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Nelson–Aalen estimator

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Nonparametric estimate of cumulative hazard

TheNelson–Aalen estimator is anon-parametric estimator of thecumulative hazard rate function in case ofcensored data orincomplete data.^[1] It is used insurvival theory,reliability engineering andlife insurance to estimate the cumulative number of expected events. An "event" can be the failure of a non-repairable component, the death of a human being, or any occurrence for which the experimental unit remains in the "failed" state (e.g., death) from the point at which it changed on. Theestimator is given by

{\tilde {H}}(t)=\sum _{t_{i}\leq t}{\frac {d_{i}}{n_{i}}},

with $d_{i}$ the number of events at time $t_{i}$ and $n_{i}$ the total individuals at risk at $t_{i}$ .^[2]

The curvature of the Nelson–Aalen estimator gives an idea of the hazard rate shape. A concave shape is an indicator forinfant mortality while a convex shape indicateswear out mortality.

It can be used for example when testing the homogeneity ofPoisson processes.^[3]

It was constructed byWayne Nelson andOdd Aalen.^[4]^[5]^[6]The Nelson-Aalen estimator is directly related to theKaplan-Meier estimator and both maximize theempirical likelihood.^[7]

References

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^"Kaplan–Meier and Nelson–Aalen Estimators". 21 September 2008.
^"Kaplan–Meier Survival Estimates".
^Kysely, Jan; Picek, Jan; Beranova, Romana (2010). "Estimating extremes in climate change simulations using the peaks-over-threshold method with a non-stationary threshold".Global and Planetary Change.72 (1–2):55–68.Bibcode:2010GPC....72...55K.doi:10.1016/j.gloplacha.2010.03.006.
^Nelson, W. (1969). "Hazard plotting for incomplete failure data".Journal of Quality Technology.1:27–52.doi:10.1080/00224065.1969.11980344.
^Nelson, W. (1972). "Theory and applications of hazard plotting for censored failure data".Technometrics.14 (4):945–965.doi:10.1080/00401706.1972.10488991.
^Aalen, Odd (1978)."Nonparametric inference for a family of counting processes".Annals of Statistics.6 (4):701–726.doi:10.1214/aos/1176344247.JSTOR 2958850.
^Zhou, M. (2015). Empirical Likelihood Method in Survival Analysis (1st ed.). Chapman and Hall/CRC.https://doi.org/10.1201/b18598,https://books.google.com/books?id=9-b5CQAAQBAJ&dq=Does+the+Nelson%E2%80%93Aalen+estimator+construct+an+empirical+likelihood%3F&pg=PA7

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Statistics

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

Dependence

Graphics

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

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 Score/Lagrange multiplier Wald

Specific tests

Z-test(normal) Student'st-test F-test
Goodness of fit	Chi-squared G-test Kolmogorov–Smirnov Anderson–Darling Lilliefors Jarque–Bera Normality(Shapiro–Wilk) Likelihood-ratio test 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

Correlation	Pearson product-moment Partial correlation Confounding variable Coefficient of determination
Regression analysis (see alsoTemplate:Least squares and regression analysis	Errors and residuals Regression validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines (MARS)
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, anova) Analysis of covariance Multivariate ANOVA Degrees of freedom

Categorical / multivariate / time-series / survival analysis

Categorical

Multivariate

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 estimation Fourier analysis Least-squares spectral analysis Wavelet Whittle likelihood

Survival

Survival function	Kaplan–Meier estimator (product limit) Proportional hazards models Accelerated failure time (AFT) model First hitting time
Hazard function	Nelson–Aalen estimator
Test	Log-rank test

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 Jurimetrics National accounts Official statistics Population statistics Psychometrics
Spatial statistics	Cartography Environmental statistics Geographic information system Geostatistics Kriging

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