The signal and the noise: inherent challenges for isotopic studies in bioarchaeology Tamsin C. O’Connell Department of Archaeology, University of Cambridge, Downing Street, CB2 3DZ, UK A R T I C L E I N F O Keywords: Carbon Nitrogen Oxygen Strontium Isotope Diet Migration Mobility Geolocation Mixing models Uncertainty A B S T R A C T Isotopic analysis as a method of assessing diet or geographical origin is now ubiquitous in archaeology, to the point where seemingly no project is complete without it. Yet despite its prevalence, it is not a straightforward technique that provides a simple answer. I argue that many researchers overlook the fact that the situation is rarely clear-cut, with many contributing factors, that each situation is usually complex and can vary depending on the nature of each study or sample. This can lead to interpretations that are overly simple or at a higher degree of precision than are warranted by the data. In this paper I outline some of the issues that confront us as we try to unravel the tangled web that is isotopic patterning in consumer data, in particular the factors that determine the limits of the technique’s resolution, and therefore our interpretations. I identify some points for best practice, and then discuss in more detail areas that I think need overt attention at the level of each and every study: confidence in specimen and data integrity; considerations of analytical scope and scale; data interrogation. 1. Introduction Isotopic analyses of biological materials are now widely used across the full temporal and global sweep of archaeology. Because we analyse tissues or biochemical fractions from an organism (be it human, sheep, bird, mollusc), we can obtain information about the individual as well as the group, a perspective not often available in archaeology. Studies using carbon, nitrogen, oxygen and strontium, as well as sulfur and hydrogen, have yielded information about diet and mobility, as well as insight into invisible aspects of past lives – quotidian activities, lifetime practices, identity, group dynamics, community structure, life histories. From its transformative beginnings (Vogel and van der Merwe, 1977; Tauber, 1981), the technique has become routine in application, but also somewhat routine in terms of the information yielded. The ease of specimen preparation and increased prevalence of isotope ratio mass spectrometers has contributed to its rapid growth. Yet despite its ease of execution, it is not a cut-and-dried technique, and data interpretation can be complex. Greater use by specialists and non-specialists has resulted in studies that range from excellent to dubious, from ground-breaking to mundane. When I read studies that make mistakes, oversimplify interpretations, overlook established relationships or display ignorance of primary literature, it makes me frustrated. It also leads me to question how have we got to this juncture – why has a field that is a success story and a growth area in archaeological science reached something of a stasis? 1.1. “A little learning is a dang’rous thing”1 In his 1973 paper, “The Loss of Innocence”, Clarke writes of the impact of expanding consciousness on a discipline. He outlines the process of a discipline’s development, identifying “significant thresholds in the transitions from consciousness through self-consciousness to critical self-consciousness and beyond” (emphasis in the original, Clarke, 1973). My view is that relatively few of us who work with isotopic analyses are critically self-conscious – who have embraced the unnerving feeling of “look how little we know and how inappropriate are our models and explanations” and sought to go beyond (Clarke, 1973). And thus I fear that we have a division in the field of archaeological isotopic analysis, between those who are (self-)conscious yet naïve and those who are critically self-conscious. This division can be drawn between those who treat the technique as providing a simple answer to a question and those who recognise that This article is part of a special issue entitled: Next generation archaeological science published in Journal of Archaeological Science. E-mail address: tco21@cam.ac.uk. 1 Alexander Pope, 1711, “An Essay on Criticism”, line 215. Contents lists available at ScienceDirect Journal of Archaeological Science journal homepage: www.elsevier.com/locate/jas https://doi.org/10.1016/j.jas.2025.106379 Received 31 May 2025; Received in revised form 8 September 2025; Accepted 15 September 2025 Journal of Archaeological Science 183 (2025) 106379 Available online 24 September 2025 0305-4403/© 2025 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ). https://orcid.org/0000-0002-4744-0332 https://orcid.org/0000-0002-4744-0332 mailto:tco21@cam.ac.uk www.sciencedirect.com/science/journal/03054403 https://www.elsevier.com/locate/jas https://doi.org/10.1016/j.jas.2025.106379 https://doi.org/10.1016/j.jas.2025.106379 http://creativecommons.org/licenses/by/4.0/ biological isotopic data are proxies for complicated processes – between those who think they understand the technique and those who really do have a solid grasp of the technique and its limitations. I suspect that this is a fairly inevitable outcome of the field’s rapid rate of expansion – there are still relatively few experts who appreciate what we do and don’t know, what are the inherent challenges, the outstanding unanswered questions and as-yet untested assumptions. Limited depth of knowledge within practitioners—as well as a lack of scientific rigour—is responsible for many of the weak isotopic studies in archaeology (see discussions in Makarewicz and Sealy, 2015). Further more, the growth of the published corpus across multiple disciplines (including ecology and geochemistry) means that few researchers can have a fully comprehensive grasp of the literature. No-one can read every isotope paper any more, but it is essential that we are all aware of some key ones – particularly foundational work that addresses basic assumptions and challenges in the field. Isotopic analysis does not give us a simple ‘answer’ about diet or migration or mobility. Just as John Evans said that we “cannot dig a hole in a peat bog and find an ancient climate” (p.96 in Evans, 2003), so we cannot measure someone’s isotopic ratios and find their diet or place of birth – we analyse a proxy and then interpret our data to arrive at our findings. From the technique’s perspective, simplicity of execution does not equate to simplicity of interpretation. From the researcher’s perspective, competence in execution does not equate to competence in interpretation. Recognition of this is fundamental in the transition to critical self-consciousness, which is essential in order to make appro priate and best use of isotopic data in archaeology (and more broadly). 1.2. This paper In this paper, I reflect on the state of the field, and consider what we can do to progress. I review the fundamentals of isotopic analysis, and in doing so identify the factors that determine the limits of our in terpretations. I then consider areas where I think we could improve or where challenges remain, around questions of specimen and data integrity, scope and scale, and data interrogation. It is a personal view, and others may see different priorities for the field. Yet I address themes that I know are of concern to others within archaeology, geochemistry and related fields. Predominantly, this paper focuses on isotopic studies of consumers (humans and other animals), but some of what I say ap plies to other biological materials, including plants and food residues. I specifically identify what I perceive as shortcomings or mistakes that are commonly made in bioarchaeological isotopic studies, recog nising that this may be confronting for some, in the hope that this pro vokes constructive reflection and debate. We have all done things that with hindsight we would now do differently – I know that I have when I look back at my work over the last 30+ years. What is critical is that we all reflect, learn and improve, that we subject ourselves to “vigorous self- scrutiny” (Gould, 1988), and strive to do our work as thoughtfully and as well as we can. I hope that what I have to say is relevant across the spectrum of re searchers. For the more expert and experienced isotopists, I aim to be thought-provoking – inviting them to reflect on how we can develop the field and disseminate best practice. For the less expert readers, be they users or next generation scholars, I aim to be both thought-provoking and informative – enabling them to increase the depth and breadth of their understanding in order to make full and effective use of the methods and concepts. 2. The technique The technique relies on two key underlying premises. Firstly, that isotopic ratios of many chemical elements vary across different pools and reservoirs in the biosphere, and therefore that different types of foods and waters may be isotopically varied across ecosystems and time. Secondly, that because an individual’s body tissues are synthesised from the food and water that they ingest, their body tissues are isotopically linked in a predictable way to their food/water intake. So an isotopic analysis of an individual’s body tissues can be related back to the foodstuffs they ate, the water they drank, and the geological sources of their intake (Kohn, 1996; Bentley, 2006; Lee-Thorp, 2008). Everything isotopic is relative, linked both to source and to process of incorporation. A simplified version of the analytical process is that isotopic ratios are measured on a specimen from the consumer(s), then these mea surements are converted (explicitly or implicitly) to the equivalent ra tios that would be expected for the intake, which are then compared to the isotopic ratios of potential sources of intakes (foodstuffs, water sources, soils, geological sources etc.), in order to identify the actual diet composition or potential geographical origin. The isotopic ratios that can be measured are dependent on the chemical elements present in the specimen analysed: carbon, nitrogen, sulfur, oxygen and hydrogen for proteinaceous material (e.g. collagen, hair); carbon, oxygen and strontium for biominerals such as bioapatite, calcite or aragonite (e.g. tooth enamel, shell); carbon for plant materials, as well as sometimes nitrogen, strontium, sulfur, oxygen and hydrogen. Other isotopic ratios, including those of calcium, lead, neodymium and zinc have also been analysed in archaeological biomaterials (Reynard et al., 2010; Montgomery et al., 2010a; Jaouen and Pons, 2017; Plomp et al., 2019; Evans et al., 2022). Depending on the element, multiple isotopic ratios may be measured (e.g. both 87Sr/86Sr and 88Sr/86Sr, both 18O/16O and 17O/16O: Knudson et al., 2010; Lehmann et al., 2022; Feng et al., 2024). The isotopic patterning in each element results from the ways in which it is cycled throughout the biosphere, underpinned by one or two dominant drivers of variation: for carbon, that is plant photosynthetic pathways and ocean-atmosphere interaction; for nitrogen, it is nitrogen cycling within soils and nitrogen metabolism within consumers; for oxygen and hydrogen, it is the global hydrological cycle; for sulfur, it is the bedrock geology, atmospheric depositional sources and soil hy drology; for strontium, it is geochemistry and age of the underlying bedrock, combined with weathering patterns (Mook, 1986; Thode, 1991; Rozanski et al., 1993; France, 1995; Robinson, 2001; Bentley, 2006; Poupin et al., 2011; Cernusak et al., 2013; Stevens et al., 2025; Wexler and Stevens, 2025). For some elements there is large isotopic variation across the biosphere, and for some there is little or none. 3. Isotopic resolution The resolution at which we can interpret any isotopic data is intrinsically tied to the two key premises stated above. Our ability to make a strong link between isotopic ratios of consumer and consumed is constrained by isotopic variability within and between ecosystems, and the fidelity of transfer of isotopic signals from intake to body tissues – our archaeological interpretations are bounded by ecological and bio logical constraints. I see it as critical to reflect explicitly on these con straints within our work, as in certain situations they can limit us more than most researchers overtly acknowledge. There are three main rele vant areas: ecosystem patterns, the transfer of signal from intake to body, and the component of time. 3.1. Isotopic patterning within ecosystems Natural isotopic variation occurs throughout the biosphere driven by physical, chemical and biochemical processes. Because of the range and complexity of these processes, isotopic ratios are environmental signals, not specific indicators (or signatures or ‘fingerprints’) of particular food or water sources. Different foodstuffs can be isotopically similar (ni trogen isotopic equivalence of meat and milk from the same animal, carbon isotopic uniformity of different ruminant species within an ecosystem), or the same foodstuffs can vary in different environments. A clear example is the ‘Big Mac® index’, where a world-wide study of Big T.C. O’Connell Journal of Archaeological Science 183 (2025) 106379 2 Mac® beef patties showed a wide range of δ13C values: compositional similarity did not equate to isotopic similarity because beef cattle are fed on isotopically different resources in different geographical locations (Martinelli et al., 2011). Whilst geography leads to spatial isotopic variation, mapping of oxygen, hydrogen and strontium across the British Isles, Europe and North America shows great swathes of space with similar isotopic ratios (Bowen, 2010a; Evans et al., 2010; Willmes et al., 2014). The very nature of isotopic ratios as environmental signals means that we cannot uniquely link specific isotopic ratios to specific food or water resources, thus equifinality and multifinality constrain the reso lution of our dietary interpretations. Similarly, our ability to use isotopic values as unique and precise geolocational signals depends on the iso topic variability of the area under study. We should identify the isotopic constraints on the precision of our interpretations – for example to ask how local is ‘local’ in isotopic terms? For any given study, the geographical range of ‘isotopically local’ may not match to our concept of ‘archaeologically local’ – failure to recognise this can lead to problems in our inferences and interpretations. 3.2. “You are what you consume” - the transfer of signal from intake to body The isotopic relationship between body tissues and food/water intake is complicated. The isotopic ratios of intake are clearly the pri mary influence on body isotopic ratios, but internal and external factors can have a secondary effect, including: diet composition, isotopic ho mogeneity of intake, metabolism, physiology, tissue type, geochemical, environmental and climatic conditions (Bentley, 2006; Boecklen et al., 2011; Nehlich, 2015; Vander Zanden et al., 2016). The importance of different factors depends on the chemical element. The measured isotopic ratio of the specimen taken from the con sumer does not equal the isotopic ratio of intake for most elements – that we transform our proxy data (the measured isotopic ratios of consumer tissue) to the target variable (isotopic ratios of diet) is encapsulated in the old truism of “you are what you eat plus a few permil”. The magnitude of that “few permil” depends on many things, such as the element, the specimen, the organism. For example, for carbon, the dif ference between diet and collagen δ13C values is typically taken as +5‰ but has been shown to vary considerably in feeding experiments, and the difference between diet and bioapatite (enamel) δ13C values varies by species (probably due to digestive physiology including methano genesis) – ca. +14.6‰ for ruminants, ca. +13.3‰ for pigs, and ca. +9‰ for rats and mice (DeNiro and Epstein, 1978; Ambrose and Norr, 1993; Tieszen and Fagre, 1993; Passey et al., 2005; Froehle et al., 2010; Cerling et al., 2021). The values used for these transformations (referred to variously as offsets, differences, fractionation factors, discrimination factors, TDFs) are not biological constants – they are empirically derived from decades of controlled feeding studies that have sought to quantify isotopic dif ferences between diet and body tissues in a range of organisms for many elemental systems (DeNiro and Epstein, 1978; DeNiro and Epstein, 1981; Richards et al., 2003a; Tuross et al., 2008; Caut et al., 2009; Reynard et al., 2010; Boecklen et al., 2011; O’Connell et al., 2012; Wolf et al., 2012; Lewis et al., 2017; Webb et al., 2017; Anders et al., 2019; Weber et al., 2020). Such fundamental work has also identified key factors that influence the relationship between the isotopic ratios of intake and body tissues – and that thereby affect our ability to use iso topic ratios of body tissues as a marker of intake and location. These include: for carbon, the effect of dietary macronutrient composition, ‘protein routing’ and methane production; for nitrogen, the effect of physiology, growth and metabolism; for oxygen, the balance of food vs. water intake, the effect of physiology, the variability of source pools; and for strontium, the relationship between bedrock geochemistry and bioavailable strontium, biopurification and isotopic fractionation during incorporation from food (Ambrose and Norr, 1993; Tieszen and Fagre, 1993; Kohn, 1996; Bentley, 2006; Jim et al., 2006; Hedges and Reynard, 2007; Poupin et al., 2011; Brettell et al., 2012; O’Connell, 2017; Cerling et al., 2021). We need to use empirically-derived values because our understand ing of metabolic isotopic fractionation in the consumer is not deduced from first principles. We still do not have a full understanding of the mechanisms that govern the transfer of the intake isotopic signal to that in the body, particularly for the lighter elements, and this is an ongoing active area of research, conceptually and empirically (e.g. Martínez del Rio and Wolf, 2005; Tuross et al., 2008; Lee et al., 2012; Cantalapie dra-Hijar et al., 2017; O’Connell, 2017; Hughes et al., 2018; Tejada-Lara et al., 2018). The current best estimates for offsets to transform our data are not perfect but they are derived from consensus based on multiple studies. Because the relationships between consumed and consumer isotopic ratios are inferred from mathematical correlations from obser vational data, there is inherent uncertainty, the magnitude of which depends on the strength of the correlations (Pollard et al., 2011; Pryor et al., 2014; Skrzypek et al., 2016). This uncertainty is carried forward into our inferences of intake from consumer isotopic data. When transforming (and analysing) our data, we must be aware of the inherent assumptions in the process, as well as any associated un certainty. Without probing the validity of our mathematical trans formations, and the validity of the assumptions that underpin them, or even that they are transformations at all, we can end up with flawed interpretations: for example, early work on human diet in North America underestimated of the importance of maize at low levels of intake, because researchers used the ‘scrambling’ or linear mixing model of dietary incorporation that did not account for dietary protein routing to body protein (Schwarcz, 1991). The concept of a fixed isotopic offset or fractionation factor is diffi cult to accept for any element or organism – such fixity is unlikely to be true for any biological system, and doesn’t match well to the data that we have across many species in many ecosystems. It is also based on the assumption that the system is in isotopic equilibrium, and that the tissue analysed can be transformed to the intake of interest using relationships derived under steady-state conditions. This assumption is violated for dynamic or shifting systems where the isotopic ratios of the intake change over time, and isotopic equilibrium may be never reached in a consumer in natural conditions (Olive et al., 2003; Martínez del Rio and Carleton, 2012; Cathelin et al., 2025). But it is highly likely that such factors can approximate to a single value under constrained conditions. A good example is that of the isotopic fractionation in plant photosyn thesis – the relationship is complicated but well-understood at a mech anistic level, and under most conditions, the measured value can be related to a very few parameters (Cernusak et al., 2013). We should recognise that any measured isotopic ratio is the integrated outcome of multiple metabolic reactions, some of which will cause isotopic frac tionation (Schoeller, 1999; Schmidt et al., 2015). I find it more helpful to think of body isotopic ratios as emergent properties rather than fixed signals of intake – a change of perspective that might not change how we handle our data, but should change our perception of what is being measured. 3.3. Time, temporality and tissue synthesis Biological systems are not static uniform systems. Different tissues form over different time periods: some forming discretely and rapidly with no remodelling (e.g. hair), some forming and remodelling quickly (e.g. liver), some forming and remodelling slowly (e.g. bone), and some turning over very slowly, if ever. Tissues are synthesised from a body pool or pools, which themselves may have a fast or slow turnover rate (Cerling et al., 2007). Thus, there is a time lag between the intake of food/water and its incorporation into tissues, which links to the turn over rate of each tissue: Slatkin et al. (1985) showed that the cerebellar DNA of European-born immigrants to the USA still had the ‘European’ (C3) isotopic signal of their childhood many decades later, whilst their T.C. O’Connell Journal of Archaeological Science 183 (2025) 106379 3 cerebellar white matter isotopically resembled that of individuals born and raised in the USA. Each tissue therefore represents a different time-slice of an in dividual’s intake, depending on growth period and turnover rate: bone with its slow turnover integrates long-term intake, hair provides a linear record of recent intake, and teeth reflect intake consumed during their formation period, as do shells, feathers, fur, antler (Thomas and Crowther, 2015; Vander Zanden et al., 2015). If we are interested in long-term patterns, then the isotopic signals in bone collagen may be sufficient to characterise an individual’s habitual intake (although the ‘ten year’ figure has little or no foundation). But at times, a temporal mismatch between analysed tissue and intake of interest may obscure or skew interpretations, particularly for dynamic or shifting systems. An excellent example of the fine detail that can be elucidated from the temporal record of different tissues is Montgomery et al.’s work on early Neolithic settlers in Shetland, where different patterns of marine con sumption could be inferred from the isotopic information in bone collagen and tooth dentine – only the sequential analyses of tooth dentine could demonstrate short-term episodes of high-level marine food consumption, rather than low-level long-term supplementation of a terrestrial-based diet that might be inferred from the slight 13C and 15N enrichment visible in bone collagen (Montgomery et al., 2013). Yet we should be wary of assuming that greater temporal or spatial analytical resolution is necessarily better. We may be able to analyse a tissue at very high resolution (micrometre profile of the isotopic varia tion in a tooth) but we must question what the isotopic/elemental profile represents if the tooth enamel forms over a longer period and draws on a larger body pool (Montgomery et al., 2010b). Similarly, we do not yet fully understand the typical expected isotopic variation within and between collagen from different skeletal elements, which is likely driven by time, turnover and biomechanical stress and is unlikely to be consistent across all individuals (Fahy et al., 2017; Anders et al., 2019; de Gruchy et al., 2024). Overall, such general biological isotopic variability or ‘noise’ is hard to capture and therefore the threshold at which isotopic variability is driven by external factors (e.g. diet) rather than internal biology is not simple to predict. 3.4. The signal and the noise The resolution of our interpretations is limited by isotopic variation in the biosphere and our imperfect comprehension of the mechanism and fidelity of isotopic signal transfer to the body. Our interpretations may be further constrained by scant comparative (‘baseline’) data for potential consumed resources and by the restricted number of measured parameters (often two or possibly three isotopic ratios). I reiterate, we must recognise that biological isotopic data are proxies for complicated processes. There is no straightforward one-to-one mapping of consumer isotopic ratios to a unique set of intake isotopic ratios, rather a trans formation that relies on a series of assumptions and previously observed correlations. At its simplest, to get the most out of isotopic data, we must tease out the signal that we think is meaningful from the other variability that we know is there. We know that there is a lot of ‘noise’ in all isotopic sys tems, and that this varies by chemical element and between different systems under examination. But we should recognise that what might be noise in one study or to one researcher could be a meaningful signal in another context – it depends on our perspective, the question, the study. To improve the resolution of our interpretations, we must maximise the signal-to-noise ratio in each study. We can achieve this by considering the nature of the system we are examining and the nature of our data, then seeking to identify our signal of interest whilst understanding and controlling for the ‘noise’. I see three areas that need more overt attention at the level of each and every study – confidence in specimen and data integrity; considerations of analytical scope and scale; robust yet imaginative data interrogation. 4. Specimen/data integrity We should ensure that we are confident as to the nature and integrity of our measured specimens as well as the data generated – lack of such confidence leads to greater uncertainty at the stage of data analysis and interpretation. The question of what is actually being measured is often overlooked, but is fundamental in all isotopic work. Due to the way that the isotopic ratio is measured in most mass spectrometers (compound- specific aside), the value obtained is the average of whatever goes in. So if we do not know what we are analysing, and how it relates to what we think we are analysing, then we sabotage ourselves from the start. When undertaking destructive analyses, there is furthermore always a trade-off between destruction of the specimen and gain of information. That trade-off is different for different specimens, depending on the specimens themselves (including their scarcity and perceived value) and the nature and quality of the information yielded. Yet the worst outcome, in scientific and ethical terms, is both loss of specimen and no useable data. We must all remain aware of the ethical implications of our destructive analyses, and aim to ensure that we obtain useful and useable data by wise sampling choices of suitable specimens. A key consideration is the question of preservation, and the likeli hood of whether the specimen has undergone any diagenetic change. What is available for analysis in archaeological studies is always con strained by preservation, and for all archaeological specimens, taphonomy and diagenesis should always be explicitly addressed, since diagenetic changes to the specimen may have altered the original iso topic ratio. Clear examples are seen in: oxygen isotopic changes during the recrystallization of shell aragonite to calcite; the unreliability of carbon and oxygen isotopic analysis in bone bioapatite; and the differ ential uptake of strontium during burial into dentine vs. enamel, with concomitant large shifts in strontium isotopic ratios (Koch et al., 1997; Chiaradia et al., 2003; Staudigel and Swart, 2016). Furthermore, diagenetic change may affect different moieties of any specimen differently, which increases the complexity (Kendall et al., 2018). A second key point is to understand the effect of preparative methods as well as sampling and analytical procedures on any isotopic mea surements. Methods to remove contamination or to isolate sub-fractions of any specimen have the potential to alter the measured isotopic ratio through physical or chemical effects – and preparative methods may affect different types of specimen materials in different ways (Lee-Thorp and van der Merwe, 1991; Balasse et al., 2002; Hoppe et al., 2003; Sweeting et al., 2006; Pestle et al., 2014; Pellegrini and Snoeck, 2016). Sampling design and analytical methods may also cause differences (Balasse, 2003; Montgomery et al., 2010b; Reade et al., 2015; Reynard et al., 2019). Whilst such effects are often unavoidable, they can be taken into consideration when examining the results, but this requires that the methods used are fully documented and reported. To understand what we are measuring, ideally, we characterise the integrity of the specimen analysed, both as a whole and for the chemical element of interest. This allows us to check the state of preservation compared to our expectations, in order to decide how confident we are that the measured signal is biogenic (or is at least the signal that we are aiming to measure). 4.1. Mass balance and isotopic mass balance Before thinking about how to characterise what we are measuring, it is worth reflecting in somewhat abstract terms on some basic concepts concerning mass balance and isotopic mass balance, and their implica tions for our measurements. At the risk of stating the obvious, isotopic ratios can only be measured when the relevant chemical element is present in a specimen – for example, we cannot measure strontium isotopic ratios in bone collagen because collagen contains no strontium. Furthermore, laws of mass balance dictate that any change (diagenetic or otherwise) in elemental abundance within the specimen can cause isotopic change. When the element is abundant and any potential T.C. O’Connell Journal of Archaeological Science 183 (2025) 106379 4 abundance change is small, then any resulting isotopic change is likely to be small – if a bone that contains lots of carbon in the collagen molecule (ca. 20% collagen in bone by dry weight, and collagen is ca. 45% carbon) is contaminated by a tiny amount of soil humic acids, then the measured carbon isotopic ratios of the bone collagen may be slightly shifted, but probably within fractions of a permil. When the element is at low abundance, and change in abundance may be large, then potential isotopic change may also be large – if a bone that contained very little strontium in vivo takes up a lot of strontium post-mortem from a strontium-rich burial environment, then the original biogenic isotopic ratio will be swamped by that of diagenetic strontium and the measured strontium isotopic ratios of the bone bioapatite will reflect those of the place of burial and not the place where the person was living when the bone was formed. To identify and minimise the impact of potential diagenetic effects on isotopic measurements, the best way is to choose isotopic systems and specimen types where the chemical element of interest is bio genically intrinsic to the specimen material during the organism’s life, preferably at high abundance, and the material’s integrity can be characterised in compositional terms. For specimen types where the chemical element is biogenically intrinsic, but the specimen material is hard to characterise, we have to work a bit harder. For specimen types where the chemical element is incidental, not intrinsic, then it is difficult to characterise the integrity of material and isotopic signal – since the biogenic abundances may vary widely – yet is even more important, as diagenesis may have far more of an impact on any isotopic measure ment. Now we can take these abstract concepts and consider what they mean for isotopic measurements of biological specimens in archaeo logical contexts. 4.2. Specimen type and measurement integrity Taking first the specimen as a whole, our ability to characterise specimen materials depends on the nature of the specimen matrix – what sort of material are we analysing, e.g. protein, biomineral, composite? Proteins are molecules composed of carbon, hydrogen, nitrogen, oxygen and a little sulfur, in known proportions in a known and discrete mo lecular structure. In the case of gelatinised ‘collagen’, its composition and structure can be characterised independently of isotopic analysis (Eastoe, 1955; Vaughan, 1981), with published quality control criteria (DeNiro, 1985; Ambrose, 1990), and reputable isotopic studies on collagen report such data. For instance, we know how much carbon to expect in any specimen of collagen, and if it is less or more, we can start to consider whether such changes are likely to have altered the isotopic measurement (perhaps by collagen degradation or the presence of carbon-rich and isotopically different lipids or humic acids). Collagen is great molecule for isotopic studies in archaeology, not only because of its known composition, as well as its robusticity to degradation, but also because of its ease of preparative isolation through demineralisation and gelatinisation (Longin, 1971). Other proteinaceous materials may not be so straightforward, depending on their amenity to extraction and preparation. The acid-insoluble non-gelatinised fraction of bone that is often referred to as ‘collagen’ includes non-trivial quantities of non-collagenous proteins and can be considered a composite material. Hair is predominantly keratin(s), but also comprises small proportions of pigments and other compounds, in a complex structure (cuticle, cortex and medulla) (Gillespie, 1991), and is usually analysed as the composite tissue, as are other keratinous tissues such as feather, fur, nail and whisker (Cherel et al., 2000; Lewis et al., 2006; Ehleringer et al., 2008; Voigt et al., 2013). Muscle and skin are composites of different proteins and lipids in varying proportions, with the attendant problems of how best to characterise and prepare them for isotopic analysis (Sweeting et al., 2006; Haun et al., 2019; Cloyed et al., 2020). Other complex biological matrices (proteinaceous and other) such as wood and cereal grains present similar difficulties in characterisation and identification of integrity or alteration (Styring et al., 2013). Specimens containing proteins and/or lipids can be analysed as bulk tissue, or can be isoto pically measured at the level of individual components – the multiple different lipids and amino acids within each specimen (com pound-specific isotopic analyses). The elemental composition of each such compound can usually be precisely characterised, because its chemical structure is known, but in most situations the isotopic mea surement process is technically more challenging, and so the analytical uncertainty on the isotopic measurements may be greater (Evershed, 2008; Whiteman et al., 2019). Biominerals offer a different challenge. In the case of bioapatite (highly substituted biological calcium phosphate apatite found in vertebrate tooth and bone), the material comprises calcium, phos phorus, oxygen, carbon, which we can characterise by abundance, but not by discrete molecular structure, since the material has a complex mixed crystalline structure (Driessens et al., 1978). A similar situation is true for mollusc shells, which are typically comprised of either one or two polymorphs of calcium carbonate – calcite and aragonite – which are chemically but not structurally or isotopically equivalent (Lécuyer et al., 2012). Chemically similar forms of a biomineral may also be physically very different (e.g. tooth enamel and bone bioapatite), which results in differences in the fidelity of preservation of isotopic ratios. We have known for a long time based on strong empirical evidence that enamel bioapatite preserves the biogenic isotopic signal fairly robustly, but bone bioapatite is an unreliable material for most isotopic analyses no matter how it is prepared (Koch et al., 1997; Bentley et al., 2004; Pellegrini and Snoeck, 2016) – yet bone bioapatite isotopic data are often presented with minimal justification. Because structural integrity is harder to determine in biominerals, although elemental abundance can be measured, other parameters regarding the nature of the material can come into play, such as histology, crystallinity, porosity or the presence of other trace elements that might reflect diagenetic change (Hedges et al., 1995; Loftus et al., 2015; Grimstead et al., 2018; Ber tacchi et al., 2024). If we do not understand the composition of a biological material, how it is produced, over what time-period, and how it may be changed by diagenesis and by preparative methods prior to analysis, then its isotopic analysis is likely to lead to intractable data of questionable utility, as we cannot understand what the measured isotopic ratio rep resents. For example, we know that dental calculus is a composite of organic molecules and biomineral in varying proportions within and between individuals, laid down over an unknown period of time (Lieverse, 1999). We also know that carbon isotopic ratios of organics and biomineral within the same individual are very different (for both chemical and biological reasons), and that failing to purify one from the other leads to uninterpretable isotopic data – as shown by decades of research in stable isotope and radiocarbon chemistry (Longin, 1971; Ambrose, 1990). Therefore, it is unsurprising that isotopic analysis of bulk calculus is of limited use when investigating dietary intake (Scott and Poulson, 2012; Salazar-Garcia et al., 2014). Similar challenges exist for analyses of other complex materials that we know have been sub jected to major physical and chemical changes, such as charred grains, and we should exercise great caution. 4.3. Chemical element and measurement integrity Moving from the specimen as a whole, we then need to consider integrity from a chemical element perspective. The lighter elements are typically intrinsic and structural in biological materials, with a known abundance. For most heavier elements such as strontium, lead or zinc (but not calcium), the situation is more difficult, as they are not struc turally intrinsic to most biological materials studied, and thus there is no predictable biogenic abundance within tooth enamel, bone bioapatite, shell calcite/aragonite, wool, hair, wood or cereal grains. In the absence of a predictable biogenic abundance, it is far harder to quantify (or even estimate) when diagenetic change may have happened. But abundance data can indicate whether measured elemental concentrations are in the T.C. O’Connell Journal of Archaeological Science 183 (2025) 106379 5 region that might be expected (biologically feasible), and can also be used to identify anomalous data points and other patterns of interest (Montgomery et al., 2010a; Evans et al., 2012; Dalle et al., 2022). This is particularly important when materials are more likely to take up con taminants from the burial environment, which depends on element-specific distribution and partition coefficients as well as burial conditions (Pike and Richards, 2002; Reynard and Balter, 2014), and when biogenic and diagenetic pools have similar isotopic values, making it even harder identify the contamination. Researchers have studied ways to identify diagenetic heavy element contamination (spatial dis tributions can show it clearly: Pike and Hedges, 2001; Chiaradia et al., 2003), and for some materials (such as cereal grains and wool) studies have shown the difficulties of removing exogenous material (von Hol stein et al., 2015; Styring et al., 2019). Practitioners should report elemental concentrations and abundance data for all isotopic analyses, allowing readers to assess the validity of the data. However, there are too many published studies of archaeological materials that report nothing other than isotopic data, leaving readers in a position where they are unable to assess validity of the data, and therefore validity of the claims and interpretations (e.g. Frei et al., 2015; Rich et al., 2016; Frei et al., 2017). 4.4. Data integrity Finally, we must ensure that our measurements are precise and ac curate. This requires reliable machine operation, with appropriate calibration and handling of the isotopic data, as has been widely dis cussed elsewhere (Szpak et al., 2017; Roberts et al., 2018). This is in many ways the most obvious aspect of analytical integrity, but can be particularly demanding where there is a disconnect between the researcher and the mass spectrometer, as when isotopic analyses are measured at ‘service’ labs, and the researcher may have little engage ment with the measurement and manipulation of their own data. I encourage all researchers to engage with their data and measurement thereof as early as possible, and this may be promoted by open-source data handling methods (e.g. Isoverse, accessed Aug 2025 https: //www.isoverse.org/). Such quality management is increasingly important in an era of ‘big data’ (Shipley et al., 2024). We don’t want to interrogate large composite datasets for patterns of interest, and end up separating the data based on analytical laboratories – to ensure data compatibility, an active data assessment is imperative. As others have stated (Reynard and Balter, 2014; Roberts et al., 2018), we need quality control indices as well as best practice reporting standards for all isotopic systems and analyte materials, as per collagen. Then we need all researchers to use them, and all journals and referees to ensure that the standards are complied with as a prerequisite to publication. 5. Scope and scale of analytical framework(s) In terms of the scope at which bioarchaeological isotopic studies are executed, we can consider three spatio-temporal scales: large scale, both geographically and temporally, such as regional- or continent-wide and multicentennial-duration; local scale, such as community or multi- community scale over more limited time and space; and intra-individ ual scale over a lifetime. The scope of the analytical framework will not only shape the study’s structure, but also play a part in determining the relevance of different factors influencing isotopic variability, and therefore the resolution of our interpretations. This is because analytical scale affects the magnitude of the signal that we want to observe vs. the other variation or ‘noise’ that is not of interest in a particular study. Large-scale studies are relatively few in number, because situations where there is likely to be a clearly visible isotopic change over a wide range of space and time and which is socially/humanly/environmen tally meaningful are comparatively rare – major changes in diet driven by choice, resource availability or the impetus of societal change, or major environmental changes, or significant shifts in location where we can clearly spot a first-generation migrant. The majority of isotopic studies in archaeology fall within the second category, those at a local scale – site-based studies over a limited timespan within a constrained geographical range, investigating variation driven by social, environ mental or subsistence-based changes. Of increasing interest are intra- individual studies – those cases where we can see changes over life time in tissues that grow sequentially (hair, teeth, shell), or where we can compare multiple tissues recording different time periods of an in dividual’s life, for a fine-scale life-history or ‘osteobiographical’ approach. There is an interplay between the analytical scale of the study, the isotopic variation we can identify as a real signal and the resolution of our possible interpretations. The study’s scale will affect what is deemed suitable as comparative isotopic material, both in geographical and temporal scope: for example, a range of 500 years and 500 km for comparative material might be appropriate for the Upper Palaeolithic but probably unacceptable for the Mediaeval period. With changing analytical scale, questions of the uncertainty in transformations and underpinning assumptions come into play. The isotopic variation that we accept as meaningful within the context of each study will differ with the study’s scale – what might be appropriately identified as notable intra-individual variation is less than that which would be recognised as reflecting significant isotopic difference within a group of contempora neous individuals, let alone a collection of individuals across space and time (and in all cases such variation is almost certainly greater than analytical precision). Thus, a δ13C range of 2‰ is noteworthy when seen within an individual’s intra-tooth dentine, transcending the typical variation we ascribe to metabolic or physiological factors, but is typical in a Balkan mediaeval cemetery population, less than that seen in across five LBK sites in central Europe, and is ten-fold greater than typical quoted analytical precision for most carbon isotopic measurements using EA-IRMS (Lightfoot et al., 2012; Hedges et al., 2013; Beaumont and Montgomery, 2016). Large-scale studies often involve both clear isotopic dichotomies and a comparative approach (time and/or space), though they also involve far more sources of isotopic variation beyond the researcher’s control. For some such studies, the targeted isotopic differences are so great that the presence of greater variation (which is likely to be noise for such a study) is unproblematic. Whilst one might not be able to obtain any form of meaningful comparative material, or control for minor environ mental, physiological and temporal fluctuations, it often does not matter – for example, the magnitude of the change in consumer δ13C with a shift from C3 to C4/marine consumption is large enough to outweigh any relatively trivial variation due to physiology or environmental stress or minor isotopic differences within resource groups (Richards et al., 2003b; Schoeninger, 2009; Sponheimer and Lee-Thorp, 2015). Yet not all large-scale studies provide a straightforward isotopic pattern, high lighting potential differences of interest that can be explored further using other strands of evidence (Stevens and Hedges, 2004; Stevens et al., 2008; Sponheimer and Lee-Thorp, 2015). Studies that take a ‘supra-regional perspective’ (Bentley et al., 2012; Hedges et al., 2013; Parker Pearson et al., 2016; Santana-Sagredo et al., 2019), or are syn thetic in nature, combining previously published data (Montgomery et al., 2010a; Evans et al., 2012; Reynard and Tuross, 2015; Lightfoot and O’Connell, 2016; Tsutaya, 2017), may not have such clear isotopic changes, but are often aimed at looking at replicability of patterns across multiple sites. At the other end of the scale, intra-individual studies are by defini tion comparative in nature. They are not without their own problems (comparability of tissues, timescales recorded, possibility of turnover or not, analytical constraints with smaller specimen size), but external factors are less important (e.g. potential environmental change is less because of the temporal limit of a lifetime). One can posit that intra- individual comparisons either hold many variables constant (sex, sta tus) or are in fact the signals that we are interested in (weaning, T.C. O’Connell Journal of Archaeological Science 183 (2025) 106379 6 https://www.isoverse.org/ https://www.isoverse.org/ ontogeny etc.) – thus the targeted isotopic signal might be smaller, but so is the noise. The data are effectively ‘internally normalised’, allowing us to identify small scale shifts with movement or dietary changes or sea sonal shifts. This has yielded nuanced insights into individual and group lifeways (Evans et al., 2006; Wilson et al., 2007; Schroeder et al., 2009; Beaumont and Montgomery, 2016; Herrscher et al., 2024), as well as questions of climatic variability and resource management and use (Balasse, et al., 2002, 2012a; Mannino et al., 2007; Prendergast et al., 2016; Towers et al., 2017; Szpak and Valenzuela, 2020). I see the mid-range local-scale studies as more challenging: targeted isotopic changes are likely to be smaller in magnitude; the study may not be comparative in nature; and other sources of isotopic variation are often less controlled (duration of a site, composition of the sampled sites and populations, availability of comparative material). In such circum stances, the magnitude of the signal may not necessarily be much greater than the expected noise. We may observe intra-group variation that could result from some degree of mobility or genuine shifts in dietary intake, or that could be due to the typically observed biological vari ability within a group living in the same location and consuming the same resources. The reference (‘baseline’) data necessary to transform consumer values to predicted intake may not be either sufficiently comprehensive (missing resources such as plants) or close enough in time and space to enable such transformations with any certainty or confidence. To do studies at this scale well, isotopic data usually requires integration with other data (archaeological, osteological, environ mental), to explore such aspects as sex and status differences, or con trasts between neighbouring sites (Muldner et al., 2009; Copeland et al., 2011; Hakenbeck et al., 2010; Knipper et al., 2017; Dahlstedt et al., 2024). The possible degree of interpretative precision has implications for the kinds of questions we can address with isotopic data: different scales of study need different types of questions, with differing integration of other relevant evidence. We must always consider how detailed we can be or need to be – broad brush or fine detail. An outwardly- straightforward question such as “what did people eat?” might be appropriate when asked of the transition to agriculture in north-west Europe, with a valid and archaeologically interesting answer along the lines of predominantly marine resources for Mesolithic people, pre dominantly terrestrial resources for early Neolithic people, depending on location (Richards et al., 2003b). But such a broad question is likely to produce an uninteresting or overly precise answer when asked of an individual or a group – especially when comparative isotopic data on fundamental resource groups (e.g. terrestrial plants) are unavailable. In such circumstances, I would argue that even posing such a question is flawed, because the scope of the answer is too ill-defined, and isotopic information alone is likely insufficient to tackle an archaeologically interesting question. As Makarewicz and Sealy (2015) put it so well: “The most successful stable isotope studies are designed in such a way as to achieve a good fit between the archaeological question and the capabilities of the stable isotope tracers.” To do this, we should reflect on the scope and scale of each study, and then tailor our question (and research design) appro priately, to ensure that it can be sufficiently well-answered by our data, and can be justified in the context of the study. 6. Data interrogation Despite yielding quantitative data, I would argue that isotopic analysis is rarely used in more than a semi-quantitative way in archae ology. The field has developed from small datasets in one or two di mensions to large multi-elemental datasets, often with many other potential comparative lines of evidence. Yet many researchers are stuck in a one- or two-dimensional world, where the predominant mode of data analysis is to treat the data as a group, to plot it in a bivariate scatterplot, and to describe the patterns seen visually (to ‘eyeball’ it). Pre-determined groupings may be compared using classical statistical techniques. There may be some effort to use Bayesian mixing models to categorise intakes, but few attempts to use modelling as a predictive tool. Rarely are hypotheses formally proposed and tested. although there are excellent examples from the earliest days of the technique’s application (e.g. Sealy and van der Merwe, 1985). We could interrogate our datasets far more thoroughly and more imaginatively, if we broke out of our one- or two-dimensional percep tion of the isotopic world, and embraced data-analytical methods from other disciplines using isotopic data (e.g. ecology, epidemiology) and large multidimensional datasets (materials science, genomics and pro teomics). I argue that developments in the areas of data exploration and representation, and data analysis (including modelling and simulation) could transform our interpretations. 6.1. Data exploration and representation For most isotopic datasets in archaeology, we usually have clusters of data that we may or may not be able to contextualise with comparative isotopic data (e.g. potential food resources, local water signals, local geological signals), as well as other information (archaeological, oste ological, biomolecular, environmental). We need to describe our (iso topic) data in a manner that allows us to interrogate it in a quantitative yet assumption-free way. Data exploration is a key step in such inter rogation (Zuur et al., 2010). Tukey, who coined the phrase ‘exploratory data analysis’ (as distinct from confirmatory data analysis), as well as many of its key tools, defined it as “an attitude, a state of flexibility, a willingness to look for those things that we believe are not there, as well as those we believe to be there”, emphasising that it should be “actively incisive rather than passively descriptive” (pp. 806 and lxii, respectively, Jones, 1987). Yet I perceive the default approach in most isotopic studies to be a passively descriptive presentation of the data, typically in a two-dimensional scatterplot as absolute values and summarised as mean and standard deviation. Such a descriptive approach may result from our measurement of one or two isotopic parameters (e.g. carbon, or strontium, or carbon and nitrogen), rarely more, which has reduced the impetus for the use of multi-dimensional data analysis techniques. But I see the descriptive approach as also indicative of most researchers’ conception of isotopic data primarily in absolute and not relative terms, linked to assumptions of the data equating directly to intake, rather than as proxy data integrating a series of complicated ecological and bio logical processes. Many statistical techniques exist to probe variation in multi- dimensional large datasets, widely used with other archaeological data (Baxter, 2008, 2009; Owen et al., 2014; Bogaard et al., 2016; Wood et al., 2017; Contreras et al., 2018; Rousseeuw and Bossche, 2018). Exploratory methods that are not specific to isotopic data, such as methods for inferring memberships or parameters of subpopulations (LDA, K-means, mixture models) or dimensionality reduction techniques (PCA) are rarely used in archaeological isotopic studies, yet can provide a way to explore patterning in the data with few prior assumptions (Honch et al., 2012; Larsen et al., 2013; Bocherens, 2015; Hutchinson et al., 2015; Horswill et al., 2016). Such an approach does not rely on making a link between the isotopic data of consumer and intake, which reduces problems arising from poor comparative data, and reduces reliance on calculated offsets (e.g. TDFs) that are more variable and with greater uncertainty than many care to admit. The need for multi-dimensional analytical techniques will only intensify due to the increasing use of compound-specific isotopic analyses (Mora et al., 2017) – it does not make sense to focus on just one or two ‘interesting’ amino acids within a larger dataset, ignoring other data (Naito et al., 2016). 6.2. Data distribution A consequence of the focus on absolute values (rather than as relative measures) is the lack of attention then paid to the spread of data within T.C. O’Connell Journal of Archaeological Science 183 (2025) 106379 7 and between groups. Given the challenges of modelling the full complexity of individual intake from a few isotopic parameters, I ask why would we ‘throw away’ information such as the nature of distri bution of the group? I believe that everyone should spend as much time examining the distribution of their data as they do on the central ten dency because it is so informative. Data distribution can indicate skew, multi-modality, dispersion, evenness. To summarise complex distributions appropriately and robustly, it is sensible to consider both parametric and non-parametric measures (e.g. reporting both mean and median, both standard devia tion and IQR), as well as ways to represent the variation and inherent uncertainty in all distributions (histograms, kernel density estimates, violin plots, heat maps etc.) (Franconeri et al., 2021). Outliers may be identified mathematically within univariate or multivariate datasets (Lightfoot and O’Connell, 2016; Rousseeuw and Bossche, 2018), but intra-group variation reveals more than just the presence of true outliers (Araujo et al., 2011; Bolnick et al., 2011). An exploration of inter-individual variation within a group is at the heart of much of the recent work in ecology on the ‘isotopic niche’ (Bearhop et al., 2004; Newsome et al., 2007). As Newsome et al. (2007) outline, the concept of the niche is “central to ecological thinking because [it] represent[s] convenient shorthand for many of the concepts used by ecologists to approach a variety of important problems, including resource use, geographic diversity, and many aspects of community composition and structure” – such a concept is also strongly relevant in archaeology. Whilst the isotopic niche does not equate directly to the ecological niche (which itself is somewhat elusive to define: Colwell and Rangel, 2009; Pocheville, 2015), isotopic data, as a time-integrated record of intake, can be used to investigate intra- and inter-individual aspects of niche width by examining the “relative position of consumers in isotopic space” (Layman et al., 2012). In palaeoecology, evidence of isotopic niche change for Pleistocene bears across the circum-Arctic shows shifts in carnivore behaviour and subsistence following changes in species competition (Bocherens, 2015). A study of human subsistence change over the period of maize introduction in the Amazon basin used niche widths to explore temporal and geographical continuity and change (Hermenegildo et al., 2025). A suite of isotopic metrics has been developed to examine aspects of community-wide and individual-level variation and specialisation, and it seems myopic to ignore these pa rameters that describe data distribution, density of clustering, evenness of distribution within isotopic space in a manner that permits compar ison across groups (Bearhop et al., 2004; Layman et al., 2007; Schmidt et al., 2007; Layman and Post, 2008; Turner et al., 2010; Jackson et al., 2011; Cucherousset and Villéger, 2015). With the recognition of the relativity and uncertainty of our proxies should come the realisation that hard boundaries have no place in iso topic representation and interpretation – there is no single precise strontium or oxygen isotopic threshold for local/non-local at a site, nor of a particular number at which nitrogen isotopic ratios represent con sumption (or presence) of manured vs. unmanured crops. A dotted line on a graph demarcating a boundary is usually a very poor representation of isotopic reality. This leads to a challenge for all isotopists about how to conceptualise and represent population distributions as well as un certainties and probabilities in our assessments and interpretations, which brings us to the topic of data analysis. 6.3. Data analytical methods and approaches Most data analyses in bioarchaeological isotopic papers are of two types: statistical comparison tests, and mathematical modelling to describe and quantify isotopic variation. Classical frequentist statistical tests offer a straightforward approach to the comparison of groups that are typically predefined based on other archaeological evidence. The interplay of multiple influencing factors limits the use of simple tests in all but the simplest cases, but linear regression in more sophisticated forms (e.g. generalised linear mixed models, hierarchical multilevel mixed models) offers the opportunity to probe complex situations, including data inter-dependence (Perri et al., 2019). Such testing may be appropriate where there is a clear hypothesis (e.g. “do these two groups differ?”), but there are a number of limita tions when used in isotopic studies. Beyond the enduring problems of sample size, statistical power and effect size, independence, false pos itives/negatives, and the tyranny of statistical significance (Wasserstein and Lazar, 2016; Amrhein et al., 2019), there is the insuperable obstacle that most classical frequentist statistical comparison tests (e.g. Student’s t-test or ANOVA) and the most common regression analyses can only be applied to one dependent variable. Studies involving two or more iso topic dimensions must be addressed with multivariate regression, rather than multiple comparisons for each isotopic dimension separately (Froehle et al., 2012; Zhu and Sealy, 2019). From the work on isotopic niche modelling, a combination of quantitative approaches with area-based and directional metrics pro vides increased quantitative rigour, enabling statistical testing of explicit hypotheses about isotopic difference over time and space (Schmidt et al., 2007; Layman and Post, 2008; Turner et al., 2010; Layman et al., 2012; Cucherousset and Villéger, 2015). Other ap proaches such as Bayesian estimation (e.g. the BEST test) also offer advantages, particularly with more than one dependent variable (Kruschke, 2013). But what is critical is to reflect in each situation as to which method of inference testing is appropriate, rather than blindly assuming that null hypothesis significance testing is the route to follow (Lecoutre et al., 2001; Stephens et al., 2007). Mathematical modelling has been a major development in isotopic work over the last two decades, in three main categories: time-series modelling; isoscape models; Bayesian mixing models for dietary assessment. The strengths of mathematical modelling are demonstrated in the application to intra-individual time-series data – the transformation of spatially/temporally dependent isotopic data from intra-individual se rial sub-sampling allows curves to be fitted to the noisy situation of real data. When applied to herbivore hypsodont teeth, it allows researchers to handle tooth growth patterns and sampling geometry (Passey and Cerling, 2002; Kohn, 2004; Zazzo, et al., 2006, 2012; Balasse et al., 2012b; Norwood et al., 2023; Yang et al., 2025). Balasse’s pioneering work on sheep birth spacing demonstrates that quantitative information can be obtained from relative isotopic patterning, providing meaningful insights into human behaviours, in this case animal management (Balasse, et al., 2020, 2024). A similar approach has been widely applied in sclerochronology to examine human foraging behaviour (Mannino et al., 2007; Prendergast et al., 2016; Bosch et al., 2018; Thompson et al., 2024). There has as yet been little exploration of ways to interrogate serial analyses of human tooth dentine and enamel that go beyond a descriptive linear representation of the data, and this area seems ripe for development. For more complex patterns of change, trajectory analysis is a useful framework for representation and analysis (Sturbois et al., 2022). Isoscape models – projections of geospatial isotopic variation – provide elegant frameworks for the interpretation of consumer isotopic data, particularly for oxygen, hydrogen and strontium (Bowen, 2010a; West et al., 2010; Spies et al., 2025). Such models give a probabilistic assignment of a match between source regions and individual – not an ‘answer’ – with a degree of uncertainty (Bowen, 2010b; Wunder, 2010) and one can argue that they are at their most powerful when excluding potential areas of origin (areas with which data are inconsistent). Derived using geostatistical interpolation techniques, isoscape models are always dependent on the quality and coverage of the source data – poor data cannot be rectified by the miracles of interpolation, although machine learning offers opportunities to handle patchy data in some environments (Bataille et al., 2020). Because of this, it is critical to understand (and report) the confidence and uncertainty of any isoscape model and subsequent mapping and probabilistic assignments (Pellegrini et al., 2016; Laffoon et al., 2017; Glew et al., 2019; Bataille T.C. O’Connell Journal of Archaeological Science 183 (2025) 106379 8 et al., 2021). There has been a rapid expansion in the use of Bayesian mixing models (e.g. simmr, SIAR, MixSIAR, IsotopeR, FRUITS) to estimate the contribution of different sources to consumer signatures (Hopkins and Ferguson, 2012; Phillips, 2012; Fernandes et al., 2014; Govan et al., 2023). I see that for most bioarchaeological isotopists, such modelling is a mechanism to provide an ‘answer’ in terms of intake, echoing my earlier comments that many researchers conceive of their data as a direct measure of intake, rather than as a proxy. These Bayesian models can be useful, but are still trying to rationalise a very complex system (with myriad biological, ecological, social, cultural factors at play), and all come with major caveats regarding impact of sample numbers, source variability, physiological factors (Phillips and Koch, 2002; Phillips and Gregg, 2003; Parnell, et al., 2010, 2013; Phillips et al., 2014). The limitation of all such models to discriminate successfully between a maximum of n+1 source groups for n measured parameters (isotopic values) is frequently ignored (Phillips, 2012). Models can only be as good as your data and your assumptions (“junk in, junk out”), and should never be blindly applied, as all model developers state clearly – yet there is often little sophisticated consideration of the sensitivity of model outputs to the input parameters, which can lead to incorrect, unduly precise or overly simple answers to complex scenarios (see dis cussions in: Cheung and Szpak, 2021; Cheung and Szpak, 2022). We should all strive to be “good Bayesians” in our use of such methods, explicitly justifying all chosen values of priors and assumptions, and reporting uncertainties in parameters and outputs (Buck and Meson, 2015; Lewis and Sealy, 2018). One must question each time as to what is the point of using such a mixing model. In many situations, I ask whether the models are genu inely informative, or do they merely tell us what is clearly visible from the isotopic data, just with more mathematical gloss? For modelling to be useful, there must be a feedback loop from real world to the mathe matical model and back to the real-world data, to test if the model is feasible, as well as informative (Lewis and Sealy, 2018). There is rarely any evidence of subsequent testing of the outcomes of such models, the lack of which could be interpreted as post-hoc storytelling (framing exploratory analyses as confirmatory analyses). Nor of simulations, which can help in constraining our interpretations, as well as testing model sensitivity (Nielsen et al., 2015; Jabot et al., 2017). If one pos tulates that consumers were having access to two different resources, or were moving between two environments, what would happen if we simulated that, and compared the simulated outcome to the observed data? Researchers may be unwilling to attempt simulation, perceiving it incorrectly as mathematically difficult. However, it is philosophically challenging, especially in a field where parameters may be insufficiently or poorly constrained, because it requires researchers to formalise their assumptions within the rigour of the simulation model and thus to explore explicitly the beliefs they hold. But such approaches offer great potential in exploring the complex nature of isotopic data (e.g. Trueman et al., 2019). Just as critical as choosing statistical approaches and executing data analyses well is to recognise when there are no appropriate statistical methods that can be applied to a dataset. When sample numbers are too small, or unbalanced, or datasets are clearly patchy, lacking compara tive data, then sometimes description and ‘eyeballing’ must suffice. But we must always make a positive choice as to the route we take. 7. Going forward It is only by acknowledging and exploring the constraints of isotopic analysis, not ignoring them, that we can best use it in archaeology. The approach yields most when we ask the right sorts of questions of the right sorts of specimens, and when we are confident in our data and interpretations. Best practice is attainable by thoughtful deliberation during the entire process, from design to execution and then publication of isotopic work, including transparent, fair and ethical approaches to sample and specimen collection, data generation and reporting, as has been highlighted elsewhere (Makarewicz and Sealy, 2015; Britton, 2017; Vaiglova et al., 2023; Stantis et al., 2025). We should couple best practice in the execution and publication of applied studies with fundamental research, including empirical studies on the effects of metabolism and diet composition, and analytical development particu larly apropos compound-specific isotopic analyses. Yet best practice for the field as a whole also requires conceptual reflection on what we are doing – to go beyond ‘processual’ isotopic applications. Thirty years ago, Sillen et al. (1989) warned that there would be “no more easy answers” in isotopic studies in archaeology – we need to take this to heart, to move our field forward, although it may be uncomfortable, requiring “painful refinement in the critical flame” (Clarke, 1973). Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Declaration of competing interest The author declares that she has no known competing financial in terests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This paper arose out of much reflection for many presentations over the years, crystallised into talks at the SAA session “Next Generation Archaeological Science” in 2018, convened by Marcos Martínon Torres and Robin Torrence, and the ISBA 2021 meeting, organised by Ludovic Orlando et al. – I thank them for the invitations to speak. I am also appreciative of the tolerance of all audiences who heard earlier half- baked versions of this, in Leiden, Oxford, Southampton, Cambridge, London, Aarhus, Dublin, Reading, and whose questions helped me refine what I was trying to say. Thanks to: the Dorothy Garrod Isotope gang for great discussions over the years; Enrico Crema for data analytical wisdom; Seren Griffiths for her thought-provoking talk at UKAS 2019; Clive Trueman for his many thoughts, including on the Holy Grail and the Pentangle of Doom; Jane Evans for useful discussions and for helping me to wrestle with the necessity of double negatives; the editors and the three reviewers for making this a better paper, as well as Cheryl Makarewicz, Robin Tor rence, and two pre-COVID reviewers for insightful comments and sug gestions on a previous iteration. 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