Variance Of Estimator Research Articles

Using interpenetrating subsampling to incorporate interpreter variability into estimation of the total variance of land cover area estimates

Reference data obtained by interpreters is a key component of sample-based estimation of area of land cover and land cover change. However, interpreters may disagree when assigning the reference class label for a given sample unit and this inconsistency between interpreters contributes to the overall uncertainty of the estimated area. Interpenetrating subsampling (IPS) offers a practical way to incorporate interpreter variability into an unbiased estimator of the total variance. This method requires partitioning the full sample into g nonoverlapping groups with the sample units in each group then evaluated by a different interpreter and each interpreter determines the reference class data for only one group. The total variance is estimated by the among group variability of the g estimates of area. IPS was applied to estimate the total variance of land cover area estimates for a sample of 300 pixels selected from the Puget Sound region of the Northwest United States. The reference land cover data were obtained by seven interpreters who each labeled all 300 pixels. These data provided a unique opportunity to explore properties of IPS such as variability over different random partitions of the sample into groups and variability over different subsets of interpreters. IPS estimates of total variance were produced for each land cover class for group sizes of g = 2 through g = 6 and all possible combinations of the seven interpreters for each group size. The estimated total variance decreased with increasing number of groups. Incorporating interpreter variance increased the estimated total variance by a factor ranging from 1.08 (agriculture) to 7.06 (grass/shrub) in simple random sampling. The total variance estimates varied substantially over the random partitions of the sample into groups, but this variability decreased as the group size increased. Compared with other total variance estimators, the IPS estimator is simpler to compute and is more cost effective because it does not require repeat interpretations

An algorithm for decoy-free false discovery rate estimation in XL-MS/MS proteomics.

Cross-linking tandem mass spectrometry (XL-MS/MS) is an established analytical platform used to determine distance constraints between residues within a protein or from physically interacting proteins, thus improving our understanding of protein structure and function. To aid biological discovery with XL-MS/MS, it is essential that pairs of chemically linked peptides be accurately identified, a process that requires: (i) database search, that creates a ranked list of candidate peptide pairs for each experimental spectrum and (ii) false discovery rate (FDR) estimation, that determines the probability of a false match in a group of top-ranked peptide pairs with scores above a given threshold. Currently, the only available FDR estimation mechanism in XL-MS/MS is the target-decoy approach (TDA). However, despite its simplicity, TDA has both theoretical and practical limitations that impact the estimation accuracy and increase run time over potential decoy-free approaches (DFAs). We introduce a novel decoy-free framework for FDR estimation in XL-MS/MS. Our approach relies on multi-sample mixtures of skew normal distributions, where the latent components correspond to the scores of correct peptide pairs (both peptides identified correctly), partially incorrect peptide pairs (one peptide identified correctly, the other incorrectly), and incorrect peptide pairs (both peptides identified incorrectly). To learn these components, we exploit the score distributions of first- and second-ranked peptide-spectrum matches for each experimental spectrum and subsequently estimate FDR using a novel expectation-maximization algorithm with constraints. We evaluate the method on ten datasets and provide evidence that the proposed DFA is theoretically sound and a viable alternative to TDA owing to its good performance in terms of accuracy, variance of estimation, and run time. https://github.com/shawn-peng/xlms.

Effects of multicollinearity and data granularity on regression models of stream temperature

Water temperature is a key factor influencing biota of stream ecosystems. Hence, it is important to comprehend the environmental drivers of stream temperature for robust prediction of conditions and effective management of stream communities. Linear regression models are commonly used for predictive purposes, but their predictive capacity and interpretability can be significantly affected by their complexity and the structure of input data. In some cases, researchers may be obligated to favor prediction power or interpretability while compromising the other. Therefore, insight into relationships between model fit, correlation among predictor variables (i.e., multicollinearity), and level of temporal aggregation of data (i.e., data granularity) may be helpful to reduce such trade-offs. In this paper, we investigated these relationships within a hierarchical set of multiple linear regression (MLR) models examining environmental factors influencing stream temperature dynamics. Our findings showed that as the number of predictor variables (i.e., model complexity) increased, the magnitude of multicollinearity in MLR models increased, but model fit also increased. The results also revealed that using data averaged over longer time frames (i.e., coarser data granularity) yielded high multicollinearity, as indexed by variance inflation factor values (VIF) for all model predictors. This led to higher variance in parameter estimates (i.e., parameter instability) and potential challenges in model interpretation as the sign of parameter estimates changed in many streams examined. Multicollinearity was not the only reason for these changes in the sign of parameter estimates as they were also observed in simple linear regression models across varying levels of data granularity. Based on our findings, we conclude that the selection of data granularity is an important consideration in multiple regression modeling, with profound implications for model interpretability.

Алгоритмы построения дискретных приближенно Q-оптимальных планов эксперимента при активной идентификации регрессионных моделей многофакторных систем

In the theory of optimal experiment, there is a group of optimality criteria, for example, such as D-, A-, E-criteria, reflecting the accuracy of estimating model parameters. There is also a group of criteria related to the accuracy of the forecast based on the model, which can be characterized by the variance of estimates of mathematical expectations of responses. For example, using the G-optimality criterion allows you to obtain designs on which the constructed models will minimize the maximum variance of the forecast. Among these is the Q-optimality criterion, which assumes minimizing the average variance of the forecast for the regression model over the planning area. Most of the theoretical and applied research is related to the use of the D-optimality criterion. This is also explained by the fact that the criteria of D- and G-optimality are interconnected. At the same time, it should be noted that minimizing the maximum variance in the general case may not lead to a decrease in the average variance of the forecast area. In this regard, the use of Q-optimal designs in practical regression modeling tasks is relevant. For the widespread introduction into practice of the active identification of regression models of the concept of Q-optimality of experimental designs, an arsenal of effective algorithms for their construction is needed. The paper proposes and describes algorithms for constructing discrete approximate Q-optimal designs. The proposed algorithms are based on the developed approach of consistently increasing the number of points in the designs, as well as procedures for replacing points in the design. The designs obtained by such algorithms are recommended for use in practice when, on average, good prediction accuracy is required for the model over the entire range of input factors.

Ballou's Ancestral Inbreeding Coefficient: Formulation and New Estimate with Higher Reliability.

Inbreeding is unavoidable in small populations. However, the deleterious effects of inbreeding on fitness-related traits (inbreeding depression) may not be an inevitable phenomenon, since deleterious recessive alleles causing inbreeding depression might be purged from populations through inbreeding and selection. Inbreeding purging has been of great interest in conservation biology and animal breeding, because populations manifesting lower inbreeding depression could be created even with a small number of breeding animals, if inbreeding purging exists. To date, many studies intending to detect inbreeding purging in captive and domesticated animal populations have been carried out using pedigree analysis. Ballou's ancestral inbreeding coefficient (FBAL-ANC) is one of the most widely used measurements to detect inbreeding purging, but the theoretical basis for FBAL-ANC has not been fully established. In most of the published works, estimates from stochastic simulation (gene-dropping simulation) have been used. In this report, the author provides a mathematical basis for FBAL-ANC and proposes a new estimate by hybridizing stochastic and deterministic computation processes. A stochastic simulation suggests that the proposed method could considerably reduce the variance of estimates, compared to ordinary gene-dropping simulation, in which whole gene transmissions in a pedigree are stochastically determined. The favorable property of the proposed method results from the bypass of a part of the stochastic process in the ordinary gene-dropping simulation. Using the proposed method, the reliability of the estimates of FBAL-ANC could be remarkably enhanced. The relationship between FBAL-ANC and other pedigree-based parameters is also discussed.

A naïve Bayes regularized logistic regression estimator for low-dimensional classification

To reduce the estimator's variance and prevent overfitting, regularization techniques have attracted great interest from the statistics and machine learning communities. Most existing regularized methods rely on the sparsity assumption that a model with fewer parameters predicts better than one with many parameters. This assumption works particularly well in high-dimensional problems. However, the sparsity assumption may not be necessary when the number of predictors is relatively small compared to the number of training instances. This paper argues that shrinking the coefficients towards a low-variance data-driven estimate could be a better regularization strategy for such situations. For low-dimensional classification problems, we propose a naïve Bayes regularized logistic regression (NBRLR) that shrinks the logistic regression coefficients toward the naïve Bayes estimate to provide a reduction in variance. Our approach is primarily motivated by the fact that naïve Bayes is functionally equivalent to logistic regression if naïve Bayes' conditional independence assumption holds. Under standard conditions, we prove the consistency of the NBRLR estimator. Extensive simulation and empirical experimental results show that NBRLR is a competitive alternative to various state-of-the-art classifiers, especially on low-dimensional datasets.

Investigating the impact of paternal age, paternal heat stress, and estimation of non-genetic paternal variance on dairy cow phenotype

BackgroundLinear models that are commonly used to predict breeding values in livestock species consider paternal influence solely as a genetic effect. However, emerging evidence in several species suggests the potential effect of non-genetic semen-mediated paternal effects on offspring phenotype. This study contributes to such research by analyzing the extent of non-genetic paternal effects on the performance of Holstein, Montbéliarde, and Normande dairy cows. Insemination data, including semen Batch Identifier (BI, a combination of bull identification and collection date), was associated with various traits measured in cows born from the insemination. These traits encompassed stature, milk production (milk, fat, and protein yields), udder health (somatic cell score and clinical mastitis), and female fertility (conception rates of heifers and cows). We estimated (1) the effects of age at collection and heat stress during spermatogenesis, and (2) the variance components associated with BI or Weekly aggregated BI (WBI).ResultsOverall, the non-genetic paternal effect estimates were small and of limited biological importance. However, while heat stress during spermatogenesis did not show significant associations with any of the traits studied in daughters, we observed significant effects of bull age at semen collection on the udder health of daughters. Indeed, cows born from bulls collected after 1500 days of age had higher somatic cell scores compared to those born from bulls collected at a younger age (less than 400 days old) in both Holstein and Normande breeds (+ 3% and + 5% of the phenotypic mean, respectively). In addition, across all breeds and traits analyzed, the estimates of non-genetic paternal variance were consistently low, representing on average 0.13% and 0.09% of the phenotypic variance for BI and WBI, respectively (ranging from 0 to 0.7%). These estimates did not significantly differ from zero, except for milk production traits (milk, fat, and protein yields) in the Holstein breed and protein yield in the Montbéliarde breed when WBI was considered.ConclusionsOur findings indicate that non-genetic paternal information transmitted through semen does not substantially influence the offspring phenotype in dairy cattle breeds for routinely measured traits. This lack of substantial impact may be attributed to limited transmission or minimal exposure of elite bulls to adverse conditions.

Estimation of Fe Grade at an Ore Deposit Using Extreme Gradient Boosting Trees (XGBoost)

Estimating the spatial distribution of ore grade is one of the most critical and important steps to continue investment decision on the deposit. Kriging is the most widely used method to estimate the ore grade while alternative techniques are being developed. Machine learning algorithms can be used as alternative methods to classical kriging. In this paper, Fe grade of a deposit is estimated with XGBoost algorithm, and results are compared with kriging estimation results. For estimation processes, samples collected from the drillholes are used. To mitigate the effect of varying sampling length, both estimations use composites of these samples. Due to the different nature of the estimation methods, different steps have been taken to perform estimations. Results show that XGBoost estimates produced higher ranged estimates which is a desired result in ore grade estimation while minimum and maximum of the estimates were lower and higher than the kriging estimates, respectively. However, like kriging estimates, estimation results were smoother than composites while variance of the XGBoost estimates were lower than variance of composites. This means that even though estimation with XGBoost mitigates the smoothing effect, estimation results suffer from smoothing effect like kriging.

Optimal two-phase sampling for comparing correlated areas under the ROC curves of two screening tests in the presence of verification bias

ABSTRACT The accuracy of a screening test is often measured by the area under the receiver characteristic (ROC) curve (AUC) of a screening test. Two-phase designs have been widely used in diagnostic studies for estimating one single AUC and comparing two AUCs where the screening test results are measured for a large sample (Phase one sample) while the disease status is only verified for a subset of Phase one sample (Phase two sample) by a gold standard. In this paper, we consider the optimal two-phase sampling design for comparing the performance of two ordinal screening tests in classifying disease status. Specifically, we derive an analytical variance formula for the AUC difference estimator and use it to find the optimal sampling probabilities that minimize the variance formula for the AUC difference estimator. According to the proposed optimal two-phase design, the strata with the levels of two tests far apart from each other should be over-sampled while the strata with the levels of two tests close to each other should be under-sampled. Simulation results indicate that two-phase sampling under optimal allocation (OA) achieves a substantial amount of variance reduction, compared with two-phase sampling under proportional allocation (PA). Furthermore, in comparison with a one-phase random sampling, two-phase sampling under OA or PA has a clear advantage in reducing the variance of AUC difference estimator when the variances of the two screening test results in the disease population differ greatly from their counterparts in non-disease population.

Negative Binomial Three Parameter Lindley Distribution and Its Properties

Many researchers have proposed mixed distributions as one of the most important methods for obtaining new probability distributions. Several studies have shown that mixed Negative Binomial distributions fits count data better than Poisson and Negative Binomial distribution itself. In this paper, we introduce a mixed distribution by mixing the distributions of negative binomial and three Parameter Lindley distribution. This new distribution has a thick tail and may be considered as an alternative for fitting count data with over dispersion. The parameters of the new distribution are estimated using MLE method and properties studied. Special cases of the new distribution and also identified. A simulation study carried out shows that the ML estimators give the parameter estimates close to the parameter when the sample is large, that is, the bias and variance of the parameter estimates decrease with increase in sample size showing the consistent nature of the new compound distribution. The study also compares the performance of the new distribution over distributions of Poisson, Negative Binomial, Negative Binomial oneParameter Lindley Distribution, Negative Binomial two Parameter distribution, three parameter Lindley distribution using a real count over dispersed dataset and the results shows that Negative Binomial three parameter Lindley distribution gave the smallest Kolmogorov Smirnov test statistic, AIC and BIC as compared to other distributions, hence the new distribution provided a better fit compared to other distributions under study for fitting over dispersed count data.

Drinking water QMRA and decision-making: Sensitivity of risk to common independence assumptions about model inputs

When assessing risk posed by waterborne pathogens in drinking water, it is common to use Monte Carlo simulations in Quantitative Microbial Risk Assessment (QMRA). This method accounts for the variables that affect risk and their different values in a given system. A common underlying assumption in such analyses is that all random variables are independent (i.e., one is not associated in any way with another). Although the independence assumption simplifies the analysis, it is not always correct. For example, treatment efficiency can depend on microbial concentrations if changes in microbial concentrations either affect treatment themselves or are associated with water quality changes that affect treatment (e.g., during/after climate shocks like extreme precipitation events or wildfires). Notably, the effects of erroneous assumptions of independence in QMRA have not been widely discussed. Due to the implications of drinking water safety decisions on public health protection, it is critical that risk models accurately reflect the context being studied to meaningfully support decision-making. This work illustrates how dependence between pathogen concentration and either treatment efficiency or water consumption can impact risk estimates using hypothetical scenarios of relevance to drinking water QMRA. It is shown that the mean and variance of risk estimates can change substantially with different degrees of correlation. Data from a water supply system in Calgary, Canada are also used to illustrate the effect of dependence on risk. Recognizing the difficulty of obtaining data to empirically assess dependence, a framework to guide evaluation of the effect of dependence is presented to enhance support for decision making. This work emphasizes the importance of acknowledging and discussing assumptions implicit to models.

Multilevel meta-analysis of single-case experimental designs using robust variance estimation.

Single-case experimental designs (SCEDs) are used to study the effects of interventions on the behavior of individual cases, by making comparisons between repeated measurements of an outcome under different conditions. In research areas where SCEDs are prevalent, there is a need for methods to synthesize results across multiple studies. One approach to synthesis uses a multilevel meta-analysis (MLMA) model to describe the distribution of effect sizes across studies and across cases within studies. However, MLMA relies on having accurate sampling variances of effect size estimates for each case, which may not be possible due to auto-correlation in the raw data series. One possible solution is to combine MLMA with robust variance estimation (RVE), which provides valid assessments of uncertainty even if the sampling variances of effect size estimates are inaccurate. Another possible solution is to forgo MLMA and use simpler, ordinary least squares (OLS) methods with RVE. This study evaluates the performance of effect size estimators and methods of synthesizing SCEDs in the presence of auto-correlation, for several different effect size metrics, via a Monte Carlo simulation designed to emulate the features of real data series. Results demonstrate that the MLMA model with RVE performs properly in terms of bias, accuracy, and confidence interval coverage for estimating overall average log response ratios. The OLS estimator corrected with RVE performs the best in estimating overall average Tau effect sizes. None of the available methods perform adequately for meta-analysis of within-case standardized mean differences. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Efficient Classes of Robust Ratio Type Estimators of Mean and Variance in Adaptive Cluster Sampling

This paper proposes two classes of robust ratio type estimators of finite population mean and two classes of robust ratio type estimators of finite population variance using a single auxiliary variable under the adaptive cluster sampling design. Seven robust ratio type estimators have been developed from each class. The generalized expressions of bias and mean square error for each class have been derived up to the first order of approximation. The exact expressions of bias and MSE for all the developed estimators have been presented. Using simulation studies conducted in R programming, the high efficiency of all the developed estimators over similar existing estimators in adaptive cluster sampling has been demonstrated. . KEYWORDS :Adaptive cluster sampling, Classes of robust ratio type estimators, Hidden clustered data, Estimation of mean, Estimation of variance.

Enhanced Estimation of Population Variance under Simple Random Sampling with an Application to Real Data

In this paper, we present an enhanced class of estimators for finite population variance under simple random sampling that incorporates an auxiliary information. The equations for bias and mean square error of the proposed estimators are produced up to the first order of approximation. Some well-known estimators are found as a special case of the proposed estimators for suitably chosen values of the scalars. The effectiveness of the proposed estimators is calculated in relation to the existing estimators both theoretically and empirically using five real data sets. The empirical results exhibit the superiority of the suggested estimators over to the existing estimators in each population.. KEYWORDS :Auxiliary variable, Mean square error, Variance estimation, Class of estimators.

Unbiased estimation of sampling variance for Simpson's diversity index.

Quantification of measurement uncertainty is crucial for robust scientific inference, yet accurate estimates of this uncertainty remain elusive for ecological measures of diversity. Here, we address this longstanding challenge by deriving a closed-form unbiased estimator for the sampling variance of Simpson's diversity index. In numerical tests the estimator consistently outperforms existing approaches, particularly for applications in which species richness exceeds sample size. We apply the estimator to quantify biodiversity loss in marine ecosystems and to demonstrate ligand-dependent contributions of T-cell-receptor chains to specificity, illustrating its versatility across fields. The novel estimator provides researchers with a reliable method for comparing diversity between samples, essential for quantifying biodiversity trends and making informed conservation decisions.

Augmented line sampling and combination algorithm for imprecise time-variant reliability analysis

Assessment of imprecise time-variant reliability in engineering is a critical task when accounting for both the variability of structural properties and loads over time and the presence of uncertainties involved in the ambiguity of parameters simultaneously. To estimate the Imprecise Time-variant Failure Probability Function (ITFPF) and derive the imprecise reliability results as a byproduct, Adaptive Combination Augmented Line Sampling (ACALS) is proposed. It consists of three integrated features: Augmented Line Sampling (ALS), adaptive strategy, and the optimal combination. ALS is adopted as an efficient analysis tool to obtain the failure probability function w.r.t. imprecise parameters. Then, the adaptive strategy iteratively applies ALS while considering both imprecise parameters and time simultaneously. Finally, the optimal combination algorithm collects all result components in an optimal manner to minimize the Coefficient of Variance (C.o.V.) of the ITFPF estimate. Overall, the proposed ACALS method outperforms the original ALS method by efficiently estimating the ITFPF while guaranteeing a minimal C.o.V. Thus, the proposed approach can serve as an effective tool for imprecise time-variant reliability analysis in real engineering applications. Several examples are presented to demonstrate the superiority of the proposed approach in addressing the challenges of estimating the ITFPF.

Borehole‐Based Interval Kriging for 3D Lithofacies Modeling

AbstractDeveloping a three‐dimensional (3D) lithofacies model from boreholes is critical for providing a coherent understanding of complex subsurface geology, which is essential for groundwater studies. This study aims to introduce a new geostatistical method—interval kriging—to efficiently conduct 3D borehole‐based lithological modeling with sand/non‐sand binary indicators. Interval kriging is a best linear unbiased estimator for irregular interval supports. Interval kriging considers 3D anisotropies between two orthogonal components—a horizontal plane and a vertical axis. A new 3D interval semivariogram is developed. To cope with the nonconvexity of estimation variance, the minimization of estimation variance is regulated with an additional regularization term. The minimization problem is solved by a global‐local genetic algorithm embedded with quadratic programming and Brent's method to obtain kriging weights and kriging length. Four numerical and real‐world case studies demonstrate that interval kriging is more computationally efficient than 3D kriging because the covariance matrix is largely reduced without sacrificing borehole data. Moreover, interval kriging produces more realistic geologic characteristics than 2.5D kriging, while conditional to spatial borehole data. Compared to the multiple‐point statistics (MPS) algorithm—SNESIM, interval kriging can reproduce the geological architecture and spatial connectivity of channel‐type features, meanwhile producing tabular‐type features with better connectivity. Because the regularization term constrains kriged value toward 0 or 1, interval kriging produces more certainty in sand/non‐sand classification than 2.5D kriging, 3D kriging, and SNESIM. In conclusion, interval kriging is an effective and efficient 3D geostatistical algorithm that can capture the 3D structural complexity while significantly reducing computational time.

Causal association of immune cells and endometriosis: a Mendelian randomization study.

To investigate the causal effect of immune cells on endometriosis (EMS), we performed a Mendelian randomization analysis. Mendelian randomization (MR) uses genetic variants as instrumental variables to investigate the causal effects of exposures on outcomes in observational data. In this study, we conducted a thorough two-sample MR analysis to investigate the causal relationship between 731 immune cells and endometriosis. We used complementary Mendelian randomization (MR) methods, including weighted median estimator (WME) and inverse variance weighted (IVW), and performed sensitivity analyses to assess the robustness of our results. Four immune phenotypes have been found to be significantly associated with the risk of developing EMS: B cell %lymphocyte (WME: OR: 1.074, p = 0.027 and IVW: OR: 1.058, p = 0.008), CD14 on Mo MDSC (WME: OR: 1.056, p =0.021 and IVW: OR: 1.047, p = 0.021), CD14+ CD16- monocyte %monocyte (WME: OR: 0.947, p = 0.024 and IVW: OR: 0.958, p = 0.011), CD25 on unsw mem (WME: OR: 1.055, p = 0.030 and IVW: OR: 1.048, p = 0.003). Sensitivity analyses confirmed the main findings, demonstrating consistency across analyses. Our MR analysis provides compelling evidence for a direct causal link between immune cells and EMS, thereby advancing our understanding of the disease. It also provides new avenues and opportunities for the development of immunomodulatory therapeutic strategies in the future.

Re-weighting the randomized controlled trial for generalization: finite-sample error and variable selection

Abstract Randomized controlled trials (RCTs) may suffer from limited scope. In particular, samples may be unrepresentative: some RCTs over- or under-sample individuals with certain characteristics compared to the target population, for which one wants conclusions on treatment effectiveness. Re-weighting trial individuals to match the target population can improve the treatment effect estimation. In this work, we establish the expressions of the bias and variance of such re-weighting procedures—also called inverse propensity of sampling weighting (IPSW)—in presence of categorical covariates for any sample size. Such results allow us to compare the theoretical performance of different versions of IPSW estimates. Besides, our results show how the performance (bias, variance, and quadratic risk) of IPSW estimates depends on the two sample sizes (RCT and target population). A by-product of our work is the proof of consistency of IPSW estimates. In addition, we analyse how including covariates that are not necessary for identifiability of the causal effect may impact the asymptotic variance. Including covariates that are shifted between the two samples but not treatment-effect modifiers increases the variance while non-shifted but treatment-effect modifiers do not. We illustrate all the takeaways in a didactic example, and on a semi-synthetic simulation inspired from critical care medicine.

Efficient designs for three-sequence stepped wedge trials with continuous recruitment.

The standard approach to designing stepped wedge trials that recruit participants in a continuous stream is to divide time into periods of equal length. But the choice of design in such cases is infinitely more flexible: each cluster could cross from the control to the intervention at any point on the continuous time-scale. We consider the case of a stepped wedge design with clusters randomised to just three sequences (designs with small numbers of sequences may be preferred for their simplicity and practicality) and investigate the choice of design that minimises the variance of the treatment effect estimator under different assumptions about the intra-cluster correlation. We make some simplifying assumptions in order to calculate the variance: in particular that we recruit the same number of participants, , from each cluster over the course of the trial, and that participants present at regularly spaced intervals. We consider an intra-cluster correlation that decays exponentially with separation in time between the presentation of two individuals from the same cluster, from a value of for two individuals who present at the same time, to a value of for individuals presenting at the start and end of the trial recruitment interval. We restrict attention to three-sequence designs with centrosymmetry - the property that if we reverse time and swap the intervention and control conditions then the design looks the same. We obtain an expression for the variance of the treatment effect estimator adjusted for effects of time, using methods for generalised least squares estimation, and we evaluate this expression numerically for different designs, and for different parameter values. There is a two-dimensional space of possible three-sequence, centrosymmetric stepped wedge designs with continuous recruitment. The variance of the treatment effect estimator for given and can be plotted as a contour map over this space. The shape of this variance surface depends on and on the parameter , but typically indicates a broad, flat region of close-to-optimal designs. The 'standard' design with equally spaced periods and 1:1:1 allocation rarely performs well, however. In many different settings, a relatively simple design can be found (e.g. one based on simple fractions) that offers close-to-optimal efficiency in that setting. There may also be designs that are robustly efficient over a wide range of settings. Contour maps of the kind we illustrate can help guide this choice. If efficiency is offered as one of the justifications for using a stepped wedge design, then it is worth designing with optimal efficiency in mind.