NONPARAMETRIC REGRESSION SPLINES FOR GENERALIZED LINEAR MODELS IN THE PRESENCE OF MEASUREMENT ERROR

Transcription

1 NONPARAMETRIC REGRESSION SPLINES FOR GENERALIZED LINEAR MODELS IN THE PRESENCE OF MEASUREMENT ERROR J. D. Maca July 1, 1997 Abstract The purpose of this manual is to demonstrate the usage of software for analyzing nonparametric regression spline models in the presence of measurement error. The software was written to be used with the Splus software package, and currently can only be used from the UNIX operating system platform.

2 1 INTRODUCTION The software package analyzes regression data where measurement error is present. With measurement error data, one does not directly observe the independent variable, X. Instead, only X measured with an assumed additive error is found. This X measured with error will be called W. Thus W = X + U, where E(U) =0,var(U)=σu 2. Therefore, the data recorded is { } Y i, (W ij ) m i j=1, m i 1; i =1,...,n. If there are replicates of W at some or all of the X values (m i > 1 for at least one i), then a consistent estimate of the σ u can be found by σ 2 u = 2 Loading the Software ni=1 mi j=1 (W ij W i ) 2 ni=1 (m i 1) (1) Before each session, the comiled C code must be dynamically linked with the Splus software. This is done the the command dyn.load( spline.o ) and will only need to be done once at the begining of each Splus session. The file spline.o contains all the C code necessary for the non parametric regression spline analysis. The Splus code which calls this C code must also be loaded. This program can be loaded by source( NPreg.S ) and will only need to be done once. After loading the program should remain in memory. 3 Mixture of Normal Estimation The first step to fitting the structural spline is to estimate the number and parameters of the mixture normal distribution. This is done with the Splus function: thetas = find.k.mix.norm(w, n1,n2, kmax = k max, sigmau= σ u ) This function will estimate the values for the mixture normal densities in W having k mixtures for all k, 1 k k max, and estimates the value for k which fits the data best, and put these into the variable thetas. To do this, you must provide the function with an ARRAY W, which is a n m array holding the independent measurements. The function will then take n1 Gibbs samples to estimate the mixture normal parameters for each k in 1 k k max. It then picks the optimal value for k, and then calculates n2 gibbs samples for estimates the parameters for a k -mixture normal distribution. 1

3 You can change the maximum number of populations that you wish the mixture normal distribution to have with the kmax = k max option. The default value is 4. If there are replicates in W, then than estimate for the measurement error σ u can be calculated using Equation 1. However, if there are no replicates for W, then and estimate for the measurement error must be provided. This is done with the sigmau= σ u option. 2

4 4 Spline estimation The full command for estimating a regression spline is out <- me.glim (W,Y,NS="cp",simex=F,type=1,pmax=1,maxreps=30, B=25, xlim=null, lambda(0,.2,.4,.6,.8,1,1.25), thetas=null,sigmau=null, nknots=10, alpha=c(100,10,1,.05,.01,.001,.0001,.00001, , )) The only required arguments are W, the independent variable measured with error, and Y, the dependent variable. Note, both of these must be matrices with the same number of rows (observations). All other arguments are optional, and will produce optional analysis. The first option is NS="cp", which controls the knot selection method. The default is cp, which is the ridge regression smoothing method( Ruppert & Carroll, 1996). An optional method is the GCV method, which is used by NS="gcv" The next argument is the simex=f option which control whether the structural method or the SIMEX algorithm is used to find the spline. The default, simex=f, specifies that the structural method is used. To use the SIMEX method, change to simex=t. The next option is the type=1, which specifies which GLIM to fit. The default is type=1 which denotes the normal is fit. This will fit the same model as if no distributional assumption are made. The final option for model fitting is the pmax=1, which controls the order of the regression spline. The values 0, 1, 2 represent a linear, quadratic, cubic model respectively. The default is to fit a quadratic model. The rest of the options determine how the models are fit. The first option is maxreps=30, which is the upper limit to the number of iterations in the scoring method to fit the generalized linear model. The next two options pertain to the SIMEX method if used. The option lambda=λ which is a vector of values to be using in the SIMEX algorithm. The default vector is λ =(0,.2,.4,.6,.8,1,1.25). At each of these λ values, there will be B = 25 generated datasets. The number of generated datasets can be changed to any value. The argument alpha = α controls the values used in the ridge regression smoothing method. This can again be changed to any vector containing the exact desired values. The ridge regression method assumes a large number fixed number of fixed knot points. This number is designated by nknots=10. This can be changed to any integer. The function will automatically calculate the mixture normal parameters. However, since this is the most time consuming part of the regression spline estimation, it is should only be done once. For this reason if you know that you are going to fit many splines to the same set of data, to first find the mixture normal parameters by thetas.est = find.k.mix.norm(w, n1,n2, kmax = k max, sigmau= σ u ) 3

5 then use the thetas = thetas.est to uses the previously calculated estimates. As in Section 3, if there is no replicates for W, then an previous estimate for σ u must be used. This estimate is given to the program through the sigmau = σ u command. If no value is given, it will assume that it is unknown, and try to calculate it. 5 Output After the spline has been found to the specifications of the user, it will return four pieces of information. The first set of information is a vector called Xhat and a vector called Yhat. These two vectors can be used for plotting the regression spline estimates. If a structural spline was fit, the range of the Xhat variable can be set with the xlim=null command. If a vector is supplied, such as xlim=c(0,1), this will force the estimates found to only have X values between 0 and 1. The default value is to use the inner 90% of the range between the highest and lowest values for W. The next information returned will be be returned if a structural spline is fit. First, is a vector betahat, which is the vector of the β coefficients for the regression spline. The second part is a vector called knots which is the vector of the knots used for the regression spline estimation. If the SIMEX algorithm was used, both of these vectors will be returned as NULL. 6 Examples For this example, I will be using the data found in the Appendix, in Section 7.1. There is 500 data values, (X), which were generated from an Uniform(0,1) distribution. The true curve was then generated as Y =sin(4πx)+ɛ, where ɛ Normal(0,.05 2 ). The measurement error was generated from the Normal(0, ), and there were 2 replications at each X point. With this data, the Structural curve using 15 knots points and the ridge regression smoothing method would be found by the following command. out <- me.glim(w,y,nknots=15) Note that was not necessary to specify the knot selection procedure, as ridge regression is the default method. The fitted model can now be plotted in Splus, by plot(out$xhat,out$yhat). To find the SIMEX estimate using this same data, and the GCV mothod of knot selection, the following command should be issued out <- me.glim(w,y,ns= gcv,simex=t,b=50) 4

6 which will simulate 50 datasets at each λ value. To find the naive estimate (for comparison), one should give the command out <- me.glim(w,y,simex=t,b=1,lambda=c(0)) 7 Appendix 7.1 Data for Example 1 The data used for the example in Section 6 can be found in Table 1. Although X values are given in the dataset, these are unobseravle in real data situations, and are not used in the spline estimation. They are present here only for simulation purposes. There is a plot of this data in Figure 1 which displays the average W value versus Y for each observation. The true cureve from which the data was generated is also displayed on the plot. 5

7 Table 1: Data for Example 1 Obs X W 1 W 2 Y Obs X W 1 W 2 Y

8 Table 2: Data for Example 1 (continued) Obs X W 1 W 2 Y Obs X W 1 W 2 Y

9 Table 3: Data for Example 1 (continued) Obs X W 1 W 2 Y Obs X W 1 W 2 Y

10 Table 4: Data for Example 1 (continued) Obs X W 1 W 2 Y Obs X W 1 W 2 Y

11 Observed data and true curve Average W value Y values Figure 1: Plot of the true curve on the observed data 10