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In a 1st-level fMRI analysis, it is unclear to me what the t-test performed for each voxel is actually comparing. I've seen this being described in one of two different ways:

  1. The t-test estimates how well the model (HRF curve convoluted with the design) fits the data (BOLD signal), in other words is a measure of the goodness of this fit.

  2. The t-test compares the BOLD signal to a baseline.

I guess both of these can be conceptualized as one-sample t-tests, but it's unclear if the two comparisons would in fact be equivalent, or if different types of t-tests are used in different situations, with the two cases given above being but two examples.

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The coefficient is the measure of how strongly the given factor predicts the dependent variable, or in other words how much of the variance in the dependent variable can be explained by that factor. So the coefficient itself, and not the t-statistic of the coefficient, is a measure of what you're calling how well the model (or, more correctly, the regressor) fits the data.

The t-statistic of the coefficient in a multiple regression is the statistical test of that coefficient, calculated as the coefficient divided by its standard error. So it would be most correct to say that this is a measure of how reliable the sample coefficient is, or how likely it is, given the coefficient we got, that the true coefficient is 0. But this is theoretically independent of the goodness of the fit, and you could have a very stable but small coefficient, which would get a high t-stat, or a large-on-average, but highly variable coefficient, which would receive a low t-stat. The results of the test will also, obviously, depend on your sample size and the degrees of freedom in your model.

Where the confusion might be is that the coefficient is more or less the same as a comparison to baseline, becuase the baseline is not (or shouldn't be) an explicit regressor in the model. A higher coefficient will mean, all else being equal, that the fit is better and/or that the effect is larger. But unless you're explicitly running a comparison of activation vs. baseline, this not what the coefficient actually means.

You can read more about this last issue here: https://github.com/jdkent/tests_and_musings/blob/master/magnitude_and_delay_in_bold.ipynb?fbclid=IwAR3H-pvYmVQtMaDBzDAfexyNeuCUHHe6vuYLsV83l957rvHawv1bWJlaB9U

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  • $\begingroup$ Thanks Henry for a very helpful answer. I think you are using the term 'baseline' in the sense of a 'comparison to zero', as e.g. in a one-sample t-test; whereas what I mean with it has to do with the "standard" contrast often reported in fMRI papers, the result of which can be that e.g. brain region X is "activated" under condition A (vs baseline). In that sense, 'baseline' often means rest, or some other condition in which parameters are controlled to be the same as in condition A. I hope I understood you correctly - in either case, please let me know if this changes your answer in any way. $\endgroup$ – z8080 Oct 31 '18 at 17:29
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    $\begingroup$ The way a regression model is standardly run is to convolve a HRF function with the timing of the stimuli presentations. So no explicit baseline is usually used. So although it's not strictly true to call this a comparison to baseline, the regression is, in essence, a comparison with activation during rest - i.e. activation that is seen when stimuli are not being presented. When a higher beta weight is achieved, it means that the activation in that region rises and falls in greater sync with presentation of stimuli and rest. The t-stat is just a measure of the reliability of that sync. $\endgroup$ – Henry Brice Nov 2 '18 at 6:30
  • $\begingroup$ Makes sense - many thanks once again! $\endgroup$ – z8080 Nov 2 '18 at 8:39

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