Equal-sized Clusters Research Articles

Barriers to help-seeking for Malaysian women with symptoms of breast cancer: a mixed-methods, two-step cluster analysis

BackgroundImproving help-seeking behaviour is a key component of down-staging breast cancer and improving survival, but the specific challenges faced by low-income women in an Asian setting remain poorly characterized. Here, we determined the extent of help-seeking delay among Malaysian breast cancer patients who presented at late stages and explored sub-groups of women who may face specific barriers.MethodsTime to help-seeking was assessed in 303 women diagnosed with advanced breast cancer between January 2015 and March 2020 at a suburban tertiary hospital in Malaysia. Two-step cluster analysis was conducted to identify subgroups of women who share similar characteristics and barriers. Barriers to help-seeking were identified from nurse interviews and were analyzed using behavioural frameworks.ResultsThe average time to help-seeking was 65 days (IQR = 250 days), and up to 44.5% of women delayed by at least 3 months. Three equal-sized clusters emerged with good separation by time to help-seeking (p < 0.001). The most reported barrier across clusters was poor knowledge about breast health or breast cancer symptoms (36.3%), regardless of help-seeking behaviour (p = 0.931). Unexpectedly, women with no delay (9 days average) and great delay (259 days average) were more similar to each other than to women with mild delays (58 days average), but, women who experienced great delay reported poor motivation due to fear and embarrassment (p = 0.066) and a lack of social support (p = 0.374) to seek help.ConclusionsDown-staging of breast cancer in Malaysia will require a multi-pronged approach aimed at modifying culturally specific social and emotional barriers, eliminating misinformation, and instilling motivation to seek help for breast health for the women most vulnerable to help-seeking delays.

Version Age of Information in Clustered Gossip Networks

We consider a network consisting of a single source and <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> receiver nodes that are grouped into equal-sized clusters. Each cluster corresponds to a distinct community such that nodes that belong to different communities cannot exchange information. We use dedicated cluster heads in each cluster to facilitate communication between the source and the nodes within that cluster. Inside clusters, nodes are connected to each other according to a given network topology. Based on the connectivity levels within clusters, each node relays its current stored version of the source update to its neighboring nodes by <i>local gossiping</i>. We consider disconnected, ring, and fully connected network topologies for each cluster. For each of these network topologies, we characterize the average version age at each node and find the average version age scaling as a function of the network size <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>. Next, by allowing gossiping among the cluster heads, we improve the version age scaling at the receiver nodes. Then, focusing on a ring network topology in each cluster, we introduce hierarchy to the considered clustered gossip network model without using dedicated cluster heads. Finally, we find the version age-optimum cluster sizes as a function of the source, cluster head, and node update rates through numerical evaluations.

Exact Recovery and Sharp Thresholds of Stochastic Ising Block Model

The stochastic block model (SBM) is a random graph model in which the edges are generated according to the underlying cluster structure on the vertices. The (ferromagnetic) Ising model, on the other hand, assigns ±1 labels to vertices according to an underlying graph structure in a way that if two vertices are connected in the graph then they are more likely to be assigned the same label. In SBM, one aims to recover the underlying clusters from the graph structure while in Ising model, an extensively-studied problem is to recover the underlying graph structure based on i.i.d. samples (labelings of the vertices). In this paper, we propose a natural composition of SBM and the Ising model, which we call the Stochastic Ising Block Model (SIBM). In SIBM, we take SBM in its simplest form, where $n$ vertices are divided into two equal-sized clusters and the edges are connected independently with probability $p$ within clusters and $q$ across clusters. Then we use the graph $G$ generated by the SBM as the underlying graph of the Ising model and draw $m$ i.i.d. samples from it. The objective is to exactly recover the two clusters in SBM from the samples generated by the Ising model, without observing the graph $G$ . As the main result of this paper, we establish a sharp threshold $m^\ast $ on the sample complexity of this exact recovery problem in a properly chosen regime, where $m^\ast $ can be calculated from the parameters of SIBM. We show that when $m\ge m^\ast $ , one can recover the clusters from $m$ samples in $O(n)$ time as the number of vertices $n$ goes to infinity. When $m , we further show that for almost all choices of parameters of SIBM, the success probability of any recovery algorithms approaches 0 as $n\to \infty $ .

Clustering and Neighbouring technique Based Energy-Efficient Routing for WSNs

Energy efficiency is the main prerequisite for the permanent and reliable operation of wireless sensor networks (WSNs). Clustering techniques are designed to build energy-efficient networks that enhance network lifetime. Clustering poses certain challenges that directly affect the network performance such as the cluster head selection and routing process. This paper proposes an approach called Clustering and Neighbouring technique Based Energy-Efficient Routing (CNBEER). The CNBEER approach utilises the clustering algorithm and neighbouring technique to prolong the network lifetime by reducing its total energy consumption. The clustering method divides a network into equal-sized clusters to mitigate inessential energy consumption. Moreover, the clustering algorithm selects a cluster head based on the G function which considers the current energy and position of the nodes of a base station. The cluster head changes dynamically depending on the G value to ensure that the energy consumed at the sensor nodes remains at an approximately equal rate. The CNBEER approach uses the neighbouring technique to discover the best multihop path to the base station. The performance of the proposed approach achieved improvement in the concept of residual energy and network lifetime compared with existing protocols.

FLCEER

Underwater acoustic sensor networks (UASN) play a crucial role in various applications such as tsunami detection, surveillance of the ocean by the defense department, monitoring offshore oil, and identifying gas basins underwater. UASNs can be one of the supporting infrastructures for the Internet of Things (IoT). UASNs have the problems of long latency, high bit error rate, and low bandwidth. These pose various challenges such as high consumption of energy, low reliability, low packet retransmission, and high delay for UASNs. To overcome the shortcomings mentioned above, various approaches are suggested. This article proposes a multi-layer fuzzy logic cluster-based energy-efficient routing protocol for UASNs. It splits the network area into equal sized rings. The priority number (PRN) is utilized for all underwater cluster heads (UCHs). Based on the highest PRN, the UCH starts communicating among UCHs. Here, the PRN makes the task very selective avoiding collisions and also reducing propagation delays. The cluster formation is done by sending a message to all underwater cluster members (UCMs) and the selection of UCH and UCM are done. Each has a threshold value. The intra-ring clustering process splits a ring into equal-sized clusters. Additionally, inter-cluster routing applies the fuzzy logic metrics to choose the optimum data route in transferring the data from the underwater cluster head (UCH) to the sink node (SN). It is tested using Aqua-Sim simulation which is based on NS2. It is compared with an existing protocol such as multi-layer cluster energy efficient (MLCEE), depth-based routing (DBR), energy efficient DBR (EEDBR). The results prove that it has improved energy efficiency, packet delivery ratio, throughput, and the network's lifetime.

Unmasking herd protection by an oral cholera vaccine in a cluster-randomized trial.

BackgroundSeveral studies have shown that inactivated, whole-cell oral cholera vaccines (OCVs) confer both direct protection on vaccinees and herd protection on populations. Because our earlier cluster-randomized effectiveness trial (CRT) in urban Bangladesh failed to detect OCV herd protection, we reanalysed the trial to assess whether herd effects were masked in our original analysis.MethodsA total of 267 270 persons were randomized to 90 approximately equal-sized clusters. In 60 clusters persons aged 1 year and older were eligible to receive OCV and in 30 clusters persons received no intervention and served as controls. We analysed OCV protection against severely dehydrating cholera for the entire clusters, as in our original analysis, and for subclusters consisting of residents of innermost households. We hypothesized that if OCV herd protection was attenuated by cholera transmission into the clusters from the outside in this densely populated setting, herd protection would be most evident in the innermost households.ResultsDuring 2 years of follow-up of all residents of the clusters, total protection (protection of OCV recipients relative to control residents) was 58% [95% confidence interval (CI): 43%, 70%; P<0.0001], indirect protection (protection of non-OCV recipients in OCV clusters relative to control participants) was 16% (95% CI: –20%, 41%; P=0.35) and overall OCV protection (protection of all residents in the OCV clusters relative to control residents) was 46% (95% CI: 30%, 59%; P<0.0001). Analyses of the inner 75% and 50% households of the clusters showed similar findings. However, total protection was 75% (95% CI: 50%, 87%, P<0.0001), indirect protection 52% (95% CI: –9%, 79%; P=0.08) and overall protection 72% (95% CI: 49%, 84%; P<0.0001) for the innermost 25% households.ConclusionConsistent with past studies, substantial OCV herd protective effects were identified, but were unmasked only by analysing innermost households of the clusters. Caution is needed in defining clusters for analysis of vaccine herd effects in CRTs of vaccines.

An impossibility result for reconstruction in the degree-corrected stochastic block model

We consider the Degree-Corrected Stochastic Block Model (DC-SBM): a random graph on $n$ nodes, having i.i.d. weights $(\phi_{u})_{u=1}^{n}$ (possibly heavy-tailed), partitioned into $q\geq2$ asymptotically equal-sized clusters. The model parameters are two constants $a,b>0$ and the finite second moment of the weights $\Phi^{(2)}$. Vertices $u$ and $v$ are connected by an edge with probability $\frac{\phi_{u}\phi_{v}}{n}a$ when they are in the same class and with probability $\frac{\phi_{u}\phi_{v}}{n}b$ otherwise. We prove that it is information-theoretically impossible to estimate the clusters in a way positively correlated with the true community structure when $(a-b)^{2}\Phi^{(2)}\leq q(a+b)$. As by-products of our proof we obtain $(1)$ a precise coupling result for local neighbourhoods in DC-SBMs, that we use in Gulikers, Lelarge and Massoulie (2016) to establish a law of large numbers for local-functionals and $(2)$ that long-range interactions are weak in (power-law) DC-SBMs.

Unmasking Herd Protection by an Oral Cholera Vaccine in a Cluster-Randomised Trial: A Reanalysis

Background: Several studies have shown that inactivated, whole-cell oral cholera vaccines (OCVs) confer both direct protection to vaccinees and herd protection to populations. Because our earlier cluster randomised, effectiveness trial (CRT) in urban, Bangladesh failed to detect OCV herd protection, we reanalysed the trial to assess whether herd effects were masked in our original analysis. Methods: 267,270 persons were randomised to 90 approximately equal-sized clusters. In 60 clusters persons aged one year and older were eligible to receive OCV and in 30 clusters persons received no intervention and served as controls. We analysed OCV protection against severely dehydrating cholera for the entire clusters, as in our original analysis, and for subclusters consisting of residents of innermost households. We hypothesised that if OCV herd protection was attenuated by cholera transmission into the clusters from the outside in this densely populated setting, herd protection would be most evident in the innermost households. Findings: During two years of follow-up of all residents of the clusters total protection (protection of OCV recipients relative to control residents) was 58% (95% CI: 43%, 70%; P<.0001), indirect protection (protection of non-OCV recipients in OCV clusters relative to control participants) was 16% (95% CI: -20%, 41%; P=.35), and overall OCV protection (protection of all residents in the OCV clusters relative to control residents) was 46% (95% CI:30%,59%; P<.0001). Analyses of the inner 75% and 50% households of the clusters showed similar findings. However, total protection was 75% (95% CI:50%,87%, P<.0001), indirect protection 52% (95% CI:-9%,79%; P=.08), and overall protection 72% (95% CI:49%,84%; P<.0001) for the innermost 25% households. Interpretation: Consistent with past studies, substantial OCV herd protective effects were identified, but were unmasked only by analysing innermost households of the clusters. Caution is needed in defining clusters for analysis of vaccine herd effects in CRTs of vaccines. Trial Registration Number: Clinical Trials.gov NCT01339845 Funding: Bill and Melinda Gates Foundation Declaration of Interest: We declare no competing interests. Ethical Approval: The study protocol was approved by the Research Review Committee and the Ethical Review Committee of the icddr,b, Dhaka, Bangladesh and the Institutional Review Board of the International Vaccine Institute. Written informed consent was obtained from residents 18 years and older and from the parents or guardians of residents aged 1 to 17 years of age. Additional assent was obtained from residents aged 11 to 17 years.

Fast automatic estimation of the number of clusters from the minimum inter-center distance for k-means clustering

Center-based clustering methods like k-Means intend to identify closely packed clusters of data points by respectively finding the centers of each cluster. However, k-Means requires the user to guess the number of clusters, instead of estimating the same on the run. Hence, the incorporation of accurate automatic estimation of the natural number of clusters present in a data set is important to make a clustering method truly unsupervised. For k-Means, the minimum of the pairwise distance between cluster centers decreases as the user-defined number of clusters increases. In this paper, we observe that the last significant reduction occurs just as the user-defined number surpasses the natural number of clusters. Based on this insight, we propose two techniques: the Last Leap (LL) and the Last Major Leap (LML) to estimate the number of clusters for k-Means. Over a number of challenging situations, we show that LL accurately identifies the number of well-separated clusters, whereas LML identifies the number of equal-sized clusters. Any disparity between the values of LL and LML can thus inform a user about the underlying cluster structures present in the data set. The proposed techniques are independent of the size of the data set, making them especially suitable for large data sets. Experiments show that LL and LML perform competitively with the best cluster number estimation techniques while imposing drastically lower computational burden.

Striatonigral neurons divide into two distinct morphological-physiological phenotypes after chronic L-DOPA treatment in parkinsonian rats

Dendritic regression of striatal spiny projection neurons (SPNs) is a pathological hallmark of Parkinson’s disease (PD). Here we investigate how chronic dopamine denervation and dopamine replacement with L-DOPA affect the morphology and physiology of direct pathway SPNs (dSPNS) in the rat striatum. We used a lentiviral vector optimized for retrograde labeling (FuG-B-GFP) to identify dSPNs in rats with 6-hydroxydopamine (6-OHDA) lesions. Changes in morphology and physiology of dSPNs were assessed through a combination of patch-clamp recordings and two photon microscopy. The 6-OHDA lesion caused a significant reduction in dSPN dendritic complexity. Following chronic L-DOPA treatment, dSPNs segregated into two equal-sized clusters. One group (here called “cluster-1”), showed sustained dendritic atrophy and a partially normalized electrophysiological phenotype. The other one (“cluster-2”) exhibited dendritic regrowth and a strong reduction of intrinsic excitability. Interestingly, FosB/∆FosB induction by L-DOPA treatment occurred preferentially in cluster-2 dSPNs. Our study demonstrates the feasibility of retrograde FuG-B-GFP labeling to study dSPNs in the rat and reveals, for the first time, that a subgroup of dSPNs shows dendritic sprouting in response to chronic L-DOPA treatment. Investigating the mechanisms and significance of this response will greatly improve our understanding of the adaptations induced by dopamine replacement therapy in PD.

NP-Hardness of balanced minimum sum-of-squares clustering

The balanced clustering problem consists of partitioning a set of n objects into K equal-sized clusters as long as n is a multiple of K. A popular clustering criterion when the objects are points of a q-dimensional space is the minimum sum of squared distances from each point to the centroid of the cluster to which it belongs. We show in this paper that this problem is NP-hard in general dimension already for triplets, i.e., when n/K=3.

Global and Local Information in Clustering Labeled Block Models

The stochastic block model is a classical cluster-exhibiting random graph model that has been widely studied in statistics, physics, and computer science. In its simplest form, the model is a random graph with two equal-sized clusters, with intracluster edge probability $p$ , and intercluster edge probability $q$ . We focus on the sparse case, i.e., $p, q = O(1/n)$ , which is practically more relevant and also mathematically more challenging. A conjecture of Decelle, Krzakala, Moore, and Zdeborova, based on ideas from statistical physics, predicted a specific threshold for clustering. The negative direction of the conjecture was proved by Mossel, Neeman, and Sly (2012), and more recently, the positive direction was independently proved by Massoulie and Mossel, Neeman, and Sly. In many real network clustering problems, nodes contain information as well. We study the interplay between node and network information in clustering by studying a labeled block model, where in addition to the edge information, the true cluster labels of a small fraction of the nodes are revealed. In the case of two clusters, we show that below the threshold, a small amount of node information does not affect recovery. On the other hand, we show that for any small amount of information, efficient local clustering is achievable as long as the number of clusters is sufficiently large (as a function of the amount of revealed information).

Achieving Exact Cluster Recovery Threshold via Semidefinite Programming: Extensions

Resolving a conjecture of Abbe, Bandeira and Hall, the authors have recently shown that the semidefinite programming (SDP) relaxation of the maximum likelihood estimator achieves the sharp threshold for exactly recovering the community structure under the binary stochastic block model of two equal-sized clusters. The same was shown for the case of a single cluster and outliers. Extending the proof techniques, in this paper it is shown that SDP relaxations also achieve the sharp recovery threshold in the following cases: (1) Binary stochastic block model with two clusters of sizes proportional to network size but not necessarily equal; (2) Stochastic block model with a fixed number of equal-sized clusters; (3) Binary censored block model with the background graph being Erd\H{o}s-R\'enyi. Furthermore, a sufficient condition is given for an SDP procedure to achieve exact recovery for the general case of a fixed number of clusters plus outliers. These results demonstrate the versatility of SDP relaxation as a simple, general purpose, computationally feasible methodology for community detection.

Achieving Exact Cluster Recovery Threshold via Semidefinite Programming

The binary symmetric stochastic block model deals with a random graph of n vertices partitioned into two equal-sized clusters, such that each pair of vertices is independently connected with probability p within clusters and q across clusters. In the asymptotic regime of p = a log n/n and q = b log n/n for fixed a, b, and n → ∞, we show that the semidefinite programming relaxation of the maximum likelihood estimator achieves the optimal threshold for exactly recovering the partition from the graph with probability tending to one, resolving a conjecture of Abbe et al. Furthermore, we show that the semidefinite programming relaxation also achieves the optimal recovery threshold in the planted dense subgraph model containing a single cluster of size proportional to n.

Drug-related causes of death: Socioeconomic and demographic characteristics of the deceased.

The aim of this study was to describe subgroups of those who died from a drug-related cause of death employing demographic and socioeconomic data. A total of 1,628 persons with registered drug-related deaths in the Norwegian Cause of Death Registry between 2003 and 2009 were matched with research registers of data on demographic and socioeconomic factors during the five years prior to their deaths. Three equal-sized clusters were identified: persons with very low socioeconomic status, disability pensioners and people on the edge of the workforce. Socioeconomic situation prior to drug-related deaths was more heterogeneous than expected. Greater knowledge about the members of the disability pensioner and the edge of the workforce clusters must be established in order to make prevention efforts towards these groups more precise and goal oriented.

Reconstruction and estimation in the planted partition model

The planted partition model (also known as the stochastic blockmodel) is a classical cluster-exhibiting random graph model that has been extensively studied in statistics, physics, and computer science. In its simplest form, the planted partition model is a model for random graphs on $$n$$ nodes with two equal-sized clusters, with an between-class edge probability of $$q$$ and a within-class edge probability of $$p$$ . Although most of the literature on this model has focused on the case of increasing degrees (ie. $$pn, qn \rightarrow \infty $$ as $$n \rightarrow \infty $$ ), the sparse case $$p, q = O(1/n)$$ is interesting both from a mathematical and an applied point of view. A striking conjecture of Decelle, Krzkala, Moore and Zdeborova based on deep, non-rigorous ideas from statistical physics gave a precise prediction for the algorithmic threshold of clustering in the sparse planted partition model. In particular, if $$p = a/n$$ and $$q = b/n$$ , then Decelle et al. conjectured that it is possible to cluster in a way correlated with the true partition if $$(a - b)^2 > 2(a + b)$$ , and impossible if $$(a - b)^2 < 2(a + b)$$ . By comparison, the best-known rigorous result is that of Coja-Oghlan, who showed that clustering is possible if $$(a - b)^2 > C (a + b)$$ for some sufficiently large $$C$$ . We prove half of their prediction, showing that it is indeed impossible to cluster if $$(a - b)^2 < 2(a + b)$$ . Furthermore we show that it is impossible even to estimate the model parameters from the graph when $$(a - b)^2 < 2(a + b)$$ ; on the other hand, we provide a simple and efficient algorithm for estimating $$a$$ and $$b$$ when $$(a - b)^2 > 2(a + b)$$ . Following Decelle et al, our work establishes a rigorous connection between the clustering problem, spin-glass models on the Bethe lattice and the so called reconstruction problem. This connection points to fascinating applications and open problems.

Effect of women's groups and volunteer peer counselling on rates of mortality, morbidity, and health behaviours in mothers and children in rural Malawi (MaiMwana): a factorial, cluster-randomised controlled trial

Women's groups and health education by peer counsellors can improve the health of mothers and children. We assessed their effects on mortality and breastfeeding rates in rural Malawi. We did a 2×2 factorial, cluster-randomised trial in 185,888 people in Mchinji district. 48 equal-sized clusters were randomly allocated to four groups with a computer-generated number sequence. 24 facilitators guided groups through a community action cycle to tackle maternal and child health problems. 72 trained volunteer peer counsellors made home visits at five timepoints during pregnancy and after birth to support breastfeeding and infant care. Primary outcomes for the women's group intervention were maternal, perinatal, neonatal, and infant mortality rates (MMR, PMR, NMR, and IMR, respectively); and for the peer counselling were IMR and exclusive breastfeeding (EBF) rates. Analysis was by intention to treat. The trial is registered as ISRCTN06477126. We monitored outcomes of 26,262 births between 2005 and 2009. In a factorial model adjusted only for clustering and the volunteer peer counselling intervention, in women's group areas, for years 2 and 3, we noted non-significant decreases in NMR (odds ratio 0.93, 0.64-1.35) and MMR (0.54, 0.28-1.04). After adjustment for parity, socioeconomic quintile, and baseline measures, effects were larger for NMR (0.85, 0.59-1.22) and MMR (0.48, 0.26-0.91). Because of the interaction between the two interventions, a stratified analysis was done. For women's groups, in adjusted analyses, MMR fell by 74% (0.26, 0.10-0.70), and NMR by 41% (0.59, 0.40-0.86) in areas with no peer counsellors, but there was no effect in areas with counsellors (1.09, 0.40-2.98, and 1.38, 0.75-2.54). Factorial analysis for the peer counselling intervention for years 1-3 showed a fall in IMR of 18% (0.82, 0.67-1.00) and an improvement in EBF rates (2.42, 1.48-3.96). The results of the stratified, adjusted analysis showed a 36% reduction in IMR (0.64, 0.48-0.85) but no effect on EBF (1.18, 0.63-2.25) in areas without women's groups, and in areas with women's groups there was no effect on IMR (1.05, 0.82-1.36) and an increase in EBF (5.02, 2.67-9.44). The cost of women's groups was US$114 per year of life lost (YLL) averted and that of peer counsellors was $33 per YLL averted, using stratified data from single intervention comparisons. Community mobilisation through women's groups and volunteer peer counsellor health education are methods to improve maternal and child health outcomes in poor rural populations in Africa. Saving Newborn Lives, UK Department for International Development, and Wellcome Trust.

More efficient unconditional tests for exchangeable binary data with equal cluster sizes

We consider exact unconditional procedures for testing independence of exchangeable binary data with equal-sized clusters. The approximate unconditional approach is recommended as a basic procedure. The exact unconditional procedure based on estimation followed by maximization is recommended for the small number of clusters.

A new dependence parameter approach to improve the design of cluster randomized trials with binary outcomes

Power and sample size calculations for cluster randomized trials require prediction of the degree of correlation that will be realized among outcomes of participants in the same cluster. This correlation is typically quantified as the intraclass correlation coefficient (ICC), defined as the Pearson correlation between two members of the same cluster or proportion of the total variance attributable to variance between clusters. It is widely known but perhaps not fully appreciated that for binary outcomes, the ICC is a function of outcome prevalence. Hence, the ICC and the outcome prevalence are intrinsically related, making the ICC poorly generalizable across study conditions and between studies with different outcome prevalences. We use a simple parametrization of the ICC that aims to isolate that part of the ICC that measures dependence among responses within a cluster from the outcome prevalence. We incorporate this parametrization into sample size calculations for cluster randomized trials and compare our method to the traditional approach using the ICC. Our dependence parameter, R, may be less influenced by outcome prevalence and has an intuitive meaning that facilitates interpretation. Estimates of R from previous studies can be obtained using simple statistics. Comparison of methods showed that the traditional ICC approach to sample size determination tends to overpower studies under many scenarios, calling for more clusters than truly required. The methods are developed for equal-sized clusters, whereas cluster size may vary in practice. The dependence parameter R is an alternative measure of dependence among binary outcomes in cluster randomized trials that has a number of advantages over the ICC.

Clustering, Spatial Correlations, and Randomization Inference

It is a standard practice in regression analyses to allow for clustering in the error covariance matrix if the explanatory variable of interest varies at a more aggregate level (e.g., the state level) than the units of observation (e.g., individuals). Often, however, the structure of the error covariance matrix is more complex, with correlations not vanishing for units in different clusters. Here, we explore the implications of such correlations for the actual and estimated precision of least squares estimators. Our main theoretical result is that with equal-sized clusters, if the covariate of interest is randomly assigned at the cluster level, only accounting for nonzero covariances at the cluster level, and ignoring correlations between clusters as well as differences in within-cluster correlations, leads to valid confidence intervals. However, in the absence of random assignment of the covariates, ignoring general correlation structures may lead to biases in standard errors. We illustrate our findings using the 5% public-use census data. Based on these results, we recommend that researchers, as a matter of routine, explore the extent of spatial correlations in explanatory variables beyond state-level clustering.