We present an update about the use of a special type of parametric designs in fMRI research that can be very useful in investigations involving natural and multi-featured stimuli such as pictures or words. This method has already been developed by Büchel and colleagues [1
] but unfortunately it has not been used as frequently as it deserves. For this reason, we present a summary of the logic behind the use of parametric designs in fMRI research, discuss shortly its mathematical background and applicability, and present an empirical example where parametric regressors carry the most relevant modulation of the fMRI signal.
In several fields of neurocognition, stimuli can be assigned to experimental conditions so that they (i) are homogeneous within each cell of an experimental design and (ii) differ only with regard to a single aspect across the different levels of an experimental factor. Each statistical contrast unequivocally isolates therefore one and only one neurocognitive process. However, in the case of natural stimuli, such as pictures (e.g. kitchen utensils vs. garage tools) or written words (e.g. varying in length, number of syllables, frequency, neighbourhoods, regularities, consistencies etc.), the task of matching different groups of items for their attributes is particularly challenging, because there is often only a finite a number of stimuli to fit into each of the different cells of the experimental design that vary simultaneously in more than one feature. In these cases, different dimensions of stimuli can only be matched on average. Words, for instance, can vary in the number of letters, the frequency of occurrence, the number of lexical neighbours as well as the frequency of occurrence of orthographic or phonological sub-lexical units. Different words may have for example 1, 2 or 8 different lexical neighbours. Therefore, for each stimulus dimension (i.e. frequency of occurrence, number of lexical neighbours, etc.) there is a non-zero distance between each single item and the average for each of the different dimensions, characterizing the amount of variation within condition.
Due to variation within condition, the statistics for the size of fMRI signal elicited by the different items pertaining to the very same condition may vary considerably. Consequently, the type-II error for detecting a difference in fMRI signal between two different conditions may be inflated. The main problem for the interpretation of the results of such an experiment is whether it is acceptable to consider the variation within condition as measurement error or not. If the variation within condition is small in comparison with the variation between different conditions, treating it as measurement error is not problematic. However, if the variation within a cell of the experimental design increases due to systematic variation in known dimensions of multi-featured stimuli, the validity of the whole study may be questioned.
In the present paper we examine a method proposed by Büchel and colleagues [1
] for dealing with systematic variation between items. The method involves the definition of parametric regressors representing each of the several dimensions of complex stimuli. These parametric regressors absorb the systematic variation inherent in different dimensions of complex stimuli such as words, sentences or arithmetic problems, and allow for separating it from genuine measurement error. In the following, we will present the method, discuss its main applications, and present an example in which the variation between items (and their exact scaling properties) was the most relevant aspect of the experimental design.
Overview of the method
Activation Yij in a particular voxel can be described as in (1) for each replication i (for every i from 1 to p) of an experimental condition j (for every j from 1 to q):
having a as the intercept, Xj as a (continuous) parameter describing the present experimental design, βj as the regression coefficient for the parameter Xj and εij as residual error. The coefficient βj describes the event- or block-specific expected BOLD-response under a given experimental condition j assuming that within an experimental condition the BOLD-response induced by event- or block-specific stimulation will be a constant. A corollary of this assumption is that variation in the BOLD response occurring within an experimental condition will be considered residual error.
When stimuli in an experimental condition are sampled from a universe of natural items, some variation among items will always be present. An artificial increase of residual error ε due to variation in the BOLD-response produced by variation within condition contributes negatively to the sensitivity of the fMRI design. Importantly, when the variance among items not only represents a confounding factor but genuine scaling properties of stimulus features, it is mandatory to deal with them appropriately by modelling this variance within conditions.
Parametric modelling always allows for the description of variation in the event- or block specific BOLD-response, when the source (or sources) of variation is known a priori and can be specified numerically as parametric regressor. Importantly, the variation within conditions may be due not only to one single stimulus feature, but rather be due to two or more features. In this case, for each of the dimensions of multi-featured stimuli a regressor can be defined, which absorbs the contribution of that dimension for the variation within a given experimental condition (but see the section on the limitations of this approach in the discussion, below). The specification of parametric regressors is given as follows: the parameter Xj described by a canonical hemodynamic function in common fMRI designs, which has exactly the same form across all replications i of a given experimental condition j, can be expressed as the average effect Xj of a predictor X on brain activation. Moreover, in parametric designs a second set of predictors may be complemented by a set of k dimensions (for every k from 1 to r) which are nested under each condition j and which absorb the variation within each condition. The full model presented in (2) contains a predictor representing the average effect of experimental condition j plus an additional parameter for each parametric regressor k considered. Xij1, Xij2, ... Xijk ... Xijr contain the variation in each of k different stimulus dimensions. Note that parameters βjk are hierarchically bound to the average parameter βj and that the number of parameters βjk associated with each average parameter βj may differ. Therefore, (1) can be generalized by assuming a set of r > 1 dimensions:
Yij = α + βjXj + βj1Xij1 + ... + βjk Xijk + ... +βjrXijr + εij
By entering parametric regressors in the fMRI design, the proportion of variance which can be accounted for by the variation within conditions is separated from the residual error εij. This extension of the model presented in (1) has two consequences: (i) the statistical test on the significance of null-order parameter βj will not be biased by variation within conditions, which can be explained by predictors βj1 to βjr. (ii) Furthermore, the relevance of regression coefficients βj1 to βjr may be assessed.
The definition of parametric regressors with the single purpose of isolating variation within conditions as a confounding factor is trivial and has been employed regularly in fMRI research. The sole purpose of this application is to control for the impact of undesired sources of variance affecting statistics about the effects of interest. In this case, variation within conditions can be considered an effect of non-interest, the impact of which on the statistics can be partialled out from residual error.
Nevertheless, parametric regressors also may be defined with the aim of directly investigating theoretical predictions with respect to the fMRI activation observed. In the following, we will concentrate on the advantages and limitations of such an application. In fact, parametric regressors make possible an investigation of the direction and actual scaling properties of variation of fMRI activation. Examination of the impact of quantitative regressors on the fMRI activation has been presented by Büchel and colleagues [1
]. In that study the authors defined one single parametric regressor and applied polynomial expansions (i.e. quadratic, cubic, etc.) to investigate non-linear relationships between the BOLD-response and this single experimental parameter. Here we use the method [1
] for two purposes: (i) instead of examining the impact of polynomial expansion of a single parametric predictor on fMRI activation, defining a set of predictors which, according to some theoretical expectation, may account for a significant amount of variability among trials produced by known and quantifiable properties of stimuli. Furthermore, the method is useful for (ii) assessing the significance of each single parameter for brain activation (i.e. one-sample t-tests) to the comparison between different models (i.e. statistics for two or more samples), which normally differ only with respect to one out of a set of parametric regressors. With this second type of application, we are able to statistically test hypotheses about the exact form of variation in fMRI activation.
In the following example, we compare the model fit obtained for different numerical compressions of the predictors employed (i.e. logarithmic vs. linear scale). Results of these comparisons may help to determine the exact form of variation and the underlying rate of neuronal response to each of the different stimulus dimensions examined.
An empirical example
Numerical cognition provides a straightforward example for the usefulness of parametric regression. Numbers do naturally differ in their parametric properties, such as, for instance, their magnitude [2
]. Since no number shares the same magnitude with another, naturally there is variation in this dimension within every experimental condition in which different numbers are used. Number magnitude is assumed to be represented in the cortex around the intraparietal sulcus (IPS) [2
]. Behavioral studies [5
] and a neural network model [6
] have indicated that numerical distance is logarithmically compressed. Some recent single-cell recording studies reported that cells in pre-frontal and parietal cortex are tuned to specific magnitudes [7
]; their signal is best described by a logarithmically compressed scale [10
]. Similar results have been obtained in fMRI studies [12
]. Furthermore, studies on two-digit number processing have shown that participants may not be able to compare the magnitude of decade digits while ignoring the unit digits, even when the units are totally irrelevant for the task at hand [see [5
] for a review, [16
Given this theoretical background, we ask two empirical questions about the fMRI signal that can be investigated more precisely by means of parametric than by conventional categorical methods. The first question is whether the fMRI signal in the intraparietal cortex is better accounted for by the overall distance when participants are asked to choose the larger from two two-digit Arabic numbers or by the distance between decade digits. Since there are no two-digit numbers "without" a unit digit to serve as stimuli for a control condition, the only way to examine this problem empirically is to compare the BOLD-response evoked by overall distance with that evoked by decade distance (decade digitlarger number – decade digitsmaller number). If the statistical fit for overall distance is better than for decade distance, one may infer that the fMRI activation in the intraparietal cortex due to the contrast (overall distance > decade distance) is associated with the processing of the overall magnitude of numbers.
A second empirical question is whether the fMRI signal in intraparietal cortex is better accounted for by the logarithm of the distance than by the linear distance between two-digit numbers. This question has been investigated first in an fMRI study by Pinel and colleagues [4
]. These authors found that in six out of seven regions of interest the percent signal change dropped in accord with the logarithm of the distance between numbers rather than with the linear distance. Nevertheless, the authors examined the effect of logarithmic scaling on fMRI signal by splitting the range of distances into three arbitrary categories (i.e. small, medium and large distances) instead of treating distance as a continuum. This approach presents disadvantages in comparison with the modelling with parametric regressors: The method employed by Pinel and colleagues [4
] may fail to distinguish between the impact of decade distance and overall distance on fMRI signal (i.e. the first empirical question examined in the present example). This may have affected the determination of the exact spatial distribution of the neurons responding more strongly to the logarithmically compressed magnitude of numbers.
In the following, we describe the results of the parametric analysis of an fMRI experiment examining the two empirical questions stated above.
Fourteen male right-handed volunteers (mean age = 27, range 21–38 years) took part in the study after giving their written consent to the imaging protocol which has been approved by the local Ethics Committee of the Medical Faculty and is in compliance with the Helsinki Declaration. Participants had to select the larger number of a pair of two-digit Arabic numbers (range: 21–98) and press a key [for further details on the design of experiment and characteristics of the task as well as on behavioural data, see [16
], including supplementary material]. Overall distance, decade distance, unit distance and problem size were matched both absolutely and logarithmically between stimulus categories [16
]. The four digits chosen as units and decades of the two-digit number pair were always different. Furthermore, in the present study unit numbers were totally irrelevant for magnitude comparison since no within decade comparisons were included.
For each participant, a high-resolution T1-weighted anatomical scan was acquired with a Philips 1.5T Gyroscan MRI system (TR = 30 ms, matrix = 256 × 256 mm, 170 slices, voxel size = 0.86 × 0.86 × 2 mm; FOV = 220 mm, TE = 4.6 ms; flip angle = 30°). The anatomical scans were normalized using the standard T1 template of SPM2.
Two functional imaging runs sensitive to blood oxygenation level-dependent (BOLD) contrast were recorded for each participant with a Philips 1.5T Gyroscan MRI system (T2*-weighted echo-planar sequence, TR = 2800 ms; TE = 50 ms; flip angle = 90°; FOV = 220 mm, 64 × 64 matrix; 30 slices, voxel size = 3.4 × 3.4 × 4 mm). In each run, 316 scans + 5 dummy scans were acquired. In a rapid event-related design, a total of 576 trials (480 experimental trials + 96 null events) were presented at a rate of 3 seconds. The fMRI time series was corrected for movement and unwarped in SPM2. Images were resampled every 4-mm and normalized to a standard EPI template using the sinc interpolation method. Moreover, functional images were co-registered with the normalized anatomical pictures. Finally, functional images were smoothed with an 8-mm Gaussian kernel.
We convolved brain activity for all experimental trials with the canonical hemodynamic response function (HRF) in a single experimental condition and defined three parametric regressors representing overall distance, decade distance, and problem size. The correlations between the different parameters and the in-line correlations (i.e. the correlations obtained after convolution with the HRF function) between the parametric regressors and the average hemodynamic response are shown in Table and Table , respectively. In order to scale the estimated regression parameters uniformly, the parametric regressors representing overall distance, decade distance, and problem size were standardized to a mean of 0 and a standard deviation of 1.
Means and correlation matrix for the parametric regressors (n = 240 items, variances in the main diagonal)
In line correlation matrix for the parametric regressors
In order to examine whether the fMRI signal in the intraparietal cortex can be better accounted for by the overall distance than by decade distance alone, we estimated two separate models. In one model, overall distance and problem size were entered as parametric regressors and, in a separate model, decade distance and problem size were entered as parametric regressors. A summary of the procedure for definition, estimation, and statistical assessment of the different parametric models is presented in Table .
Summary of model definition, estimation and statistical comparison using parametric predictors
To avoid the problem of multiple comparisons typical for whole brain analysis when assessing the empirical hypotheses about the amount of signal captured by parametric predictors, small volume analysis was carried out in specific sub-regions of parietal cortex. For the analysis of the regions of interest (ROI), 6 mm-spheres in the left and right parietal cortex were extracted from the brain images using the toolbox MARSBAR. Selection of these ROIs was based on regions showing significant differences in the experimental contrasts in the whole brain analysis.