Question on approximations of full logistic regression model
I tried to make a histogram of bootstrap distribution of my logistic
model according to "Regression Model Strategy" (pp197-200). Attached is
the histogram I made. The figure demonstrates bootstrap distribution of
log odds ratio from my logistic model. The solid curve is a kernel
density estimate and dashed curve is a normal density with the dame mean
and standard deviation as the bootstrapped values. Vertical lines
indicate asymmetric 0.9, 0.95, and 0,99 two-sided confidence limits for
the log odds ratio based on quantiles of the bootstrap values.
It seems to me that bootstrap distribution is normal and that estimation
of confidence interval is, ummm, accurate.
Am I right?
The codes I used are followings;
> library(rms)
> b <- bootcov(MyModel.penalized, B=1000, coef.reps=T)
> s <- summary(MyModel.penalized, x1=c(1.0, 1.5), x2=c(1.0, 1.5),
ClinicalScore=c(4,6), procedure=c("E", "A"))
> X <- predict(MyModel.penalized, data.frame(T1=c(1.0, 1.5), T2=c(1.0,
1.5), ClinicalScore=c(4,6), procedure=c("E", "A")), type="x")
> X
Intercept x1 x2 ClinicalScore procedure=E
1 1 1.0 1.0 4 1
2 1 1.5 1.5 6 0
> Xdif <- X[2, drop=F] -X[1, drop=F]
> Xdif
Intercept x1 x2 ClinicalScore procedure=E
2 0 0.5 0.5 2 -1
> conf <- c(.9, .95, .99)
> bootplot(b, X=Xdif, conf.int=conf, xlim=c(0, 6))
> boot.log.odds.ratio <- b$boot.Coef %*% t(Xdif)
> sd <- sqrt(var(boot.log.odds.ratio))
> sd
2
2 0.7412509
> z <- seq(0, 6, length=104)
> lines(z, dnorm(z, mean=mean(boot.log.odds.ratio), sd = sd), lty=2)
(11/05/16 22:01), Frank Harrell wrote:
The choice is not clear, and requires some simulations to estimate the average absolute error of the covariance matrix estimators. Frank ???? wrote:
Thank you for your reply, Prof. Harrell. I agree with you. Dropping only one variable does not actually help a lot. I have one more question. During analysis of this model I found that the confidence intervals (CIs) of some coefficients provided by bootstrapping (bootcov function in rms package) was narrower than CIs provided by usual variance-covariance matrix and CIs of other coefficients wider. My data has no cluster structure. I am wondering which CIs are better. I guess bootstrapping one, but is it right? I would appreciate your help in advance. -- KH (11/05/16 12:25), Frank Harrell wrote:
I think you are doing this correctly except for one thing. The validation and other inferential calculations should be done on the full model. Use the approximate model to get a simpler nomogram but not to get standard errors. With only dropping one variable you might consider just running the nomogram on the entire model. Frank KH wrote:
Hi, I am trying to construct a logistic regression model from my data (104 patients and 25 events). I build a full model consisting of five predictors with the use of penalization by rms package (lrm, pentrace etc) because of events per variable issue. Then, I tried to approximate the full model by step-down technique predicting L from all of the componet variables using ordinary least squares (ols in rms package) as the followings. I would like to know whether I am doing right or not.
library(rms) plogit<- predict(full.model) full.ols<- ols(plogit ~ stenosis+x1+x2+ClinicalScore+procedure, sigma=1) fastbw(full.ols, aics=1e10)
Deleted Chi-Sq d.f. P Residual d.f. P AIC R2 stenosis 1.41 1 0.2354 1.41 1 0.2354 -0.59 0.991 x2 16.78 1 0.0000 18.19 2 0.0001 14.19 0.882 procedure 26.12 1 0.0000 44.31 3 0.0000 38.31 0.711 ClinicalScore 25.75 1 0.0000 70.06 4 0.0000 62.06 0.544 x1 83.42 1 0.0000 153.49 5 0.0000 143.49 0.000 Then, fitted an approximation to the full model using most imprtant variable (R^2 for predictions from the reduced model against the original Y drops below 0.95), that is, dropping "stenosis".
full.ols.approx<- ols(plogit ~ x1+x2+ClinicalScore+procedure) full.ols.approx$stats
n Model L.R. d.f. R2 g Sigma 104.0000000 487.9006640 4.0000000 0.9908257 1.3341718 0.1192622 This approximate model had R^2 against the full model of 0.99. Therefore, I updated the original full logistic model dropping "stenosis" as predictor.
full.approx.lrm<- update(full.model, ~ . -stenosis)
validate(full.model, bw=F, B=1000)
index.orig training test optimism index.corrected n Dxy 0.6425 0.7017 0.6131 0.0887 0.5539 1000 R2 0.3270 0.3716 0.3335 0.0382 0.2888 1000 Intercept 0.0000 0.0000 0.0821 -0.0821 0.0821 1000 Slope 1.0000 1.0000 1.0548 -0.0548 1.0548 1000 Emax 0.0000 0.0000 0.0263 0.0263 0.0263 1000
validate(full.approx.lrm, bw=F, B=1000)
index.orig training test optimism index.corrected n Dxy 0.6446 0.6891 0.6265 0.0626 0.5820 1000 R2 0.3245 0.3592 0.3428 0.0164 0.3081 1000 Intercept 0.0000 0.0000 0.1281 -0.1281 0.1281 1000 Slope 1.0000 1.0000 1.1104 -0.1104 1.1104 1000 Emax 0.0000 0.0000 0.0444 0.0444 0.0444 1000 Validatin revealed this approximation was not bad. Then, I made a nomogram.
full.approx.lrm.nom<- nomogram(full.approx.lrm,
fun.at=c(0.05,0.1,0.2,0.4,0.6,0.8,0.9,0.95), fun=plogis)
plot(full.approx.lrm.nom)
Another nomogram using ols model,
full.ols.approx.nom<- nomogram(full.ols.approx,
fun.at=c(0.05,0.1,0.2,0.4,0.6,0.8,0.9,0.95), fun=plogis)
plot(full.ols.approx.nom)
These two nomograms are very similar but a little bit different. My questions are; 1. Am I doing right? 2. Which nomogram is correct I would appreciate your help in advance. -- KH
-------------- next part -------------- A non-text attachment was scrubbed... Name: x5factor_final-CI-histogram.pdf Type: application/pdf Size: 103774 bytes Desc: not available URL: <https://stat.ethz.ch/pipermail/r-help/attachments/20110518/00b34a96/attachment.pdf>