely attributed to A. However, if A and B occur randomly,
then there should be no systematic error."
The key is to understand what is meant by "occur randomly".
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Below I present some results from various Monte Carlo simulations. The
"true" IRF's for conditions A and B were:
hA = { 1 10 5 1 }
hB = { 1 10 5 1 }
The "measured data" were created by convolving the input stimulus functions
with their respective IRF's, and adding white gaussian noise.
y(t) = b0 + b1*t + hA[0]*A(t) + hA[1]*A(t-1) + hA[2]*A(t-2) + hA[3]*A(t-3)
........+ hB[0]*B(t) + hB[1]*B(t-1) + hB[2]*B(t-2) + hB[3]*B(t-3) + w(t)
where w(t) is iid N(0,25).
For the following, 64 experimental designs, similar to the above, were randomly
generated, and for each of these experimental designs, 512 time series were
created using the above model, for a total of 32768 time series.
A typical experimental design is as follows:
A(t) = { 0 1 0 0 0 0 1 1 0 1 1 0 1 0 1 0 0 0 0 1 0 1 0 1 0 1 0 0 1 0 0 0 etc. }
B(t) = { 1 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1 etc. }
The following table lists the mean and std. dev. for each of the estimated
parameters from 3dDeconvolve:
Param ...Mean ..Std.Dev.
...b0 0.06954 1.34059
...b1 -0.0005 0.00617
hA[0] 1.00112 0.89448
hA{1] 9.99714 0.88960
hA[2] 4.99984 0.89454
hA[3] 0.98969 0.88481
hB[0] 1.00404 0.88261
hB[1] 10.0052 0.89136
hB[2] 5.00196 0.88768
hB[3] 0.99402 0.88748
Now, what happens if we leave out condition B from the model? That is, if we
run 3dDeconvolve with the same input data, but using the following model:
y(t) = b0 + b1*t + hA[0]*A(t) + hA[1]*A(t-1) + hA[2]*A(t-2) + hA[3]*A(t-3) + w(t)
Then we get the following results:
Param ...Mean ..Std.Dev.
...b0 9.20719 1.33998
...b1 -0.0065 0.01003
hA[0] 0.53160 1.04324
hA[1] 4.96837 0.84265
hA[2] 2.52043 1.06362
hA[3] 0.49198 1.06469
Obviously, the estimates for the hA parameters are biased downward. But why?
If A and B occur randomly, then why should there be a bias? The answer is that
A and B do NOT occur randomly with respect to each other. They are, in fact,
mutually exclusive. Since A and B never occur simultaneously, there is a great
deal of structure in this "random" design.
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Now, let's change the experimental design so that the input stimuli for the
two conditions occur INDEPENDENTLY.
A typical experimental design is as follows:
A(t) = { 0 1 0 0 0 0 1 1 0 1 1 0 1 0 1 0 0 0 0 1 0 1 0 1 0 1 0 0 1 0 0 0 etc. }
B(t) = { 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 1 1 0 0 0 1 0 0 1 0 1 0 1 0 0 1 1 etc. }
Again, 64 experimental designs were randomly created according to this
paradigm, and for each of these designs, 512 time series were created
using the full model, for a total of 32768 time series.
If we use the full model to analyze the data, we get the expected results:
Param ...Mean ..Std.Dev.
...b0 0.07082 1.09084
...b1 -0.0005 0.00617
hA[0] 1.00025 0.77643
hA[1] 9.99533 0.77111
hA[2] 5.00048 0.77472
hA[3] 0.99347 0.76537
hB[0] 0.99972 0.77174
hB[1] 9.99910 0.77403
hB[2] 4.99714 0.77006
hB[3] 1.00325 0.76943
Now, repeating the analysis, but leaving out condition B:
Param ...Mean ..Std.Dev.
...b0 6.32744 1.69108
...b1 -0.0072 0.01179
hA[0] 1.02400 1.06498
hA[1] 10.0510 1.14755
hA[2] 5.04815 1.11795
hA[3] 1.17939 1.06535
We see that the mean parameter estimates are very close to the true values,
in spite of the fact that condition B has been left out of the model.
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Of course, I realize that it is not always possible for a particular
experimental paradigm to allow simultaneous presentation of different
stimuli. In any case, it is very important that the researcher carefully
select the model that is used to represent the data.
Doug Ward