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s in operations while the second stage to obtain a random sample about the characteristics of those businesses advantages edit can be cheaper than other sampling plans e g fewer travel expenses and administration cost feasibility this sampling plan takes large populations into account since these groups are so small deploying any other sampling plan would be very costly economy the regular two major concerns of expenditure i e traveling and listing are greatly reduced in this method for example compiling research information about every household in a city would be very costly whereas compiling information about various blocks of the city will be more economical here traveling as well as listing efforts will be greatly reduced reduced variability in the rare case of a negative intraclass correlation between subjects within a cluster the estimators produced by cluster sampling will yield more accurate estimates than data obtained from a simple random sample i e the design effect will be larger than 1 this is not a commonplace scenario major use when the sampling frame of all elements is not available we can resort only to cluster sampling disadvantages edit higher sampling error which can be expressed by the design effect the ratio between the variance of an estimator made from the samples of the cluster study and the variance of an estimator obtained from a sample of subjects in an equally reliable randomly sampled unclustered study 5 the larger the intraclass correlation is between subjects within a cluster the worse the design effect becomes i e the larger it gets from 1 indicating a larger expected increase in the variance of the estimator in other words the more there is heterogeneity between clusters and more homogeneity between subjects within a cluster the less accurate our estimators become this is because in such cases we are better on sampling as many clusters as we can and making do with a small sample of subjects from within each cluster i e two stage cluster sampling complexity cluster sampling is more sophisticated and requires more attention with how to plan and how to analyze i e to take into account the weights of subjects during the estimation of parameters confidence intervals etc more on cluster sampling edit two stage cluster sampling edit two stage cluster sampling a simple case of multistage sampling is obtained by selecting cluster samples in the first stage and then selecting a sample of elements from every sampled cluster consider a population of n clusters in total in the first stage n clusters are selected using the ordinary cluster sampling method in the second stage simple random sampling is usually used 6 it is used separately in every cluster and the numbers of elements selected from different clusters are not necessarily equal the total number of clusters n the number of clusters selected n and the numbers of elements from selected clusters need to be pre determined by the survey designer two stage cluster sampling aims at minimizing survey costs and at the same time controlling the uncertainty related to estimates of interest 7 this method can be used in health and social sciences for instance researchers used two stage cluster sampling to generate a representative sample of the iraqi population to conduct mortality surveys 8 sampling in this method can be quicker and more reliable than other methods which is why this method is now used frequently inference when the number of clusters is small edit cluster sampling methods can lead to significant bias when working with a small number of clusters for instance it can be necessary to cluster at the state or city level units that may be small and fixed in number microeconometrics methods for panel data often use short panels which is analogous to having few observations per clusters and many clusters the small cluster problem can be viewed as an incidental parameter problem 9 while the point estimates can be reasonably precisely estimated if the number of observations per cluster is sufficiently high we need the number of clusters g displaystyle g rightarrow infty for the asymptotics to kick in if the number of clusters is low the estimated covariance matrix can be downward biased 10 small numbers of clusters are a risk when there is serial correlation or when there is intraclass correlation as in the moulton context when having few clusters we tend to underestimate serial correlation across observations when a random shock occurs or the intraclass correlation in a moulton setting 11 several studies have highlighted the consequences of serial correlation and highlighted the small cluster problem 12 13 in the framework of the moulton factor an intuitive explanation of the small cluster problem can be derived from the formula for the moulton factor assume for simplicity that the number of observations per cluster is fixed at n below v c β displaystyle v_ c beta stands for the covariance matrix adjusted for clustering v β displaystyle v beta stands for the covariance matrix not adjusted for clustering and ρ stands for the intraclass correlation v c β v β 1 n 1 ρ displaystyle frac v_ c hat beta v hat beta 1 n 1 rho the ratio on the left hand side indicates how much the unadjusted scenario overestimates the precision therefore a high number means a strong downward bias of the estimated covariance matrix a small cluster problem can be interpreted as a large n when the data is fixed and the number of clusters is low the number of data within a cluster can be high it follows that inference when the number of clusters is small will not have the correct coverage 11 several solutions for the small cluster problem have been proposed one can use a bias corrected cluster robust variance matrix make t distribution adjustments or use bootstrap methods with asymptotic refinements such as the percentile t or wild bootstrap that can lead to improved finite sample inference 10 cameron gelbach and miller 2008 provide microsimulations for different methods and find that the wild bootstrap performs well in the face of a small number of clusters 14 see also edit multistage sampling sampling statistics simple random sampling stratified sampling references edit handbook on master sampling frames for agricultural statistics pdf free download docplayer net retrieved 2024 01 10 david brown study claims iraq s excess death toll has reached 655 000 washington post wednesday october 11 2006 retrieved september 14 2010 nelson gary a july 2014 cluster sampling a pervasive yet little recognized survey design in fisheries research transactions of the american fisheries society 143 4 926 938 bibcode 2014trafs 143 926n doi 10 1080 00028487 2014 901252 aga gemechu a francis david c jolevski filip rodriguez meza jorge l wimpey joshua s january 2023 an application of adaptive cluster sampling to surveying informal businesses journal of survey statistics and methodology 11 5 1246 1266 doi 10 1093 jssam smac037 kerry and bland 1998 statistics notes the intracluster correlation coefficient in cluster randomization british medical journal 316 1455 1460 ahmed saifuddin 2009 methods in sample surveys pdf the johns hopkins university and saifuddin ahmed archived pdf from the original on 2013 09 28 daniel pfeffermann c radhakrishna rao 2009 handbook of statistics vol 29a sample surveys theory methods and infernece elsevier b v isbn 978 0 444 53124 7 lp galway nathaniel bell al s sae amy hagopian gilbert burnham abraham flaxman wiliam m weiss julie rajaratnam tim k takaro 27 april 2012 a two stage cluster sampling method using gridded population data a gis and google earthtm imagery in a population based mortality survey in iraq international journal of health geographics 11 1 12 bibcode 2012ijhgg 11 12g doi 10 1186 1476 072x 11 12 pmc 3490933 pmid 22540266 cameron a c and p k trivedi 2005 microeconometrics methods and applications cambridge university press new york 1 2 cameron c and d l miller 2015 a practitioner s guide to cluster robust inference journal of human resources 50 2 pp 317 372 1 2 angrist j d and j s pischke 2009 mostly harmless econometrics an empiricist s companion princeton university press new jersey bertrand m e duflo and s mullainathan 2004 how much should we trust differences in differences estimates quarterly journal of economics 119 1 pp 249 275 kezdi g 2004 robust standard error estimation in fixed effect panel models hungarian statistical review 9 pp 95 116 cameron c j gelbach and d l miller 2008 bootstrap based improvements for inference with clustered errors the review of economics and statistics 90 pp 414 427 v t e statistics outline index descriptive statistics continuous data center mean arithmetic arithmetic geometric contraharmonic cubic generalized power geometric harmonic heronian heinz lehmer median mode dispersion average absolute deviation coefficient of variation interquartile range percentile range standard deviation variance shape central limit theorem moments kurtosis l moments skewness count data index of dispersion summary tables contingency table frequency distribution grouped data dependence partial correlation pearson product moment correlation rank correlation kendall s τ spearman s ρ scatter plot graphics bar chart biplot box plot control chart correlogram fan chart forest plot histogram pie chart q q plot radar chart run chart scatter plot stem and leaf display violin plot heatmap scatter plot matrix ecdf plot line chart statistical data processing transformations data transformation log transformation power transform box cox transformation yeo johnson transformation variance stabilizing transformation anscombe transform fisher transformation scaling and normalization feature scaling normalization standardization z score min max normalization unit vector normalization data cleaning data cleaning outlier winsorizing truncation missing data data reduction dimensionality reduction principal component analysis factor analysis time series preprocessing differencing detrending seasonal adjustment stationarity transformation data collection study design effect size missing data optimal design population replication sample size determination statistic statistical power survey methodology sampling cluster stratified opinion poll questionnaire standard error controlled experiments blocking factorial experiment interaction random assignment randomized controlled trial randomized experiment scientific control adaptive designs adaptive clinical trial stochastic approximation up and down designs observational studies cohort study cross sectional study natural experiment quasi experiment statistical inference statistical theory population statistic probability distribution sampling distribution order statistic empirical distribution density estimation statistical model model specification l p space parameter location scale shape parametric family likelihood monotone location scale family exponential family completeness sufficiency statistical functional bootstrap u v optimal decision loss function efficiency statistical distance divergence asymptotics robustness sensitivity analysis frequentist inference point estimation estimating equations maximum likelihood method of moments m estimator minimum distance unbiased estimators mean unbiased minimum variance rao blackwellization lehmann scheffé theorem median unbiased plug in interval estimation confidence interval pivot likelihood interval prediction interval tolerance interval resampling bootstrap jackknife testing hypotheses 1 2 tails power uniformly most powerful test permutation test randomization test multiple comparisons parametric tests likelihood ratio g test score lagrange multiplier wald z test normal specific tests parametric student s t test f test goodness of fit chi squared kolmogorov smirnov anderson darling lilliefors jarque bera normality shapiro wilk model selection cross validation aic bic rank statistics sign sample median signed rank wilcoxon hodges lehmann estimator rank sum mann whitney nonparametric anova 1 way kruskal wallis 2 way friedman ordered alternative jonckheere terpstra van der waerden test bayesian inference bayesian probability prior posterior credible interval bayes factor bayesian estimator maximum posterior estimator correlation regression analysis correlation pearson product moment partial correlation confounding variable coefficient of determination regression analysis errors and residuals regression validation mixed effects models simultaneous equations models multivariate adaptive regression splines mars template least squares and regression analysis linear regression simple linear regression ordinary least squares general linear model bayesian regression non standard predictors nonlinear regression nonparametric semiparametric isotonic robust homoscedasticity and heteroscedasticity generalized linear model exponential families logistic bernoulli binomial poisson regressions partition of variance analysis of variance anova analysis of variance ancova manova degrees of freedom categorical multivariate time series survival analysis categorical cohen s kappa contingency table graphical model log linear model mcnemar s test cochran mantel haenszel statistics multivariate regression manova principal components canonical correlation discriminant analysis cluster analysis classification structural equation model factor analysis multivariate distributions elliptical distributions normal time series general decomposition trend stationarity seasonal adjustment exponential smoothing cointegration structural break granger causality specific tests dickey fuller johansen q statistic ljung box durbin watson breusch godfrey time domain autocorrelation acf partial pacf cross correlation xcf arma model arima model box jenkins autoregressive conditional heteroskedasticity arch vector autoregression var autoregressive model ar frequency domain spectral density 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