More statistics

Figure 1

Having put in a plea for serious statisticians at reference 1 a couple of weeks ago, in the course of perusing the helpful and accessible reference 2, I came across the figure above, presenting some changes in brain content with age, in humans. One of the points at issue being whether certain regions went up or down with age and whether such changes were linear, monotonic, quadratic or what. A question which is much easier to answer in these days of MRI – say from the year 2000 – than it would have been when one would have had to dissect a large number of brains, assuming a sufficient and disposable supply of same. Also a question which clearly needs statistical input.

Being keen on taking my social statistics raw and being suspicious of wheezes like seasonal adjustment, I was puzzled by what might be meant by standardized residuals and I was tempted to dig. This led me to reference 3, bottom right in the figure above, a response to references 4 and 5, all three papers appearing in Elsevier’s journal ‘Neurobiology of Aging’, to be found at reference 6. A transformative journal, whatever that might mean.

The data plotted in reference 3, for the purpose of giving a better span of age, is drawn from two previous studies, which added together give N=113. While the comparator at reference 4 has N=73 and that at reference 5 has N=87. So quite decent sized samples compared with some involving scanning brains that one comes across.

However, I seem to have missed something, as the two studies which have been added together – references 2 and 5 in the Jernigan and Gamst  paper at reference 3 – cover the age ranges 5-15 (N=35) and 30-100 (N=78), with a gap in the range 15-30 – while the scatterplots presented show no such gap. I tried counting the dots in Figure 2 below, getting first to around 140 and second to around 130, with the former count including a few dots which seemed to be under the lines. So both counts well in excess of the 113 expected, but well short of adding everything together too. A disturbing failure of understanding.

Nevertheless, the general idea seems to be to take various measurements of brains from a sample which gives reasonable coverage of age, and then to see how those measurements change with age across the sample. That is to say, just one set of measurements from any one subject, with all the measurements being taken at roughly the same time: an attempt to derive understanding of what happens over time from a snapshot in time. All in support of better describing, better modelling of, the processes underlying brain cell birth, migration, growth and death. 

It’s not been spelt out, but I now guess that the variable in question, say TV1, the volume of the thalamus, has been linearly regressed against CV, the cranial volume. We then have it that TV1 = α × CV + β + TV2, where α and β are the regression coefficients that have been calculated, either from the sample of heads to hand, or perhaps from some large sample, perhaps from some digital library of same, and TV2 is the residual, that is to say what is plotted on the vertical axis of the second panel from the left in the figure above. That part of the volume of the thalamus which is not accounted for by the volume of the host brain. I also guess that residuals have been adjusted to have means of zero, presumably not what happens anyway when one does a least squares regression. The point being to validate, to improve comparisons between subjects with different brain sizes – with the assumption here being that the size of a well defined region of the brain is not of some fixed size, doing some fixed job, but with a size which will vary with the size of the host brain. 

Aside, a quick Bing turns up reference 7, which suggests that while the size of the eye does indeed vary, I did not spot any suggestion that this variation is linked to height or brain size, although there is some correlation with host orbit size. And from my dental background, I recall that size of teeth is not tied to size of jaw – a failing which keeps orthodontists in business.

This residual proceeding reduced the scatter and reduced, if not eliminated, any difference between the sexes. And if we then do a quadratic rather than a linear fit we get some nice curves. Maybe these relationships are not linear after all.

Figure 2

Figure 2 is the original from reference 3 from which the right hand panel of Figure 1, cerebral white matter, is derived. The dots are the sample points, the straight line is the linear regression, the dashed line both the best fit quadratic curve and the dot-dash line the best fit cubic curve – part of the point being that the cubics do not seem to add much to the quadratics.

But even to the eye, this scatterplot does not look to reflect a linear relationship underlying all the noise; there is a hump. From which the authors quite reasonably deduce that whatever is going on is more complicated than simple decay of white matter with age – which is the story one might tell if one only looked at ages 30 or 40 to 100, rather than the 0 to 100 we actually have here.

Figure 3

Figure 3 is another of the other diagrams in reference 3. Again, the dots are the sample points, the straight line is the linear regression, the curved line both the best fit quadratic and cubic curves – this time, more or less identical. But there is another hump, albeit not as strong as the first.

The question for me being, given the scatter, given all the manipulations that have got us from the underlying MRI scans to that scatter, how sure one can be sure about this? And how can one be sure that the three left hand panels of Figure 1 have not levelled off in middle age, with the steady decline into old age being no more than a mathematical artefact of fitting a quadratic to a scatter plot which starts high?

There is talk in reference 3 of Spearman's rank correlation coefficient or Spearman's ρ, which Wikipedia tells me is about testing whether a relation between two variables can be described as monotonic. Completely monotonic will give a value for ρ of plus or minus 1. But I did not spot anything stronger than the observation that quadratic gave a better fit than linear – not that I would have understood a statistical argument. And the paper ends on a properly cautious note.

While reference 2 goes no further than observing that the graphs in Figure 1 above ‘illustrate that during childhood and adolescence changes in brain structure are at least as dramatic as those at the end of life’. And noting that ‘all volumes are normalized for cranial volume –which does not change appreciably over this age range’. 

As noted above, this last is what makes it possible to compare one subject at one age with another subject at another age, assuming, as it were, that other things are equal. But maybe one day there will be enough data in digital libraries to do longitudinal studies, which look at the same subjects over a period of time, which would probably provide stronger evidence. Maybe that has already been done, given that the work reported here was done between 15 and 20 years ago.

Conclusions

Enthusiasm in drawing conclusions from nice curves which have been superimposed on scatter plots needs to be restrained by a bit of statistical common sense. Better still, apply some statistical skills.

PS: clicking on the figures should make any small print visible.

References

Reference 1: https://psmv4.blogspot.com/2021/07/teach-myself-all-about-fmri.html. 

Reference 2: The Basics of Brain Development - Joan Stiles and Terry L. Jernigan – 2010.

Reference 3: Changes in volume with age: consistency and interpretation of observed effects – Terry L. Jernigan, Anthony C. Gamst – 2005. 

Reference 4: Effects of age on volumes of cortex, white matter and subcortical structures – Kristine B. Walhovd, Anders M. Fjell, Ivar Reinvang, Arvid Lundervold, Anders M. Dale, Dag E. Eilertsen, Brian T. Quinn, David Salat, Nikos Makris, Bruce Fisch – 2005. N=73.

Reference 5: Normal neuroanatomical variation due to age: The major lobes and a parcellation of the temporal region - John S. Allen, Joel Bruss, C. Kice Brown, Hanna Damasio – 2005. N=87.

Reference 6: https://www.journals.elsevier.com/neurobiology-of-aging. 

Reference 7: Variations in Eyeball Diameters of the Healthy Adults – Inessa Bekerman, Paul Gottlieb, Michael Vaiman – 2014.