PLmixed: Estimate (Generalized) Linear Mixed Models with Factor Structures

A simulation study was conducted to assess Type I error and statistical power of linear mixed models, generalized linear mixed models, and linear quantile mixed models for the analysis of longitudinal cytometry by time-of-flight data. Findings indicate that, while generalized linear mixed models have superior statistical power, they also suffer from inflated Type I error rates. Linear mixed models can have substantially lower statistical power than the other methods at larger effect sizes. While linear quantile mixed models have slightly lower statistical power at smaller effect sizes, they have intermediate statistical power at larger effect sizes and good Type I error control. Taken altogether, these results generally recommend the use of linear quantile mixed models for longitudinal datasets such as for cytometry by time-of-flight data, although linear mixed models may be slightly more useful at small effect sizes and sample sizes.

A central point of disagreement, in certain long-standing discussions about a particular whaling dataset in the Scientific Committee of the International Whaling Commission, has directly involved model selection issues for linear mixed effect models. The biological question under discussion is associated with a clearly defined parameter of primary interest, a focus parameter, which makes model selection with the Focused Information Criterion (FIC) more appropriate than other selection methods. Since the existing FIC methodology has not covered the case of linear mixed effects models, this article sets up the required framework and develops the necessary formulae for the relevant FIC. Our new criterion requires the asymptotic distribution of estimators derived for a given candidate linear mixed model but with behaviour examined under a wider linear mixed model. These results, needed here to build our FIC, also have independent interest.

Earlier studies to assess the effects of risk factors on child nutritional status in Bangladesh have used conventional regression models that are inadequate to capture a complete scenario of effects. Therefore, this study aimed to evaluate the heterogeneous effects of factors at different points of conditional height-for-age Z-score (HAZ) distribution accounting for cluster-level variation using linear quantile mixed model (LQMM) and to compare them with a linear mixed model (LMM). In addition, an unconditional quantile model (UQM) was used to measure the effect of factors on the unconditional (marginal) HAZ distribution. A total of 6340 children aged 0–59 months extracted from the 2014 Bangladesh Demographic and Health Survey. Different factors – maternal characteristics (age, occupation, nutritional status, parity, birth interval), parental education, child age, breastfeeding status, and morbidity had significant heterogeneous effects on HAZ distribution. For example, secondary or higher educated parents had substantial differential impacts on the lower tail and upper tail of the child HAZ distribution, which was masked by LMM estimate. Moreover, significant cluster-level variations found across all quantiles of child HAZ. During intervention design, heterogeneous effects of factors and cluster variation ought to consider addressing the undernutrition problem in Bangladesh.

We study model selection for clustered data, when the focus is on cluster specific inference. Such data are often modelled using random effects, and conditional Akaike information was proposed in Vaida & Blanchard (2005) and used to derive an information criterion under linear mixed models. Here we extend the approach to generalized linear and proportional hazards mixed models. Outside the normal linear mixed models, exact calculations are not available and we resort to asymptotic approximations. In the presence of nuisance parameters, a profile conditional Akaike information is proposed. Bootstrap methods are considered for their potential advantage in finite samples. Simulations show that the performance of the bootstrap and the analytic criteria are comparable, with bootstrap demonstrating some advantages for larger cluster sizes. The proposed criteria are applied to two cancer datasets to select models when the cluster-specific inference is of interest.

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

Linear mixed effects models are highly flexible in handling a broad range of data types and are therefore widely used in applications. A key part in the analysis of data is model selection, which often aims to choose a parsimonious model with other desirable properties from a possibly very large set of candidate statistical models. Over the last 5–10 years the literature on model selection in linear mixed models has grown extremely rapidly. The problem is much more complicated than in linear regression because selection on the covariance structure is not straightforward due to computational issues and boundary problems arising from positive semidefinite constraints on covariance matrices. To obtain a better understanding of the available methods, their properties and the relationships between them, we review a large body of literature on linear mixed model selection. We arrange, implement, discuss and compare model selection methods based on four major approaches: information criteria such as AIC or BIC, shrinkage methods based on penalized loss functions such as LASSO, the Fence procedure and Bayesian techniques.

Pharmacokinetic-Pharmacodynamic Modeling and Simulation

Privileged by rapid increase in available epigenomic data, epigenome‐wide association studies (EWAS) are to make a profound contribution to understand the molecular mechanism of DNA methylation in cognitive aging. Current statistical methods used in EWAS are dominated by models based on multiple assumptions, for example, linear relationship between molecular profiles and phenotype, normal distribution for the methylation data and phenotype. In this study, we applied an assumption‐free method, the generalized correlation coefficient (GCC), and compare it to linear models, namely the linear mixed model and kinship model. We use DNA methylation associated with a cognitive score in 400 and 206 twins as discovery and replication samples respectively. DNA methylation associated with cognitive function using GCC, linear mixed model, and kinship model, identified 65 CpGs (p < 1e‐04) from discovery sample displaying both nonlinear and linear correlations. Replication analysis successfully replicated 9 of these top CpGs. When combining results of GCC and linear models to cover diverse patterns of relationships, we identified genes like KLHDC4, PAPSS2, and MRPS18B as well as pathways including focal adhesion, axon guidance, and some neurological signaling. Genomic region‐based analysis found 15 methylated regions harboring 11 genes, with three verified in gene expression analysis, also the 11 genes were related to top functional clusters including neurohypophyseal hormone and maternal aggressive behaviors. The GCC approach detects valuable methylation sites missed by traditional linear models. A combination of methylation markers from GCC and linear models enriched biological pathways sensible in neurological function that could implicate cognitive performance and cognitive aging.

Background: Advanced breast cancer (ABC) is affecting substantial number of Chinese women in Hong Kong. Understanding their unmet supportive care needs (SCNs) is important for health care system to precisely allocate resources to areas demanded for help the most and for patients to achieve better quality of life.&#13;\n&#13;\nObjectives: (1) To validate Chinese version of Supportive Care Needs Survey Questionnaire (SCNS-SF34), (2) to address the prevalence of unmet SCNs at baseline (newly diagnosis of ABC) and explore relevant factors associating with baseline unmet SCNs, (3) to identify the trajectories of unmet SCNs from baseline, 6-week, 12-week, 18-week to one-year post-diagnosis of ABC among Hong Kong Chinese women and identify the predictors related to individual resources.&#13;\n&#13;\nMethods: The study consisted of two phases. In Phase I, breast cancer (BC) patients were recruited from six public hospitals and the SCNS-SF34 (which covers five domains of needs) was administered concurrently with measures of psychological distress CHQ-12 (Chinese Health Questionnaire-12), HADS (Hospital Anxiety and Depression Scale), symptom distress (MSAS-SF), and patient satisfaction (ChPSQ-9) to explore factor structure by using Exploratory Factor Analysis (EFA) and to examine internal consistency, and convergent, divergent and discriminant validities of the identified factor structure. In phase II, women newly diagnosed with ABC were recruited and followed up to assess their unmet SCN trajectories one year after diagnosis. Prevalence of initial baseline SCNs and associated demographic, medical and psychological factors were identified. Linear Mixed Modeling (LMM) was performed to assess trajectories for each SCNs domain. Hypothesized variables reflecting personal and social resources (optimism, trait hope, social support, psychological distress and patient satisfaction) were examined for association with the changes of unmet SCNs after adjusting for demographic and medical characteristics.&#13;\n&#13;\nResults: Instead of five-factor structure identified in the original SCNS-SF34, a four-factor structure with 33 items was identified, comprising: 1) Health system, information and patient support needs (HSIPS), 2) Psychological needs (PSYC), 3) Physical and daily living needs (PDL) and 4) Sexuality needs (SEX). The SCNS-SF33-C demonstrated moderate-to-good internal consistency (Cronbach’s alphas=0.75-0.92) across all domains. Acceptable convergent and divergent validity were demonstrated. Discriminant validity was demonstrated in the SCNS-SF33-C’s ability to differentiate between clinically distinct patient groups (ABC vs. localized BC and active treatment vs. no active treatment). Of the top 15 unmet SCNs, all belonged to the HSIPS domain. There were significant linear declines in unmet HSIPS and PSYC needs over the year after diagnosis, but not in PDL and SEX. After adjusting for demographic and medical factors, LMM identified symptom distress, patient satisfaction and patient satisfaction x time are predictors of HSIP. Total symptom distress, optimism, anxiety and anxiety x time predicted PSYC. Total symptom distress was predictor of PDL. Anxiety was predictor of SEX.&#13;\n&#13;\nConclusions: The SCNS-SF33-C has a suitable factor structure and psychometric properties for the use in assessing unmet psychosocial SCN among Chinese women with BC. Generally, unmet HSIP and PSYC tended to decline, while levels of unmet PDL and SEX tended to persist over time. Specific individual resources predicted the future change of unmet SCNs.

In this paper, we discuss two different data transformation techniques for dealing with censored data: Kaplan-Meier weights and the k-nearest neighbor imputation method. The main objective of this paper is to find penalized spline estimates for the components of a linear mixed effect model with right-censored data. In the context of a mixed model setting, the estimation procedure is performed based on the modified or transformed dataset obtained via these transformation techniques. In order to compare the outcomes from a linear mixed model using these two approaches, a Monte Carlo simulation and two real data examples are presented. According to our results, the k-nearest neighbor imputation is very successful in dealing with censored observations.

Hyperspectral Compressed Sensing(HCS) is crucial for data storage and the real-time transmission of airborne-or spacebornebased imaging platforms. The Linear Mixing Model(LMM) has been successfully applied to HCS reconstruction. However, the obtained spectrum may be disturbed, thereby limiting the improvement of reconstruction quality due to the influence of illumination conditions,topographic changes, and atmospheric effects. Spectral disturbance is corrected on the basis of LMM by introducing the spectral correction term, and a linear mixing model for spectral perturbation correction is proposed. Moreover, an improved HCS method based on modified LMM is proposed. This proposed model only performs spectral compressed sampling on the original hyperspectral images at the sampling end. The proposed method uses the proposed spectral perturbation correction model to reconstruct the original hyperspectral images based on the compressed sampling data. The alternating direction multiplier method is used to estimate the optimal values of each component in the modified LMM to obtain the optimal reconstruction quality. Experimental results show that the proposed method can achieve better reconstructed performance compared with other classical HCS methods.

Bootstrap is a widely applicable computational statistical method. The focus of this paper lies in developing a bootstrap method for linear mixed models under homoscedasticity violation (heteroscedasticity) in variance of errors. We assume that the form of heteroscedasticity is unknown. To generate bootstrap response data as close as possible to the actual response data, we transform the marginal residuals to ensure that the variance of the modified marginal residuals in bootstrap samples is an unbiased estimator for the variance of the response variable in the linear mixed models. The consistency is proved, implying that the parameter estimators by the proposed method are asymptotically unbiased. The simulations are conducted with varying error terms and sample sizes to show improvement of the proposed method in parameter estimation. The simulation results are compared to demonstrate that the proposed method outperforms the other considered approaches including the wild method in the heteroscedastic linear mixed models under small samples and performs competitively with the parametric method and both methods outperform the other considered methods in homoscedastic linear mixed models. The proposed bootstrap method is applied to squid data for illustration of its effectiveness.

In the previous chapters, we discussed autoregressive linear mixed effects models. In this section, we discuss the relationships between the autoregressive linear mixed effects models and nonlinear mixed effects models, growth curves, and differential equations. The autoregressive model shows a profile approaching an asymptote, where the change is proportional to the distance remaining to the asymptote. Autoregressive models in discrete time correspond to monomolecular curves in continuous time. Autoregressive linear mixed effects models correspond to monomolecular curves with random effects in the baseline and asymptote, and special error terms. The autoregressive coefficient is a nonlinear parameter, but all random effects parameters in the model are linear. Therefore, autoregressive linear mixed effects models are nonlinear mixed effects models without nonlinear random effects and have a closed form of likelihood. When there are time-dependent covariates, autoregressive linear mixed effects models are represented by a differential equation and random effects. The monomolecular curve is one of the popular growth curves. We introduce other growth curves, such as the logistic curves and von Bertalanffy curves, and generalizations of growth curves. Re-parameterization is often performed in nonlinear models, and various representations of re-parameterization in monomolecular and other curves are provided herein.

Water availability is the primary constraint to sorghum production worldwide. The crop is grown in water-limited environments of varying extent and timing with water stress during grain filling being common. Varying extent and timing of water availability generates substantial genotype × environment interaction (GEI) for sorghum production, which restricts genetic improvement of grain yield. The objective of this thesis is to better understand sorghum performance under drought conditions and identify potential strategies to improve the rate of genetic gain for grain yield. To better understand the impacts and to deconstruct GEI by removing the confounding effects of flowering time (DTF) and tillering (FTN) on grain yield, linear mixed models were used to evaluate the impacts of DTF and FTN on grain yield of 1741 unique test hybrids. Hybrids were derived from crosses between 1078 elite male parents and 3 female testers and grown in 21 yield testing trials at 15 locations across the major sorghum production regions in Australia in three seasons. DTF made a significant contribution to genetic variation in grain yield of male lines in 14 of the 48 tester/trial combinations, explaining 0.2% to 61.0% variation, and FTN in 12 tester/trial combinations, explaining 1.4% to 56.9% variation. The relationship of DTF and FTN with grain yield of hybrids in the whole dataset was frequently positive, but varied depending on the genetic background of testers and growing environments. However, between trial genetic correlations for grain yield were not significantly improved after accounting for the effects of DTF and FTN using linear models. These results suggests that other genetic factors affecting canopy development dynamics and grain yield might contribute more of the genetic variation in grain yield. It is possible that linear mixed models did not capture the non-linear effects of flowering time and fertile tiller number on yield. One of the consequences of water stress during grain filling is lodging, with major consequences on grain yield and quality. Hence, resistance to lodging has been a major target of selection in sorghum breeding programs in Australia. A retrospective analysis was conducted on 37 commercial hybrids grown in 83 yield testing trials in major sorghum production regions in Australia in 14 seasons to quantify the geographical and seasonal variations in lodging occurrence and severity. Lodging occurred more frequently in Central Highlands and less frequently in Liverpool Plains, in comparison to the overall average across regions. The severity of lodging also varied across regions, with the most severe lodging (>20%) occurring in the Central Highlands and Western Downs. Lockyer Valley was the only region free from lodging. In addition, seasonal patterns of lodging frequency and severity were also observed. Over the 14 growing seasons, the frequency of lodging varied from 0 to 100%, with the most severe lodging (>20%) observed in 2005, 2016, and 2017. The Southern Oscillation Index explained 29% of the seasonal variation in lodging frequency. Results also showed that the levels of lodging resistance in commercial hybrids were not improved. This is possibly because sorghum breeders trade of improvements in lodging resistance to increase grain yield due to the complex nature of lodging. Currently, sorghum breeders use strategies of direct selection against lodging susceptibility and direct selection for stay-green (delayed leaf senescence) rather than selection for lodging resistance to improved lodging resistance. A better understanding of the genetic architecture of lodging and its association with other traits would enable faster and greater genetic gain in lodging resistance and grain yield. Large scale genome-wide association studies on lodging were conducted using data from 17 Australian sorghum trials over three seasons involving 2308 unique test hybrids. The identification of 213 QTL for lodging clearly demonstrate the complexity of lodging. The majority (92%) of the lodging QTL showed a significant association with leaf senescence and plant height, suggesting lodging in grain sorghum is primarily driven by carbohydrate remobilisation and plant height. Additionally, the thesis found that one third of the previously reported QTL for stalk rots co-located with lodging QTL identified in this study, 11 of which were associated with leaf senescence and/or plant height. These results indicate that genes providing classical resistance to stalk rots are likely to be of limited effectiveness in breeding for enhanced lodging resistance. However, the observation of significant enrichment of genes in lignin biosynthesis pathways within lodging QTL identified in this thesis indicates that lodging resistance is likely to be improved by selecting for stem strength and composition traits such as lignin content.

Background: With the demand of the growing population, the broiler industry has grown up rapidly over the last few decades and it plays as an affordable source of good quality nutritious animal protein. This broiler industry focuses mainly on optimizing the profit through improving body weight and feed efficiency but the health issues of consumers are not taken into consideration seriously. It is important to know the changing pattern of concentration level of the biochemical parameter (total cholesterol) due to different feeds as well as different ages of chicken. Methods: This experimental study through longitudinal data was conducted using repeated measurements from each of seventy randomly selected broilers, partitioned into two groups according to two types of feed, at four-time points. Since measurements from the same subject were taken at four time periods, traditional approach of analysis may not be appropriate as it ignore the correlation between repeated measurements. Therefore, linear mixed model was adopted for the analysis of our obtained dataset. Result: Linear mixed effect model did not reveal any significant difference of standard and hatcher’s supplied feeds over time on the evolution of total cholesterol level. This might be due to little difference in different compositions of both feeds. However, both exploratory data analysis and modelling confirmed that irrespective of the available feed types, total cholesterol level of broiler serum increased significantly over time (age) which leads to a recommendation for the consumers to eat younger age (lower weight) broiler chicken.