We study causal inference in randomized experiments (or quasi-experiments) following a 2 x 2 factorial design. There are two treatments, denoted A and B, and units are randomly assigned to one of four categories: treatment A alone, treatment B alone, joint treatment, or none. Allowing for endogenous non-compliance with the two binary instruments representing the intended assignment, as well as unrestricted interference across the two treatments, we derive the causal interpretation of various instrumental variable estimands under more general compliance conditions than in the literature. In general, if treatment takeup is driven by both instruments for some units, it becomes difficult to separate treatment interaction from treatment effect heterogeneity. We provide auxiliary conditions and various bounding strategies that may help zero in on causally interesting parameters. We apply our results to a program randomly offering two different treatments to first-year college students, namely, tutoring and financial incentives, in order to assess the effect of the treatments on academic performance.

Chapter 1 (Caliper Matching): Caliper matching is used to estimate causal effects of a binary treatment from observational data by comparing matched treated and control units. Units are matched when their propensity scores, the conditional probability of receiving treatment given pretreatment covariates, are within a certain distance called caliper. So far, theoretical results on caliper matching are lacking, leaving practitioners with ad-hoc caliper choices and inference procedures. We bridge this gap by proposing a caliper that balances the quality and the number of matches. We prove that the resulting estimator of the average treatment effect, and average treatment effect on the treated, is asymptotically unbiased and normal at parametric rate. We describe the conditions under which semiparametric efficiency is obtainable, and show that when the parametric propensity score is estimated, the variance is increased for both estimands. Finally, we construct asymptotic confidence intervals for the two estimands.

Chapter 2 (Combining Experimental And Observational Data: The APOLLO Trial): In the APOLLO trial (Tol et al., 2022, 2024), we inferred the causal effects of two hip fracture treatments, Posterolateral Approach (PLA) and Direct Lateral Approach (DLA), on health outcomes of patients in the Netherlands. The starting point of the inference was a Randomised Experiment (RE), where patients were randomly assigned to PLA or DLA, independently of their baseline characteristics. In addition, data from a `Natural Experiment' (NE, or observational data) were also collected, under the plausible assumption that therein the allocation to PLA or DLA can be considered as good as random conditional on the patients' baseline characteristics. We estimated the average treatment effects of DLA versus PLA in the RE and the NE data separately, using a flexible and asymptotically efficient estimation strategy. We found no significant difference between PLA and DLA in any of the RE or the NE datasets. To improve the precision of the inference by increasing the sample size, we tested whether the RE and the NE datasets can be combined. Having found no evidence against combination, we estimated the average treatment effect on the combined dataset as well. Despite the improved precision, there was still no significant difference between PLA and DLA. Our conclusions were weakened by missing data, but they proved to be robust to our approach in handling the missingness and to our estimation strategy.

Chapter 3 (Private Double Robust Inference): Privacy mechanisms preserve the privacy of individuals in a sample by injecting noise into their sensitive data in a controlled manner, revealing only the noisy, privatised data to the statistician for inference purposes. The inference of a parameter exhibits a rate double robustness property when the large-sample bias of an estimator of the parameter is characterised by the product of the estimation errors of two other, auxiliary (or nuisance), often infinite-dimensional, parameters. We propose a novel class of rate double robust parameters whose novelty lies in the potentially nonlinear but smooth dependence on a low-dimensional regression parameter. Among others, this includes average treatment effects. We show that the properties of the sensitive-data model carry over to the privatised-data model by a suitable choice of the privacy mechanism, which, in general, means a total-variationally private mechanism. In particular, the double robustness property is retained, enabling efficient estimation from the privatised sample. We also find that the estimation of the nuisance parameters is not harder, albeit possibly less efficient and computationally more demanding, from the privatised sample compared to the sensitive sample for a given, suitable privacy mechanism. Indeed, if the estimation is feasible from the sensitive sample by some procedure, we can directly transport that procedure to the privatised setting. Lastly, we develop a private method of moments estimator for parametric models. This shows that in the private setting a parametric assumption about one nuisance parameter affords more flexible modelling and slower estimation of the other one, as is the well-known case in the nonprivate setting.

Importance: Hip fractures in older adults are serious injuries that result in disability, higher rates of illness and death, and a substantial strain on health care resources. High-quality evidence to improve hip fracture care regarding the surgical approach of hemiarthroplasty is lacking. Objective: To compare 6-month outcomes of the posterolateral approach (PLA) and direct lateral approach (DLA) for hemiarthroplasty in patients with acute femoral neck fracture. Design, Setting, and Participants: This multicenter, randomized clinical trial (RCT) comparing DLA and PLA was performed alongside a natural experiment (NE) at 14 centers in the Netherlands. Patients aged 18 years or older with an acute femoral neck fracture were included, with or without dementia. Secondary surgery of the hip, pathological fractures, or patients with multitrauma were excluded. Recruitment took place between February 2018 and January 2022. Treatment allocation was random or pseudorandom based on geographical location and surgeon preference. Statistical analysis was performed from July 2022 to September 2022. Exposure: Hemiarthroplasty using PLA or DLA. Main Outcome and Measures: The primary outcome was health-related quality of life 6 months after surgery, quantified with the EuroQol Group 5-Dimension questionnaire (EQ-5D-5L). Secondary outcomes included dislocations, fear of falling and falls, activities of daily living, pain, and reoperations. To improve generalizability, a novel technique was used for data fusion of the RCT and NE. Results: A total of 843 patients (542 [64.3%] female; mean [SD] age, 82.2 [7.5] years) participated, with 555 patients in the RCT (283 patients in the DLA group; 272 patients in the PLA group) and 288 patients in the NE (172 patients in the DLA group; 116 patients in the PLA group). In the RCT, mean EQ-5D-5L utility scores at 6 months were 0.50 (95% CI, 0.45-0.55) after DLA and 0.49 (95% CI, 0.44-0.54) after PLA, with 77% completeness. The between-group difference (-0.04 [95% CI, -0.11 to 0.04]) was not statistically significant nor clinically meaningful. Most secondary outcomes were comparable between groups, but PLA was associated with more dislocations than DLA (RCT: 15 of 272 patients [5.5%] in PLA vs 1 of 283 patients [0.4%] in DLA; NE: 6 of 113 patients [5.3%]) in PLA vs 2 of 175 patients [1.1%] in DLA). Data fusion resulted in an effect size of 0.00 (95% CI, -0.04 to 0.05) for the EQ-5D-5L and an odds ratio of 12.31 (95% CI, 2.77 to 54.70) for experiencing a dislocation after PLA. Conclusions and Relevance: This combined RCT and NE found that among patients treated with a cemented hemiarthroplasty after an acute femoral neck fracture, PLA was not associated with a better quality of life than DLA. Rates of dislocation and reoperation were higher after PLA. Randomized and pseudorandomized data yielded similar outcomes, which suggests a strengthening of these findings. Trial Registration: ClinicalTrials.gov Identifier: NCT04438226.

Background and purpose — The posterolateral and direct lateral surgical approach are the 2 most common surgical approaches for performing a hemiarthroplasty in patients with a hip fracture. It is unknown which surgical approach is preferable in terms of (cost-)effectiveness and quality of life. Methods and analysis — We designed a multicenter randomized controlled trial (RCT) with an economic evaluation and a natural experiment (NE) alongside. We will include 555 patients ≥ 18 years with an acute femoral neck fracture. The primary outcome is patient-reported health-related quality of life assessed with the EQ-5D-5L. Secondary outcomes include healthcare costs, complications, mortality, and balance (including fear of falling, actual falls, and injuries due to falling). An economic evaluation will be performed for quality adjusted life years (QALYs). We will use variable block randomization stratified for hospital. For continuous outcomes, we will use linear mixed-model analysis. Dichotomous secondary outcome measures will be analyzed using chi-square statistics and logistic regression models. Primary analyses are based on the intention-to-treat principle. Additional as treated analyses will be performed to evaluate the effect of protocol deviations. Study summary — (i) Largest RCT addressing the health-related patient outcome of the main surgical approaches of hemiarthroplasty. (ii) Focus on outcomes that are important for the patient. (iii) Pragmatic and inclusive RCT with few exclusion criteria, e.g., patients with dementia can participate. (iv) Natural experiment alongside to amplify the generalizability. (v) The first study conducting a costutility analysis comparing both surgical approaches.