crrQR: Competing Risks Quantile Regression
In cmprskQR: Analysis of Competing Risks Using Quantile Regressions
Description
Usage
Arguments
Details
Value
References
See Also
Examples
Description
quantile regression modeling of subdistribution functions in competing risks
Usage
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Arguments
ftime
vector of failure/censoring times
fstatus
vector with a unique code for each failure type and a separate code for
censored observations
X
matrix (nobs x ncovs) of covariates
failcode
code of fstatus that denotes the failure type of interest
cencode
code of fstatus that denotes censored observations
tau.range
vector of length 2 denoting the range of quantiles
tau.step
grid size on tau.range (spacing between two grid points)
subset
a logical vector specifying a subset of cases to include in the
analysis
na.action
a function specifying the action to take for any cases missing any of
ftime, fstatus, cov1, cov2, cengroup, or subset.
rq.method
method of computation for quantile regressions. (cf. documentation of method
rq.fit in package quantreg for details.)
variance
if FALSE, then suppresses computation of asymptotic variances
x
crrQR object (output from crrQR()) for method print
...
included for compatibility with the generic functions. Not
currently used.
Details
Fits the competing risks quantile regression model described in Peng and Fine (2009).
While the use of model formulas is not supported, the
model.matrix function can be used to generate suitable matrices
of covariates from factors, eg
model.matrix(~factor1+factor2)[,-1] will generate the variables
for the factor coding of the factors factor1 and factor2.
The final [,-1] removes the constant term from the output of
model.matrix.
If variance=FALSE, then
some of the functionality in summary.crrQR and print.crrQR
will be lost. This option can be useful in situations where crrQR is
called repeatedly for point estimates, but standard errors are not
required, such as in bootstrapping the cumulative incidence function for
confidence intervals.
The print method prints the estimated coefficients, the estimated standard errors,
and the two-sided p-values for the test of the individual coefficients equal to 0.
A first implementation of the estimation procedure was prepared by Limin Peng and Ruosha Li.
Value
Returns a list of class crrQR, with components
$beta.seq
the estimated regression coefficients
$tau.seq
the sequence of quantiles computed
$var.seq
estimated variance covariance matrix of coef
$inf.func
list of estimated influence functions
$call
the call to crr
$n
the number of observations used in fitting the model
$n.missing
the number of observations removed from the input data
due to missing values
$cvt.length
number of covariates (columns of X)
References
Peng L and Fine JP (2009) Competing risks quantile regression. JASA 104:1440-1453.
See Also
predict.crrQR plot.predict.crrQR
summary.crrQR rq.fit
Examples
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16
# simulated data to test
set.seed(10)
ftime <- rexp(200)
fstatus <- sample(0:2,200,replace=TRUE)
X <- matrix(runif(600),nrow=200)
dimnames(X)[[2]] <- c('x1','x2','x3')
#compute model
print(z <- crrQR(ftime,fstatus,X))
summary(z)
# predict and plot cumulative incedences
reference <- as.matrix(rbind(c(.1,.5,.8),c(.1,.5,.2)))
dimnames(reference)[[2]] <- c('x1','x2','x3')
z.p <- predict(z,reference)
print(z.p)
plot(z.p,lty=1,color=2:3)
crrQR(ftime,fstatus,X,failcode=2)
Example output
..............................................done.
stopping at tau=0.47
coefficients:
const x1 x2 x3
0.01 0.10182752 -0.018225105 -0.012531227 -0.094305017
0.02 0.10182752 -0.018225105 -0.012531227 -0.094305017
0.03 0.10182752 -0.018225105 -0.012531227 -0.094305017
0.04 0.10182752 -0.018225105 -0.012531227 -0.094305017
0.05 0.10182752 -0.018225105 -0.012531227 -0.094305017
0.06 0.10182752 -0.018225105 -0.012531227 -0.094305017
0.07 0.10182752 -0.018225105 -0.012531227 -0.094305017
0.08 0.10182752 -0.018225105 -0.012531227 -0.094305017
0.09 0.09800512 -0.024441965 -0.054057490 0.007253677
0.1 0.09800512 -0.024441965 -0.054057490 0.007253677
0.11 0.11379768 -0.029151110 -0.002214549 -0.020370088
0.12 0.11379768 -0.029151110 -0.002214549 -0.020370088
0.13 0.11379768 -0.029151110 -0.002214549 -0.020370088
0.14 0.11918642 0.076045495 0.067364208 -0.088625912
0.15 0.24738188 0.020930280 0.025710897 -0.188980805
0.16 0.46248714 0.209349053 -0.099809892 -0.484074634
0.17 0.46248714 0.209349053 -0.099809892 -0.484074634
0.18 0.41850933 0.245702966 0.031812194 -0.466285131
0.19 0.48559154 0.243799783 0.036648872 -0.537053272
0.2 0.51973703 0.242831044 0.039110788 -0.573074939
0.21 0.73404731 -0.001658431 0.060126014 -0.635865537
0.22 0.82822302 -0.079380771 0.129324773 -0.730829482
0.23 0.80156517 -0.040721666 0.193158766 -0.747392969
0.24 0.71544604 0.038695999 0.246457650 -0.667620473
0.25 0.71544604 0.038695999 0.246457650 -0.667620473
0.26 0.79979695 0.374428325 0.446281597 -0.977901207
0.27 0.79979695 0.374428325 0.446281597 -0.977901207
0.28 0.99227976 0.473917633 0.142796834 -0.930645862
0.29 0.97728103 0.476649403 0.209194371 -0.950003861
0.3 0.98888541 0.470575873 0.155864900 -0.887732299
0.31 1.03049009 0.483916156 -0.034812071 -0.718908367
0.32 1.18024445 0.650108552 -0.039885312 -0.933239627
0.33 1.16719980 0.682730720 -0.040727224 -0.924857761
0.34 1.16312333 0.744629560 -0.081655359 -0.887869942
0.35 1.30163888 0.726496565 -0.238553017 -0.866082983
0.36 2.14020578 0.535072162 -0.995720386 -0.947352883
0.37 2.70576266 0.220447035 -1.307020321 -1.225897233
0.38 2.67490153 0.304599027 -1.411308215 -1.034736321
0.39 2.75332855 0.230875384 -1.418286761 -1.069285111
0.4 2.74408485 0.231414811 -1.371026364 -1.068610087
0.41 2.65720411 0.236484839 -0.926830005 -1.062265599
0.42 2.58521945 0.270466165 -1.003210549 -0.600030208
0.43 3.24934957 0.577840490 -1.774431903 -0.967981569
0.44 3.16357582 1.123887002 -1.110083051 -1.327990852
0.45 3.14667300 2.828755629 0.371481099 -2.968600553
0.46 3.48448670 2.450244875 0.790407158 -3.299424751
standard errors:
const x1 x2 x3
0.01 2.056e-01 2.217e-01 2.883e-02 1.038e-01
0.02 1.388e-17 3.469e-18 0.000e+00 0.000e+00
0.03 0.000e+00 0.000e+00 0.000e+00 0.000e+00
0.04 0.000e+00 0.000e+00 0.000e+00 0.000e+00
0.05 2.776e-17 6.939e-18 3.469e-18 0.000e+00
0.06 0.000e+00 0.000e+00 0.000e+00 0.000e+00
0.07 2.776e-17 3.469e-18 9.179e-18 1.388e-17
0.08 6.621e-03 1.077e-02 7.193e-02 1.759e-01
0.09 3.103e-17 7.758e-18 5.004e-17 1.321e-17
0.1 2.089e-02 2.198e-02 1.257e-02 1.523e-02
0.11 1.326e-01 4.965e-02 1.625e-01 1.069e-01
0.12 1.418e-01 5.016e-02 1.695e-01 1.359e-01
0.13 8.683e-02 2.270e-01 1.844e-01 1.617e-01
0.14 1.371e-01 5.700e-02 6.909e-02 1.425e-01
0.15 2.953e-01 3.011e-01 1.731e-01 3.407e-01
0.16 1.820e-01 1.100e-01 1.767e-01 1.469e-01
0.17 2.043e-01 1.613e-01 2.283e-01 1.307e-01
0.18 8.683e-02 1.167e-01 4.706e-02 1.155e-01
0.19 6.892e-02 1.136e-01 3.827e-02 4.550e-02
0.2 3.701e-01 4.723e-01 2.359e-01 1.858e-01
0.21 4.268e-01 4.754e-01 2.471e-01 1.913e-01
0.22 4.774e-01 4.922e-01 1.863e-01 2.329e-01
0.23 3.051e-01 2.590e-01 1.414e-01 2.352e-01
0.24 4.009e-01 3.873e-01 2.956e-01 2.241e-01
0.25 3.403e-01 6.138e-01 3.654e-01 5.414e-01
0.26 3.307e-01 2.736e-01 2.781e-01 5.703e-01
0.27 2.590e-01 1.674e-01 4.668e-01 3.033e-01
0.28 1.220e-01 1.433e-01 3.366e-01 3.906e-01
0.29 9.559e-02 7.660e-02 4.327e-01 3.695e-01
0.3 3.126e-01 4.138e-01 4.091e-01 3.139e-01
0.31 1.771e-01 3.879e-01 3.606e-01 3.575e-01
0.32 1.967e-01 3.857e-01 1.678e-01 2.728e-01
0.33 4.506e-01 5.067e-01 2.656e-01 2.058e-01
0.34 1.113e+00 8.247e-01 1.138e+00 5.726e-01
0.35 1.428e+00 9.237e-01 1.228e+00 6.121e-01
0.36 1.562e+00 1.108e+00 1.183e+00 6.614e-01
0.37 2.265e+00 1.744e+00 1.482e+00 7.820e-01
0.38 6.983e-01 6.096e-01 5.679e-01 2.794e-01
0.39 5.205e-01 3.668e-01 5.523e-01 8.739e-01
0.4 1.040e+00 1.678e+00 6.947e-01 1.287e+00
0.41 9.466e-01 1.523e+00 1.225e+00 1.410e+00
0.42 8.391e-01 1.716e+00 1.130e+00 9.605e-01
0.43 1.648e+00 2.785e+00 3.082e+00 3.198e+00
0.44 1.131e+00 2.574e+00 3.139e+00 4.021e+00
0.45 1.193e+00 1.745e+00 2.912e+00 3.301e+00
0.46 1.193e+00 1.745e+00 2.912e+00 3.301e+00
two-sided p-values:
const x1 x2 x3
0.01 6.2e-01 9.3e-01 6.6e-01 3.6e-01
0.02 0.0e+00 0.0e+00 0.0e+00 0.0e+00
0.03 0.0e+00 0.0e+00 0.0e+00 0.0e+00
0.04 0.0e+00 0.0e+00 0.0e+00 0.0e+00
0.05 0.0e+00 0.0e+00 0.0e+00 0.0e+00
0.06 0.0e+00 0.0e+00 0.0e+00 0.0e+00
0.07 0.0e+00 0.0e+00 0.0e+00 0.0e+00
0.08 0.0e+00 9.1e-02 8.6e-01 5.9e-01
0.09 0.0e+00 0.0e+00 0.0e+00 0.0e+00
0.1 2.7e-06 2.7e-01 1.7e-05 6.3e-01
0.11 3.9e-01 5.6e-01 9.9e-01 8.5e-01
0.12 4.2e-01 5.6e-01 9.9e-01 8.8e-01
0.13 1.9e-01 9.0e-01 9.9e-01 9.0e-01
0.14 3.8e-01 1.8e-01 3.3e-01 5.3e-01
0.15 4.0e-01 9.4e-01 8.8e-01 5.8e-01
0.16 1.1e-02 5.7e-02 5.7e-01 9.8e-04
0.17 2.4e-02 1.9e-01 6.6e-01 2.1e-04
0.18 1.4e-06 3.5e-02 5.0e-01 5.4e-05
0.19 1.8e-12 3.2e-02 3.4e-01 0.0e+00
0.2 1.6e-01 6.1e-01 8.7e-01 2.0e-03
0.21 8.5e-02 1.0e+00 8.1e-01 8.9e-04
0.22 8.3e-02 8.7e-01 4.9e-01 1.7e-03
0.23 8.6e-03 8.8e-01 1.7e-01 1.5e-03
0.24 7.4e-02 9.2e-01 4.0e-01 2.9e-03
0.25 3.6e-02 9.5e-01 5.0e-01 2.2e-01
0.26 1.6e-02 1.7e-01 1.1e-01 8.6e-02
0.27 2.0e-03 2.5e-02 3.4e-01 1.3e-03
0.28 4.4e-16 9.4e-04 6.7e-01 1.7e-02
0.29 0.0e+00 4.9e-10 6.3e-01 1.0e-02
0.3 1.6e-03 2.6e-01 7.0e-01 4.7e-03
0.31 5.9e-09 2.1e-01 9.2e-01 4.4e-02
0.32 2.0e-09 9.2e-02 8.1e-01 6.2e-04
0.33 9.6e-03 1.8e-01 8.8e-01 7.0e-06
0.34 3.0e-01 3.7e-01 9.4e-01 1.2e-01
0.35 3.6e-01 4.3e-01 8.5e-01 1.6e-01
0.36 1.7e-01 6.3e-01 4.0e-01 1.5e-01
0.37 2.3e-01 9.0e-01 3.8e-01 1.2e-01
0.38 1.3e-04 6.2e-01 1.3e-02 2.1e-04
0.39 1.2e-07 5.3e-01 1.0e-02 2.2e-01
0.4 8.4e-03 8.9e-01 4.8e-02 4.1e-01
0.41 5.0e-03 8.8e-01 4.5e-01 4.5e-01
0.42 2.1e-03 8.7e-01 3.7e-01 5.3e-01
0.43 4.9e-02 8.4e-01 5.6e-01 7.6e-01
0.44 5.2e-03 6.6e-01 7.2e-01 7.4e-01
0.45 8.3e-03 1.1e-01 9.0e-01 3.7e-01
0.46 3.5e-03 1.6e-01 7.9e-01 3.2e-01
Competing Risks Quantile Regression
Call:
crrQR.int(ftime = ftime, fstatus = fstatus, X = X, tau.L = tau.range[1],
tau.U = tau.range[2], tau.step = tau.step, outcome = failcode,
cencode = cencode, variance = variance, rq.method = rq.method,
orig.num = n)
ave.eff se.ave.eff p.signf.test cnst.test var.cnst.test p.cnst.test
const 1.002 0.292 0.0006 -0.722 0.0628 0.0039
x1 0.281 0.368 0.4400 -0.245 0.1032 0.4500
x2 -0.203 0.299 0.5000 0.210 0.0770 0.4500
x3 -0.617 0.335 0.0650 0.368 0.1002 0.2500
Num. cases = 200
[,1] [,2] [,3]
0.01 1.018464 1.077753 0.01
0.02 1.018464 1.077753 0.02
0.03 1.018464 1.077753 0.03
0.04 1.018464 1.077753 0.04
0.05 1.018464 1.077753 0.05
0.06 1.018464 1.077753 0.06
0.07 1.018464 1.077753 0.07
0.08 1.018464 1.077753 0.08
0.09 1.077168 1.072490 0.09
0.1 1.077168 1.072490 0.10
0.11 1.097988 1.111490 0.11
0.12 1.097988 1.111490 0.12
0.13 1.097988 1.111490 0.13
0.14 1.093706 1.153439 0.14
0.15 1.117558 1.251740 0.15
0.16 1.047344 1.400322 0.16
0.17 1.047344 1.400322 0.17
0.18 1.089760 1.441564 0.18
0.19 1.103684 1.523310 0.19
0.2 1.110839 1.566683 0.20
0.21 1.290787 1.890368 0.21
0.22 1.350242 2.093383 0.22
0.23 1.344683 2.105586 0.23
0.24 1.361311 2.032000 0.24
0.25 1.361311 2.032000 0.25
0.26 1.320565 2.374532 0.26
0.27 1.320565 2.374532 0.27
0.28 1.442753 2.521719 0.28
0.29 1.447069 2.558810 0.29
0.3 1.497339 2.550601 0.30
0.31 1.626367 2.503511 0.31
0.32 1.614009 2.825442 0.32
0.33 1.608373 2.801452 0.33
0.34 1.626552 2.770934 0.34
0.35 1.754433 2.949971 0.35
0.36 2.554791 4.510384 0.36
0.37 2.984939 6.228389 0.37
0.38 3.228179 6.006007 0.38
0.39 3.359676 6.381580 0.39
0.4 3.410385 6.475275 0.40
0.41 3.926016 7.425979 0.41
0.42 5.107125 7.320325 0.42
0.43 5.183769 9.265718 0.43
0.44 5.251587 11.650217 0.44
0.45 3.457026 20.523468 0.45
0.46 4.415738 31.971019 0.46
attr(,"class")
[1] "predict.crr"
Warning messages:
1: In plot.window(...) : "color" is not a graphical parameter
2: In plot.xy(xy, type, ...) : "color" is not a graphical parameter
3: In axis(side = side, at = at, labels = labels, ...) :
"color" is not a graphical parameter
4: In axis(side = side, at = at, labels = labels, ...) :
"color" is not a graphical parameter
5: In box(...) : "color" is not a graphical parameter
6: In title(...) : "color" is not a graphical parameter
............................................done.
stopping at tau=0.45
coefficients:
const x1 x2 x3
0.01 0.1136544 -0.149740034 0.095648004 -0.05974612
0.02 0.1136544 -0.149740034 0.095648004 -0.05974612
0.03 0.1136544 -0.149740034 0.095648004 -0.05974612
0.04 0.1136544 -0.149740034 0.095648004 -0.05974612
0.05 0.1136544 -0.149740034 0.095648004 -0.05974612
0.06 0.1136544 -0.149740034 0.095648004 -0.05974612
0.07 0.1136544 -0.149740034 0.095648004 -0.05974612
0.08 0.1136544 -0.149740034 0.095648004 -0.05974612
0.09 0.1514410 -0.222035184 0.123730413 -0.05758450
0.1 0.1753381 -0.250749452 0.156195664 -0.09018926
0.11 0.1828001 -0.268184132 0.159010559 -0.08345396
0.12 0.2214426 -0.039232204 -0.018863269 -0.16920285
0.13 0.2214426 -0.039232204 -0.018863269 -0.16920285
0.14 0.1976801 -0.028785258 -0.007860142 -0.04166458
0.15 0.2001035 -0.031231192 -0.008859357 -0.04116054
0.16 0.2086544 -0.009563356 -0.030910590 -0.04605576
0.17 0.2165703 -0.020115491 -0.032642954 -0.04319447
0.18 0.2134285 -0.050191610 -0.013862444 0.01555718
0.19 0.2158502 -0.061555646 -0.014066513 0.10336149
0.2 0.1931813 -0.079862915 0.093389578 0.15809703
0.21 0.1828348 -0.078289704 0.094078265 0.25864195
0.22 0.2371564 -0.146372335 0.071361614 0.40514899
0.23 0.2907137 -0.192132094 0.025535674 0.41881913
0.24 0.2968880 -0.197940744 0.022057938 0.41921917
0.25 0.2377862 0.029015655 0.073450483 0.42741924
0.26 0.1759698 0.086608002 0.205923922 0.39632640
0.27 0.2211745 0.255422481 0.114309825 0.32885165
0.28 0.2170630 0.269912670 0.116475170 0.33036898
0.29 0.2072494 0.348639929 0.024121875 0.56147390
0.3 0.1054773 0.530380305 -0.009157201 0.73683803
0.31 0.1234108 0.488267721 -0.009283091 0.76325323
0.32 0.3844318 0.311601276 -0.038600349 0.68907100
0.33 0.3638754 0.235049224 0.142298986 0.75605330
0.34 0.2892624 0.668033531 0.404153003 0.62525688
0.35 0.2965483 0.662128610 0.394571070 0.63179339
0.36 0.3352983 0.589601739 0.506320935 0.63072661
0.37 0.3110196 0.533857268 0.795145760 0.59811059
0.38 0.8766168 0.106704640 0.549547444 0.46168676
0.39 0.7708872 0.309421726 0.592695661 0.58430723
0.4 1.3997638 -0.472971165 0.169560265 1.05171476
0.41 1.4086611 -0.480011894 0.162140620 1.05329787
0.42 2.2887286 -1.136179579 -0.586485783 1.20315904
0.43 3.6029411 -3.258529813 -1.310357895 4.19807054
0.44 3.5669129 -3.568542211 -1.167651920 5.45808770
standard errors:
const x1 x2 x3
0.01 0.000e+00 0.000e+00 0.000e+00 0.000e+00
0.02 0.000e+00 0.000e+00 0.000e+00 0.000e+00
0.03 0.000e+00 0.000e+00 0.000e+00 0.000e+00
0.04 0.000e+00 0.000e+00 0.000e+00 0.000e+00
0.05 0.000e+00 0.000e+00 2.776e-17 0.000e+00
0.06 0.000e+00 0.000e+00 0.000e+00 0.000e+00
0.07 0.000e+00 1.110e-16 3.672e-17 2.404e-17
0.08 5.344e-02 1.022e-01 3.971e-02 3.057e-03
0.09 2.776e-17 2.776e-17 1.963e-17 2.502e-17
0.1 4.328e-02 1.570e-01 1.424e-01 4.656e-02
0.11 4.511e-02 2.652e-01 2.046e-01 1.034e-01
0.12 4.196e-03 1.682e-01 1.058e-01 6.349e-02
0.13 2.277e-02 8.722e-02 4.347e-02 2.584e-02
0.14 2.423e-03 2.446e-03 9.992e-04 5.040e-04
0.15 2.423e-03 2.446e-03 9.992e-04 5.040e-04
0.16 1.431e-02 4.729e-02 2.690e-02 2.918e-02
0.17 7.805e-03 5.811e-02 4.839e-02 6.723e-02
0.18 6.272e-02 7.236e-02 3.052e-02 9.310e-02
0.19 3.977e-02 3.148e-02 1.170e-01 4.166e-02
0.2 1.277e-01 1.722e-01 1.049e-01 1.270e-01
0.21 7.369e-02 1.089e-01 1.217e-01 2.258e-01
0.22 1.352e-01 3.127e-01 2.851e-01 2.353e-01
0.23 2.052e-01 3.904e-01 2.875e-01 1.728e-01
0.24 2.382e-01 4.877e-01 2.873e-01 1.372e-01
0.25 1.289e-01 4.786e-01 2.312e-01 3.069e-01
0.26 8.995e-02 2.236e-01 1.976e-01 2.056e-01
0.27 9.858e-02 2.398e-01 9.176e-02 2.533e-01
0.28 1.757e-01 4.727e-01 2.308e-01 4.783e-01
0.29 2.091e-01 4.848e-01 1.859e-01 3.249e-01
0.3 2.015e-01 2.268e-01 5.332e-01 3.549e-01
0.31 3.474e-01 5.994e-01 6.452e-01 4.462e-01
0.32 5.906e-01 7.654e-01 8.533e-01 7.310e-01
0.33 6.602e-01 1.111e+00 6.187e-01 6.115e-01
0.34 1.287e-01 2.872e-01 4.375e-01 9.236e-01
0.35 2.152e-01 2.449e-01 6.859e-01 1.137e+00
0.36 5.866e-01 7.010e-01 6.853e-01 1.142e+00
0.37 5.972e-01 7.266e-01 1.031e+00 1.224e+00
0.38 1.398e+00 1.595e+00 1.276e+00 1.868e+00
0.39 1.191e+00 1.366e+00 1.022e+00 1.646e+00
0.4 1.786e+00 1.834e+00 1.568e+00 6.341e-01
0.41 4.060e+00 5.007e+00 2.551e+00 3.539e+00
0.42 3.748e+00 5.308e+00 2.952e+00 6.666e+00
0.43 6.766e+00 7.086e+00 4.928e+00 6.177e+00
0.44 6.766e+00 7.086e+00 4.928e+00 6.177e+00
two-sided p-values:
const x1 x2 x3
0.01 0.0e+00 0.0000 0.0e+00 0.0e+00
0.02 0.0e+00 0.0000 0.0e+00 0.0e+00
0.03 0.0e+00 0.0000 0.0e+00 0.0e+00
0.04 0.0e+00 0.0000 0.0e+00 0.0e+00
0.05 0.0e+00 0.0000 0.0e+00 0.0e+00
0.06 0.0e+00 0.0000 0.0e+00 0.0e+00
0.07 0.0e+00 0.0000 0.0e+00 0.0e+00
0.08 3.3e-02 0.1400 1.6e-02 0.0e+00
0.09 0.0e+00 0.0000 0.0e+00 0.0e+00
0.1 5.1e-05 0.1100 2.7e-01 5.3e-02
0.11 5.1e-05 0.3100 4.4e-01 4.2e-01
0.12 0.0e+00 0.8200 8.6e-01 7.7e-03
0.13 0.0e+00 0.6500 6.6e-01 5.8e-11
0.14 0.0e+00 0.0000 3.6e-15 0.0e+00
0.15 0.0e+00 0.0000 0.0e+00 0.0e+00
0.16 0.0e+00 0.8400 2.5e-01 1.1e-01
0.17 0.0e+00 0.7300 5.0e-01 5.2e-01
0.18 6.7e-04 0.4900 6.5e-01 8.7e-01
0.19 5.7e-08 0.0510 9.0e-01 1.3e-02
0.2 1.3e-01 0.6400 3.7e-01 2.1e-01
0.21 1.3e-02 0.4700 4.4e-01 2.5e-01
0.22 7.9e-02 0.6400 8.0e-01 8.5e-02
0.23 1.6e-01 0.6200 9.3e-01 1.5e-02
0.24 2.1e-01 0.6800 9.4e-01 2.3e-03
0.25 6.5e-02 0.9500 7.5e-01 1.6e-01
0.26 5.0e-02 0.7000 3.0e-01 5.4e-02
0.27 2.5e-02 0.2900 2.1e-01 1.9e-01
0.28 2.2e-01 0.5700 6.1e-01 4.9e-01
0.29 3.2e-01 0.4700 9.0e-01 8.4e-02
0.3 6.0e-01 0.0190 9.9e-01 3.8e-02
0.31 7.2e-01 0.4200 9.9e-01 8.7e-02
0.32 5.2e-01 0.6800 9.6e-01 3.5e-01
0.33 5.8e-01 0.8300 8.2e-01 2.2e-01
0.34 2.5e-02 0.0200 3.6e-01 5.0e-01
0.35 1.7e-01 0.0069 5.7e-01 5.8e-01
0.36 5.7e-01 0.4000 4.6e-01 5.8e-01
0.37 6.0e-01 0.4600 4.4e-01 6.3e-01
0.38 5.3e-01 0.9500 6.7e-01 8.0e-01
0.39 5.2e-01 0.8200 5.6e-01 7.2e-01
0.4 4.3e-01 0.8000 9.1e-01 9.7e-02
0.41 7.3e-01 0.9200 9.5e-01 7.7e-01
0.42 5.4e-01 0.8300 8.4e-01 8.6e-01
0.43 5.9e-01 0.6500 7.9e-01 5.0e-01
0.44 6.0e-01 0.6100 8.1e-01 3.8e-01
cmprskQR documentation built on Sept. 15, 2019, 5:02 p.m.
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Description Usage Arguments Details Value References See Also Examples
Description
quantile regression modeling of subdistribution functions in competing risks
Usage
1 2 3 4 5 6 |
Arguments
ftime |
vector of failure/censoring times |
fstatus |
vector with a unique code for each failure type and a separate code for censored observations |
X |
matrix (nobs x ncovs) of covariates |
failcode |
code of fstatus that denotes the failure type of interest |
cencode |
code of fstatus that denotes censored observations |
tau.range |
vector of length 2 denoting the range of quantiles |
tau.step |
grid size on tau.range (spacing between two grid points) |
subset |
a logical vector specifying a subset of cases to include in the analysis |
na.action |
a function specifying the action to take for any cases missing any of ftime, fstatus, cov1, cov2, cengroup, or subset. |
rq.method |
method of computation for quantile regressions. (cf. documentation of method
|
variance |
if |
x |
crrQR object (output from |
... |
included for compatibility with the generic functions. Not currently used. |
Details
Fits the competing risks quantile regression model described in Peng and Fine (2009).
While the use of model formulas is not supported, the
model.matrix function can be used to generate suitable matrices
of covariates from factors, eg
model.matrix(~factor1+factor2)[,-1] will generate the variables
for the factor coding of the factors factor1 and factor2.
The final [,-1] removes the constant term from the output of
model.matrix.
If variance=FALSE, then
some of the functionality in summary.crrQR and print.crrQR
will be lost. This option can be useful in situations where crrQR is
called repeatedly for point estimates, but standard errors are not
required, such as in bootstrapping the cumulative incidence function for
confidence intervals.
The print method prints the estimated coefficients, the estimated standard errors, and the two-sided p-values for the test of the individual coefficients equal to 0.
A first implementation of the estimation procedure was prepared by Limin Peng and Ruosha Li.
Value
Returns a list of class crrQR, with components
$beta.seq |
the estimated regression coefficients |
$tau.seq |
the sequence of quantiles computed |
$var.seq |
estimated variance covariance matrix of coef |
$inf.func |
list of estimated influence functions |
$call |
the call to crr |
$n |
the number of observations used in fitting the model |
$n.missing |
the number of observations removed from the input data due to missing values |
$cvt.length |
number of covariates (columns of X) |
References
Peng L and Fine JP (2009) Competing risks quantile regression. JASA 104:1440-1453.
See Also
predict.crrQR plot.predict.crrQR
summary.crrQR rq.fit
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | # simulated data to test
set.seed(10)
ftime <- rexp(200)
fstatus <- sample(0:2,200,replace=TRUE)
X <- matrix(runif(600),nrow=200)
dimnames(X)[[2]] <- c('x1','x2','x3')
#compute model
print(z <- crrQR(ftime,fstatus,X))
summary(z)
# predict and plot cumulative incedences
reference <- as.matrix(rbind(c(.1,.5,.8),c(.1,.5,.2)))
dimnames(reference)[[2]] <- c('x1','x2','x3')
z.p <- predict(z,reference)
print(z.p)
plot(z.p,lty=1,color=2:3)
crrQR(ftime,fstatus,X,failcode=2)
|
Example output
..............................................done.
stopping at tau=0.47
coefficients:
const x1 x2 x3
0.01 0.10182752 -0.018225105 -0.012531227 -0.094305017
0.02 0.10182752 -0.018225105 -0.012531227 -0.094305017
0.03 0.10182752 -0.018225105 -0.012531227 -0.094305017
0.04 0.10182752 -0.018225105 -0.012531227 -0.094305017
0.05 0.10182752 -0.018225105 -0.012531227 -0.094305017
0.06 0.10182752 -0.018225105 -0.012531227 -0.094305017
0.07 0.10182752 -0.018225105 -0.012531227 -0.094305017
0.08 0.10182752 -0.018225105 -0.012531227 -0.094305017
0.09 0.09800512 -0.024441965 -0.054057490 0.007253677
0.1 0.09800512 -0.024441965 -0.054057490 0.007253677
0.11 0.11379768 -0.029151110 -0.002214549 -0.020370088
0.12 0.11379768 -0.029151110 -0.002214549 -0.020370088
0.13 0.11379768 -0.029151110 -0.002214549 -0.020370088
0.14 0.11918642 0.076045495 0.067364208 -0.088625912
0.15 0.24738188 0.020930280 0.025710897 -0.188980805
0.16 0.46248714 0.209349053 -0.099809892 -0.484074634
0.17 0.46248714 0.209349053 -0.099809892 -0.484074634
0.18 0.41850933 0.245702966 0.031812194 -0.466285131
0.19 0.48559154 0.243799783 0.036648872 -0.537053272
0.2 0.51973703 0.242831044 0.039110788 -0.573074939
0.21 0.73404731 -0.001658431 0.060126014 -0.635865537
0.22 0.82822302 -0.079380771 0.129324773 -0.730829482
0.23 0.80156517 -0.040721666 0.193158766 -0.747392969
0.24 0.71544604 0.038695999 0.246457650 -0.667620473
0.25 0.71544604 0.038695999 0.246457650 -0.667620473
0.26 0.79979695 0.374428325 0.446281597 -0.977901207
0.27 0.79979695 0.374428325 0.446281597 -0.977901207
0.28 0.99227976 0.473917633 0.142796834 -0.930645862
0.29 0.97728103 0.476649403 0.209194371 -0.950003861
0.3 0.98888541 0.470575873 0.155864900 -0.887732299
0.31 1.03049009 0.483916156 -0.034812071 -0.718908367
0.32 1.18024445 0.650108552 -0.039885312 -0.933239627
0.33 1.16719980 0.682730720 -0.040727224 -0.924857761
0.34 1.16312333 0.744629560 -0.081655359 -0.887869942
0.35 1.30163888 0.726496565 -0.238553017 -0.866082983
0.36 2.14020578 0.535072162 -0.995720386 -0.947352883
0.37 2.70576266 0.220447035 -1.307020321 -1.225897233
0.38 2.67490153 0.304599027 -1.411308215 -1.034736321
0.39 2.75332855 0.230875384 -1.418286761 -1.069285111
0.4 2.74408485 0.231414811 -1.371026364 -1.068610087
0.41 2.65720411 0.236484839 -0.926830005 -1.062265599
0.42 2.58521945 0.270466165 -1.003210549 -0.600030208
0.43 3.24934957 0.577840490 -1.774431903 -0.967981569
0.44 3.16357582 1.123887002 -1.110083051 -1.327990852
0.45 3.14667300 2.828755629 0.371481099 -2.968600553
0.46 3.48448670 2.450244875 0.790407158 -3.299424751
standard errors:
const x1 x2 x3
0.01 2.056e-01 2.217e-01 2.883e-02 1.038e-01
0.02 1.388e-17 3.469e-18 0.000e+00 0.000e+00
0.03 0.000e+00 0.000e+00 0.000e+00 0.000e+00
0.04 0.000e+00 0.000e+00 0.000e+00 0.000e+00
0.05 2.776e-17 6.939e-18 3.469e-18 0.000e+00
0.06 0.000e+00 0.000e+00 0.000e+00 0.000e+00
0.07 2.776e-17 3.469e-18 9.179e-18 1.388e-17
0.08 6.621e-03 1.077e-02 7.193e-02 1.759e-01
0.09 3.103e-17 7.758e-18 5.004e-17 1.321e-17
0.1 2.089e-02 2.198e-02 1.257e-02 1.523e-02
0.11 1.326e-01 4.965e-02 1.625e-01 1.069e-01
0.12 1.418e-01 5.016e-02 1.695e-01 1.359e-01
0.13 8.683e-02 2.270e-01 1.844e-01 1.617e-01
0.14 1.371e-01 5.700e-02 6.909e-02 1.425e-01
0.15 2.953e-01 3.011e-01 1.731e-01 3.407e-01
0.16 1.820e-01 1.100e-01 1.767e-01 1.469e-01
0.17 2.043e-01 1.613e-01 2.283e-01 1.307e-01
0.18 8.683e-02 1.167e-01 4.706e-02 1.155e-01
0.19 6.892e-02 1.136e-01 3.827e-02 4.550e-02
0.2 3.701e-01 4.723e-01 2.359e-01 1.858e-01
0.21 4.268e-01 4.754e-01 2.471e-01 1.913e-01
0.22 4.774e-01 4.922e-01 1.863e-01 2.329e-01
0.23 3.051e-01 2.590e-01 1.414e-01 2.352e-01
0.24 4.009e-01 3.873e-01 2.956e-01 2.241e-01
0.25 3.403e-01 6.138e-01 3.654e-01 5.414e-01
0.26 3.307e-01 2.736e-01 2.781e-01 5.703e-01
0.27 2.590e-01 1.674e-01 4.668e-01 3.033e-01
0.28 1.220e-01 1.433e-01 3.366e-01 3.906e-01
0.29 9.559e-02 7.660e-02 4.327e-01 3.695e-01
0.3 3.126e-01 4.138e-01 4.091e-01 3.139e-01
0.31 1.771e-01 3.879e-01 3.606e-01 3.575e-01
0.32 1.967e-01 3.857e-01 1.678e-01 2.728e-01
0.33 4.506e-01 5.067e-01 2.656e-01 2.058e-01
0.34 1.113e+00 8.247e-01 1.138e+00 5.726e-01
0.35 1.428e+00 9.237e-01 1.228e+00 6.121e-01
0.36 1.562e+00 1.108e+00 1.183e+00 6.614e-01
0.37 2.265e+00 1.744e+00 1.482e+00 7.820e-01
0.38 6.983e-01 6.096e-01 5.679e-01 2.794e-01
0.39 5.205e-01 3.668e-01 5.523e-01 8.739e-01
0.4 1.040e+00 1.678e+00 6.947e-01 1.287e+00
0.41 9.466e-01 1.523e+00 1.225e+00 1.410e+00
0.42 8.391e-01 1.716e+00 1.130e+00 9.605e-01
0.43 1.648e+00 2.785e+00 3.082e+00 3.198e+00
0.44 1.131e+00 2.574e+00 3.139e+00 4.021e+00
0.45 1.193e+00 1.745e+00 2.912e+00 3.301e+00
0.46 1.193e+00 1.745e+00 2.912e+00 3.301e+00
two-sided p-values:
const x1 x2 x3
0.01 6.2e-01 9.3e-01 6.6e-01 3.6e-01
0.02 0.0e+00 0.0e+00 0.0e+00 0.0e+00
0.03 0.0e+00 0.0e+00 0.0e+00 0.0e+00
0.04 0.0e+00 0.0e+00 0.0e+00 0.0e+00
0.05 0.0e+00 0.0e+00 0.0e+00 0.0e+00
0.06 0.0e+00 0.0e+00 0.0e+00 0.0e+00
0.07 0.0e+00 0.0e+00 0.0e+00 0.0e+00
0.08 0.0e+00 9.1e-02 8.6e-01 5.9e-01
0.09 0.0e+00 0.0e+00 0.0e+00 0.0e+00
0.1 2.7e-06 2.7e-01 1.7e-05 6.3e-01
0.11 3.9e-01 5.6e-01 9.9e-01 8.5e-01
0.12 4.2e-01 5.6e-01 9.9e-01 8.8e-01
0.13 1.9e-01 9.0e-01 9.9e-01 9.0e-01
0.14 3.8e-01 1.8e-01 3.3e-01 5.3e-01
0.15 4.0e-01 9.4e-01 8.8e-01 5.8e-01
0.16 1.1e-02 5.7e-02 5.7e-01 9.8e-04
0.17 2.4e-02 1.9e-01 6.6e-01 2.1e-04
0.18 1.4e-06 3.5e-02 5.0e-01 5.4e-05
0.19 1.8e-12 3.2e-02 3.4e-01 0.0e+00
0.2 1.6e-01 6.1e-01 8.7e-01 2.0e-03
0.21 8.5e-02 1.0e+00 8.1e-01 8.9e-04
0.22 8.3e-02 8.7e-01 4.9e-01 1.7e-03
0.23 8.6e-03 8.8e-01 1.7e-01 1.5e-03
0.24 7.4e-02 9.2e-01 4.0e-01 2.9e-03
0.25 3.6e-02 9.5e-01 5.0e-01 2.2e-01
0.26 1.6e-02 1.7e-01 1.1e-01 8.6e-02
0.27 2.0e-03 2.5e-02 3.4e-01 1.3e-03
0.28 4.4e-16 9.4e-04 6.7e-01 1.7e-02
0.29 0.0e+00 4.9e-10 6.3e-01 1.0e-02
0.3 1.6e-03 2.6e-01 7.0e-01 4.7e-03
0.31 5.9e-09 2.1e-01 9.2e-01 4.4e-02
0.32 2.0e-09 9.2e-02 8.1e-01 6.2e-04
0.33 9.6e-03 1.8e-01 8.8e-01 7.0e-06
0.34 3.0e-01 3.7e-01 9.4e-01 1.2e-01
0.35 3.6e-01 4.3e-01 8.5e-01 1.6e-01
0.36 1.7e-01 6.3e-01 4.0e-01 1.5e-01
0.37 2.3e-01 9.0e-01 3.8e-01 1.2e-01
0.38 1.3e-04 6.2e-01 1.3e-02 2.1e-04
0.39 1.2e-07 5.3e-01 1.0e-02 2.2e-01
0.4 8.4e-03 8.9e-01 4.8e-02 4.1e-01
0.41 5.0e-03 8.8e-01 4.5e-01 4.5e-01
0.42 2.1e-03 8.7e-01 3.7e-01 5.3e-01
0.43 4.9e-02 8.4e-01 5.6e-01 7.6e-01
0.44 5.2e-03 6.6e-01 7.2e-01 7.4e-01
0.45 8.3e-03 1.1e-01 9.0e-01 3.7e-01
0.46 3.5e-03 1.6e-01 7.9e-01 3.2e-01
Competing Risks Quantile Regression
Call:
crrQR.int(ftime = ftime, fstatus = fstatus, X = X, tau.L = tau.range[1],
tau.U = tau.range[2], tau.step = tau.step, outcome = failcode,
cencode = cencode, variance = variance, rq.method = rq.method,
orig.num = n)
ave.eff se.ave.eff p.signf.test cnst.test var.cnst.test p.cnst.test
const 1.002 0.292 0.0006 -0.722 0.0628 0.0039
x1 0.281 0.368 0.4400 -0.245 0.1032 0.4500
x2 -0.203 0.299 0.5000 0.210 0.0770 0.4500
x3 -0.617 0.335 0.0650 0.368 0.1002 0.2500
Num. cases = 200
[,1] [,2] [,3]
0.01 1.018464 1.077753 0.01
0.02 1.018464 1.077753 0.02
0.03 1.018464 1.077753 0.03
0.04 1.018464 1.077753 0.04
0.05 1.018464 1.077753 0.05
0.06 1.018464 1.077753 0.06
0.07 1.018464 1.077753 0.07
0.08 1.018464 1.077753 0.08
0.09 1.077168 1.072490 0.09
0.1 1.077168 1.072490 0.10
0.11 1.097988 1.111490 0.11
0.12 1.097988 1.111490 0.12
0.13 1.097988 1.111490 0.13
0.14 1.093706 1.153439 0.14
0.15 1.117558 1.251740 0.15
0.16 1.047344 1.400322 0.16
0.17 1.047344 1.400322 0.17
0.18 1.089760 1.441564 0.18
0.19 1.103684 1.523310 0.19
0.2 1.110839 1.566683 0.20
0.21 1.290787 1.890368 0.21
0.22 1.350242 2.093383 0.22
0.23 1.344683 2.105586 0.23
0.24 1.361311 2.032000 0.24
0.25 1.361311 2.032000 0.25
0.26 1.320565 2.374532 0.26
0.27 1.320565 2.374532 0.27
0.28 1.442753 2.521719 0.28
0.29 1.447069 2.558810 0.29
0.3 1.497339 2.550601 0.30
0.31 1.626367 2.503511 0.31
0.32 1.614009 2.825442 0.32
0.33 1.608373 2.801452 0.33
0.34 1.626552 2.770934 0.34
0.35 1.754433 2.949971 0.35
0.36 2.554791 4.510384 0.36
0.37 2.984939 6.228389 0.37
0.38 3.228179 6.006007 0.38
0.39 3.359676 6.381580 0.39
0.4 3.410385 6.475275 0.40
0.41 3.926016 7.425979 0.41
0.42 5.107125 7.320325 0.42
0.43 5.183769 9.265718 0.43
0.44 5.251587 11.650217 0.44
0.45 3.457026 20.523468 0.45
0.46 4.415738 31.971019 0.46
attr(,"class")
[1] "predict.crr"
Warning messages:
1: In plot.window(...) : "color" is not a graphical parameter
2: In plot.xy(xy, type, ...) : "color" is not a graphical parameter
3: In axis(side = side, at = at, labels = labels, ...) :
"color" is not a graphical parameter
4: In axis(side = side, at = at, labels = labels, ...) :
"color" is not a graphical parameter
5: In box(...) : "color" is not a graphical parameter
6: In title(...) : "color" is not a graphical parameter
............................................done.
stopping at tau=0.45
coefficients:
const x1 x2 x3
0.01 0.1136544 -0.149740034 0.095648004 -0.05974612
0.02 0.1136544 -0.149740034 0.095648004 -0.05974612
0.03 0.1136544 -0.149740034 0.095648004 -0.05974612
0.04 0.1136544 -0.149740034 0.095648004 -0.05974612
0.05 0.1136544 -0.149740034 0.095648004 -0.05974612
0.06 0.1136544 -0.149740034 0.095648004 -0.05974612
0.07 0.1136544 -0.149740034 0.095648004 -0.05974612
0.08 0.1136544 -0.149740034 0.095648004 -0.05974612
0.09 0.1514410 -0.222035184 0.123730413 -0.05758450
0.1 0.1753381 -0.250749452 0.156195664 -0.09018926
0.11 0.1828001 -0.268184132 0.159010559 -0.08345396
0.12 0.2214426 -0.039232204 -0.018863269 -0.16920285
0.13 0.2214426 -0.039232204 -0.018863269 -0.16920285
0.14 0.1976801 -0.028785258 -0.007860142 -0.04166458
0.15 0.2001035 -0.031231192 -0.008859357 -0.04116054
0.16 0.2086544 -0.009563356 -0.030910590 -0.04605576
0.17 0.2165703 -0.020115491 -0.032642954 -0.04319447
0.18 0.2134285 -0.050191610 -0.013862444 0.01555718
0.19 0.2158502 -0.061555646 -0.014066513 0.10336149
0.2 0.1931813 -0.079862915 0.093389578 0.15809703
0.21 0.1828348 -0.078289704 0.094078265 0.25864195
0.22 0.2371564 -0.146372335 0.071361614 0.40514899
0.23 0.2907137 -0.192132094 0.025535674 0.41881913
0.24 0.2968880 -0.197940744 0.022057938 0.41921917
0.25 0.2377862 0.029015655 0.073450483 0.42741924
0.26 0.1759698 0.086608002 0.205923922 0.39632640
0.27 0.2211745 0.255422481 0.114309825 0.32885165
0.28 0.2170630 0.269912670 0.116475170 0.33036898
0.29 0.2072494 0.348639929 0.024121875 0.56147390
0.3 0.1054773 0.530380305 -0.009157201 0.73683803
0.31 0.1234108 0.488267721 -0.009283091 0.76325323
0.32 0.3844318 0.311601276 -0.038600349 0.68907100
0.33 0.3638754 0.235049224 0.142298986 0.75605330
0.34 0.2892624 0.668033531 0.404153003 0.62525688
0.35 0.2965483 0.662128610 0.394571070 0.63179339
0.36 0.3352983 0.589601739 0.506320935 0.63072661
0.37 0.3110196 0.533857268 0.795145760 0.59811059
0.38 0.8766168 0.106704640 0.549547444 0.46168676
0.39 0.7708872 0.309421726 0.592695661 0.58430723
0.4 1.3997638 -0.472971165 0.169560265 1.05171476
0.41 1.4086611 -0.480011894 0.162140620 1.05329787
0.42 2.2887286 -1.136179579 -0.586485783 1.20315904
0.43 3.6029411 -3.258529813 -1.310357895 4.19807054
0.44 3.5669129 -3.568542211 -1.167651920 5.45808770
standard errors:
const x1 x2 x3
0.01 0.000e+00 0.000e+00 0.000e+00 0.000e+00
0.02 0.000e+00 0.000e+00 0.000e+00 0.000e+00
0.03 0.000e+00 0.000e+00 0.000e+00 0.000e+00
0.04 0.000e+00 0.000e+00 0.000e+00 0.000e+00
0.05 0.000e+00 0.000e+00 2.776e-17 0.000e+00
0.06 0.000e+00 0.000e+00 0.000e+00 0.000e+00
0.07 0.000e+00 1.110e-16 3.672e-17 2.404e-17
0.08 5.344e-02 1.022e-01 3.971e-02 3.057e-03
0.09 2.776e-17 2.776e-17 1.963e-17 2.502e-17
0.1 4.328e-02 1.570e-01 1.424e-01 4.656e-02
0.11 4.511e-02 2.652e-01 2.046e-01 1.034e-01
0.12 4.196e-03 1.682e-01 1.058e-01 6.349e-02
0.13 2.277e-02 8.722e-02 4.347e-02 2.584e-02
0.14 2.423e-03 2.446e-03 9.992e-04 5.040e-04
0.15 2.423e-03 2.446e-03 9.992e-04 5.040e-04
0.16 1.431e-02 4.729e-02 2.690e-02 2.918e-02
0.17 7.805e-03 5.811e-02 4.839e-02 6.723e-02
0.18 6.272e-02 7.236e-02 3.052e-02 9.310e-02
0.19 3.977e-02 3.148e-02 1.170e-01 4.166e-02
0.2 1.277e-01 1.722e-01 1.049e-01 1.270e-01
0.21 7.369e-02 1.089e-01 1.217e-01 2.258e-01
0.22 1.352e-01 3.127e-01 2.851e-01 2.353e-01
0.23 2.052e-01 3.904e-01 2.875e-01 1.728e-01
0.24 2.382e-01 4.877e-01 2.873e-01 1.372e-01
0.25 1.289e-01 4.786e-01 2.312e-01 3.069e-01
0.26 8.995e-02 2.236e-01 1.976e-01 2.056e-01
0.27 9.858e-02 2.398e-01 9.176e-02 2.533e-01
0.28 1.757e-01 4.727e-01 2.308e-01 4.783e-01
0.29 2.091e-01 4.848e-01 1.859e-01 3.249e-01
0.3 2.015e-01 2.268e-01 5.332e-01 3.549e-01
0.31 3.474e-01 5.994e-01 6.452e-01 4.462e-01
0.32 5.906e-01 7.654e-01 8.533e-01 7.310e-01
0.33 6.602e-01 1.111e+00 6.187e-01 6.115e-01
0.34 1.287e-01 2.872e-01 4.375e-01 9.236e-01
0.35 2.152e-01 2.449e-01 6.859e-01 1.137e+00
0.36 5.866e-01 7.010e-01 6.853e-01 1.142e+00
0.37 5.972e-01 7.266e-01 1.031e+00 1.224e+00
0.38 1.398e+00 1.595e+00 1.276e+00 1.868e+00
0.39 1.191e+00 1.366e+00 1.022e+00 1.646e+00
0.4 1.786e+00 1.834e+00 1.568e+00 6.341e-01
0.41 4.060e+00 5.007e+00 2.551e+00 3.539e+00
0.42 3.748e+00 5.308e+00 2.952e+00 6.666e+00
0.43 6.766e+00 7.086e+00 4.928e+00 6.177e+00
0.44 6.766e+00 7.086e+00 4.928e+00 6.177e+00
two-sided p-values:
const x1 x2 x3
0.01 0.0e+00 0.0000 0.0e+00 0.0e+00
0.02 0.0e+00 0.0000 0.0e+00 0.0e+00
0.03 0.0e+00 0.0000 0.0e+00 0.0e+00
0.04 0.0e+00 0.0000 0.0e+00 0.0e+00
0.05 0.0e+00 0.0000 0.0e+00 0.0e+00
0.06 0.0e+00 0.0000 0.0e+00 0.0e+00
0.07 0.0e+00 0.0000 0.0e+00 0.0e+00
0.08 3.3e-02 0.1400 1.6e-02 0.0e+00
0.09 0.0e+00 0.0000 0.0e+00 0.0e+00
0.1 5.1e-05 0.1100 2.7e-01 5.3e-02
0.11 5.1e-05 0.3100 4.4e-01 4.2e-01
0.12 0.0e+00 0.8200 8.6e-01 7.7e-03
0.13 0.0e+00 0.6500 6.6e-01 5.8e-11
0.14 0.0e+00 0.0000 3.6e-15 0.0e+00
0.15 0.0e+00 0.0000 0.0e+00 0.0e+00
0.16 0.0e+00 0.8400 2.5e-01 1.1e-01
0.17 0.0e+00 0.7300 5.0e-01 5.2e-01
0.18 6.7e-04 0.4900 6.5e-01 8.7e-01
0.19 5.7e-08 0.0510 9.0e-01 1.3e-02
0.2 1.3e-01 0.6400 3.7e-01 2.1e-01
0.21 1.3e-02 0.4700 4.4e-01 2.5e-01
0.22 7.9e-02 0.6400 8.0e-01 8.5e-02
0.23 1.6e-01 0.6200 9.3e-01 1.5e-02
0.24 2.1e-01 0.6800 9.4e-01 2.3e-03
0.25 6.5e-02 0.9500 7.5e-01 1.6e-01
0.26 5.0e-02 0.7000 3.0e-01 5.4e-02
0.27 2.5e-02 0.2900 2.1e-01 1.9e-01
0.28 2.2e-01 0.5700 6.1e-01 4.9e-01
0.29 3.2e-01 0.4700 9.0e-01 8.4e-02
0.3 6.0e-01 0.0190 9.9e-01 3.8e-02
0.31 7.2e-01 0.4200 9.9e-01 8.7e-02
0.32 5.2e-01 0.6800 9.6e-01 3.5e-01
0.33 5.8e-01 0.8300 8.2e-01 2.2e-01
0.34 2.5e-02 0.0200 3.6e-01 5.0e-01
0.35 1.7e-01 0.0069 5.7e-01 5.8e-01
0.36 5.7e-01 0.4000 4.6e-01 5.8e-01
0.37 6.0e-01 0.4600 4.4e-01 6.3e-01
0.38 5.3e-01 0.9500 6.7e-01 8.0e-01
0.39 5.2e-01 0.8200 5.6e-01 7.2e-01
0.4 4.3e-01 0.8000 9.1e-01 9.7e-02
0.41 7.3e-01 0.9200 9.5e-01 7.7e-01
0.42 5.4e-01 0.8300 8.4e-01 8.6e-01
0.43 5.9e-01 0.6500 7.9e-01 5.0e-01
0.44 6.0e-01 0.6100 8.1e-01 3.8e-01
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