lecture 8: nelson aalen estimator and smoothing kernel smoothing smoothing splines

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Lecture X: Nelson Aalen Estimator and Smoothing

Lecture 8: Nelson Aalen Estimator and SmoothingKernel SmoothingSmoothing Splines1Nelson Aalen vs. Kaplan MeierApproximately the same when risk sets (Yi) large relative to number of subjects who experience the event

Larger difference when ties present

Different standard errors when ties present

Smaller MSE than KM for S(t) > 0.20 and larger otherwise

NA biased upward when survival estimated close to 0

2Why consider NA over KMNA provides estimate of hazard rateRecall hazard rate can be estimated as slope of the cumulative hazard

NA extended to more complicated situationsNon-parametric baseline hazard estimationFailure to meet proportional hazard assumption3Kernel SmoothingNA estimator of H(t) provides efficient estimate of cumulative hazard function

More often interested in hazard rate h(t)

Slope of H(t) provides crude estimate of h(t)

Can use kernel smoothing based on H(t) and var(H(t)) to get this estimate of h(t)

4Kernel SmoothingRecall the NA estimate:

Crude hazard rate estimate:

Can take advantage of the weighted average of crude estimates over event times close to tCloseness determined by bandwidth bAverage estimates over (t b, t + b)

5Kernel Smoothed HR EstimatorHR estimator

t > b (greater than bandwidth)A pointwise CI for the smoothed hazard rate is constructed similarly as for the hazard rate

6Kernel FunctionControls weights of nearby pointsK(x) simplest class are symmetric PDFs with

The most common choice are kernels defined on [-1,1] and are polynomial functions related to the beta distribution

7Kernel FunctionExamples:Rectangle (uniform)

Epanechnikov

Biquadratic (Biweight)

Triquadratic

8

A Few More Notes on Kernel FunctionsIssues with symmetric kernelsWhen b > t, symmetric kernels are no longer valid since there are no event times less than 0

Occurs at the left part of the graphThere are suggested modified kernels

When data are sparse, the variability of the kernel estimator is large and results can be quite biased

There are boundary corrections that can be applied10Choice of BandwidthMay chose to get desired smoothness of H(t)May choose to minimize some criteria For example MSEMay choose using cross-validation Computationally intensive11StepsCalculate DH(ti) at each ti Calculate kernel at each ti for chosen kernel function based on chosen bTake product of kernel at ti and DH(ti) for each tiSum over D (total number of events)12Bone Marrow Transplant for LeukemiaPatient undergoing bone marrow transplant (BMT) for acute leukemiaThree types of leukemiaALLAML low riskAML high riskWere focusing on the ALL patients (first 10 events only)Using the Epanechnikov kernel

13Bone Marrow Transplant for LeukemiaFocusing on the ALL patients (first 10 events only)Using the Epanechnikov kernelBandwidth of 50Want to estimate hazard rate at 75 weeks:

14Time (ti)10.0263550.0533740.0811860.10971040.13911070.16941090.20071100.23291220.29961290.3353

15Potential IssuesChoice of bandwidthHow to select appropriate band widthN used to estimate hazard decreases as t increasesUnexpected noise at later time points

Tail ProblemPrefer symmetric kernels that integrate to 1 over support [-1,1]All observations s within the same distance from t are weighted the same (i.e. in interval [t b, t + b])Problem if t < b or t > Tmax - b16Selection of BandwidthMagnitudeSmall Bandwidthless smooth curve and larger variance but small biasLarger bandwidthSmoother and smaller variance but more bias

Choice most crucial (influential) near boundary

17Selection of BandwidthGlobal bandwidthConstant bandwidth across all tChosen for simplicityOptimized by:

Problematic towards right tail (variance HUGE)

18Selection of BandwidthLocal Bandwidth Vary depending on where you are in timeVariance dominates bias in right tail so account for with larger bandwidthOptimal local bandwidth obtained

19Choosing bGlobalPlug-in (or choose optimum based on minimizing ISME)Cross-validationBootstrappingLocalVary by risk set size

kth nearest neighborBandwidth = distance of t to the kth nearest uncensored neighborChoice of k effects smoothness

Consider constraining boundary areas when using global bandwidth

20R Package muhazFunction: muhazEstimate hazard function from right-censored data using kernel-based methods

Options 3 bandwidth functionsGlobal, local, kth nearest neighbor

3 types of boundary correctionNone, left, both

4 four shapes kernel functionRectangle, epanechnikov, biquadratic, triquadratic

R Package muhazArguments (some anyway)times: event timesdelta: censoring indicatormin/max.time: boundary timesbw.grid: bandwidth grid used in MSE minimizationbw.method: local or global bandwidth?b.cor: specifies the type of boundary correctionskern: choice of kernel

R Code (BMT data)library(muhaz)data

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