Learn and talk about Replication (statistics), Design of experiments, Sampling (statistics)

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In engineering, science, and statistics, replication is the repetition of an experimental condition so that the variability associated with the phenomenon can be estimated. ASTM, in standard E1847, defines replication as "the repetition of the set of all the treatment combinations to be compared in an experiment. Each of the repetitions is called a replicate."

Replication is not the same as repeated measurements of the same item: they are dealt with differently in statistical experimental design and data analysis.

For proper sampling, a process or batch of products should be in reasonable statistical control; inherent random variation is present but variation due to assignable (special) causes is not. Evaluation or testing of a single item does not allow for item-to-item variation and may not represent the batch or process. Replication is needed to account for this variation among items and treatments.

Example [edit]

As an example, consider a continuous process which produces items. Batches of items are then processed or treated. Finally, tests or measurements are conducted. Several options might be available to obtain ten test values. Some possibilities are:

One finished and treated item might be measured repeatedly to obtain ten test results. Only one item was measured so there is no replication. The repeated measurements help identify observational error.
Ten finished and treated items might be taken from a batch and each measured once. This is not full replication because the ten samples are not random and not representative of the continuous nor batch processing.
Five items are taken from the continuous process based on sound statistical sampling. These are processed in a batch and tested twice each. This includes replication of initial samples but does not allow for batch-to-batch variation in processing. The repeated tests on each provide some measure and control of testing error.
Five items are taken from the continuous process based on sound statistical sampling. These are processed in five different batches and tested twice each. This plan includes proper replication of initial samples and also includes batch-to-batch variation. The repeated tests on each provide some measure and control of testing error.

Each option would call for different data analysis methods and yield different conclusions.

Bibliography [edit]

ASTM E122-07 Standard Practice for Calculating Sample Size to Estimate, With Specified Precision, the Average for a Characteristic of a Lot or Process
"Engineering Statistics Handbook", NIST/SEMATEK
Pyzdek, T, "Quality Engineering Handbook", 2003, ISBN 0-8247-4614-7.
Godfrey, A. B., "Juran's Quality Handbook", 1999, ISBN 9780070340039.

Statistics

Descriptive statistics

Continuous data

Location	Mean (Arithmetic, Geometric, Harmonic) Median Mode

Dispersion	Range Standard deviation Coefficient of variation Percentile Interquartile range

Shape	Variance Skewness Kurtosis Moments L-moments

Count data

Index of dispersion

Summary tables

Dependence

Statistical graphics

Data collection

Designing studies	Effect size Standard error Statistical power Sample size determination

Survey methodology	Sampling Stratified sampling Opinion poll Questionnaire

Controlled experiment	Design of experiments Randomized experiment Random assignment Replication Blocking Factorial experiment Optimal design

Uncontrolled studies	Natural experiment Quasi-experiment Observational study

Statistical inference

Statistical theory	Sampling distribution Order statistics Sufficiency Completeness Exponential family Permutation test (Randomization test) Empirical distribution Bootstrap U statistic Efficiency Asymptotics Robustness

Frequentist inference	Unbiased estimator (Mean unbiased minimum variance, Median unbiased) Biased estimators (Maximum likelihood, Method of moments, Minimum distance, Density estimation) Confidence interval Testing hypotheses Power Parametric tests (Likelihood-ratio, Wald, Score)

Specific tests	Z (normal) Student's t-test F Chi-squared Signed-rank (1-sample, 2-sample, 1-way anova) Shapiro–Wilk Kolmogorov–Smirnov

Bayesian inference	Bayesian probability Prior Posterior Credible interval Bayes factor Bayesian estimator Maximum posterior estimator

Correlation and regression analysis

Correlation	Pearson product–moment correlation Partial correlation Confounding variable Coefficient of determination

Regression analysis	Errors and residuals Regression model validation Mixed effects models Simultaneous equations models

Linear regression	Simple linear regression Ordinary least squares General linear model Bayesian regression

Non-standard predictors	Nonlinear regression Nonparametric Semiparametric Isotonic Robust

Generalized linear model	Exponential families Logistic (Bernoulli) Binomial Poisson

Partition of variance	Analysis of variance (ANOVA) Analysis of covariance Multivariate ANOVA Degrees of freedom

Categorical, multivariate, time-series, or survival analysis

Categorical data

Multivariate statistics

Time series analysis

General	Decomposition Trend Stationarity Seasonal adjustment

Time domain	ACF PACF XCF ARMA model ARIMA model Vector autoregression

Frequency domain	Spectral density estimation

Survival analysis

Applications

Biostatistics	Bioinformatics Clinical trials & studies Epidemiology Medical statistics

Engineering statistics	Chemometrics Methods engineering Probabilistic design Process & Quality control Reliability System identification

Social statistics	Actuarial science Census Crime statistics Demography Econometrics National accounts Official statistics Population Psychometrics

Spatial statistics	Cartography Environmental statistics Geographic information system Geostatistics Kriging

v t e Design of experiments

Scientific Method	Scientific experiment Statistical design Control Internal & external validity Experimental unit Blinding Optimal design: Bayesian Random assignment Randomization Restricted randomization Replication versus subsampling Sample size

Treatment & Blocking	Treatment Effect size Contrast Interaction Confounding Orthogonality Blocking Covariate Nuisance variable

Models & Inference	Linear regression Ordinary least squares Bayesian Random effect Mixed model Hierarchical model: Bayesian Analysis of variance (Anova) Cochran's theorem Manova (multivariate) Ancova (covariance) Compare means Multiple comparison

Designs: Completely Randomized	Factorial Fractional factorial Plackett-Burman Taguchi Response surface methodology Polynomial & rational modeling Box-Behnken Central composite Block Generalized randomized block design (GRBD) Latin square Graeco-Latin square Orthogonal array Latin hypercube Repeated measures design Crossover study Randomized controlled trial Sequential analysis Sequential probability ratio test

Glossary Category Statistics portal Statistical outline Statistical topics

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Replication (statistics)

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