1. Lecture 8: Nelson Aalen Estimator and Smoothing
Kernel Smoothing
Smoothing Splines

2. Nelson Aalen vs. Kaplan Meier
Approximately 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

3. Why consider NA over KM NA provides estimate of hazard rate
Recall hazard rate can be estimated as slope of the cumulative hazard
NA extended to more complicated situations
Non-parametric baseline hazard estimation
Failure to meet proportional hazard assumption

4. Kernel Smoothing
NA 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)

5. Kernel Smoothing Recall the NA estimate: Crude hazard rate estimate:
Can take advantage of the weighted average of crude estimates over event times close to t
“Closeness” determined by bandwidth b
Average estimates over (t – b, t + b)

6. Kernel Smoothed HR Estimator
t > b (greater than bandwidth)
A pointwise CI for the smoothed hazard rate is constructed similarly as for the hazard rate

7. Kernel Function Controls weights of nearby points
K(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

8. Kernel Function Examples: Rectangle (uniform) Epanechnikov
Biquadratic (Biweight)
Triquadratic

10. A Few More Notes on Kernel Functions
Issues with symmetric kernels
When b > t, symmetric kernels are no longer valid since there are no event times less than 0
Occurs at the left part of the graph
There 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 applied

11. Choice of Bandwidth May chose to get desired smoothness of H(t)
May choose to minimize some criteria
For example MSE
May choose using cross-validation
Computationally intensive

12. Steps Calculate DH(ti) at each ti
Calculate kernel at each ti for chosen kernel function based on chosen b
Take product of kernel at ti and DH(ti) for each ti
Sum over D (total number of events)

13. Bone Marrow Transplant for Leukemia
Patient undergoing bone marrow transplant (BMT) for acute leukemia
Three types of leukemia
ALL
AML low risk
AML high risk
We’re focusing on the ALL patients (first 10 events only)
Using the Epanechnikov kernel

14. Bone Marrow Transplant for Leukemia
Focusing on the ALL patients (first 10 events only)
Using the Epanechnikov kernel
Bandwidth of 50
Want to estimate hazard rate at 75 weeks:

15. Time (ti) 1
0.0263  55
 0.0533 74
0.0811  86
 0.1097
104
 0.1391
107
 0.1694
109
0.2007
110
0.2329
122
0.2996
129
0.3353

16. Potential Issues Choice of bandwidth Tail Problem
How to select appropriate band width
N used to estimate hazard decreases as t increases
Unexpected noise at later time points
Tail Problem
Prefer 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 - b

17. Selection of Bandwidth
Magnitude
Small Bandwidth
less smooth curve and larger variance but small bias
Larger bandwidth
Smoother and smaller variance but more bias
Choice most crucial (influential) near boundary

18. Selection of Bandwidth
Global bandwidth
Constant bandwidth across all t
Chosen for simplicity
Optimized by:
Problematic towards right tail (variance HUGE)

19. Selection of Bandwidth
Local Bandwidth
Vary depending on where you are in time
Variance dominates bias in right tail so account for with larger bandwidth
Optimal local bandwidth obtained

20. Choosing b Global Local
Plug-in (or choose optimum based on minimizing ISME)
Cross-validation
Bootstrapping
Local
Vary by risk set size
kth nearest neighbor
Bandwidth = distance of t to the kth nearest uncensored neighbor
Choice of k effects smoothness
Consider constraining boundary areas when using global bandwidth

21. R Package “muhaz” Function: muhaz Options
Estimate hazard function from right-censored data using kernel-based methods
Options
3 bandwidth functions
Global, local, kth nearest neighbor
3 types of boundary correction
None, left, both
4 four shapes kernel function
Rectangle, epanechnikov, biquadratic, triquadratic

22. R Package “muhaz” Arguments (some anyway) times: event times
delta: censoring indicator
min/max.time: boundary times
bw.grid: bandwidth grid used in MSE minimization
bw.method: local or global bandwidth?
b.cor: specifies the type of boundary corrections
kern: choice of kernel

23. R Code (BMT data)
library(muhaz) data<-read.csv("H:\\public_html\\BMRTY722_Summer2015\\Data\\BMT_1_3.csv") times<-ifelse(data$Death==0 & data$Relapse==0, data$TTR, NA) times<-ifelse(data$Relapse==1, data$TTR, times) times<-ifelse(data$Death==1 & data$TTR>=data$TTD, data$TTD, times) event<-ifelse(data$Death==1 | data$Relapse==1, 1, 0) data<-cbind(data, times, event) ALL<-data[data$Disease==1,] AMLlo<-data[data$Disease==2,] AMLhi<-data[data$Disease==3,]

24. R Code (Comparing Kernels)
### Comparing Kernel functions with set bandwidth ALLe<-muhaz(ALL$times/365, ALL$event, bw.grid=0.33) ALLr<-muhaz(ALL$times/365, ALL$event, bw.grid=0.33, kern="r") ALLb<-muhaz(ALL$times/365, ALL$event, bw.grid=0.33, kern="b") ALLt<-muhaz(ALL$times/365, ALL$event, bw.grid=0.33, kern="t") par(cex.axis=0.8) plot(ALLr, lty=1, lwd=2, col=1, xlab="Time to Relapse or Death (Years)", ylab="Estimated Hazard Rate", xlim=c(0, 2), ylim=c(0, 1)) lines(ALLe, lty=3, lwd=2, col=2) lines(ALLb, lty=2, lwd=2, col=3) lines(ALLt, lty=4, lwd=2, col=4) legend(1.5, 1, c("Rectangular","Epanechnikov", "Biweight","Triquadratic"), col=1:4, lty=c(1,3,2,4), lwd=2, cex=0.8, bty="n")

25. Comparing Kernels

26. R Code (Comparing global bandwidth)
### Comparing different set global bandwidth ALLe1<-muhaz(ALL$times/365, ALL$event, bw.grid=0.25) ALLe2<-muhaz(ALL$times/365, ALL$event, bw.grid=0.5) ALLe3<-muhaz(ALL$times/365, ALL$event, bw.grid=1) ALLe4<-muhaz(ALL$times/365, ALL$event, bw.method="global") plot(ALLe1, lty=1, lwd=2, col=1, xlab="Time to Relapse or Death (Years)", ylab="Estimated Hazard Rate", xlim=c(0, 2), ylim=c(0, 1)) lines(ALLe2, lty=3, lwd=2, col=2) lines(ALLe3, lty=2, lwd=2, col=3) lines(ALLe4, lty=4, lwd=2, col=4) legend(1.5, 1, c("Rectangular","Epanechnikov", "Biweight","Triquadratic"), col=1:4, lty=c(1,3,2,4), lwd=2, cex=0.8, bty="n")

27. Comparing Global Bandwidths

28. R Code (bandwidth methods)
### Comparing different set global bandwidth ALLe1<-muhaz(ALL$times/365, ALL$event, bw.grid=0.25) ALLe2<-muhaz(ALL$times/365, ALL$event, bw.grid=0.5) ALLe3<-muhaz(ALL$times/365, ALL$event, bw.grid=1) ALLe4<-muhaz(ALL$times/365, ALL$event, bw.method="global") plot(ALLe1, lty=1, lwd=2, col=1, xlab="Time to Relapse or Death (Years)", ylab="Estimated Hazard Rate", xlim=c(0, 2), ylim=c(0, 1)) lines(ALLe2, lty=3, lwd=2, col=2) lines(ALLe3, lty=2, lwd=2, col=3) lines(ALLe4, lty=4, lwd=2, col=4) legend(1.5, 1, c("Rectangular","Epanechnikov", "Biweight","Triquadratic"), col=1:4, lty=c(1,3,2,4), lwd=2, cex=0.8, bty="n")

29. Comparing Bandwidth Methods

30. Alternatively…

31. Smoothing Splines
Alternative way to estimate the smooth hazard function is via smoothing splines
But first, a basic discussion of smoothing splines

32. Motivation for Splines
Why use smoothing splines in any kind of modeling
Though we often treat relationships as linear, this is not generally reality
Convenient and easy to interpret
May be a reasonable under constraints
Sometimes necessary
Large p, small N (avoid overfitting)
Sometimes however, it makes sense to move beyond linearity

33. Motivation
-So consider the plot shown here. Let’s say we wanted to develop a function describing the relationship between the variables x and y. In this case we could fit a straight line through the data (though there is a lot of “noise” in such a case)

34. Motivation
-So the line shown here represents the linear fit where Y is treated as our dependent outcome and X is our independent predictor. Thus, we can write an expression describing this function.
-In this case we see there is a lot of variability around our fitted line and probabily, if we were to conduct a statitical test, we would not find any association between Y and X.
-So if we look at the data, do we see any other possible non-linear pattern?

35. Motivation
-It turns in this case that the dependent variable Y is associated with the independent variable X but he association is not linear. In fact Y is associated with sin(x) such that these data are generated from the relationship y=sin(12*(x+0.2))/(x+0.2)+error.
-The error free function is shown in turquoise

36. Linear Basis Expansion
Linear Regression
Question: How do you find ? x1 x2 x3 y 1 -3 6 12 …

37. Linear Basis Expansion
Non-linear Model
Question: Now how do you find ? x1 x2 x3 y 1 3 6 12 …

38. Linear Basis Expansion
Non-linear model
Apply linear regression to obtain x1 x2 x3 u1 u2 u3 u4 y 1 -3 6 3
1201
0.052 12 …

39. Linear Basis Expansion
We can easily fit any model of type
i.e. we can easily do a linear basis expansion in X
However, if the model is thought to be nonlinear, but exact form unknown, we can try to introduce interaction terms

40. Piecewise Polynomial Functions
Assume X is one-dimensional
Definition: Assume the domain [a, b] of X is split into intervals [a, x1], [x2, x3],…, [a, xm, b]. Then f (X) is said to be a piecewise polynomial if f (X) is represented by separate polynomials in the different intervals. The points x1, x2, …, xm are called knots

41. Motivation

42. Piecewise Polynomials

43. Piecewise Polynomials with Constraints
For a continuous piecewise linear function we need to add constraints…
linear functions on each interval with constraints

44. With Constraints

45. Splines
A piecewise polynomial is called an order-M spline if it has continuous derivatives up to order M-1
Can be represented by a basis function (K = # knots)
4th order spline is called a cubic spline
Note, in cubic splines, knot-discontinuity is not visible

46. Cubic Splines
Given a < x1 < x2 < … < xn < b, a function h is a cubic spline if
On each interval (a , x1), (x1, x2), …, (xn,b), h is a cubic polynomial
The polynomial pieces fit together at points xi (knots) s.t. h itself and its first and second derivatives are continuous at each xi, and hence on the whole [a,b]

47. Natural Cubic Splines
A cubic spline f is called a natural cubic spline if its 2nd and 3rd derivatives are 0 at a and b
Implies that f is linear on extreme intervals

49. Fitting Smooth Functions to Data
Minimize a penalized sum of squared residuals
The function f that minimizes RRS for a given l is a natural cubic spline with knots at all unique values of xi (Note- this means there are N knots)
Where l is a smoothing parameter
= 0: any function interpolating data
= +∞: least squares line fit

50. R Package “gss” R smoothing splines for hazard function
Description: Estimate hazard function using smoothing spline ANOVA models. The symbolic model specification via formula follows the same rules as in lm, but with response of special form

51. “sshzd” Function
sshzd takes standard right-censored lifetime data, with possible left-truncation and covariates; in Surv(futime,status,start=0)~..., futime is the follow-up time, status is the censoring indicator, and start is the optional left-truncation time. The main effect of futime must appear in the model terms specified via ....
Parallel to those in a ssanova object, the model terms are sums of unpenalized and penalized terms. Attached to every penalized term there is a smoothing parameter, and the model complexity is largely determined by the number of smoothing parameters.
The selection of smoothing parameters is through a cross-validation mechanism described in Gu (2002, Sec. 7.2), with a parameter alpha; alpha=1 is "unbiased" for the minimization of Kullback-Leibler loss but may yield severe undersmoothing, whereas larger alpha yields smoother estimates.
A subset of the observations are selected as "knots." Unless specified via id.basis or nbasis, the number of "knots" q is determined by max(30,10n^{2/9}), which is appropriate for the default cubic splines for numerical vectors.

52. Marrow Transplant
### Smoothed hazards data<-read.csv("H:\\BMTRY_722_Summer2013\\BMT_1_3.csv") times<-ifelse(data$Death==0 & data$Relapse==0, data$TTR, NA) times<-ifelse(data$Relapse==1, data$TTR, times) times<-ifelse(data$Death==1 & data$TTR>=data$TTD, data$TTD, times) event<-ifelse(data$Death==1 | data$Relapse==1, 1, 0) data<-cbind(data, times, event) km.fit<-survfit(Surv(data$times, data$event)~data$Disease) h1<--log(km.fit$surv[1:37]); t1<-km.fit$time[1:37] h2<--log(km.fit$surv[38:91]); t2<-km.fit$time[38:91] h3<--log(km.fit$surv[92:135]); t3<-km.fit$time[92:135] plot(t1, h1, xlab="Time", ylab="H(t)", xlim=c(0, 2640), ylim=c(0,1.5), type="s", col=1, lwd=2) lines(t2, h2, type="s", col=2, lwd=2) lines(t3, h3, type="s", col=3, lwd=2) legend(1750, .3, c("ALL","AML low","AML high"), col=1:3, lwd=2, cex=0.8)

54. R for Smoothing Splines Hazard
hazfit<-sshzd(Surv(times, event)~disease*times) haz<-hzdrate.sshzd(hazfit, data.frame(times=times, disease=disease)) h1<-haz[disease==1]; id1<-order(h1); t1<-time[disease==1] h2<-haz[disease==2]; id2<-order(h2); t2<-time[disease==2] h3<-haz[disease==3]; id3<-order(h3); t3<-time[disease==3] plot(time[disease==1], haz[disease==1], xlim=c(0, max(times)), ylim=range(haz), xlab="Time", type="l",ylab="hazard", lwd=2, col=1) lines(times[disease==2], haz[disease==2], lwd=2, col=2) lines(times[disease==3], haz[disease==3], lwd=2, col=3) legend(1750, .004, c("ALL","AML low","AML high"), col=1:3, lwd=2, cex=0.8)

56. Non-Intersecting Hazards
hazfit<-sshzd(Surv(times, event)~disease+times) haz<-hzdrate.sshzd(hazfit, data.frame(times=times, disease=disease)) h1<-haz[disease==1]; id1<-order(h1); t1<-time[disease==1] h2<-haz[disease==2]; id2<-order(h2); t2<-time[disease==2] h3<-haz[disease==3]; id3<-order(h3); t3<-time[disease==3] plot(time[disease==1], haz[disease==1], xlim=c(0, max(times)), ylim=range(haz), xlab="Time", type="l",ylab="hazard", lwd=2, col=1) lines(times[disease==2], haz[disease==2], lwd=2, col=2) lines(times[disease==3], haz[disease==3], lwd=2, col=3) legend(1750, .004, c("ALL","AML low","AML high"), col=1:3, lwd=2, cex=0.8)