Confounding and regression adjustment in difference‐in‐differences studies

BackgroundThe classical linear model is widely used in the analysis of clinical trials with continuous outcomes. However, required model assumptions are frequently not met, resulting in estimates of treatment effect that can be inefficient and biased. In addition, traditional models assess treatment effect only on the mean response, and not on other aspects of the response, such as the variance. Distributional regression modelling overcomes these limitations. The purpose of this paper is to demonstrate its usefulness for the analysis of clinical trials, and superior performance to that of traditional models.MethodsDistributional regression models are demonstrated, and contrasted with normal linear models, on data from the LIPID randomized controlled trial, which compared the effects of pravastatin with placebo in patients with coronary heart disease. Systolic blood pressure (SBP) and the biomarker midregional pro-adrenomedullin (MR-proADM) were analysed. Treatment effect was estimated in models that used response distributions more appropriate than the normal (Box-Cox-t and Johnson’s Su for MR-proADM and SBP, respectively), applied censoring below the detection limit of MR-proADM, estimated treatment effect on distributional parameters other than the mean, and included random effects for longitudinal observations. A simulation study was conducted to compare the performance of distributional regression models with normal linear regression, under conditions mimicking the LIPID study. The R package gamlss (Generalized Additive Models for Location, Scale and Shape), which implements maximum likelihood estimation for distributional regression modelling, was used throughout.ResultsIn all cases the distributional regression models fit the data well, in contrast to poor fits obtained for traditional models; for MR-proADM a small but significant treatment effect on the mean was detected by the distributional regression model and not the normal model; and for SBP a beneficial treatment effect on the variance was demonstrated. In the simulation study distributional models strongly outperformed normal models when the response variable was non-normal and heterogeneous; and there was no disadvantage introduced by the use of distributional regression modelling when the response satisfied the normal linear model assumptions.ConclusionsDistributional regression models are a rich framework, largely untapped in the clinical trials world. We have demonstrated a sample of the capabilities of these models for the analysis of trials. If interest lies in accurate estimation of treatment effect on the mean, or other distributional features such as variance, the use of distributional regression modelling will yield superior estimates to traditional normal models, and is strongly recommended.Trial registrationThe LIPID trial was retrospectively registered on ANZCTR on 27/04/2016, registration number ACTRN12616000535471.

BackgroundMulticentre randomized controlled trials (RCTs) routinely use randomization and analysis stratified by centre to control for differences between centres and to improve precision. No consensus has been reached on how to best analyze correlated continuous outcomes in such settings. Our objective was to investigate the properties of commonly used statistical models at various levels of clustering in the context of multicentre RCTs.MethodsAssuming no treatment by centre interaction, we compared six methods (ignoring centre effects, including centres as fixed effects, including centres as random effects, generalized estimating equation (GEE), and fixed- and random-effects centre-level analysis) to analyze continuous outcomes in multicentre RCTs using simulations over a wide spectrum of intraclass correlation (ICC) values, and varying numbers of centres and centre size. The performance of models was evaluated in terms of bias, precision, mean squared error of the point estimator of treatment effect, empirical coverage of the 95% confidence interval, and statistical power of the procedure.ResultsWhile all methods yielded unbiased estimates of treatment effect, ignoring centres led to inflation of standard error and loss of statistical power when within centre correlation was present. Mixed-effects model was most efficient and attained nominal coverage of 95% and 90% power in almost all scenarios. Fixed-effects model was less precise when the number of centres was large and treatment allocation was subject to chance imbalance within centre. GEE approach underestimated standard error of the treatment effect when the number of centres was small. The two centre-level models led to more variable point estimates and relatively low interval coverage or statistical power depending on whether or not heterogeneity of treatment contrasts was considered in the analysis.ConclusionsAll six models produced unbiased estimates of treatment effect in the context of multicentre trials. Adjusting for centre as a random intercept led to the most efficient treatment effect estimation across all simulations under the normality assumption, when there was no treatment by centre interaction.

Finding a new drug which is both safe and efficient is an expensive and time-consuming endeavour. In particular, establishing the ‘dose-effect relationship’ – how beneficial a drug is at different dosages – can be challenging. Predicting this curve requires gathering experimental data by exposing and recording how cells respond to various levels of the drug. However, extreme values are often observed at low and high dosages, potentially introducing errors that are hard to correct in the prediction process. Yet, these extreme observations are sometimes genuine so researchers cannot just ignore them. To improve dose-effect estimation, Zhou, Liu, Fang et al. developed a new general-purpose approach. It uses advanced statistical modelling to account for extremes in lab data. This strategy outperformed other methods when dealing with these observations while also providing higher efficiency in data analysis with more uniform data in experiments. To facilitate implementation, Zhou, Liu, Fang et al. set up a user-friendly tool baptized ‘REAP’; this free online resource allows scientists without advanced statistical experience to harness the new approach and to perform dose-effect analysis more easily and accurately. This could boost research across many different disciplines that examine the effects of chemicals on cells.

In this paper two causal models that approximate the nearly frequency-independent cyclic behaviour of soils are analysed in detail. The study was motivated by the need to conduct time-domain viscoelastic analysis on soil structures without adopting the ad hoc assumption of Rayleigh damping. First, the causal hysteretic model is introduced in which its imaginary part is frequency independent the same way that is the imaginary part of the popular non-causal constant hysteretic model. The adoption of an imaginary part that is frequency independent even at the zero-frequency limit, in conjunction with the condition that the proposed model should be causal, yields a real part that is frequency dependent and singular at zero frequency. The paper shows that the causal hysteretic model, although pathological at the static limit, is the mathematical connection between the non-causal constant hysteretic model and the physically realizable Biot model. The mathematical structure of the two causal models is examined and it is shown that the causal hysteretic model is precisely the high-frequency limit of the Biot model. Although both models have a closed-form time-domain representation, only the Biot model is suitable for a time-domain viscoelastic analysis with commercially available computer software. The paper demonstrates that the simplest, causal and physically realizable linear hysteretic model that can approximate the cyclic behaviour of soil is the Biot model. The proposed study elucidates how the dynamic analysis of soil structures can be conducted rigorously in terms of the viscoelastic properties of the soil material and not with the ad hoc Rayleigh damping approach which occasionally has been criticized that tends to overdamp the higher vibration modes. The study concludes that under pulse-type motions the Rayleigh damping approximation tends to overestimate displacements because of the inappropriate viscous type of dissipation that is imposed. Under longer motions that induce several cycles, the concept of equivalent viscous damping is more appropriate and the Rayleigh damping approximation results to a response that is comparable to the response computed with a rigorous time-domain viscoelastic finite element analysis. Copyright © 2000 John Wiley & Sons, Ltd.

Steady state analysis techniques use linear phasor circuit models, but power flow calculations require the solution of nonlinear algebraic equations. Small-signal linear models form the basis of solution algorithms such as the Newton-Raphson method and decoupled load flows. Since linear models are attractive from the viewpoint of gaining significant insight into system behavior under different operating scenarios, large-signal models such as 'DC models' are commonly employed. Results obtained from linear models are approximate, but they provide easy comparison between different operating conditions. Contingency ranking using a DC model, and generation shift and line outage distribution factors is highly satisfactory in many instances. Studies requiring steady state analysis such as security analysis, optimal power flow and economic dispatch, and voltage stability can significantly reduce computational burdens by assuming linear models. The main objective of this paper is to study quantitatively the relative merits of commonly used linear models in steady state power system analysis. The 'DC model' assumes negligible resistances and flat voltage profiles, and relates the bus phase angles to real power injections. A new linear model which accounts for the voltage and reactive power flows is developed, and compared to the results from 'DC model' and 'small-signal' models. The figure of merit for comparison is taken as the root mean square difference of similar quantities such as bus voltages and line active and reactive flows. IEEE 14-bus is used for the testing. Results show significant potential for the new linear model.

Visualizing External Validity: Graphical Displays to Inform the Extension of Treatment Effects from Trials to Clinical Practice.

This study aimed to formulate a suitable statistical model to determine acetylcholinesterase enzyme's half-maximal inhibitory (IC<sub>50</sub>) by a series of synthetic compounds. It was done with the same core structure for acetylcholinesterase inhibition for anti-Alzheimer’s disease (AD). The IC<sub>50</sub> of eighteen synthesized compounds on anticholinesterase activities was obtained and statistical methods were applied. Regression models were fitted to the dose-response curve to look for their IC<sub>50</sub>. Simple linear regression is the simplest model for the dose-response curve. However, polynomial regression models or non-linear regression models fit the data more accurately. The adjusted coefficient of determination (𝑅<sup>2</sup><sub>𝑎𝑑𝑗</sub>) was used to determine the best model among the linear models, while the root mean square error (RMSE) is more suitable in determining the goodness of fit between linear and non-linear model. Four-parameter logistic (4-PLR) regression often fits the dose-response data closely. Based on the RMSE value, a polynomial regression fitted better than 4-PLR with the IC<sub>50</sub> of 245.52.

For the purpose of control health expenditures, there are some papers investigating the characteristics of patients who may incur high expenditures. However fewer papers are found which are based on the overall medical conditions, so this chapter was to find a relationship among the prevalence of medical conditions, utilization of healthcare services, and average expenses per person. The authors used bootstrapping simulation for data preprocessing and then used linear regression and random forest methods to train several models. The metrics root mean square error (RMSE), mean absolute percent error (MAPE), mean absolute error (MAE) all showed that the selected linear regression model performs slightly better than the selected random forest regression model, and the linear model used medical conditions, type of services, and their interaction terms as predictors.

Article Figures and data Abstract eLife digest Introduction Results Discussion Materials and methods Data availability References Decision letter Author response Article and author information Metrics Abstract End-stage kidney disease (ESKD) patients are at high risk of severe COVID-19. We measured 436 circulating proteins in serial blood samples from hospitalised and non-hospitalised ESKD patients with COVID-19 (n = 256 samples from 55 patients). Comparison to 51 non-infected patients revealed 221 differentially expressed proteins, with consistent results in a separate subcohort of 46 COVID-19 patients. Two hundred and three proteins were associated with clinical severity, including IL6, markers of monocyte recruitment (e.g. CCL2, CCL7), neutrophil activation (e.g. proteinase-3), and epithelial injury (e.g. KRT19). Machine-learning identified predictors of severity including IL18BP, CTSD, GDF15, and KRT19. Survival analysis with joint models revealed 69 predictors of death. Longitudinal modelling with linear mixed models uncovered 32 proteins displaying different temporal profiles in severe versus non-severe disease, including integrins and adhesion molecules. These data implicate epithelial damage, innate immune activation, and leucocyte–endothelial interactions in the pathology of severe COVID-19 and provide a resource for identifying drug targets. eLife digest COVID-19 varies from a mild illness in some people to fatal disease in others. Patients with severe disease tend to be older and have underlying medical problems. People with kidney failure have a particularly high risk of developing severe or fatal COVID-19. Patients with severe COVID-19 have high levels of inflammation, causing damage to tissues around the body. Many drugs that target inflammation have already been developed for other diseases. Therefore, to repurpose existing drugs or design new treatments, it is important to determine which proteins drive inflammation in COVID-19. Here, Gisby, Clarke, Medjeral-Thomas et al. measured 436 proteins in the blood of patients with kidney failure and compared the levels between patients who had COVID-19 to those who did not. This revealed that patients with COVID-19 had increased levels of hundreds of proteins involved in inflammation and tissue injury. Using a combination of statistical and machine learning analyses, Gisby et al. probed the data for proteins that might predict a more severe disease progression. In total, over 200 proteins were linked to disease severity, and 69 with increased risk of death. Tracking how levels of blood proteins changed over time revealed further differences between mild and severe disease. Comparing this data with a similar study of COVID-19 in people without kidney failure showed many similarities. This suggests that the findings may apply to COVID-19 patients more generally. Identifying the proteins that are a cause of severe COVID-19 – rather than just correlated with it – is an important next step that could help to select new drugs for severe COVID-19. Introduction Coronavirus disease (COVID-19), caused by the SARS-CoV-2 virus, displays wide clinical heterogeneity from asymptomatic to fatal disease. Patients with severe disease exhibit marked inflammatory responses and immunopathology. The mechanisms underlying this remain incompletely characterised, and the key molecular mediators are yet to be determined. The first treatment shown to reduce mortality from COVID-19 in randomised trials was dexamethasone (Horby et al., 2020), a corticosteroid that has broad non-specific effects on the immune system. Even with corticosteroid treatment, mortality in severe COVID-19 remains significant. There is a wide armamentarium of existing drugs that target inflammation more selectively, providing potential repurposing opportunities for the treatment of COVID-19. Recently, the REMAP-CAP trial has demonstrated efficacy of anti-IL6 receptor blockade in patients admitted to intensive care units with severe disease (Gordon et al., 2021). In order to select the most promising agents for future trials, we urgently need to better understand the molecular drivers of severe disease. Proteins are the effector molecules of biology and the targets of most drugs. Therefore, proteomic profiling to identify the key mediators of severe disease provides a valuable tool for identifying and prioritising potential drug targets (Suhre et al., 2021). Risk factors for severe or fatal COVID-19 include age, male sex, non-European ancestry, obesity, diabetes mellitus, cardiovascular disease, and immunosuppression (Williamson et al., 2020). End-stage kidney disease (ESKD) is one of the strongest risk factors for severe COVID-19 (estimated hazard ratio for death 3.69) (Williamson et al., 2020), and ESKD patients hospitalised with COVID-19 have a mortality of approximately 30% (Docherty et al., 2020; Corbett et al., 2020; Ng et al., 2020; Valeri et al., 2020). ESKD patients have a high prevalence of vascular and cardiometabolic disease (e.g. hypertension, ischaemic heart disease, diabetes), either as a result of the underlying cause of their renal disease or as a consequence of renal failure. In addition, ESKD results in both relative immunosuppression and chronic low-grade inflammation, which may impact viral defence and the host inflammatory response. Here we performed proteomic profiling of serial blood samples of ESKD patients with COVID-19, leveraging the unique opportunity for longitudinal sampling in both the outpatient and inpatient settings afforded by a large multi-ethnic haemodialysis cohort (Figure 1a). These data revealed 221 proteins that are dysregulated in COVID-19 versus matched non-infected ESKD patients. Using linear mixed models, joint models, and machine learning, we identified proteins that are markers of COVID-19 severity and risk of death. Finally, we characterised the temporal dynamics of the blood proteomic response during COVID-19 infection in ESKD patients, uncovering 32 proteins that display altered trajectories in patients with severe versus non-severe disease. Figure 1 with 1 supplement see all Download asset Open asset Study design. (a) Schematic representing a summary of the patient cohorts, sampling, and the major analyses. Blue and red stick figures represent outpatients and hospitalised patients, respectively. (b) Timing of serial blood sampling in relation to clinical course of COVID-19 (subcohort A). Black asterisks indicate when samples were obtained. Three patients were already in hospital prior to COVID-19 diagnosis (indicated by red bars). Results We recruited 55 ESKD patients with COVID-19 (subcohort A; Table 1). All patients were receiving haemodialysis prior to acquiring COVID-19. Blood samples were taken as soon as feasible following COVID-19 diagnosis. At time of initial sample, 30 patients were outpatients attending haemodialysis sessions and 25 were hospitalised inpatients (see Materials and methods, Figure 1). Following the initial blood sample, serial sampling was performed for 51/55 patients. We also recruited 51 non-infected haemodialysis patients as ESKD controls, mirroring the age, sex, and ethnicity distribution of the COVID-19 cases (Figure 1—figure supplement 1a–c). We used the Olink proteomics platform to measure 436 proteins (Supplementary file 1a) in 256 plasma samples from the COVID-19 patients and the 51 control samples. The proteins measured consisted of five multiplex 'panels' focussed on proteins relevant to immuno-inflammation, cardiovascular, and cardiometabolic disease. The 436 proteins assayed showed strong enrichment for immune-related proteins (Supplementary file 1b). Table 1 Characteristics of subcohort A. COVID-19-positive ESKD patients (n = 55)ESKD controls (n = 51) OverallPeak severity mild or moderate (n = 28)Peak severity severe or critical (n = 27)Age Median (IQR)72.2 62.5–77.373.4 65.5–76.468.5 61.8–78.870.1 62.2–75.1Sex M F39 (70.9%) 16 (29.1%)18 (64.3%) 10 (35.7%)21 (77.8%) 6 (22.2%)36 (70.6%) 15 (29.4%)Ethnicity White Black South Asian Asian (other) Other16 (29.1%) 8 (14.5%) 18 (32.7%) 4 (7.3%) 9 (16.4%)5 (17.9%) 5 (17.9%) 10 (35.7%) 1 (3.6%) 7 (25.0%)11 (40.7%) 3 (11.1%) 8 (29.6%) 3 (11.1%) 2 (7.4%)13 (25.5%) 8 (15.7%) 20 (39.2%) 3 (5.9%) 7 (13.7%)Diabetes34 (61.8%)*16 (57.1%)18 (66.7%)24 (47.1%)*Current smoker1 (1.8%)1 (3.6%)00ESKD cause DN Genetic GN HTN/vascular Other Unknown29 (52.7%) 1 (1.8%) 3 (5.5%) 5 (9.1%) 8 (14.5%) 9 (16.4%)14 (50.0%) 1 (3.6%) 1 (3.6%) 3 (10.7%) 5 (17.9%) 4 (14.3%)15 (55.6%) 0 2 (7.4%) 2 (7.4%) 3 (11.1%) 5 (18.5%)20 (39.2%) 1 (2.0%) 9 (17.6%) 7 (13.7%) 4 (7.8%) 10 (19.6%)Hospitalisation due to COVID-19†33 (60%)6 (21.4%)27 (100%)N/AFatal COVID-199 (16.3%)0 (0%)9 (33.3%)N/A DN = diabetic nephropathy. GN = glomerulonephritis. HTN = hypertension. IQR = inter-quartile range. 'South Asian' represents individuals with Indian, Pakistani, or Bangladeshi ancestry. Subsets defined according to peak WHO severity over the course of the illness. N/A = not applicable. *One patient had type 1 diabetes, the remainder type 2. †3 patients were hospitalised prior to COVID-19 diagnosis. 8 patients diagnosed with COVID-19 as outpatients subsequently deteriorated were hospitalised. In addition, we performed the Olink proteomic assays in 52 serum samples from a separate set of 46 COVID-19-positive ESKD patients (subcohort B) and 11 serum samples from ESKD COVID-19-negative controls (a subset of the controls described above). For the large majority of patients in subcohort B, only a single timepoint was available. A higher proportion of these patients (41/46, 89%) were hospitalised and had severe disease (Table 2) than in subcohort A (Figure 1, Table 1). Table 2 Characteristics of subcohort B. COVID-19-positive ESKD patients (n = 46)COVID-19-negative ESKD controls (n = 11)*Age Median (IQR)64.3 60.3–73.071.6 (61.7–73.9)Sex M F32 (69.6%) 14 (30.4%)8 (72.3%) 3 (27.3%)Ethnicity White Black South Asian Asian (other) Other11 (23.9%) 8 (17.4%) 12 (26.1%) 7 (15.2%) 8 (17.4%)3 (27.3%) 3 (27.3%) 3 (27.3%) 0 2 (18.2%)Diabetes29 (63.0%)6 (54.5%)Current smoker2 (4.3%)0 (%)ESKD cause DN Genetic GN HTN/vascular Other Unknown19 (41.3%) 1 (2.2%) 7 (15.2%) 3 (6.5%) 3 (6.5%) 13 (28.3%)5 (45.5%) 0 1 (9.1%) 1 (9.1%) 2 (18.2%) 2 (18.2%)Hospitalisation due to COVID-1941 (89.1%)N/ASevere or critical COVID-1933 (71.7%)N/AFatal COVID-199 (19.6%)N/A DN = diabetic nephropathy. GN = glomerulonephritis. HTN = hypertension. IQR = inter-quartile range. 'South Asian' represents individuals with Indian, Pakistani, or Bangladeshi ancestry. Subsets defined according to peak WHO severity over the course of the illness. N/A = not applicable. *These 11 controls are a subset of the control patients used in subcohort A. Proteomic differences between COVID-19-positive and -negative ESKD patients Principal component analysis (PCA) of proteomic data from subcohort A demonstrated differences between samples from COVID-19-positive cases and controls, although the two groups did not separate into discrete clusters (Figure 2a,b). To examine the effects of COVID-19 on the plasma proteome, we performed a differential expression analysis in subcohort A between COVID-19 cases (n = 256 samples passing quality control [QC] from 55 patients) and non-infected ESKD controls (n = 51) using linear mixed models, which account for serial samples from the same individual (see Materials and methods). This revealed 221 proteins associated with COVID-19 (5% false discovery rate, FDR); the vast majority were upregulated, with only 40 downregulated (Figure 3a, Supplementary file 1c). In order to provide a succinct and standardised nomenclature, we report proteins by the symbols of the genes encoding them (see Supplementary file 1a for a mapping of symbols to full protein names). The most strongly upregulated proteins (in terms of fold change) were DDX58, CCL7, IL6, CXCL11, KRT19, and CXCL10, and the most strongly downregulated were SERPINA5, CCL16, FABP2, PON3, ITGA11, and MMP12 (Figure 3—figure supplement 1). Notably, many of the upregulated proteins were chemotaxins. Figure 2 with 2 supplements see all Download asset Open asset Principal component analysis. PC = principal component. Each point represents a sample. Colouring indicates COVID-19 status. The directions and relative sizes of the six largest PC loadings are plotted as arrows (middle column). (a, b) Subcohort A. Due to serial sampling, there are multiple samples for most patients. The proportion of variance explained in subcohort A by each PC is shown in parentheses on the axis labels. (c, d) Subcohort B. Samples are projected into the PCA coordinates from subcohort A. Figure 3 with 4 supplements see all Download asset Open asset Identification of dysregulated proteins. (a) Proteins upregulated (red) or downregulated (blue) in COVID-19-positive patients versus COVID-19-negative ESKD patients n = 256 plasma samples from 55 COVID-19-positive patients, versus n = 51 ESKD controls (one sample per control patient). (b) Proteins associated with disease severity associations of protein levels against WHO severity score at the time of sampling. Linear gradient indicates the effect size. A positive effect size (red) indicates that an increase in protein level is associated with increasing disease severity and a negative gradient (blue) the opposite. n = 256 plasma samples from 55 COVID-19-positive patients. For (a, b), p-values from linear mixed models after Benjamini–Hochberg adjustment; significance threshold = 5% FDR; dark-grey = non-significant. (c) Heatmap showing protein levels for selected proteins with strong associations with severity. Each column represents a sample (n = 256 COVID-19 samples and 51 non-infected samples). Each row represents a protein. Proteins are annotated using the symbol of their encoding gene. For the purposes of legibility, not all significantly associated proteins are shown; the heatmap is limited to the 17% most up- or downregulated proteins (by effect size) of those with a significant association. Proteins are ordered by hierarchical clustering. Samples are ordered by WHO severity at the time of blood sample ('Severity'). 'Overall course' indicates the peak WHO severity over the course of the illness. We observed that a high proportion of the measured proteins were associated with COVID-19. Given the highly targeted nature of the Olink panels that we used (enriched for immune and inflammation-related proteins), this was not surprising. Nevertheless, to ensure that the Benjamini–Hochberg adjustment of p-values was controlling the FDR at the 5% level, we performed two additional analyses (see Materials and methods). First, we estimated the FDR using an alternative method (the plug-in procedure ; Hastie et al., 2001); this confirmed appropriate FDR control. Second, we used permutation to estimate the distribution of the number of proteins expected to be declared significant under the null hypothesis of no association between any proteins and COVID-19. This showed that the probability of observing the number of differentially abundant proteins we identified was highly unlikely under the null (empirical p<1×10−5; Figure 3—figure supplement 2). Although our COVID-19-negative controls were well matched in terms of age, sex, and ethnicity (Figure 1—figure supplement 1a–c), perfect matching of comorbidities was not feasible in the context of the healthcare emergency at the time of patient recruitment. There was a higher prevalence of diabetes in the COVID-19 cases compared to the controls (61.8% versus 47.1%, respectively; Table 1). To evaluate whether differing rates of diabetes had impacted the proteins identified as differentially abundant between cases and controls, we performed a sensitivity analysis adding diabetes as an additional covariate in the linear mixed model. This did not materially affect our findings; estimated effect sizes and –log10 p-values from models with and without the inclusion of diabetes were highly correlated (Pearson r > 0.99, and r = 0.95, respectively; Figure 3—figure supplement 3a,b). Full results from both models are shown in Supplementary file 1c. Similarly, there were also differences in the underlying cause of ESKD in cases compared to controls (Table 1). We therefore performed a further sensitivity analysis adjusting for underlying cause of renal failure. This did not make any meaningful difference to our results (Figure 3—figure supplement 3c,d, Supplementary file 1c). We also considered the possibility that timing of haemodialysis might affect the plasma proteome. To minimise the impact of this, all samples were taken prior to haemodialysis. For the large majority (86.6%) of samples, the most recent haemodialysis was between 48 and 72 hr prior to blood draw. This consistency in timing of blood sampling reduces the potential for impact of this issue. Nevertheless, to evaluate whether timing of haemodialysis might have impacted our results, we performed a sensitivity analysis including time from last haemodialysis as a covariate. Our results were not materially affected by this, with −log10 p-values and estimated effect sizes very highly correlated with those obtained without inclusion of this covariate (Pearson r > 0.99 for effect size estimates and for −log10 p-values; Figure 3—figure supplement 4a,b, Supplementary file 1c). We used the smaller subcohort B (n = 52 serum samples from 46 patients with COVID-19; see Materials and methods) for validation. We first projected the data from subcohort B into the PCA space of subcohort A to examine how well the separation of cases and controls in the PCA space replicated (see Materials and methods). This revealed clearer separation of infected and non-infected patients than in subcohort A (Figure 2c,d), perhaps reflecting the higher proportion of hospitalised patients (41 of 46 patients) in subcohort B (Table 2). We next performed differential abundance analysis in subcohort B and found 201 proteins that were dysregulated in cases versus controls (5% FDR) (Supplementary file 1c). Of the 221 differentially abundant proteins from subcohort A, 150 (69.7%) were also identified in subcohort B at 5% FDR (Figure 4a). Effect sizes in each dataset showed a strong correlation (r = 0.80, Figure 4b). This demonstrates that our findings are highly reproducible despite differences in sample sizes and blood materials (plasma versus serum in subcohorts A and B, respectively). Figure 4 with 1 supplement see all Download asset Open asset Validation. (a) Overlap between the significant associations in the differential abundance analysis between ESKD patients with and without COVID-19 in subcohorts A and B. 5% FDR was used as the significance threshold in both analyses. (b) Comparison of estimated effect sizes for all 436 proteins in the differential abundance analyses (COVID-19 positive versus negative) in subcohort A and B. Each point represents a protein. Pearson's r is shown. Differential abundance analyses were performed using linear mixed models. Subcohort A analysis (plasma samples): 256 samples from 55 COVID-19 patients versus 51 non-infected patient samples (single time-point). Subcohort B (serum samples): 52 samples from 55 COVID-19 patients and 11 non-infected patient samples (single timepoint). Proteins associated with COVID-19 severity Examination of the principal components plot labelling samples by clinical severity at the time of sampling (defined by WHO severity scores, graded as mild, moderate, severe, or critical) demonstrated a gradient of COVID-19 severity, best captured by principal components 1 and 3 (Figure 2—figure supplement 1a). To determine the proteomic effects of COVID-19 severity, we tested for associations between proteins and WHO severity score at the time of blood sampling, using linear mixed models with severity encoded as an ordinal predictor (see Materials and methods). This analysis revealed 203 proteins associated with severity (Figure 3b, Supplementary file 1d). The majority of these were upregulated in more severe disease, with only 42 downregulated. A sensitivity analysis adjusting for time since last haemodialysis made no significant impact on our results (Figure 3—figure supplement 4c,d, Supplementary file 1d). Consistent with previous reports, we found that severe COVID-19 was characterised by elevated IL6. In addition, we observed a signature of upregulated monocyte chemokines (e.g. CCL2, CCL7, CXCL10), neutrophil activation and degranulation (e.g. PRTN3, MPO), and epithelial injury (e.g. KRT19, AREG, PSIP1, GRN). (Figures 3b,c and 5). SERPINA5 and leptin showed the greatest downregulation as COVID-19 severity increased (Figure 3b,c). Figure 5 Download asset Open asset proteins strongly associated with COVID-19 severity. showing distribution of plasma protein levels according to COVID-19 at the time of blood draw. indicate and inter-quartile range. n = 256 samples from 55 COVID-19 patients and 51 samples from non-infected patients. WHO severity indicates the clinical severity score of the patient at the time the sample was n = moderate n = severe n = critical n = 15 samples. monocyte markers of epithelial injury. two neutrophil and IL6. We next how the COVID-19 severity protein signature to the proteins that are differentially abundant between cases and The majority of proteins were also identified as differentially abundant in the COVID-19-positive versus -negative analysis (Figure fold for proteins in COVID-19 versus non-infected patients were correlated with effect sizes in the severity that the proteins most upregulated in cases versus controls also to the greatest in severe disease (Figure there were some (e.g. that were strongly associated with severity, not differentially expressed in infected versus non-infected patients (Figure Figure 6 Download asset Open asset Comparison of proteins differentially expressed in COVID-19 with those associated with clinical severity. (a) Overlap between the proteins significantly differentially expressed in COVID-19 (n = 256 COVID-19 samples and 51 non-infected versus those associated with severity n = 256 (subcohort A). 5% FDR was used as the significant in both analyses. (b) Comparison of effect sizes for each protein in the COVID-19-positive versus -negative analysis and severity analysis Each point represents a protein. Pearson's r is shown. (c) of proteins associated with severity, not significantly differentially abundant in the of all cases versus showing distribution of plasma protein levels according to COVID-19 at the time of blood draw. indicate and inter-quartile range. n = 256 samples from 55 COVID-19 patients and 51 samples from non-infected patients. WHO severity indicates the clinical severity score of the patient at the time the sample was n = moderate n = severe n = critical n = 15 samples. learning to predict COVID-19 severity PCA revealed that some samples from patients who had mild or moderate disease at the time of sampling with samples from patients with severe disease (Figure 2—figure supplement 1a). Examination of the same PCA plot labelling samples according to the clinical course by peak WHO severity score over the of the (Figure 2—figure supplement revealed that these samples from individuals who subsequently developed severe or critical disease. This that molecular may clinical To evaluate this we used learning to whether the proteomic signature of the first blood sample for each patient in our dataset could identify whether the patient either had severe COVID-19 at the time of sampling or severe disease in the differential expression analyses each protein of all proteins in the Using we a on the first sample for each COVID-19 patient to predict the clinical defined by peak WHO severity. For the purposes of this we clinical course into either WHO or The method in peak severity. using only predictors and clinical the method in peak severity. clinical proteins did not compared to using proteomic predictors that the information in the clinical predictors is captured at the proteomic we not that proteomic profiling is to clinical for risk during this the selected by the proteins of We therefore the to identify key proteins by (see Materials and methods, Supplementary file The most important proteins for the of or future severe disease were IL18BP, GDF15, KRT19, and (Figure is that this as a key of severe disease. Figure 7 with 1 supplement see all Download asset Open asset of severe COVID-19 and death. (a) The 12 most important proteins for clinical course (defined by peak COVID-19 WHO using a is important for it is to in many of and be to the have a The all was used as the (b) Proteins that are significant predictors of death n = 256 samples from 55 COVID-19-positive patients, of Risk estimates are from a joint model. indicate For proteins with a positive risk a higher to a high risk of and for proteins with negative Proteins associated with risk of death of 55 patients in subcohort A We therefore to identify proteins associated with risk of death. To the nature of protein for of we joint models, which linear mixed models and models et al., (see Materials and methods). This analysis identified proteins for which increased was associated with increased risk of death (Figure Supplementary file including and and 25 proteins for which increased was associated with risk of including and with clinical A number of clinical have associations with COVID-19 (e.g. elevated inflammatory and et al., 2020). We therefore compared our proteomic data from COVID-19 patients at each timepoint to clinical using linear mixed models (see Materials and methods). We found associations between plasma proteins and all clinical (Figure Supplementary file Many of these proteins were also markers of severity (e.g. IL6, KRT19, and were strongly associated with and and Of CCL7, a monocyte that was also identified as an important of severity by the was associated with monocyte and inflammatory neutrophil was associated with which IL6, and and with the and Figure 8 Download asset Open asset of clinical markers with plasma proteins. Proteins that are (red) or (blue) associated with clinical (5% p-values from differential abundance analysis using linear mixed models after Benjamini–Hochberg = non-significant. Two associations were found for shown – see Supplementary file Longitudinal analysis proteins with temporal profiles according to severity The immune response to infection is and therefore provide only the serial sampling in our dataset (Figure we the temporal of each protein and whether or not any protein trajectories in patients with a versus clinical This was a

BackgroundThe E-value, a measure that has received recent attention in the comparative effectiveness literature, reports the minimum strength of association between an unmeasured confounder and the treatment and outcome that would explain away the estimated treatment effect. This study contributes to the literature on the applications and interpretations of E-values by examining how the E-value is impacted by data with varying levels of association of unobserved covariates with the treatment and outcome measure when covariate adjustment is applied. We calculate the E-value after using regression and propensity score methods (PSMs) to adjust for differences in observed covariates. Propensity score methods are a common observational research method used to balance observed covariates between treatment groups. In practice, researchers may assume propensity score methods that balance treatment groups across observed characteristics will extend to balance of unobserved characteristics. However, that assumption is not testable and has been shown to not hold in realistic data settings. We assess the E-value when covariate adjustment affects the imbalance in unobserved covariates.MethodsOur study uses Monte Carlo simulations to evaluate the impact of unobserved confounders on the treatment effect estimates and to evaluate the performance of the E-Value sensitivity test with the application of regression and propensity score methods under varying levels of unobserved confounding. Specifically, we compare observed and unobserved confounder balance, odds ratios of treatment vs. control, and E-Value sensitivity test statistics from generalized linear model (GLM) regression models, inverse-probability weighted models, and propensity score matching models, over correlations of increasing strength between observed and unobserved confounders.ResultsWe confirm previous findings that propensity score methods – matching or weighting – may increase the imbalance in unobserved confounders. The magnitude of the effect depends on the strength of correlation between the confounder, treatment, and outcomes. We find that E-values calculated after applying propensity score methods tend to be larger when unobserved confounders result in more biased treatment effect estimates.ConclusionsThe E-Value may misrepresent the size of the unobserved effect needed to change the magnitude of the association between treatment and outcome when propensity score methods are used. Thus, caution is warranted when interpreting the E-Value in the context of propensity score methods.

This article explores the implications of bias cancellation on the estimate of average treatment effects using ordinary least squares (OLS) and Rubin-style matching methods. Bias cancellation (offsetting biases at high and low propensities for treatment in estimates of treatment effects that are uncorrected for nonrandom selection) has been observed when job training is the treatment variable and earnings is the outcome variable. Contrary to published assertions in the literature, bias cancellation is not explainable in terms of the standard selection model, which assumes a symmetric distribution for the errors in the structural and assignment equations. A substantive rationale for bias cancellation is offered, which conceptualizes bias cancellation as the result of a mixture process based on two distinct individual-level decision-making models. While the general properties are unknown, the existence of bias cancellation appears to reduce the average bias in both OLS and matching methods relative to the symmetric distribution case.

It is well recognized that statistical linear interpolation models are computationally inexpensive and applicable to any climate data compared to the dynamic simulation method at regional scales. Using five different statistical linear interpolation models, we characterized each model’s performance to predict a climate variable of interest. General linear model, generalized additive model, spatial linear model, and bayesian spatial regression model (BSM) were analyzed. The climate variable of interest was the monthly precipitation, where the spatial variability was explained using terrain information: latitude, longitude, elevation, topographic aspect, and costal proximity. We used the root mean squared error, the mean absolute error and correlation coefficient as the performance. The BSM showed better performance in reflecting the spatial pattern of monthly precipitation compared to the other models. The monthly precipitation and its 95 % prediction interval on a 1 × 1 km grid spacing were generated through a spatial interpolation of 441 point observations.

BackgroundCausal inference has seen an increasing popularity in medical research. Estimation of causal effects from observational data allows to draw conclusions from data when randomized controlled trials cannot be conducted. Although the identification of structural causal models (SCM) and the calculation of structural coefficients has received much attention, a key requirement for valid causal inference is that conclusions are drawn based on the true data-generating model.MethodsIt remains widely unknown how large the probability is to reject the true structural causal model when observational data from it is sampled. The latter probability – the causal false-positive risk – is crucial, as rejection of the true causal model can induce bias in the estimation of causal effects. In this paper, the widely used causal models of confounders and colliders are studied regarding their causal false-positive risk in linear Markovian models. A simulation study is carried out which investigates the causal false-positive risk in Gaussian linear Markovian models. Therefore, the testable implications of the DAG corresponding to confounders and colliders are analyzed from a Bayesian perspective. Furthermore, the induced bias in estimating the structural coefficients and causal effects is studied.ResultsResults show that the false-positive risk of rejecting a true SCM of even simple building blocks like confounders and colliders is substantial. Importantly, estimation of average, direct and indirect causal effects can become strongly biased if a true model is rejected. The causal false-positive risk may thus serve as an indicator or proxy for the induced bias.ConclusionWhile the identification of structural coefficients and testable implications of causal models have been studied rigorously in the literature, this paper shows that causal inference also must develop new concepts for controlling the causal false-positive risk. Although a high risk cannot be equated with a substantial bias, it is indicative of the induced bias. The latter fact calls for the development of more advanced risk measures for committing a causal type I error in causal inference.

Pharmacokinetic-Pharmacodynamic Modeling and Simulation

Complete-case analyses can be biased if missing data are not missing completely at random. We propose simple sensitivity analyses that apply to complete-case estimates of treatment effects; these analyses use only simple summary data and obviate specifying the precise mechanism of missingness and making distributional assumptions. Bias arises when treatment effects differ between retained and nonretained participants or, among retained participants, the estimate is biased because conditioning on retention has induced a noncausal path between the treatment and outcome. We thus bound the overall treatment effect on the difference scale by specifying: 1) the unobserved treatment effect among nonretained participants; and 2) the strengths of association that unobserved variables have with the exposure and with the outcome among retained participants ("induced confounding associations"). Working with the former sensitivity parameter subsumes certain existing methods of worst-case imputation while also accommodating less-conservative assumptions (e.g., that the treatment is not detrimental on average even among nonretained participants). As an analog to the E-value for confounding, we propose the M-value, which represents, for a specified treatment effect among nonretained participants, the strength of induced confounding associations required to reduce the treatment effect to the null or to any other value. These methods could help characterize the robustness of complete-case analyses to potential bias due to missing data.