Expectation Maximization Research Articles

Towards Effective Clustered Federated Learning: A Peer-to-Peer Framework With Adaptive Neighbor Matching

In federated learning (FL), clients may have diverse objectives, and merging all clients' knowledge into one global model will cause negative transfer to local performance. Thus, clustered FL is proposed to group similar clients into clusters and maintain several global models. In the literature, centralized clustered FL algorithms require the assumption of the number of clusters and hence are not effective enough to explore the latent relationships among clients. In this paper, without assuming the number of clusters, we propose a peer-to-peer (P2P) FL algorithm named PANM. In PANM, clients communicate with peers to adaptively form an effective clustered topology. Specifically, we present two novel metrics for measuring client similarity and a two-stage neighbor matching algorithm based Monte Carlo method and Expectation Maximization under the Gaussian Mixture Model assumption. We have conducted theoretical analyses of PANM on the probability of neighbor estimation and the error gap to the clustered optimum. We have also implemented extensive experiments under both synthetic and real-world clustered heterogeneity. Theoretical analysis and empirical experiments show that the proposed algorithm is superior to the P2P FL counterparts, and it achieves better performance than the centralized cluster FL method. PANM is effective even under extremely low communication budgets.

A Bayesian Approach Toward Robust Multidimensional Ellipsoid-Specific Fitting.

This work presents a novel and effective method for fitting multidimensional ellipsoids (i.e., ellipsoids embedded in [Formula: see text]) to scattered data in the contamination of noise and outliers. Unlike conventional algebraic or geometric fitting paradigms that assume each measurement point is a noisy version of its nearest point on the ellipsoid, we approach the problem as a Bayesian parameter estimate process and maximize the posterior probability of a certain ellipsoidal solution given the data. We establish a more robust correlation between these points based on the predictive distribution within the Bayesian framework, i.e., considering each model point as a potential source for generating each measurement. Concretely, we incorporate a uniform prior distribution to constrain the search for primitive parameters within an ellipsoidal domain, ensuring ellipsoid-specific results regardless of inputs. We then establish the connection between measurement point and model data via Bayes' rule to enhance the method's robustness against noise. Due to independent of spatial dimensions, the proposed method not only delivers high-quality fittings to challenging elongated ellipsoids but also generalizes well to multidimensional spaces. To address outlier disturbances, often overlooked by previous approaches, we further introduce a uniform distribution on top of the predictive distribution to significantly enhance the algorithm's robustness against outliers. Thanks to the uniform prior, our maximum a posterior probability coincides with a more tractable maximum likelihood estimation problem, which is subsequently solved by a numerically stable Expectation Maximization (EM) framework. Moreover, we introduce an ε-accelerated technique to expedite the convergence of EM considerably. We also investigate the relationship between our algorithm and conventional least-squares-based ones, during which we theoretically prove our method's superior robustness. To the best of our knowledge, this is the first comprehensive method capable of performing multidimensional ellipsoid-specific fitting within the Bayesian optimization paradigm under diverse disturbances. We evaluate it across lower and higher dimensional spaces in the presence of heavy noise, outliers, and substantial variations in axis ratios. Also, we apply it to a wide range of practical applications such as microscopy cell counting, 3D reconstruction, geometric shape approximation, and magnetometer calibration tasks. In all these test contexts, our method consistently delivers flexible, robust, ellipsoid-specific performance, and achieves the state-of-the-art results.

Optimization and application of renal depth measurement method in the cadmium-zinc-telluride‑based SPECT/CT renal dynamic imaging.

This study aims to evaluate the accuracy of four kidney depth measurement methods-nuclear medicine tomography, nuclear medicine lateral scanning, ultrasound, and Tonnesen's formula-based estimation-using CT measurements as the reference standard. Additionally, it investigates the feasibility of utilizing nuclear medicine tomography and lateral scanning for measuring kidney depth in 99mTc-DTPA renal dynamic imaging. Hollow kidney phantoms mimicking the shape and volume of adult kidneys were 3D printed and filled with 99mTcO4- solution. These phantoms were then subjected to lateral scanning and nuclear medicine tomography using CZT (cadmium-zinc-telluride) SPECT/CT to determine the optimal post-processing method. Forty patients who underwent renal dynamic imaging were recruited for the study. Renal depths were derived from ultrasound, lateral imaging, nuclear medicine tomography, formula-based estimation, and CT measurements. The renal depths obtained through these four methods were for correlation with CT-measured renal depths. Additionally, the absolute differences between renal depths obtained by each method and the CT standard were analyzed and compared across groups. Using kidney phantoms, nuclear medicine tomography images were processed with a Butterworth filter (cutoff frequency = 0.6), and renal outlines in lateral images was manually delineated. In the clinical validation phase, correlation coefficients indicated strong associations between renal depths measured by nuclear medicine tomography (left kidney: R = 0.885, P < 0.05; right kidney: R = 0.927, P < 0.05) and lateral scanning (left kidney: R = 0.933, P < 0.05; right kidney: R = 0.956, P < 0.05) compared to CT measurements. The difference in kidney depth between nuclear medicine tomography and CT measurements were the smallest and statistically significant (left kidney: 0.69 ± 0.51; right kidney: 0.58 ± 0.41, P < 0.05). Using ordered subset expectation maximization (OSEM) in conjunction with a Butterworth filter (fc = 0.6) as the post-processing method, nuclear medicine tomography enables more accurate renal depth measurements without increasing the radiation dose to patients.

Sieve Maximum Likelihood Estimation of Partially Linear Transformation Models With Interval-Censored Data.

Partially linear models provide a valuable tool for modeling failure time data with nonlinear covariate effects. Their applicability and importance in survival analysis have been widely acknowledged. To date, numerous inference methods for such models have been developed under traditional right censoring. However, the existing studies seldom target interval-censored data, which provide more coarse information and frequently occur in many scientific studies involving periodical follow-up. In this work, we propose a flexible class of partially linear transformation models to examine parametric and nonparametric covariate effects for interval-censored outcomes. We consider the sieve maximum likelihood estimation approach that approximates the cumulative baseline hazard function and nonparametric covariate effect with the monotone splines and -splines, respectively. We develop an easy-to-implement expectation-maximization algorithm coupled with three-stage data augmentation to facilitate maximization. We establish the consistency of the proposed estimators and the asymptotic distribution of parametric components based on the empirical process techniques. Numerical results from extensive simulation studies indicate that our proposed method performs satisfactorily in finite samples. An application to a study on hypobaric decompression sickness suggests that the variable TR360 exhibits a significant dynamic and nonlinear effect on the risk of developing hypobaric decompression sickness.

Data discretization impact on deep learning for missing value imputation of continuous data

In various fields of information examination, for example, AI, profoundly getting the hang of missing information is a typical issue. Missing qualities should be tended to since they can adversely affect the exactness and adequacy of prescient models. This research investigates how data discretization affects deep learning methods for filling the missing values in datasets with continuous features. They provide a unique method for imputing missing values using deep neural networks (DNNs) called extravagant expectation maximization-deep neural network (EEM-DNN). This approach discretizes continuous features into separate intervals initially. This is justified by treating the issue of missing value imputation as a classification work, with the missing values being considered a distinct class. A DNN, designed explicitly for imputation, is then trained using the discretized data. The expectation maximization concepts are incorporated into the network architecture, and as a result, the network iteratively improves its imputation predictions. They run comprehensive experiments on several datasets from different fields to gauge the efficacy of the suggested strategy. The effectiveness of EEM-DNN is compared to that of other imputation approaches, such as traditional imputation techniques and deep learning methods without data discretization. Our findings show that data discretization significantly enhances imputation accuracy. In terms of imputation accuracy and prediction performance on downstream tasks, the EEM-DNN method regularly performs better than alternative methods. It also examines if various discretization techniques affect the overall imputation process. They find that the trade-off between bias and variance in imputed data depends on the discretization method selected. This highlights the significance of choosing a suitable discretization approach depending on the unique properties of the dataset.

Fast standard error estimation for joint models of longitudinal and time-to-event data based on stochastic EM algorithms.

Maximum likelihood inference can often become computationally intensive when performing joint modeling of longitudinal and time-to-event data, due to the intractable integrals in the joint likelihood function. The computational challenges escalate further when modeling HIV-1 viral load data, owing to the nonlinear trajectories and the presence of left-censored data resulting from the assay's lower limit of quantification. In this paper, for a joint model comprising a nonlinear mixed-effect model and a Cox Proportional Hazards model, we develop a computationally efficient Stochastic EM (StEM) algorithm for parameter estimation. Furthermore, we propose a novel technique for fast standard error estimation, which directly estimates standard errors from the results of StEM iterations and is broadly applicable to various joint modeling settings, such as those containing generalized linear mixed-effect models, parametric survival models, or joint models with more than two submodels. We evaluate the performance of the proposed methods through simulation studies and apply them to HIV-1 viral load data from six AIDS Clinical Trials Group studies to characterize viral rebound trajectories following the interruption of antiretroviral therapy (ART), accounting for the informative duration of off-ART periods.

Regularized Bayesian algorithms for Q-matrix inference based on saturated cognitive diagnosis modelling.

Q-matrices are crucial components of cognitive diagnosis models (CDMs), which are used to provide diagnostic information and classify examinees according to their attribute profiles. The absence of an appropriate Q-matrix that correctly reflects item-attribute relationships often limits the widespread use of CDMs. Rather than relying on expert judgment for specification and post-hoc methods for validation, there has been a notable shift towards Q-matrix estimation by adopting Bayesian methods. Nevertheless, their dependency on Markov chain Monte Carlo (MCMC) estimation requires substantial computational burdens and their exploratory tendency is unscalable to large-scale settings. As a scalable and efficient alternative, this study introduces the partially confirmatory framework within a saturated CDM, where the Q-matrix can be partially defined by experts and partially inferred from data. To address the dual needs of accuracy and efficiency, the proposed framework accommodates two estimation algorithms-an MCMC algorithm and a Variational Bayesian Expectation Maximization (VBEM) algorithm. This dual-channel approach extends the model's applicability across a variety of settings. Based on simulated and real data, the proposed framework demonstrated its robustness in Q-matrix inference.

An investigation to improve a nonlinear mixed-effects approach for EC50 estimation based on multi-donor dose–response data

ABSTRACT Dose–response relationships are important in assessing the efficacy and potency of compounds, which can usually be characterized by a 4-parameter logistic (4-PL) model estimating EC50, slope factor, lower asymptote, and upper asymptote. EC50, the concentration of a compound that induces a response halfway between the baseline and maximum, is a key quantity to evaluate compound potency. For multi-donor dose–response data, it is often of interest to estimate the overall EC50 (i.e. the average EC50 of the population of donors) and its 95% confidence interval (CI). A few multi-donor EC50 estimation methods have been proposed in the literature. Jiang and Kopp-Schneider (2014) systematically compared the meta-analysis approach and the nonlinear mixed-effects approach and concluded that the meta-analysis approach is simple and robust to summarize EC50 estimates from multiple experiments, especially suited in the case of a small number of experiments, while the nonlinear mixed-effects approach has the issue of convergence failures probably due to overparameterization. In this article, we propose a modification of the nonlinear mixed-effects approach by using the stochastic approximation expectation-maximization (SAEM) algorithm to estimate model parameters and using multiple starting points to search for globally optimal values, which can substantially alleviate the issue of convergence failures even for small number of donors (e.g. n = 3), and achieve a smaller absolute median bias and better coverage probability of 95% confidence interval than the meta-analysis approach when the number of donors is not too small (e.g. n ≥ 7).

Estimation of Jacquard's genetic identity coefficients with bi-allelic variants by constrained least-squares.

The Jacquard genetic identity coefficients are of fundamental importance in relatedness research. We address the estimation of these coefficients as well as other relationship parameters that derive from them such as kinship and inbreeding coefficients using a concise matrix framework. Estimation of the Jacquard coefficients via likelihood methods and the expectation-maximization algorithm is computationally very demanding for large numbers of polymorphisms. We propose a constrained least squares approach to estimate the Jacquard coefficients. A simulation study shows constrained least squares achieves root-mean-squared errors that are comparable with those of the maximum likelihood approach, in particular when founder allele frequencies are unknown, while obtaining enormous computational savings.

Optimizing Parameters for Enhanced Iterative Image Reconstruction Using Extended Power Divergence

In this paper, we propose a method for optimizing the parameter values in iterative reconstruction algorithms that include adjustable parameters in order to optimize the reconstruction performance. Specifically, we focus on the power divergence-based expectation-maximization algorithm, which includes two power indices as adjustable parameters. Through numerical and physical experiments, we demonstrate that optimizing the evaluation function based on the extended power-divergence and weighted extended power-divergence measures yields high-quality image reconstruction. Notably, the optimal parameter values derived from the proposed method produce reconstruction results comparable to those obtained using the true image, even when using distance functions based on differences between forward projection data and measured projection data, as verified by numerical experiments. These results suggest that the proposed method effectively improves reconstruction quality without the need for machine-learning techniques in parameter selection. Our findings also indicate that this approach is useful for enhancing the performance of iterative reconstruction algorithms, especially in medical imaging, where high-accuracy reconstruction under noisy conditions is required.

Annealed adaptive importance sampling method in PINNs for solving high dimensional partial differential equations

Physics-informed neural networks (PINNs) have emerged as powerful tools for solving a wide range of partial differential equations (PDEs). However, despite their user-friendly interface and broad applicability, PINNs encounter challenges in accurately resolving PDEs, especially when dealing with singular cases that may lead to unsatisfactory local minima. To address these challenges and improve solution accuracy, we propose an innovative approach called Annealed Adaptive Importance Sampling (AAIS) for computing the discretized PDE residuals of the cost functions, inspired by the Expectation Maximization (EM) algorithm used in finite mixtures to mimic target density. Our objective is to approximate discretized PDE residuals by strategically sampling additional points in regions with elevated residuals, thus enhancing the effectiveness and accuracy of PINNs. Implemented together with a straightforward resampling strategy within PINNs, our AAIS algorithm demonstrates significant improvements in efficiency across a range of tested PDEs, even with limited training datasets. Moreover, our proposed AAIS-PINNs method shows promising capabilities in solving high-dimensional singular PDEs. The adaptive sampling framework introduced here can be integrated into various PINN frameworks.

Gamma Process Based Degradation Model with Fractional Gaussian Noise

In modern industrial and engineering systems, stochastic degradation models are widely used for reliability analysis and maintenance decision-making. However, due to imperfect sensors and environmental influences, it is difficult to directly observe the latent degradation states. Traditional degradation models typically assume that measurement errors have simple statistical properties, but this assumption often does not hold in practical applications. To address this issue, this paper constructs a degradation model based on the Gamma process (GP) and assumes that measurement noise can be characterized by the fractional Gaussian noise (FGN). Furthermore, this paper proposes a method combining Gibbs sampling with the stochastic expectation-maximization (SEM) algorithm to achieve efficient estimation of the model parameters and accurate inference of the latent degradation states. Simulation results indicate that the proposed model exhibits better generalizability compared to the GP model with Gaussian noise.

Filtering Useful App Reviews Using Naïve Bayes—Which Naïve Bayes?

App reviews provide crucial feedback for software maintenance and evolution, but manually extracting useful reviews from vast volumes is time-consuming and challenging. This study investigates the effectiveness of six Naïve Bayes variants for automatically filtering useful app reviews. We evaluated these variants on datasets from five popular apps, comparing their performance in terms of accuracy, precision, recall, F-measure, and processing time. Our results show that Expectation Maximization-Multinomial Naïve Bayes with Laplace smoothing performed best overall, achieving up to 89.2% accuracy and 0.89 F-measure. Complement Naïve Bayes with Laplace smoothing demonstrated particular effectiveness for imbalanced datasets. Generally, incorporating Laplace smoothing and Expectation Maximization improved performance, albeit with increased processing time. This study also examined the impact of data imbalance on classification performance. Our findings suggest that these advanced Naïve Bayes variants hold promise for filtering useful app reviews, especially when dealing with limited labeled data or imbalanced datasets. This research contributes to the body of evidence around app review mining and provides insights for enhancing software maintenance and evolution processes.

Color image watermarking using vector SNCM-HMT

An image watermarking scheme is typically evaluated using three main conflicting characteristics: imperceptibility, robustness, and capacity. Developing a good image watermarking method is challenging because it requires a trade-off between these three basic characteristics. In this paper, we proposed a statistical color image watermarking based on robust discrete nonseparable Shearlet transform (DNST)-fast quaternion generic polar complex exponential transform (FQGPCET) magnitude and vector skew-normal-Cauchy mixtures (SNCM)-hidden Markov tree (HMT). The proposed watermarking system consists of two main parts: watermark inserting and watermark extraction. In watermark inserting, we first perform DNST on R, G, and B components of color host image, respectively. We then compute block FQGPCET of DNST domain color components, and embed watermark signal in DNST-FQGPCET magnitudes using multiplicative approach. In watermark extraction, we first analyze the robustness and statistical characteristics of local DNST-FQGPCET magnitudes of color image. We then observe that, vector SNCM-HMT model can capture accurately the marginal distribution and multiple strong dependencies of local DNST-FQGPCET magnitudes. Meanwhile, vector SNCM-HMT parameters can be computed effectively using variational expectation–maximization (VEM) parameter estimation. Motivated by our modeling results, we finally develop a new statistical color image watermark decoder based on vector SNCM-HMT and maximum likelihood (ML) decision rule. Experimental results on extensive test images demonstrate that the proposed statistical color image watermarking provides a performance better than that of most of the state-of-the-art statistical methods and some deep learning approaches recently proposed in the literature.

Consensus Guidelines for Delineation of Clinical Target Volumes for Intensity-Modulated Radiotherapy for Intact Cervical Cancer: An Update.

Accurate target delineation is essential when using intensity-modulated radiotherapy (IMRT) for intact cervical cancer. In 2011, the Radiation Therapy Oncology Group (RTOG) published a consensus guideline using magnetic resonance imaging (MRI). The current project expands on the previous atlas by including computed tomography (CT)-based contours, contours with MRI and positron- emission- tomography (PET) registrations, the addition of common and complex scenarios, and to incorporate information on simulation and treatment planning techniques. Twenty-eight experts in gynecologic radiation oncology contoured three cases, first on a non-contrast CT simulation scan, then with registered diagnostic scans. The cases included (1) FIGO IIIC1 with a bulky tumor and vaginal metastasis, (2) FIGO IIB with calcified uterine fibromas, and (3) FIGO IIIC2 with large lymph nodes. The contours on all six datasets (three CT simulations without diagnostic images and three with registered images) were analyzed for consistency of delineation using an expectation-maximization algorithm for simultaneous truth and performance level estimation (STAPLE) with kappa statistics as a measure of agreement. The contours were reviewed, discussed, and edited in a group meeting prior to finalizing. Contours showed considerable agreement between experts in each of the cases with kappa statistics of 0.67-0.72. For each case, diagnostic PET±MRI was associated with an increase in volume. The largest increase was the CTV primary for Case 2 with a 20% increase in volume and 54% increase in STAPLE estimate volume, which may be due to variance in registration priorities. For the third case, 92.9% increased their CTVs based on the addition of the diagnostic PET scan. The main areas of variance were in determining the superior extent of CTV coverage, coverage of the mesorectum, and simulation and planning protocols. This study shows the value as well as the challenges of using co-registered diagnostic imaging, with an average increase in volumes when incorporating MRI and PET.

Reliability modeling and statistical analysis of accelerated degradation process with memory effects and unit-to-unit variability

A reasonable description of the degradation process is essential for credible reliability assessment in accelerated degradation testing. Existing methods usually use Markovian stochastic processes to describe the degradation process. However, degradation processes of some products are non-Markovian due to the interaction with environments. Misinterpretation of the degradation pattern may lead to biased reliability evaluations. Besides, owing to the differences in materials and manufacturing processes, products from the same population exhibit diverse degradation paths, further increasing the difficulty of accurate reliability estimation. To address the above issues, this paper proposes an accelerated degradation model incorporating memory effects and unit-to-unit variability. The memory effect in the degradation process is captured by the fractional Brownian motion, which reflects the non-Markovian characteristic of degradation. The unit-to-unit variability is considered in the acceleration model to describe diverse degradation paths. Then, lifetime and reliability under normal operating conditions are presented. Furthermore, to give an accurate estimation of the memory effect, a new statistical analysis method based on the expectation maximization algorithm is devised. The effectiveness of the proposed method is verified by a simulation case and a real-world tuner reliability analysis case. The simulation case shows that the estimation of the memory effect obtained by the proposed statistical analysis method is much more accurate than the traditional one. Moreover, ignoring unit-to-unit variability can lead to a highly biased estimation of the memory effect and reliability. From the tuner reliability analysis case, the proposed model is superior in both deterministic degradation trend predictions and degradation boundary quantification compared to existing models, which can provide more credible reliability assessment.

Research on the P‐S‐N Curve Fitting Method of Notched Specimens Considering Small Sample Properties

ABSTRACTAiming at the issue of fatigue test data for large‐scale mechanical components of building steel are very limited, a method for fitting P‐S‐N curves under small sample data of notched specimens is proposed to predict fatigue life. First, a fatigue life subsample augmented and its reliability assessment method are established, based on Bayesian hierarchical modeling and modified Monte Carlo method. Second, a clustering combination weighting method is proposed, to define weights of hidden variables of the binomial mixture Weibull distribution, and the expectation–maximization algorithm is used to determine probability density function of the distribution. Finally, the P‐S‐N curves under various failure probabilities are fitted with Weibull distributed life models, and the convergence and prediction accuracy of the different models are compared. The results show that the fatigue data of small samples can be predicted better by using mixed Weibull distribution, and the fitting P‐S‐N curve is more reliable and accurate.

The Fault Diagnosis Method for Photovoltaic Modules Based on Machine Learning

Based on the urban natural gas pipeline accident statistics and semi-quantitative risk evaluation index system, this paper applies Bayesian network to establish a network model between various types of risk factors and the risk of natural gas pipeline failure. The EM algorithm was used to learn from the statistical accident data to obtain the parameters of the model. Based on the principle of evidential reasoning in reverse, the probability of occurrence of all risk indicators can be obtained when the probability of occurrence of urban natural gas pipeline accidents is 100%, the index weight is obtained by normalizing the occurrence probability. On this basis, this paper develops an efficient urban natural gas pipeline integrity risk identification and management software. The software can realize the basic data management of urban natural gas pipeline system, pipeline relative risk value calculation, pipeline risk level calculation and other functions, and the results are visualized. Finally, the practicability and effectiveness of the model and software are verified by a case of natural gas pipeline evaluation in a block.

Parallel-way: Multi-modality-based brain tumor segmentation using parallel capsule network

ABSTRACT Brain tumors present a formidable diagnostic challenge due to their aberrant cell growth. Accurate determination of tumor location and size is paramount for effective diagnosis. Magnetic Resonance Imaging (MRI) and Positron Emission Tomography (PET) are pivotal tools in clinical diagnosis, yet tumor segmentation within their images remains challenging, particularly at boundary pixels, owing to limited sensitivity. Recent endeavors have introduced fusion-based strategies to refine segmentation accuracy, yet these methods often prove inadequate. In response, we introduce the Parallel-Way framework to surmount these obstacles. Our approach integrates MRI and PET data for a holistic analysis. Initially, we enhance image quality by employing noise reduction, bias field correction, and adaptive thresholding, leveraging Improved Kalman Filter (IKF), Expectation Maximization (EM), and Improved Vibe Algorithm (IVib), respectively. Subsequently, we conduct multi-modality image fusion through the Dual-Tree Complex Wavelet Transform (DTWCT) to amalgamate data from both modalities. Following fusion, we extract pertinent features using the Advanced Capsule Network (ACN) and reduce feature dimensionality via Multi-objective Diverse Evolution-based selection. Tumor segmentation is then executed utilizing the Twin Vision Transformer with dual attention mechanism. Implemented our Parallel-Way framework which exhibits heightened model performance. Evaluation across multiple metrics, including accuracy, sensitivity, specificity, F1-Score, and AUC, underscores its superiority over existing methodologies.

Missing Data Imputation in Balanced Construction for Incomplete Block Designs

Observational data with massive sample sizes are often distributed on many local machines. From an experimental design perspective, investigators often desire to identify the effect of new treatments (even ML algorithms) on many blocks of experimental data. With time requirements or budget constraints, assigning all treatments to each block is not always feasible. This creates incomplete responses with respect to a randomized complete block design (RCBD). These incomplete responses are missing by design. However, whether they can be estimated with missing imputation methods is not well understood. Thus, it is challenging to correctly identify the treatment effects with missing data. To this end, this paper provides a method for imputation and analysis of the responses with missing data. The proposed method consists of three steps: Reconstruction, Imputation, and ‘Complete’-data Analysis (RICA). The incomplete responses are imputed with the expectation-maximization (EM) algorithm. The RCBD model is then fitted by the resulting dataset. The identifiability result suggests that the missing may be nonignorable for each block, but the whole data of an incomplete design are missing by design when the design is balanced. Theoretical results on relative efficiency also inform us when the missingness should be imputed for incomplete designs with the role of balanced variance. Applications on real-world data verify the efficacy of this method.