A lean additive frailty model: With an application to clustering of melanoma in Norwegian families.

Additive frailty models are used to model correlated survival data. However, the complexity of the models increases with cluster size to the extent that practical usage becomes increasingly challenging. We present a modification of the additive genetic gamma frailty (AGGF) model, the lean AGGF (L-AGGF) model, which alleviates some of these challenges by using a leaner additive decomposition of the frailty. The performances of the models were compared and evaluated in a simulation study. The L-AGGF model was used to analyze population-wide data on clustering of melanoma in 2 391 125 two-generational Norwegian families, 1960-2015. Using this model, we could analyze the complete data set, while the original model limited the analysis to a restricted data set (with cluster sizes ≤ 7 $$ \le 7 $$ ). We found a substantial clustering of melanoma in Norwegian families and large heterogeneity in melanoma risk across the population, where 52% of the frailty was attributed to the 10% of the population at highest unobserved risk. Due to the improved scalability, the L-AGGF model enables a wider range of analyses of population-wide data compared to the AGGF model. Moreover, the methods outlined here make it possible to perform these analyses in a computationally efficient manner.

Imaging predictors of response to cardiac resynchronization therapy: left ventricular work asymmetry by echocardiography and septal viability by cardiac magnetic resonance.

AIMS: Left ventricular (LV) failure in left bundle branch block is caused by loss of septal function and compensatory hyperfunction of the LV lateral wall (LW) which stimulates adverse remodelling. This study investigates if septal and LW function measured as myocardial work, alone and combined with assessment of septal viability, identifies responders to cardiac resynchronization therapy (CRT). METHODS AND RESULTS: In a prospective multicentre study of 200 CRT recipients, myocardial work was measured by pressure-strain analysis and viability by cardiac magnetic resonance (CMR) imaging (n = 125). CRT response was defined as ≥15% reduction in LV end-systolic volume after 6 months. Before CRT, septal work was markedly lower than LW work (P < 0.0001), and the difference was largest in CRT responders (P < 0.001). Work difference between septum and LW predicted CRT response with area under the curve (AUC) 0.77 (95% CI: 0.70-0.84) and was feasible in 98% of patients. In patients undergoing CMR, combining work difference and septal viability significantly increased AUC to 0.88 (95% CI: 0.81-0.95). This was superior to the predictive power of QRS morphology, QRS duration and the echocardiographic parameters septal flash, apical rocking, and systolic stretch index. Accuracy was similar for the subgroup of patients with QRS 120-150 ms as for the entire study group. Both work difference alone and work difference combined with septal viability predicted long-term survival without heart transplantation with hazard ratio 0.36 (95% CI: 0.18-0.74) and 0.21 (95% CI: 0.072-0.61), respectively. CONCLUSION: Assessment of myocardial work and septal viability identified CRT responders with high accuracy.

A hybrid landmark Aalen-Johansen estimator for transition probabilities in partially non-Markov multi-state models.

Multi-state models are increasingly being used to model complex epidemiological and clinical outcomes over time. It is common to assume that the models are Markov, but the assumption can often be unrealistic. The Markov assumption is seldomly checked and violations can lead to biased estimation of many parameters of interest. This is a well known problem for the standard Aalen-Johansen estimator of transition probabilities and several alternative estimators, not relying on the Markov assumption, have been suggested. A particularly simple approach known as landmarking have resulted in the Landmark-Aalen-Johansen estimator. Since landmarking is a stratification method a disadvantage of landmarking is data reduction, leading to a loss of power. This is problematic for "less traveled" transitions, and undesirable when such transitions indeed exhibit Markov behaviour. Introducing the concept of partially non-Markov multi-state models, we suggest a hybrid landmark Aalen-Johansen estimator for transition probabilities. We also show how non-Markov transitions can be identified using a testing procedure. The proposed estimator is a compromise between regular Aalen-Johansen and landmark estimation, using transition specific landmarking, and can drastically improve statistical power. We show that the proposed estimator is consistent, but that the traditional variance estimator can underestimate the variance of both the hybrid and landmark estimator. Bootstrapping is therefore recommended. The methods are compared in a simulation study and in a real data application using registry data to model individual transitions for a birth cohort of 184 951 Norwegian men between states of sick leave, disability, education, work and unemployment.

Time-dependent mediators in survival analysis: Modeling direct and indirect effects with the additive hazards model.

We discuss causal mediation analyses for survival data and propose a new approach based on the additive hazards model. The emphasis is on a dynamic point of view, that is, understanding how the direct and indirect effects develop over time. Hence, importantly, we allow for a time varying mediator. To define direct and indirect effects in such a longitudinal survival setting we take an interventional approach (Didelez, 2018) where treatment is separated into one aspect affecting the mediator and a different aspect affecting survival. In general, this leads to a version of the nonparametric g-formula (Robins, 1986). In the present paper, we demonstrate that combining the g-formula with the additive hazards model and a sequential linear model for the mediator process results in simple and interpretable expressions for direct and indirect effects in terms of relative survival as well as cumulative hazards. Our results generalize and formalize the method of dynamic path analysis (Fosen, Ferkingstad, Borgan, & Aalen, 2006; Strohmaier et al., 2015). An application to data from a clinical trial on blood pressure medication is given.

The surprising implications of familial association in disease risk.

BACKGROUND: A wide range of diseases show some degree of clustering in families; family history is therefore an important aspect for clinicians when making risk predictions. Familial aggregation is often quantified in terms of a familial relative risk (FRR), and although at first glance this measure may seem simple and intuitive as an average risk prediction, its implications are not straightforward. METHODS: We use two statistical models for the distribution of disease risk in a population: a dichotomous risk model that gives an intuitive understanding of the implication of a given FRR, and a continuous risk model that facilitates a more detailed computation of the inequalities in disease risk. Published estimates of FRRs are used to produce Lorenz curves and Gini indices that quantifies the inequalities in risk for a range of diseases. RESULTS: We demonstrate that even a moderate familial association in disease risk implies a very large difference in risk between individuals in the population. We give examples of diseases for which this is likely to be true, and we further demonstrate the relationship between the point estimates of FRRs and the distribution of risk in the population. CONCLUSIONS: The variation in risk for several severe diseases may be larger than the variation in income in many countries. The implications of familial risk estimates should be recognized by epidemiologists and clinicians.

Exploring Selection Bias by Causal Frailty Models: The Magnitude Matters.

Counter-intuitive associations appear frequently in epidemiology, and these results are often debated. In particular, several scenarios are characterized by a general risk factor that appears protective in particular subpopulations, for example, individuals suffering from a specific disease. However, the associations are not necessarily representing causal effects. Selection bias due to conditioning on a collider may often be involved, and causal graphs are widely used to highlight such biases. These graphs, however, are qualitative, and they do not provide information on the real life relevance of a spurious association. Quantitative estimates of such associations can be obtained from simple statistical models. In this study, we present several paradoxical associations that occur in epidemiology, and we explore these associations in a causal, frailty framework. By using frailty models, we are able to put numbers on spurious effects that often are neglected in epidemiology. We discuss several counter-intuitive findings that have been reported in real life analyses, and we present calculations that may expand the understanding of these associations. In particular, we derive novel expressions to explain the magnitude of bias in index-event studies.

Prostate-specific antigen testing for prostate cancer: Depleting a limited pool of susceptible individuals?

After the introduction of the prostate specific antigen (PSA) test in the 1980s, a sharp increase in the incidence rate of prostate cancer was seen in the United States. The age-specific incidence patterns exhibited remarkable shifts to younger ages, and declining rates were observed at old ages. Similar trends were seen in Norway. We investigate whether these features could, in combination with PSA testing, be explained by a varying degree of susceptibility to prostate cancer in the populations. We analyzed incidence data from the United States' Surveillance, Epidemiology, and End Results program for 1973-2010, comprising 511,027 prostate cancers in men ≥40 years old, and Norwegian national incidence data for 1953-2011, comprising 113,837 prostate cancers in men ≥50 years old. We developed a frailty model where only a proportion of the population could develop prostate cancer, and where the increased risk of diagnosis due to the massive use of PSA testing was modelled by encompassing this heterogeneity in risk. The frailty model fits the observed data well, and captures the changing age-specific incidence patterns across birth cohorts. The susceptible proportion of men is [Formula: see text] in the United States and [Formula: see text] in Norway. Cumulative incidence rates at old age are unchanged across birth cohort exposed to PSA testing at younger and younger ages. The peaking cohort-specific age-incidence curves of prostate cancer may be explained by the underlying heterogeneity in prostate cancer risk. The introduction of the PSA test has led to a larger number of diagnosed men. However, no more cases are being diagnosed in total in birth cohorts exposed to the PSA era at younger and younger ages, even though they are diagnosed at younger ages. Together with the earlier peak in the age-incidence curves for younger cohorts, and the strong familial association of the cancer, this constitutes convincing evidence that the PSA test has led to a higher proportion, and an earlier timing, of diagnoses in a limited pool of susceptible individuals.

Dynamic path analysis - a useful tool to investigate mediation processes in clinical survival trials.

When it comes to clinical survival trials, regulatory restrictions usually require the application of methods that solely utilize baseline covariates and the intention-to-treat principle. Thereby, much potentially useful information is lost, as collection of time-to-event data often goes hand in hand with collection of information on biomarkers and other internal time-dependent covariates. However, there are tools to incorporate information from repeated measurements in a useful manner that can help to shed more light on the underlying treatment mechanisms. We consider dynamic path analysis, a model for mediation analysis in the presence of a time-to-event outcome and time-dependent covariates to investigate direct and indirect effects in a study of different lipid-lowering treatments in patients with previous myocardial infarctions. Further, we address the question whether survival in itself may produce associations between the treatment and the mediator in dynamic path analysis and give an argument that because of linearity of the assumed additive hazard model, this is not the case. We further elaborate on our view that, when studying mediation, we are actually dealing with underlying processes rather than single variables measured only once during the study period. This becomes apparent in results from various models applied to the study of lipid-lowering treatments as well as our additionally conducted simulation study, where we clearly observe that discarding information on repeated measurements can lead to potentially erroneous conclusions.

Causal inference in multi-state models-sickness absence and work for 1145 participants after work rehabilitation.

BACKGROUND: Multi-state models, as an extension of traditional models in survival analysis, have proved to be a flexible framework for analysing the transitions between various states of sickness absence and work over time. In this paper we study a cohort of work rehabilitation participants and analyse their subsequent sickness absence using Norwegian registry data on sickness benefits. Our aim is to study how detailed individual covariate information from questionnaires explain differences in sickness absence and work, and to use methods from causal inference to assess the effect of interventions to reduce sickness absence. Examples of the latter are to evaluate the use of partial versus full time sick leave and to estimate the effect of a cooperation agreement on a more inclusive working life. METHODS: Covariate adjusted transition intensities are estimated using Cox proportional hazards and Aalen additive hazards models, while the effect of interventions are assessed using methods of inverse probability weighting and G-computation. RESULTS: Results from covariate adjusted analyses show great differences in sickness absence and work for patients with assumed high risk and low risk covariate characteristics, for example based on age, type of work, income, health score and type of diagnosis. Causal analyses show small effects of partial versus full time sick leave and a positive effect of having a cooperation agreement, with about 5 percent points higher probability of returning to work. CONCLUSIONS: Detailed covariate information is important for explaining transitions between different states of sickness absence and work, also for patient specific cohorts. Methods for causal inference can provide the needed tools for going from covariate specific estimates to population average effects in multi-state models, and identify causal parameters with a straightforward interpretation based on interventions.

Does Cox analysis of a randomized survival study yield a causal treatment effect?

Statistical methods for survival analysis play a central role in the assessment of treatment effects in randomized clinical trials in cardiovascular disease, cancer, and many other fields. The most common approach to analysis involves fitting a Cox regression model including a treatment indicator, and basing inference on the large sample properties of the regression coefficient estimator. Despite the fact that treatment assignment is randomized, the hazard ratio is not a quantity which admits a causal interpretation in the case of unmodelled heterogeneity. This problem arises because the risk sets beyond the first event time are comprised of the subset of individuals who have not previously failed. The balance in the distribution of potential confounders between treatment arms is lost by this implicit conditioning, whether or not censoring is present. Thus while the Cox model may be used as a basis for valid tests of the null hypotheses of no treatment effect if robust variance estimates are used, modeling frameworks more compatible with causal reasoning may be preferrable in general for estimation.

A hierarchical frailty model for familial testicular germ-cell tumors.

Using a 2-level hierarchical frailty model, we analyzed population-wide data on testicular germ-cell tumor (TGCT) status in 1,135,320 two-generational Norwegian families to examine the risk of TGCT in family members of patients. Follow-up extended from 1954 (cases) or 1960 (unaffected persons) to 2008. The first-level frailty variable was compound Poisson-distributed. The underlying Poisson parameter was randomized to model the frailty variation between families and was decomposed additively to characterize the correlation structure within a family. The frailty relative risk (FRR) for a son, given a diseased father, was 4.03 (95% confidence interval (CI): 3.12, 5.19), with a borderline significantly higher FRR for nonseminoma than for seminoma (P = 0.06). Given 1 affected brother, the lifetime FRR was 5.88 (95% CI: 4.70, 7.36), with no difference between subtypes. Given 2 affected brothers, the FRR was 21.71 (95% CI: 8.93, 52.76). These estimates decreased with the number of additional healthy brothers. The estimated FRRs support previous findings. However, the present hierarchical frailty approach allows for a very precise definition of familial risk. These FRRs, estimated according to numbers of affected/nonaffected family members, provide new insight into familial TGCT. Furthermore, new light is shed on the different familial risks of seminoma and nonseminoma.

Early fetal binge alcohol exposure predicts high behavioral symptom scores in 5.5-year-old children.

BACKGROUND: Fetal binge alcohol exposure has been associated with neurobehavioral and cognitive symptoms. This study explored whether binge drinking mainly before recognition of pregnancy predicted high symptom scores on the Strengths and Difficulties Questionnaire (SDQ) in 5.5-year-old children. METHODS: In a population-based, longitudinal study representative of pregnant women in Oslo, Norway, questionnaires were answered at 17 and 30 weeks of pregnancy, 6 months after term, and at child age 5.5 years (n = 1,116, constituting 66% of the original cohort). Logistic regression analyses identified factors predicting high SDQ scores, and multiple regression analyses identified direct effects on the SDQ Total. RESULTS: Binge exposure (≥5 standard units per occasion [SUpo]) during pregnancy week 0 to 6, that is, 0 to 4 weeks after conception, predicted scores in the Abnormal and Borderline range on the SDQ in 5.5-year-olds, after adjusting for other confounding variables. Very early binge exposure less often than once a week predicted high symptom scores on the SDQ Total (p = 0.05) and Hyperactivity/Inattention (significant), while exposure at least once a week demonstrated a 3- to 5-fold significant increase in high symptom scores on Total, Emotional, and Conduct problems. Reporting ≥8 SUpo had stronger predictive power than reporting 5 to 7 SUpo. The results were not explained by participants reporting major lifetime depression. Other predictive factors, although weaker, were maternal symptoms of depression and anxiety during the child's infancy. High education (mother and father), high income (maternal partner), higher child birth weight, and child female sex reduced the likelihood of high SDQ symptom scores. Path analysis demonstrated early binge exposure to have a direct effect on the SDQ Total score. CONCLUSIONS: Binge drinking up to 4 weeks after conception had a strong and direct predictive effect on SDQ symptom scores in 5.5-year-olds. These results strongly support the advice to avoid binge drinking when planning pregnancy.

A multi-state model to estimate incidence of heroin use.

BACKGROUND: Existing incidence estimates of heroin use are usually based on one information source. This study aims to incorporate more sources to estimate heroin use incidence trends in Spain between 1971 and 2005. METHODS: A multi-state model was constructed, whereby the initial state "heroin consumer" is followed by transition to either "admitted to first treatment" or to "left heroin use" (i.e. permanent cessation or death). Heroin use incidence and probabilities of entering first treatment ever were estimated following a back-calculation approach. RESULTS: The highest heroin use incidence rates in Spain, around 1.5 per 1,000 inhabitants aged 10-44, occurred between 1985 and 1990; subdividing by route of administration reveals higher incidences of injection between 1980 and 1985 (a mean of 0.62 per 1.000) and a peak for non-injectors in 1990 (0.867 per 1,000). CONCLUSIONS: A simple conceptual model for heroin users' trajectories related to treatment admission, provided a broader view of the historical trend of heroin use incidence in Spain.

Armitage lecture 2010: Understanding treatment effects: the value of integrating longitudinal data and survival analysis.

There is a single-minded focus on events in survival analysis, and we often ignore longitudinal data that are collected together with the event data. This is due to a lack of methodology but also a result of the artificial distinction between survival and longitudinal data analyses. Understanding the dynamics of such processes is important but has been hampered by a lack of appreciation of the difference between confirmatory and exploratory causal inferences. The latter represents an attempt at elucidating mechanisms by applying mediation analysis to statistical data and will usually be of a more tentative character than a confirmatory analysis. The concept of local independence and the associated graphs are useful. This is related to Granger causality, an important method from econometrics that is generally undervalued by statisticians. This causality concept is different from the counterfactual one since it lacks lacks the intervention aspect. The notion that one can intervene at will in naturally occurring processes, which seems to underly much of modern causal inference, is problematic when studying mediation and mechanisms. It is natural to assume a stochastic process point of view when analyzing dynamic relationships. We present some examples to illustrate this. It is not clear how survival analysis must be developed to handle the complex life-history data that are increasingly being collected today. We give some suggestions.

Frailty modeling of age-incidence curves of osteosarcoma and Ewing sarcoma among individuals younger than 40 years.

The Armitage-Doll model with random frailty can fail to describe incidence rates of rare cancers influenced by an accelerated biological mechanism at some, possibly short, period of life. We propose a new model to account for this influence. Osteosarcoma and Ewing sarcoma are primary bone cancers with characteristic age-incidence patterns that peak in adolescence. We analyze Surveillance, Epidemiology and End Result program incidence data for whites younger than 40 years diagnosed during the period 1975-2005, with an Armitage-Doll model with compound Poisson frailty. A new model treating the adolescent growth spurt as the accelerated mechanism affecting cancer development is a significant improvement over that model. We also model the incidence rate conditioning on the event of having developed the cancers before the age of 40 years and compare the results with those predicted by the Armitage-Doll model. Our results support existing evidence of an underlying susceptibility for the two cancers among a very small proportion of the population. In addition, the modeling results suggest that susceptible individuals with a rapid growth spurt acquire the cancers sooner than they otherwise would have if their growth had been slower. The new model is suitable for modeling incidence rates of rare diseases influenced by an accelerated biological mechanism.

A dynamic approach for reconstructing missing longitudinal data using the linear increments model.

Missing observations are commonplace in longitudinal data. We discuss how to model and analyze such data in a dynamic framework, that is, taking into consideration the time structure of the process and the influence of the past on the present and future responses. An autoregressive model is used as a special case of the linear increments model defined by Farewell (2006. Linear models for censored data, [PhD Thesis]. Lancaster University) and Diggle and others (2007. Analysis of longitudinal data with drop-out: objectives, assumptions and a proposal. Journal of the Royal Statistical Society, Series C (Applied Statistics, 56, 499-550). We wish to reconstruct responses for missing data and discuss the required assumptions needed for both monotone and nonmonotone missingness. The computational procedures suggested are very simple and easily applicable. They can also be used to estimate causal effects in the presence of time-dependent confounding. There are also connections to methods from survival analysis: The Aalen-Johansen estimator for the transition matrix of a Markov chain turns out to be a special case. Analysis of quality of life data from a cancer clinical trial is analyzed and presented. Some simulations are given in the supplementary material available at Biostatistics online.