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Background: On average, educated people are healthier, wealthier and have higher life expectancy than those with less education. Numerous studies have attempted to determine whether education causes differences in later health outcomes or whether another factor ultimately causes differences in education and subsequent outcomes. Previous studies have used a range of natural experiments to provide causal evidence. Here we compare two natural experiments: a policy reform, raising the school leaving age in the UK in 1972; and Mendelian randomization., Methods: We used data from 334 974 participants of the UK Biobank, sampled between 2006 and 2010. We estimated the effect of an additional year of education on 25 outcomes, including mortality, measures of morbidity and health, ageing and income, using multivariable adjustment, the policy reform and Mendelian randomization. We used a range of sensitivity analyses and specification tests to assess the plausibility of each method's assumptions., Results: The three different estimates of the effects of educational attainment were largely consistent in direction for diabetes, stroke and heart attack, mortality, smoking, income, grip strength, height, body mass index (BMI), intelligence, alcohol consumption and sedentary behaviour. However, there was evidence that education reduced rates of moderate exercise and increased alcohol consumption. Our sensitivity analyses suggest that confounding by genotypic or phenotypic confounders or specific forms of pleiotropy are unlikely to explain our results., Conclusions: Previous studies have suggested that the differences in outcomes associated with education may be due to confounding. However, the two independent sources of exogenous variation we exploit largely imply consistent causal effects of education on outcomes later in life., (© The Author(s) 2023. Published by Oxford University Press on behalf of the International Epidemiological Association.)

Most studies of continuous health-related outcomes examine differences in mean levels (location) of the outcome by exposure. However, identifying effects on the variability (scale) of an outcome, and combining tests of mean and variability (location-and-scale), could provide additional insights into biological mechanisms. A joint test could improve power for studies of high-dimensional phenotypes, such as epigenome-wide association studies of DNA methylation at CpG sites. One possible cause of heterogeneity of variance is a variable interacting with exposure in its effect on outcome, so a joint test of mean and variability could help in the identification of effect modifiers. Here, we review a scale test, based on the Brown-Forsythe test, for analysing variability of a continuous outcome with respect to both categorical and continuous exposures, and develop a novel joint location-and-scale score (JLSsc) test. These tests were compared to alternatives in simulations and used to test associations of mean and variability of DNA methylation with gender and gestational age using data from the Accessible Resource for Integrated Epigenomics Studies (ARIES). In simulations, the Brown-Forsythe and JLSsc tests retained correct type I error rates when the outcome was not normally distributed in contrast to the other approaches tested which all had inflated type I error rates. These tests also identified > 7500 CpG sites for which either mean or variability in cord blood methylation differed according to gender or gestational age. The Brown-Forsythe test and JLSsc are robust tests that can be used to detect associations not solely driven by a mean effect., (© 2021. The Author(s).)

Estimates from Mendelian randomization studies of unrelated individuals can be biased due to uncontrolled confounding from familial effects. Here we describe methods for within-family Mendelian randomization analyses and use simulation studies to show that family-based analyses can reduce such biases. We illustrate empirically how familial effects can affect estimates using data from 61,008 siblings from the Nord-Trøndelag Health Study and UK Biobank and replicated our findings using 222,368 siblings from 23andMe. Both Mendelian randomization estimates using unrelated individuals and within family methods reproduced established effects of lower BMI reducing risk of diabetes and high blood pressure. However, while Mendelian randomization estimates from samples of unrelated individuals suggested that taller height and lower BMI increase educational attainment, these effects were strongly attenuated in within-family Mendelian randomization analyses. Our findings indicate the necessity of controlling for population structure and familial effects in Mendelian randomization studies.

Background: Smoking is the leading avoidable cause of illness and premature mortality. The first-line treatments for smoking cessation are nicotine replacement therapy and varenicline. Meta-analyses of experimental studies have shown that participants allocated to the varenicline group were 1.57 times (95% confidence interval 1.29 to 1.91 times) as likely to be abstinent 6 months after treatment as those allocated to the nicotine replacement therapy group. However, there is limited evidence about the effectiveness of varenicline when prescribed in primary care. We investigated the effectiveness and rate of adverse events of these medicines in the general population., Objective: To estimate the effect of prescribing varenicline on smoking cessation rates and health outcomes., Data Sources: Clinical Practice Research Datalink., Methods: We conducted an observational cohort study using electronic medical records from the Clinical Practice Research Datalink. We extracted data on all patients who were prescribed varenicline or nicotine replacement therapy after 1 September 2006 who were aged ≥ 18 years. We investigated the effects of varenicline on smoking cessation, all-cause mortality and cause-specific mortality and hospitalisation for: (1) chronic lung disease, (2) lung cancer, (3) coronary heart disease, (4) pneumonia, (5) cerebrovascular disease, (6) diabetes, and (7) external causes; primary care diagnosis of myocardial infarction, chronic obstructive pulmonary disease, depression, or prescription for anxiety; weight in kg; general practitioner and hospital attendance. Our primary outcome was smoking cessation 2 years after the first prescription. We investigated the baseline differences between patients prescribed varenicline and patients prescribed nicotine replacement therapy. We report results using multivariable-adjusted, propensity score and instrumental variable regression. Finally, we developed methods to assess the relative bias of the different statistical methods we used., Results: People prescribed varenicline were healthier at baseline than those prescribed nicotine replacement therapy in almost all characteristics, which highlighted the potential for residual confounding. Our instrumental variable analysis results found little evidence that patients prescribed varenicline had lower mortality 2 years after their first prescription (risk difference 0.67, 95% confidence interval -0.11 to 1.46) than those prescribed nicotine replacement therapy. They had similar rates of all-cause hospitalisation, incident primary care diagnoses of myocardial infarction and chronic obstructive pulmonary disease. People prescribed varenicline subsequently attended primary care less frequently. Patients prescribed varenicline were more likely (odds ratio 1.46, 95% confidence interval 1.42 to 1.50) to be abstinent 6 months after treatment than those prescribed nicotine replacement therapy when estimated using multivariable-adjusted for baseline covariates. Patients from more deprived areas were less likely to be prescribed varenicline. However, varenicline had similar effectiveness for these groups., Conclusion: Patients prescribed varenicline in primary care were more likely to quit smoking than those prescribed nicotine replacement therapy, but there was little evidence that they had lower rates of mortality or morbidity in the 4 years following the first prescription. There was little evidence of heterogeneity in effectiveness across the population., Future Work: Future research should investigate the decline in prescribing of smoking cessation products; develop an optimal treatment algorithm for smoking cessation; use methods for using instruments with survival outcomes; and develop methods for comparing multivariable-adjusted and instrumental variable estimates., Limitations: Not all of our code lists were validated, body mass index and Index of Multiple Deprivation had missing values, our results may suffer from residual confounding, and we had no information on treatment adherence., Trial Registration: This trial is registered as NCT02681848., Funding: This project was funded by the National Institute for Health Research (NIHR) Health Technology Assessment programme and will be published in full in Health Technology Assessment ; Vol. 24, No. 9. See the NIHR Journals Library website for further project information., Competing Interests: Neil M Davies, Amy E Taylor, Taha Itani and Marcus R Munafò report a grant from the Global Research Awards for Nicotine Dependence, which is an Independent Competitive Grants Programme supported by Pfizer Inc. (New York, NY, USA), the maker of varenicline. Marcus R Munafò reports grants from Rusan Pharma Ltd (Mumbai, India), and non-financial support from GlaxoSmithKline (Brentford, UK) outside the submitted work. Richard M Martin was a member of the Independent Scientific Advisory Committee of the Medicines and Healthcare products Regulatory Agency, which approves applications for Clinical Practice Research Datalink studies.

Objective: We conducted a prospective cohort study of the Clinical Practice Research Database to estimate rates of varenicline and nicotine replacement therapy (NRT) prescribing and the relative effects on smoking cessation, and mental health., Methods: We used multivariable logistic regression, propensity score matched regression, and instrumental variable analysis. Exposure was varenicline or NRT prescription. Mental disorders were bipolar, depression, neurotic disorder, schizophrenia, or prescriptions of antidepressants, antipsychotics, hypnotics/anxiolytics, mood stabilizers. Outcomes were smoking cessation, and incidence of neurotic disorder, depression, prescription of antidepressants, or hypnotics/anxiolytics. Follow-ups were 3, 6, and 9 months, and at 1, 2, and 4 years., Results: In all patients, NRT and varenicline prescribing declined during the study period. Seventy-eight thousand four hundred fifty-seven smokers with mental disorders aged ≥18 years were prescribed NRT (N = 59 340) or varenicline (N = 19 117) from September 1, 2006 to December 31, 2015. Compared with smokers without mental disorders, smokers with mental disorders had 31% (95% CI: 29% to 33%) lower odds of being prescribed varenicline relative to NRT, but had 19% (95% CI: 15% to 24%) greater odds of quitting at 2 years when prescribed varenicline relative to NRT. Overall, varenicline was associated with decreased or similar odds of worse mental health outcomes than NRT in patients both with and without mental disorders, although there was some variation when analyses were stratified by mental disorder subgroup., Conclusions: Smoking cessation medication prescribing may be declining in primary care. Varenicline was more effective than NRT for smoking cessation in patients with mental disorders and there is not clear consistent evidence that varenicline is adversely associated with poorer mental health outcomes., Implications: Patients with mental disorders were less likely to be prescribed varenicline than NRT. We triangulated results from three analytical techniques. We found that varenicline was more effective than NRT for smoking cessation in patients with mental disorders. Varenicline was generally associated with similar or decreased odds of poorer mental health outcomes (ie, improvements in mental health) when compared with NRT. We report these findings cautiously as our data are observational and are at risk of confounding., (© The Author(s) 2019. Published by Oxford University Press on behalf of the Society for Research on Nicotine and Tobacco.)

Background: Mendelian randomization (MR) is a powerful tool in epidemiology that can be used to estimate the causal effect of an exposure on an outcome in the presence of unobserved confounding, by utilizing genetic variants that are instrumental variables (IVs) for the exposure. This has been extended to multivariable MR (MVMR) to estimate the effect of two or more exposures on an outcome., Methods and Results: We use simulations and theory to clarify the interpretation of estimated effects in a MVMR analysis under a range of underlying scenarios, where a secondary exposure acts variously as a confounder, a mediator, a pleiotropic pathway and a collider. We then describe how instrument strength and validity can be assessed for an MVMR analysis in the single-sample setting, and develop tests to assess these assumptions in the popular two-sample summary data setting. We illustrate our methods using data from UK Biobank to estimate the effect of education and cognitive ability on body mass index., Conclusion: MVMR analysis consistently estimates the direct causal effect of an exposure, or exposures, of interest and provides a powerful tool for determining causal effects in a wide range of scenarios with either individual- or summary-level data., (© The Author(s) 2018. Published by Oxford University Press on behalf of the International Epidemiological Association.)

We investigate the behavior of the Lasso for selecting invalid instruments in linear instrumental variables models for estimating causal effects of exposures on outcomes, as proposed recently by Kang et al. Invalid instruments are such that they fail the exclusion restriction and enter the model as explanatory variables. We show that for this setup, the Lasso may not consistently select the invalid instruments if these are relatively strong. We propose a median estimator that is consistent when less than 50% of the instruments are invalid, and its consistency does not depend on the relative strength of the instruments, or their correlation structure. We show that this estimator can be used for adaptive Lasso estimation, with the resulting estimator having oracle properties. The methods are applied to a Mendelian randomization study to estimate the causal effect of body mass index (BMI) on diastolic blood pressure, using data on individuals from the UK Biobank, with 96 single nucleotide polymorphisms as potential instruments for BMI. Supplementary materials for this article are available online., (© 2018 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.)

Aims: To investigate whether smokers prescribed varenicline had lower risks of serious ill-health during the 4 years following treatment compared with those prescribed nicotine replacement therapy (NRT)., Design: Observational cohort study of electronic medical records., Setting: A total of 370 UK general practices sampled from the Clinical Practice Research Datalink., Participants: A total of 126 718 patients aged 18 and over who were issued smoking cessation prescriptions between 1 September 2006 and 31 March 2014., Measurements: Our primary outcome was all-cause mortality within 2 years of first prescription as indicated by linked Office of National Statistics data. Our secondary outcomes were cause-specific mortality, all-cause, cause-specific hospitalization, primary care diagnosis of myocardial infarction or chronic obstructive pulmonary disease (COPD), body mass index and attendance rate to primary care within 2 years of first prescription. Risk differences and 95% confidence intervals were estimated by multivariable adjusted regression and propensity score matched regression. We used instrumental variable analysis to overcome residual confounding., Findings: People prescribed varenicline were healthier at baseline than those prescribed NRT in almost all characteristics, highlighting the potential for residual confounding. Our instrumental variable analysis results found that people prescribed varenicline had a similar risk of mortality at 2 years [risk difference per 100 patients treated = 0.67, 95% confidence interval (CI) = -0.11 to 1.46)] to those prescribed NRT, and there were similar rates of all-cause hospitalization, incident primary-care diagnoses of myocardial infarction and COPD. People prescribed varenicline subsequently attended primary care less frequently., Conclusions: Smokers prescribed varenicline in primary care in the United Kingdom do not appear to be less likely to die, be hospitalized or experience a myocardial infarction or chronic obstructive pulmonary disease during the following 2 years compared with smokers prescribed nicotine replacement therapy, but they gain more weight and attend primary care less frequently., (© 2018 The Authors. Addiction published by John Wiley & Sons Ltd on behalf of Society for the Study of Addiction.)

Competing Interests: Conflicts of interest We report no conflicts of interest.

Background: There is limited evidence about the effectiveness of varenicline and nicotine replacement therapy (NRT) for long-term smoking cessation in primary care, or whether the treatment effectiveness differs by socioeconomic position (SEP). Therefore, we estimated the long-term effectiveness of varenicline versus NRT (> 2 years) on smoking cessation, and investigated whether effectiveness differs by SEP., Methods: This is a prospective cohort study of electronic medical records from 654 general practices in England, within the Clinical Practice Research Datalink, using three different analytical methods: multivariable logistic regression, propensity score matching and instrumental variable analyses. Exposure was prescription of varenicline versus NRT, and the primary outcome was smoking cessation at 2 years' follow-up; outcome was also assessed at 3, 6, and 9 months, and at 1 and 4 years after exposure. SEP was defined using the Index of Multiple Deprivation., Results: At 2 years, 28.8% (N = 20 362/70 610) of participants prescribed varenicline and 24.3% (N = 36 268/149 526) of those prescribed NRT quit; adjusted odds ratio was 1.26 [95% confidence interval (CI): 1.23 to 1.29], P < 0.0001. The association persisted for up to 4 years and was consistent across all analyses. We found little evidence that the effectiveness of varenicline differed greatly by SEP. However, patients from areas of higher deprivation were less likely to be prescribed varenicline; adjusted odds ratio was 0.91 (95% CI: 0.90 to 0.92), P < 0.0001., Conclusions: Patients prescribed varenicline were more likely to be abstinent up to 4 years after first prescription than those prescribed NRT. In combination with other evidence, the results from this study may be used to update clinical guidelines on the use of varenicline for smoking cessation., (© The Author 2017. Published by Oxford University Press on behalf of the International Epidemiological Association)

There is increasing interest in the use of instrumental variable analysis to overcome unmeasured confounding in observational pharmacoepidemiological studies. This is partly because instrumental variable analyses are potentially less biased than conventional regression analyses. However, instrumental variable analyses are less precise, and regulators and clinicians find it difficult to interpret conflicting evidence from instrumental variable compared with conventional regression analyses. In this paper, we describe three techniques to assess which approach (instrumental variable versus conventional regression analyses) is least biased. These techniques are negative control outcomes, negative control populations and tests of covariate balance. We illustrate these methods using an analysis of the effects of smoking cessation therapies (varenicline) prescribed in primary care., (© The Author 2017. Published by Oxford University Press on behalf of the International Epidemiological Association.)

Background: Instrumental variable analysis, for example with physicians' prescribing preferences as an instrument for medications issued in primary care, is an increasingly popular method in the field of pharmacoepidemiology. Existing power calculators for studies using instrumental variable analysis, such as Mendelian randomization power calculators, do not allow for the structure of research questions in this field. This is because the analysis in pharmacoepidemiology will typically have stronger instruments and detect larger causal effects than in other fields. Consequently, there is a need for dedicated power calculators for pharmacoepidemiological research., Methods and Results: The formula for calculating the power of a study using instrumental variable analysis in the context of pharmacoepidemiology is derived before being validated by a simulation study. The formula is applicable for studies using a single binary instrument to analyse the causal effect of a binary exposure on a continuous outcome. An online calculator, as well as packages in both R and Stata, are provided for the implementation of the formula by others., Conclusions: The statistical power of instrumental variable analysis in pharmacoepidemiological studies to detect a clinically meaningful treatment effect is an important consideration. Research questions in this field have distinct structures that must be accounted for when calculating power. The formula presented differs from existing instrumental variable power formulae due to its parametrization, which is designed specifically for ease of use by pharmacoepidemiologists., (© The Author 2017. Published by Oxford University Press on behalf of the International Epidemiological Association)

The Two-Sample Two-Stage Least Squares (TS2SLS) data combination estimator is a popular estimator for the parameters in linear models when not all variables are observed jointly in one single data set. Although the limiting normal distribution has been established, the asymptotic variance formula has only been stated explicitly in the literature for the case of conditional homoskedasticity. By using the fact that the TS2SLS estimator is a function of reduced form and first-stage OLS estimators, we derive the variance of the limiting normal distribution under conditional heteroskedasticity. A robust variance estimator is obtained, which generalises to cases with more general patterns of variable (non-)availability. Stata code and some Monte Carlo results are provided in an Appendix. Stata code for a nonlinear GMM estimator that is identical to the TS2SLS estimator in just identified models and asymptotically equivalent to the TS2SLS estimator in overidentified models is also provided there.

Despite condoms being cheap and effective in preventing HIV, there remains an 8billion shortfall in condom use in risky sex-acts. Social marketing organisations apply private sector marketing approaches to sell public health products. This paper investigates the impact of marketing tools, including promotion and pricing, on demand for male and female condoms in 52 countries between 1997 and 2009. A static model differentiates drivers of demand between products, while a dynamic panel data estimator estimates their short- and long-run impacts. Products are not equally affected: female condoms are not affected by advertising, but highly affected by interpersonal communication and HIV prevalence. Price and promotion have significant short- and long-run effects, with female condoms far more sensitive to price than male condoms. The design of optimal distribution strategies for new and existing HIV prevention technologies must consider both product and target population characteristics., (Copyright © 2016 The Authors. Published by Elsevier B.V. All rights reserved.)

We consider testing for weak instruments in a model with multiple endogenous variables. Unlike Stock and Yogo (2005), who considered a weak instruments problem where the rank of the matrix of reduced form parameters is near zero, here we consider a weak instruments problem of a near rank reduction of one in the matrix of reduced form parameters. For example, in a two-variable model, we consider weak instrument asymptotics of the form [Formula: see text] where [Formula: see text] and [Formula: see text] are the parameters in the two reduced-form equations, [Formula: see text] is a vector of constants and [Formula: see text] is the sample size. We investigate the use of a conditional first-stage [Formula: see text]-statistic along the lines of the proposal by Angrist and Pischke (2009) and show that, unless [Formula: see text], the variance in the denominator of their [Formula: see text]-statistic needs to be adjusted in order to get a correct asymptotic distribution when testing the hypothesis [Formula: see text]. We show that a corrected conditional [Formula: see text]-statistic is equivalent to the Cragg and Donald (1993) minimum eigenvalue rank test statistic, and is informative about the maximum total relative bias of the 2SLS estimator and the Wald tests size distortions. When [Formula: see text] in the two-variable model, or when there are more than two endogenous variables, further information over and above the Cragg-Donald statistic can be obtained about the nature of the weak instrument problem by computing the conditional first-stage [Formula: see text]-statistics.

The use of genetic markers as instrumental variables (IV) is receiving increasing attention from economists, statisticians, epidemiologists and social scientists. Although IV is commonly used in economics, the appropriate conditions for the use of genetic variants as instruments have not been well defined. The increasing availability of biomedical data, however, makes understanding of these conditions crucial to the successful use of genotypes as instruments. We combine the econometric IV literature with that from genetic epidemiology, and discuss the biological conditions and IV assumptions within the statistical potential outcomes framework. We review this in the context of two illustrative applications., (Copyright © 2015 The Authors. Published by Elsevier B.V. All rights reserved.)

We investigated the role of common genetic variation in educational attainment and household income. We used data from 5,458 participants of the National Child Development Study to estimate: 1) the associations of rs9320913, rs11584700 and rs4851266 and socioeconomic position and educational phenotypes; and 2) the univariate chip-heritability of each phenotype, and the genetic correlation between each phenotype and educational attainment at age 16. The three SNPs were associated with most measures of educational attainment. Common genetic variation contributed to 6 of 14 socioeconomic background phenotypes, and 17 of 29 educational phenotypes. We found evidence of genetic correlations between educational attainment at age 16 and 4 of 14 social background and 8 of 28 educational phenotypes. This suggests common genetic variation contributes both to differences in educational attainment and its relationship with other phenotypes. However, we remain cautious that cryptic population structure, assortative mating, and dynastic effects may influence these associations.

Introduction: Smoking is a major avoidable cause of ill-health and premature death. Treatments that help patients successfully quit smoking have an important effect on health and life expectancy. Varenicline is a medication that can help smokers successfully quit smoking. However, there are concerns that it may cause adverse effects, such as increase in the occurrence of depression, self-harm and suicide and cardiovascular disease. In this study we aim to examine the effects of varenicline versus other smoking cessation pharmacotherapies on smoking cessation, health service use, all-cause and cause-specific mortality and physical and mental health conditions., Methods: In this project we will investigate the effects of varenicline compared to nicotine replacement therapies on: (1) long-term smoking cessation and whether these effects differ by area level deprivation; and (2) the following clinically-important outcomes: rate of general practice and hospital attendance; all-cause mortality and death due to diseases of the respiratory system and cardiovascular disease; and a primary care diagnosis of respiratory illness, myocardial infarction or depression and anxiety. The study is based on a cohort of patients prescribed these smoking cessation medications from the Clinical Practice Research Datalink (CPRD). We will use three methods to overcome confounding: multivariable adjusted Cox regression, propensity score matched Cox regression, and instrumental variable regression. The total expected sample size for analysis will be at least 180,000. Follow-up will end with the earliest of either an 'event' or censoring due to the end of registration or death., Ethics and Dissemination: Ethics approval was not required for this study. This project has been approved by the CPRD's Independent Scientific Advisory Committee (ISAC). We will disseminate our findings via publications in international peer-reviewed journals and presentations at international conferences., (Published by the BMJ Publishing Group Limited. For permission to use (where not already granted under a licence) please go to http://www.bmj.com/company/products-services/rights-and-licensing/)

Instrumental variable estimates of causal effects can be biased when using many instruments that are only weakly associated with the exposure. We describe several techniques to reduce this bias and estimate corrected standard errors. We present our findings using a simulation study and an empirical application. For the latter, we estimate the effect of height on lung function, using genetic variants as instruments for height. Our simulation study demonstrates that, using many weak individual variants, two-stage least squares (2SLS) is biased, whereas the limited information maximum likelihood (LIML) and the continuously updating estimator (CUE) are unbiased and have accurate rejection frequencies when standard errors are corrected for the presence of many weak instruments. Our illustrative empirical example uses data on 3631 children from England. We used 180 genetic variants as instruments and compared conventional ordinary least squares estimates with results for the 2SLS, LIML, and CUE instrumental variable estimators using the individual height variants. We further compare these with instrumental variable estimates using an unweighted or weighted allele score as single instruments. In conclusion, the allele scores and CUE gave consistent estimates of the causal effect. In our empirical example, estimates using the allele score were more efficient. CUE with corrected standard errors, however, provides a useful additional statistical tool in applications with many weak instruments. The CUE may be preferred over an allele score if the population weights for the allele score are unknown or when the causal effects of multiple risk factors are estimated jointly., (© 2014 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.)

The most widely used measure of segregation is the so-called dissimilarity index. It is now well understood that this measure also reflects randomness in the allocation of individuals to units (i.e. it measures deviations from evenness, not deviations from randomness). This leads to potentially large values of the segregation index when unit sizes and/or minority proportions are small, even if there is no underlying systematic segregation. Our response to this is to produce adjustments to the index, based on an underlying statistical model. We specify the assignment problem in a very general way, with differences in conditional assignment probabilities underlying the resulting segregation. From this, we derive a likelihood ratio test for the presence of any systematic segregation, and bias adjustments to the dissimilarity index. We further develop the asymptotic distribution theory for testing hypotheses concerning the magnitude of the segregation index and show that the use of bootstrap methods can improve the size and power properties of test procedures considerably. We illustrate these methods by comparing dissimilarity indices across school districts in England to measure social segregation.