As it is known that distributions of FA values (and indeed all eigenvalue-based measures (Babamoradi et al. 2013)) deviate from normality (Cascio et al. 2013; Clement-Spychala et al. 2010), robust statistical methods were used throughout. All reported confidence intervals (c.i.) are bias corrected adjusted (bca), based on 1000 bootstrap samples. In respect of correlations, these were converted to z scores before bootstrap resampling (Gorsuch and Lehmann 2010), the bca statistics were weighted by sample size (Karyawati et al. 2020), and the inverse transform then applied.

As a first step we calculated the Kendall rank correlation coefficient, to characterise the ordinal association of FA and streamline length. The magnitude of this coefficient expresses the similarity of the orderings of the two samples. This was done in two ways. In the first, the coefficient was calculated separately for each of the 43 participants. As streamlines were not resolved for all of the 120 tracts, the mean number of tract observations included in each calculation was 82 (95% c.i. 80–83). The mean value of tau – the Kendall correlation coefficient, was then obtained across participants. The mean magnitude of the correlation was 0.35 (95% c.i. 0.30–0.39), corresponding to an effect of “moderate” size. In other words, within individual brains, tracts with longer streamlines are characterised by moderately larger FA values. In the second method of analysis, the coefficient was calculated separately for each of the tracts (e.g., left M1a to right M1a), using the sample of 43 participants. The mean value of tau across all tracts was then derived. In this case, each bootstrapping sample comprised a random selection of tracts (rather than participants). The mean magnitude of the correlation was 0.28 (95% c.i. 0.24–0.31). Thus, when any given tract is considered across different brains, individuals with longer streamlines tend to exhibit larger FA values.

It should not however be assumed that the magnitude of the association between FA (or AFD etc.) and streamline length remains constant across all streamline lengths. A simple model derived from basic assumptions illustrates this point (Fig. 1). We emphasise that this is a statistical, rather than a biophysical, model. It encompasses the reality of magnitude thresholding, whereby tracking is terminated when the value obtained for a voxel falls below a pre-defined threshold. It also includes the empirical observation that estimates of FA diminish progressively towards the ends of a streamline (Zhang et al. 2018), It can further be assumed that these estimates are constrained to a finite range (e.g., 0–1). Given this set of assumptions, it is predicted that average (or median) estimates of some idealised quantitative metric (i.e., for an entire streamline) will increase in a linear fashion with extensions of streamline length (Fig. 1, Panels A to J), up to the length at which the upper limit of the range of possible values (e.g., 1) is reached. For streamlines that exceed this length (Fig. 1, Panels K to T), the average estimate of the metric (for an entire streamline) will increase as a power function with further increments in streamline length (Fig. 1, Panel U). But a key caveat applies. Real biological specimens do not yield values with a magnitude equal to the upper limit of the potential range. Rather, for each specimen (i.e., for each individual brain) there will exist an asymptotic value for the derived estimate (Clark et al. 2001) (Fig. 1, Panel V). At streamline lengths beyond that at which the asymptote is obtained, no further increase in the average estimate of the quantitative metric (FA, AFD etc.), i.e., no dependency on streamline length, is to be expected.

For illustrative purposes, the theoretical estimate of a quantitative metric, such as FA, along a streamline is modelled as increasing with streamline length from a termination magnitude threshold of 0.2, at a rate equivalent to an increase of 1.6 for every change of 100 units of length. It is further assumed that the quantitative metric is constrained to a range of 0 to 1. Given these parameters, the terminal sections (less than 50 units of length) – at both ends of the streamline, are characterised by all values being lower than 1. It is specified that in these regions the slope is constant, and independent of the overall length of the streamline. Within centre (non-terminal) sections, all values are equal to 1 (the theoretical asymptote). Average estimates of the quantitative metric of (i.e., for an entire streamline) will therefore increase in a linear fashion with increases in streamline length (Panels A to J), up to the length at which the upper limit of 1 is reached (i.e., encompassing only the two terminal sections). For streamlines in excess of this length (i.e., also encompassing a centre section) (Panels K to T), the estimate of the quantitative metric will then increase as a power function with further increments in greater streamline length (Panel U). Crucially however, the tissue of real specimens will be characterised by estimates of the quantitative metric (e.g., FA) that are lower than the theoretical average. That is, the values obtained empirically will not continue to increase towards the theoretical limit (1) with increases in streamline length. In the present illustration, a maximum value of 0.7 is assumed. It follows that for this specimen the empirical average will remain 0.7 for all streamline lengths for which the theoretical model predicts a value of 0.7 or above. It would be anticipated therefore that the average estimate of the quantitative metric will increase in a linear fashion until the streamline length at which a value of 0.7 is obtained and remain constant (slope equal to zero) for streamlines in excess of this length (Panel V). For Panels A to T, the length (l) of the streamline and the proportion (p̂) of the streamline comprised of terminal sections (less than 50 units of length) are indicated

In line with this characterisation, non-linear curve fitting using the Blackman (also known as a linear-plateau) function (Archontoulis and Miguez 2015) (Fig. 2) to characterise the relationship between FA and streamline length indicated that in 41 of the 43 participants, the fit to the Blackman function was superior to that achieved using a linear model. It is however an intrinsic feature of the Blackman response that the slope is constrained to be zero at streamline lengths beyond that of the asymptote. To accommodate the possibility that the slope obtained empirically differs from zero, a piecewise linear model can therefore also be applied. The modelling steps described were undertaken in R (R Core Team 2019), through quantile regression, using the rq() and nlrq() functions provided in the “quantreg” package (Version 5.51, Koenker et al. 2018). The SSlinp() function in the “nlraa” package (Version 0.89, Miguez et al. 2021) was used as a self-starter for the coefficients of the Blackman function. Piecewise linear models were fitted using the “segmented” package (Version 1.3–4, Muggeo 2021). Mathematical descriptions of the functions are provided in the Supporting Information.

Theoretical fits to the Blackman (also known as a linear-plateau) function are illustrated. In this example, notional data from three specimens are shown to differ only with respect to the asymptotic value of the derived estimate of a quantitative metric (e.g., FA). Specimen 1 (orange *) exhibits an asymptotic value of 0.72; Specimen 2 (blue ×) exhibits an asymptotic value of 0.80; and Specimen 3 (green +) exhibits an asymptotic value of 0.88. The streamline length corresponding to the start of the plateau region (the point of inflection) varies accordingly. In these examples, the slope of the initial segment is equivalent in each case. Empirical fits derived using the Blackman function can however also vary with respect to the slope of the initial segment. Furthermore, variations in the asymptotic value can covary with the streamline length at which the start of the plateau region occurs (Archontoulis and Miguez 2015)

One of our goals was to derive a means of compensating for the association between streamline length and FA (and other quantitative estimates derived from diffusion tractography). To this end, we employed model-averaging. It has been demonstrated that model-averaged estimators improve precision and reduce bias, relative to estimators derived from a selected “best model” (Burnham and Anderson 2002). When the aim is prediction and the calculation of residuals – which is central to the present undertaking, model averaging is always superior to a best-model strategy (Burnham and Anderson 2004). Recognising that any model is merely an approximation to the truth, Akaike’s information criterion (AIC) was used to quantify the three candidate models (linear, Blackman, and piecewise-linear) in terms of information loss. In practice, the modified version (AICc), that is suitable for small sample sizes, was used. An Akaike weight was then calculated for each model (i.e., separately for each data sample/participant). The Akaike weight takes a value between 0 and 1. Importantly, the weights of all models in a candidate set sum to 1. The Akaike weight is analogous to the probability that any given model is the best approximating model of the data (Symonds and Moussalli 2011). A weighted average of model predictions is obtained by multiplying the predicted values generated by each model by the corresponding Akaike weight, and then summing across the (in this case three) sets of weighted predicted values (Burnham and Anderson 2002; Symonds and Moussalli 2011). A model with a low Akaike weight thus has little influence on prediction (Fig. 3A).

A Empirical fits to each of the three candidate models (linear (red), Blackman (orange), and piecewise-linear (blue), and the model averaged fit (black), are shown for the 84 tracts delineated for a single individual. The Akaike weights for the respective models were as follows: linear – 0.007; Blackman – 0.564; piecewise-linear – 0.270. The point of inflection was determined to be 103.0 mm for the Blackman model (FA = 0.390), 102.0 mm for the piecewise-linear model (FA = 0.383), and 102.7 mm for the averaged model (FA = 0.388). The slope of the initial segment was estimated to be 0.003 for the Blackman model, the piecewise-linear model, and the averaged model. The slope of the second segment was estimated to be 0.0002 for the averaged model. B The FA values, adjusted for the influence of streamline length (through the application of a model averaging approach), are plotted for the same individual. In both panels, the colours are assigned to tracts in the order in which they are listed in the source data. Values corresponding to streamline lengths shorter than the inflection point are assigned closed symbols, and values corresponding to streamline lengths longer than the inflection point are assigned open symbols. In both panels, the horizontal dotted line corresponds to the median FA value of tracts with streamline lengths longer than the inflection point. In each case, the horizontal dashed line corresponds to the median FA value of tracts with streamline lengths shorter than the inflection point

The model-averaging approach can also be applied to the individual parameter estimates generated by candidate models. The Blackman and the piecewise-linear models yield estimates of the inflection point (or “breakpoint”) that marks the termination of the initial segment. In addition, both models provide estimates of the slope of the initial segment, and of the slope of the second segment (which for the Blackman model is equal to zero). Model averaged estimates for these parameters were obtained in the manner described above (the Akaike weights were rescaled such that with the exclusion of the linear model the sum remained equal to 1).

The mean value of the inflection point (i.e., the transition between the two segments for the Blackman and piecewise models) calculated across participants was 102.1 mm (95% c.i. 98.6–104.8 mm). The confidence interval (c.i. = 6.2 mm) obtained for the inflection point was considerably smaller than that of the range of streamline lengths (c.i. = 26.0 mm) observed across participants. The mean value of the FA obtained at the inflection point was 0.45 (95% c.i. 0.44–0.46). The slope of the first segment was 0.0041 (95% c.i. 0.0034–0.0050). In other words, each 10 mm increase in streamline length was associated with an increase in FA of approximately 0.04. This indicates that over the range of streamline values spanned by the initial segment (i.e., up to approximately 102 mm), the FA estimate covaries with the length of the streamline for which it was derived. Consistent with this supposition, the Kendall tau values of correlations between streamline length and FA, calculated for streamlines shorter than that of the inflection point (derived separately for each individual) corresponded to large effect sizes (mean = 0.51, 95% c.i. 0.44–0.55). It is very clearly the case therefore that below the inflection point, FA estimates are in linear association with streamline length (Fig. 4).

A summary representation of the results obtained through the application of a model averaging approach to the predicted FA values generated by the three candidate models (linear, Blackman, and piecewise-linear), for the 43 participants included in Ruddy et al. (2017). For each participant, the models were evaluated, and the predictions weighted and averaged, at a range of nominal streamline lengths (at 1 mm intervals). This range spanned the median minimum streamline length and the median maximum streamline length observed across the 43 participants. The solid line corresponds to the means of the weighted, averaged, predicted values derived from 1000 bootstrapped samples. The dashed line was generated using the lower 95% confidence interval of the bootstrapped samples. The dotted line was generated using the upper 95% confidence interval. It is apparent that, for streamlines shorter than approximately 100 mm, there is an association between FA and streamline length

In contrast, the mean slope of the second segment (Fig. 4) did not differ reliably from zero (mean =  – 0.0006, 95% c.i.  – 0.0014 to 0.00008). In this case, a 10 mm increase in streamline length was associated with a decrease in FA of 0.006. The correlation between streamline length and FA in this region was similarly weak (mean =  – 0.09, 95% c.i.  – 0.15 to  – 0.013).

The model-averaged (predicted) values represent the optimal fit to the relationship between streamline length and the quantitative metric (in this case FA), at the values of streamline length present in the data available for each individual participant. The residuals (Fig. 3B) correspond to the variations in the quantitative metric that are not predicted by streamline length. A positive value of the residual indicates that, for a given tract, the estimate is larger than that which would be predicted by streamline length alone – to an extent corresponding to its magnitude. A negative residual value for a tract indicates an estimate (e.g., of FA) smaller than would be predicted for its streamline length. The mean magnitude of the Kendall correlation between the residuals and streamline length, when calculated across participants, was  – 0.009 95% (c.i.  – 0.022 to 0.005). Calculated across tracts also, the correlation between the residuals and streamline length did not differ reliably from zero (mean =  – 0.003, 95% c.i.  – 0.054 to 0.045). The model averaging approach was therefore successful in compensating for the association between streamline length and FA.

The use of the residuals in inferential analyses may prove to be sufficient in circumstances in which the research question can be addressed by dealing with variations in the relative magnitude of the quantitative metric obtained for a defined set of tracts within each brain. There are however many instances in which it is desirable to compare the estimates (e.g., of FA) obtained for a specific tract, across groups of individuals, or to examine changes that occur in individuals over time. To address this requirement, it is necessary that the residuals are expressed relative to an appropriate reference value. The model-averaging approach affords a means of addressing this requirement. Specifically, the model averaged estimate of FA (or AFD etc.) at the inflection point provides an appropriate, individual specific, reference value. The magnitudes of the estimates obtained in the present analysis (median of 0.45 for FA across all individuals) have face validity. In so much as the fitted values of the model do not tend to vary appreciably over the range of streamline lengths extending beyond the inflection point (as indicated by the negligible slopes of the second segment), the model averaged estimate at the inflection point can reasonably be taken as an asymptotic value. Thus, it also has construct validity. It is therefore proposed that the adjusted (for streamline length) measure be obtained (i.e., separately for each brain) as the sum of the model averaged estimate of FA (or AFD etc.) at the inflection point, and the residual derived for each tract.

The adjusted values derived in the manner described, differ systematically from the original FA values that are analysed routinely in tractography studies. Necessarily the magnitude of the difference varies as a function of streamline length (Fig. 5). Estimates generated for tracts with the shortest streamlines (typically < 70 mm) are characterised by adjusted FA values that exceed the original values by more than 0.1. In respect of tracts with streamline lengths greater than the mean inflection point of 102 mm (estimated across participants), the magnitude of the difference tends to be smaller, and frequently cannot be distinguished reliably from zero (Fig. 5).

Separately for each of the 43 participants included in Ruddy et al. (2017), and for each tract, the difference between the original FA value, and the FA value adjusted for the influence of streamline length (through the application of a model averaging approach) was calculated. The filled symbols correspond to the means of these difference values, when derived from 1000 bootstrapped samples drawn from the set of 43 participants (i.e., calculated separately for each tract). For a tract to be included, it was necessary that at least half of the participants must have contributed data (i.e., streamlines were resolved). The error bars correspond to the 95% confidence intervals of the bootstrapped samples. The tracts are plotted in order of mean streamline length (calculated across participants)

The broader implications of the distortions in FA values that arise from streamline length dependence are readily demonstrated. The panel on the left of Fig. 6 illustrates the rank ordering of unadjusted FA values obtained for streamlines (n = 26 tracts) that originate and terminate within the right cerebral hemisphere. The panel on the right of Fig. 6 illustrates the rank ordering of FA values that have been adjusted for streamline length in the manner described above. It is apparent that the rank ordering of the adjusted values is dramatically different from that of the unadjusted values. This is reflected in a correlation of the rankings for the unadjusted and unadjusted FA values that is markedly lower than unity (Kendall’s tau = 0.34).

The left portion of the figure (“Unadjusted FA values”) displays the rank ordering of unadjusted FA values obtained for streamlines (n = 26 tracts) that originate and terminate within the right cerebral hemisphere. The right portion of the figure (“Adjusted FA values”) displays the rank ordering of FA values that have been adjusted for streamline length in the manner described in the text. The size of each symbol (in the legend shown in the range 0.30–0.55) corresponds to the associated FA value (i.e., derived for that tract). The position of each symbol in relation to the y axis scale corresponds to the ranking of the FA value with respect to the set of 26 tracts. Those with a higher ranking (larger FA value) appear above those with a lower ranking (smaller FA value). While the assignation of fill colour to the individual tracts is arbitrary, as it remains consistent across panels A and B, it aids in the identification of differences in ranking (i.e., unadjusted versus adjusted). In both panels, the colours (which do not relate to FA value) are assigned to tracts in the order in which they are listed in the source data. The x axis is categorical. Tracts are plotted with respect to the x axis in the order in which they are listed in the source data. It is apparent that the rank ordering of the adjusted values (Panel B) is dramatically different from that of the unadjusted values (Panel A). M1a anterior primary motor cortex, M1p posterior and primary motor cortex, PMd dorsal premotor cortex, PMv and ventral premotor cortex, SMA proper—supplementary motor area proper, pre-SMA pre-supplementary motor area, S1- primary sensory cortex, CMA cingulate motor area. Three exemplars are highlighted via arrows between the unadjusted and adjusted values. The adjusted ranking (and FA value) for the S1-SMA tract is markedly higher than the unadjusted ranking (and value). The adjusted ranking for the CMA-M1a tract is lower than the unadjusted ranking. In respect of the PMd-preSMA tract, the ranking of the adjusted and unadjusted FA values is the same

It is not difficult to envisage specific research questions for which the outcomes may depend on whether the data have first been adjusted to compensate for the association with streamline length. For example, an investigator may wish to determine whether tracts that originate and terminate within one hemisphere differ from “inter-hemispheric” tracts, with respect to a metric such as FA. For the data set described, the central tendency (derived using trimmed means) of the unadjusted FA values (Fig. 7, Panel A) obtained for “intra-hemispheric” tracts within the right hemisphere is 0.404, whereas, for “inter-hemispheric” tracts connecting the left and right hemispheres, the corresponding value is 0.427 (difference =  – 0.022, 95% c.i.  – 0.031 to  – 0.015). The outcomes of a robust inferential test (Yuen’s t) appropriate for paired observations (i.e., intra-hemispheric versus inter-hemispheric) indicate that this is a large (in terms of standardised effect size) and reliable effect (t(26) =  – 5.17, p = 2.13E-05, δt =  – 0.59, 95% c.i.  – 0.81 to  – 0.39). In marked contrast, the difference between the adjusted FA values (Fig. 7, Panel B) obtained for the intra-hemispheric (0.448) and inter-hemispheric (0.453) tracts is – 0.004 (95% c.i.  – 0.009 – 0.0008). The associated inferential test (t(26) =  – 1.29, p = 0.175, δt =  – 0.08, 95% c.i.  – 0.22 to 0.01) supports the conclusion that, when adjusted for streamline length, the FA values obtained for streamlines defined for intra-hemispheric tracts do not differ reliably from those defined for inter-hemispheric tracts.

The trimmed mean (trimming = 20%) of the FA values derived for all inter-hemispheric tracts for which streamlines were resolved for each participant, and the trimmed mean of the FA values derived for all right hemisphere intra-hemispheric tracts for which streamlines were resolved for each participant, were calculated. Unadjusted FA values are shown in Panel A. FA values that have been adjusted for streamline length in the manner described in the text are shown in Panel B. Each point represents the data for a single participant, with the colour coding determined by the order in which the 43 participants are listed in the source data. In both panels, the x coordinate for each point corresponds to the trimmed mean FA value for the right hemisphere tracts, and the y coordinate corresponds to the trimmed mean FA value for the inter-hemispheric tracts. Points lying close to the line of equality indicate similarity in the FA values obtained for inter-hemispheric and right hemisphere streamlines. Points lying below the line of equality indicate FA values for the right hemisphere streamlines that are lower than those for the inter-hemispheric streamlines. Comparison of the plots generated for the unadjusted data (Panel A) and the adjusted data (Panel B) makes apparent that differences between right hemisphere and inter-hemispheric FA values present for the unadjusted data, are not apparent for the adjusted data. The results of corresponding inferential tests are reported in the text

The model-averaging approach described is based on determining the optimal fit to the relationship between streamline length and the quantitative metric, at the values of streamline length present in the data available for each person. It is also possible to model the relationship between streamline length and a quantitative metric (e.g., FA), using the data available for each tract. In many instances however, this approach will be insufficient to provide an adequate characterisation of the relationship. This is because the extent of the variation across individuals in the length of the streamlines identified for a specific tract, will typically be much smaller than the range of the streamline lengths that characterise the entire set of tracts defined for each person. For the data considered herein, the mean (i.e., across 43 individuals) of the range of streamline lengths detected for all tracts (defined using the cortical motor network atlas) was 106.4 mm. In contrast, the mean for all tracts, of the range of streamline lengths obtained for each tract, was 63.9 mm. Necessarily also, the range of streamline lengths obtained (across persons) for a given tract will span only a subset of the range of streamline lengths defined for any given person.

The impact of these particulars on the data modelling is illustrated by Fig. 8. In Panel A, the left cingulate motor area to left dorsal premotor cortex tract (PMd) is represented. For 39 of the 43 individuals, the length of the streamlines defined for this tract was shorter than the point of inflection defined for the FA values (102.1 mm). For this range of streamline lengths, the relationship with FA is predominantly linear. In Panel B, the left CMA to right anterior primary motor cortex (M1a) tract is represented. For 18 of the 42 individuals for whom streamlines were detected, these were shorter than the FA point of inflection. Whereas, for 24 individuals, the streamlines were longer than the inflection point. In this case, a piecewise linear relationship is apparent. In Panel C, the left PMd to right PMd tract is shown. For all 43 individuals, the lengths of the streamlines detected for this tract exceeded the point of inflection defined for FA. With respect to this tract, there is no apparent relationship between streamline length and FA. These examples serve to highlight that, an adequate characterisation of the relationship that exists between streamline length and this tractography-derived metric (i.e., FA), demands a range of streamline lengths beyond that which is available for a single tract.

Empirical fits to each of the three candidate models (linear (red), Blackman (orange), and piecewise-linear (blue), and the model averaged fit (black), are shown for three example tracts. Each point represents the data for a single participant, with the colour coding determined by the order in which the 43 participants are listed in the source data. A. Between left cingulate motor area (CMA) and left dorsal premotor cortex (PMd). B Between left CMA and right anterior primary motor cortex (M1a). C Between left PMd and right PMd