CSF β-amyloid and Tau peptides showed distinctive non-linear relationships with cognitive scores. These relationships were strongest in the "normal" CAMCOG range and every biomarker's change-point was near the lower boundary of this range. These non-linearities are inconsistent with a simple linear relationship between central amyloid and Tau levels and cognitive impairment which underlies the view that modulating central amyloid and Tau may prevent cognitive decline in dementia [37, 38]. Instead, they suggest that most changes in central amyloid and Tau precede the development of clinical dementia .
Reliability of our findings
The findings of our change-point analyses are consistent with published data. First, our re-analysis of data shown in Figure 2 in Maruyama  shows a change-point in the relationship between MMSE scores and CSF Aβ1-42. This resembles our present finding (see our Figure 2) in that there is no relation between MMSE and Aβ1-42 across MMSE scores below 20 (t = 0.21, NS), but a significant relation across MMSE scores above 20 (t = 2.08, 76 df, P = 0.041). Second, our findings that relations between CSF biomarkers and CAMCOG were weak or absent below the change-points parallel the observations of Vemuri  in AD patients; the stronger relations that we found above the change-points may also parallel Vemuri's report  that biomarker levels differed in their Controls and people with MCI. Third, our finding of an inverted-U relation between Aβ1-40 and CAMCOG scores parallels the report that Aβ1-40 levels were high in MCI patients who progressed to AD . Additional factors that strengthen our change-point results are (a) the strong resemblance between the change-point-based robust regression models and the model-free Lowess fits (see Figures 1, 2, 3, 4); (b) the similarity of the change-points (in absolute CAMCOG scores) over all four biomarkers, despite the different forms of each biomarker's relation with cognitive function; (c) the significant differences between the variability and overall levels of the biomarkers on each side of the change-points. Overall, therefore, the change-points we describe are unlikely to be artefactual and relations between CSF biomarkers and cognition are probably non-linear.
The change-points that we defined are near the bottom of the range of CAMCOG scores in our non-demented volunteers. Hence, our findings are consistent with many previous reports of higher levels of CSF Tau moieties and lower levels of CSF amyloid moieties in AD [22, 25, 41, 42]. However, we prefer our change-point categories to the diagnostic classification because they derive from a model-free method, whereas the definition of Alzheimer's disease can vary [43, 44]. Moreover, our results indicate that neither the diagnostic classification, nor the diagnostically agnostic linear regression fully describes the relations between cognitive function and CSF biomarkers. Figures 1, 2, 3, 4 each show the four models that assume that the relations between CSF biomarker levels and cognitive function are either (a) simply linear (green dotted lines) or (b) simply categorical (blue dot-dash lines). In each case these simple relations fit the data quite well, but the non-linear relations based on the change-point analyses (c - black solid lines) fit better and more closely resemble the model-free robust Lowess fits (d - red dashed lines). These improvements in fit are not large, but their theoretical importance outweighs their actual size, as we discuss below.
CSF Aβ1-40 showed biphasic dependence on cognitive scores (Figure 1), with maximal levels near the lower limit of our non-demented participants' CAMCOG scores. This is consistent with the findings that MCI patients who progressed to AD had high CSF Aβ1-40  and CSF BACE activity is higher in MCI than in either controls or AD patients . It also fits with findings of biphasic changes in plasma amyloid levels . Zhong et al.  interpreted their findings as supporting the amyloid cascade hypothesis. However, the biphasic relationship of Aβ1-40 admits at least two other overlapping interpretations. First, it is consistent with Combrinck's hypothesis  that a pathogenic mechanism may 'burn out' early in AD. Second, it is consistent with the hypothesis that Aβ may be protective [45, 46], and AD occurs only when pathogenic mechanisms overwhelm the amyloid response. Further studies are necessary to explore these three possibilities.
CSF Aβ1-42 showed an overall inverse relationship with the degree of cognitive impairment (Figure 2). This fits with many reports of low Aβ1-42 in AD patients [40–42]. Our findings extend those reports by showing that Aβ1-42 depends on cognitive level only in the range of non-demented volunteers' CAMCOG scores, not in the patients' range. The absence of any relation between CSF Aβ1-42 and CAMCOG scores below the change-point (in the patient range) mirrors the recent report of Vemuri et al. . Together, these results indicate that low Aβ1-42 may provide an early marker of likely progression to AD, rather than an index of the severity of pathology in established AD. The contrast between the monotonic relation of Aβ1-42 with cognitive level and the biphasic relation of Aβ1-40 is striking (compare Figures 1 and 2). This contrast fits with observations that a low ratio of Aβ1-42/Aβ1-40 associates with imminent risk of mild cognitive impairment and Alzheimer's disease [47–49]. Together, these findings highlight a need for further studies to explain why the ratio of Aβ1-42/Aβ1-40 may vary .
CSF Tau and phospho-Tau
We found high CSF Tau moieties in AD patients . Our results extend earlier reports by showing that, like Aβ1-42, Tau and phospho-Tau levels relate to CAMCOG mainly above the change-point (in the non-demented range of scores). Again, paralleling Vemuri et al.'s report , we found no simple relation between CSF phospho-Tau and CAMCOG scores below the change-point, in the patient range (Figures 3, 4). Together, these results reinforce the view that phospho-Tau may provide an early marker of likely progression to AD, but not an index of the severity of pathology in established AD. The findings that Aβ1-42 and phospho-Tau show opposite relations with cognitive scores within the non-demented range fits with reports that the ratio of Aβ1-42/Tau may be a sensitive indicator of progression to AD [41, 52, 53].
Heteroscedasticity and measurement technicalities
The variances of the CSF biomarkers differed above and below their change-points. The simplest explanation of this is that the variance related to the mean, as often occurs in biological variables. It may also represent individual differences , or variations in pathogenic processes or cognitive profiles . For example, the heteroscedasticity may possibly relate to ApoE status. We did not test relations of ApoE alleles with change-points, since we had no a priori hypothesis about this. Further studies should address this potentially important possibility. Whatever its explanation, the heteroscedasticity in every biomarker indicates a need for caution when interpreting biomarker levels in patient and control groups, or in individuals . For now, we note that our use of highly robust linear modelling with a breakdown point of 0.5  guards against potential bias in our analyses due to heteroscedasticity.
We collected CSF samples in polystyrene tubes and stored them in polypropylene tubes. Both kinds of tube can adsorb large amounts of biomarker molecules [57, 58]. If such adsorption saturates asymptotically, and so depends non-linearly on initial biomarker concentration, then this might possibly contribute to both the heteroscedasticity and the non-linear relations between biomarker levels and cognitive function that we found. However, we think this unlikely, as follows. If adsorption of biomarkers tends to cause floor effects in their apparent levels, then cognitive function might relate to biomarker levels only when they exceed this artefactual floor. Such floor effects could possibly explain our finding that CAMCOG scores did not relate to Aβ1-42 in the patient range of CAMCOG scores (because Aβ1-42 levels are lowest in this range and so most susceptible to adsorption-mediated floor effects). However, it cannot simultaneously explain the inverted-U biphasic relation of CAMCOG with Aβ1-40; nor can it explain the direct relationships of the CAMCOG scores with tau and p-tau in the non-demented range of CAMCOG scores (where lower Tau/p-Tau levels are potentially more susceptible to adsorption-mediated floor effects). Even for Aβ1-42, this account appears tenuous, in view of Bjerke's observation that detergent treatment released similar percentages of Aβ1-42 from adsorption in both control and patient samples . In summary, we think it unlikely that measurement technicalities can account for the non-linear dependence of CSF biomarkers on cognitive function that we observed.
We related CSF biomarker levels to raw CAMCOG scores. Hence, apparent non-linearities in the relationships may in part reflect non-linearities in the metric properties of the CAMCOG (for example ). In particular, the narrow range of CAMCOG scores above the change-points may contribute to the differences in the slopes of their relations with biomarkers, since CAMCOG scores here may relate less strongly to true cognitive ability. However, this cannot account for (a) the inverted-U relation of Aβ1-40 with CAMCOG scores; nor for (b) the differences in Spearman's rank correlation coefficients (which is independent of the metric) for biomarkers and CAMCOG scores above and below the change-points; nor for (c) the differences in variance of biomarkers above and below the change-points. Therefore, while the slopes of the relationships that we found on each side of the change-points may vary under non-linear transformations of the CAMCOG scores, it seems unlikely that our use of raw CAMCOG scores can account for the existence of the change-points.
Use of CAMCOG scores as metric for dementia
The main limitation of our change-point analyses is that they used only cross-sectional data. Their use of CAMCOG scores as the metric for dementia removes time from the analysis of progression. The absolute cognitive level is clinically meaningful regardless of age or the duration of symptoms. Therefore, using it as the metric for progression of pathological mechanisms may be preferable to using time. We know that individuals' CAMCOG scores can decline over time through the normal and patient ranges . Hence, it is tempting to view the cross-sectional dependence of CSF biomarkers on CAMCOG scores as a model of an individual's likely progression over time. However, the heteroscedasticity that we observed (see above) means that individuals might show important variations from this model. Consequently, it would be inappropriate at this stage to conclude that the non-linear cross-sectional relationships that we observed can indicate which non-demented people will progress to AD, or which AD patients will decline faster. Conversely, the cognitive stability of many non-demented participants implies that the cross-sectional relations of CSF biomarkers with their cognitive function may index long-term adaptations to factors that pre-dispose to AD . Alternatively, these cross-sectional relationships may reflect Vemuri et al.'s  observation that levels of biomarkers differed between controls and MCI groups, since we did not distinguish these. MCI may be stable, remit, or progress to AD [17, 18]. Hence, the cross-sectional relations of biomarkers with CAMCOG scores in the non-demented range may reflect a mix of long-term adaptations and of vulnerabilities to progression. Further longitudinal studies relating CSF biomarker levels to cognitive function are necessary to define more precisely the links between CSF biomarkers and the putative primary pathogenic processes of AD.
The generalizabilty of our study may be comparable with other reports of CSF biomarkers. Participants in all such studies are partly self-selected, both for entry to the cohort and for consenting to LP. OPTIMA is a convenience cohort from a relatively small geographical area in and around Oxford. Our study group was relatively homogeneous and most non-demented volunteers were cognitively stable, with few converting to AD, despite our long follow-up. Together, these two considerations indicate that our study group is unlikely to be representative of the general population. Even so, the consistency of our results with previous reports, both with regard to differences in CSF biomarkers between patients and non-demented controls and to non-linearity [cf. [21–24]], even in a Japanese sample , implies that our findings may reflect general phenomena. Two further aspects of our study may improve its generalizability. First, we confirmed the diagnosis of AD and excluded non-Alzheimer dementias via neuropathological examination in most patients (though people who consent to autopsy are non-representative of the general population ). Second, we related CSF biomarker levels directly to cognitive scores. Neuropathological designations and cognitive test scores may provide a firmer basis for generalization than clinical diagnoses, whose boundaries are uncertain [16–18, 43, 44]. Overall, then, our study compares favourably with other reports in this field.