[R] Questions about lrm, validate, pentrace (Re: BMA, logistic regression, odds ratio, model reduction etc)

From: <khosoda_at_med.kobe-u.ac.jp>
Date: Sun, 24 Apr 2011 00:53:40 +0900

According to the advice, I tried rms package. Just to make sure, I have data of 104 patients (x6.df), which consists of 5 explanatory variables and one binary outcome (poor/good) (previous model 2 strategy). The outcome consists of 25 poor results and 79 good results. Therefore, My events per variable (EPV) is only 5 (much less than the rule of thumb of 10).

My questions are about validate and pentrace in rms package. I present some codes and results.
I appreciate anybody's help in advance.

> x6.lrm <- lrm(outcome ~ stenosis+x1+x2+procedure+ClinicalScore,
data=x6.df, x=T, y=T)

> x6.lrm

...

Obs  104    LR chi2      29.24    R2       0.367    C       0.816
  negative 79    d.f.         5    g        1.633    Dxy     0.632
  positive 25    Pr(> chi2) <0.0001   gr    5.118    gamma   0.632
max |deriv| 1e-08                    gp    0.237    tau-a   0.233
                                      Brier   0.127

                Coef    S.E.   Wald Z Pr(>|Z|)
Intercept      -5.5328 2.6287 -2.10  0.0353
stenosis       -0.0150 0.0284 -0.53  0.5979
x1              3.0425 0.9100  3.34  0.0008
x2             -0.7534 0.4519 -1.67  0.0955
procedure       1.2085 0.5717  2.11  0.0345
ClinicalScore 0.3762 0.2287 1.65 0.0999

It seems not too bad. Next, validation by bootstrap ...

> validate(x6.lrm, B=200, bw=F)

           index.orig training    test optimism index.corrected   n
Dxy           0.6324   0.6960  0.5870   0.1091          0.5233 200
R2            0.3668   0.4370  0.3154   0.1216          0.2453 200
Intercept     0.0000   0.0000 -0.2007   0.2007         -0.2007 200
Slope         1.0000   1.0000  0.7565   0.2435          0.7565 200
Emax          0.0000   0.0000  0.0999   0.0999          0.0999 200
D             0.2716   0.3368  0.2275   0.1093          0.1623 200
U            -0.0192  -0.0192  0.0369  -0.0561          0.0369 200
Q             0.2908   0.3560  0.1906   0.1654          0.1254 200
B             0.1272   0.1155  0.1384  -0.0229          0.1501 200
g             1.6328   2.0740  1.4647   0.6093          1.0235 200
gp            0.2367   0.2529  0.2189   0.0341          0.2026 200

The apparent Dxy is 0.63, and bias-corrected Dxy is 0.52. The maximum absolute error is estimated to be 0.099. The changes in slope and intercept are also more substantial. In all, there is evidence that I am somewhat overfitting the data, right?.

Furthermore, using step-down variable selection ...

> validate(x6.lrm, B=200, bw=T)

                Backwards Step-down - Original Model

  Deleted        Chi-Sq d.f. P      Residual d.f. P      AIC
  stenosis       0.28   1    0.5979 0.28     1    0.5979 -1.72
  ClinicalScore  2.60   1    0.1068 2.88     2    0.2370 -1.12
  x2             2.86   1    0.0910 5.74     3    0.1252 -0.26

Approximate Estimates after Deleting Factors

              Coef S.E. Wald Z P

Intercept  -5.865 1.4136 -4.149 3.336e-05
x1          2.915 0.8685  3.357 7.889e-04
procedure   1.072 0.5590  1.918 5.508e-02

Factors in Final Model

[1] x1         procedure
           index.orig training    test optimism index.corrected   n
Dxy           0.5661   0.6755  0.5559   0.1196          0.4464 200
R2            0.2876   0.4085  0.2784   0.1301          0.1575 200
Intercept     0.0000   0.0000 -0.2459   0.2459         -0.2459 200
Slope         1.0000   1.0000  0.7300   0.2700          0.7300 200
Emax          0.0000   0.0000  0.1173   0.1173          0.1173 200
D             0.2038   0.3130  0.1970   0.1160          0.0877 200
U            -0.0192  -0.0192  0.0382  -0.0574          0.0382 200
Q             0.2230   0.3323  0.1589   0.1734          0.0496 200
B             0.1441   0.1192  0.1452  -0.0261          0.1702 200
g             1.2628   1.9524  1.3222   0.6302          0.6326 199
gp            0.2041   0.2430  0.2043   0.0387          0.1654 199

If I select only two variables (x1 and procedure), bias-corrected Dxy goes down to 0.45.

[Question 1]
I have EPV problem. Even so, should I keep the full model (5-variable model)? or can I use the 2-variable (x1 and procedure) model which the validate() with step-down provides?

[Question 2]
If I use 2-variable model, should I do
x2.lrm <- lrm(postopDWI_HI ~ T1+procedure2, data=x6.df, x=T, y=T)? or keep the value showed above by validate function?

Next, shrinkage ...

> pentrace(x6.lrm, seq(0, 5.0, by=0.05))
Best penalty:
penalty df

    3.05 4.015378

The best penalty is 3.05. So, I update it with this penalty to obtain the corresponding penalized model:

> x6.lrm.pen <- update(x6.lrm, penalty=3.05, x=T, y=T)
> x6.lrm.pen

.....
Penalty factors

  simple nonlinear interaction nonlinear.interaction

    3.05      3.05        3.05                  3.05
Final penalty on -2 log L
      [,1]

[1,] 3.8
Obs     104    LR chi2      28.18    R2       0.313    C       0.818
  negative    79    d.f.     4.015    g        1.264    Dxy     0.635
  positive    25   Pr(> chi2) <0.0001 gr       3.538    gamma   0.637
max |deriv| 3e-05                    gp       0.201    tau-a   0.234
                                      Brier    0.129

                Coef    S.E.   Wald Z Pr(>|Z|) Penalty Scale
Intercept      -4.7246 2.2429 -2.11  0.0352    0.0000
stenosis       -0.0105 0.0240 -0.44  0.6621   17.8021
x1              2.3605 0.7254  3.25  0.0011    0.6054
x2             -0.5385 0.3653 -1.47  0.1404    1.2851
procedure       0.9247 0.4844  1.91  0.0563    0.8576
ClinicalScore 0.3046 0.1874 1.63 0.1041 2.4779

Arrange the coefficients of the two models side by side, and also list the difference between the two:

> cbind(coef(x6.lrm), coef(x6.lrm.pen), abs(coef(x6.lrm)-coef(x6.lrm.pen)))

                       [,1]        [,2]        [,3]
Intercept      -5.53281808 -4.72464766 0.808170417
stenosis       -0.01496757 -0.01050797 0.004459599
x1              3.04248257  2.36051833 0.681964238
x2             -0.75335619 -0.53854750 0.214808685
procedure       1.20847252  0.92474708 0.283725441
ClinicalScore 0.37623189 0.30457557 0.071656322

[Question 3]
Is this penalized model the one I should present for my colleagues? I still have EPV problem. Or is EPV problem O.K. if I use penalization?

I am still wondering about what I can do to avoid EPV problem. Collecting new data would be a long-time and huge work...

(11/04/22 1:46), khosoda_at_med.kobe-u.ac.jp wrote:

> Thank you for your comment.
> I forgot to mention that varclus and pvclust showed similar results for
> my data.
>
> BTW, I did not realize rms is a replacement for the Design package.
> I appreciate your suggestion.
> --
> KH
>
> (11/04/21 8:00), Frank Harrell wrote:
>> I think it's OK. You can also use the Hmisc package's varclus function.
>> Frank
>>
>>
>> 細田弘吉 wrote:
>>>
>>> Dear Prof. Harrel,
>>>
>>> Thank you very much for your quick advice.
>>> I will try rms package.
>>>
>>> Regarding model reduction, is my model 2 method (clustering and recoding
>>> that are blinded to the outcome) permissible?
>>>
>>> Sincerely,
>>>
>>> --
>>> KH
>>>
>>> (11/04/20 22:01), Frank Harrell wrote:
>>>> Deleting variables is a bad idea unless you make that a formal part of
>>>> the
>>>> BMA so that the attempt to delete variables is penalized for.
>>>> Instead of
>>>> BMA I recommend simple penalized maximum likelihood estimation (see the
>>>> lrm
>>>> function in the rms package) or pre-modeling data reduction that is
>>>> blinded
>>>> to the outcome variable.
>>>> Frank
>>>>
>>>>
>>>> 細田弘吉 wrote:
>>>>>
>>>>> Hi everybody,
>>>>> I apologize for long mail in advance.
>>>>>
>>>>> I have data of 104 patients, which consists of 15 explanatory
>>>>> variables
>>>>> and one binary outcome (poor/good). The outcome consists of 25 poor
>>>>> results and 79 good results. I tried to analyze the data with logistic
>>>>> regression. However, the 15 variables and 25 events means events per
>>>>> variable (EPV) is much less than 10 (rule of thumb). Therefore, I
>>>>> used R
>>>>> package, "BMA" to perform logistic regression with BMA to avoid this
>>>>> problem.
>>>>>
>>>>> model 1 (full model):
>>>>> x1, x2, x3, x4 are continuous variables and others are binary data.
>>>>>
>>>>>> x16.bic.glm<- bic.glm(outcome ~ ., data=x16.df,
>>>>> glm.family="binomial", OR20, strict=FALSE)
>>>>>> summary(x16.bic.glm)
>>>>> (The output below has been cut off at the right edge to save space)
>>>>>
>>>>> 62 models were selected
>>>>> Best 5 models (cumulative posterior probability = 0.3606 ):
>>>>>
>>>>> p!=0 EV SD model 1 model2
>>>>> Intercept 100 -5.1348545 1.652424 -4.4688 -5.15
>>>>> -5.1536
>>>>> age 3.3 0.0001634 0.007258 .
>>>>> sex 4.0
>>>>> .M -0.0243145 0.220314 .
>>>>> side 10.8
>>>>> .R 0.0811227 0.301233 .
>>>>> procedure 46.9 -0.5356894 0.685148 . -1.163
>>>>> symptom 3.8 -0.0099438 0.129690 . .
>>>>> stenosis 3.4 -0.0003343 0.005254 .
>>>>> x1 3.7 -0.0061451 0.144084 .
>>>>> x2 100.0 3.1707661 0.892034 3.2221 3.11
>>>>> x3 51.3 -0.4577885 0.551466 -0.9154 .
>>>>> HT 4.6
>>>>> .positive 0.0199299 0.161769 . .
>>>>> DM 3.3
>>>>> .positive -0.0019986 0.105910 . .
>>>>> IHD 3.5
>>>>> .positive 0.0077626 0.122593 . .
>>>>> smoking 9.1
>>>>> .positive 0.0611779 0.258402 . .
>>>>> hyperlipidemia 16.0
>>>>> .positive 0.1784293 0.512058 . .
>>>>> x4 8.2 0.0607398 0.267501 . .
>>>>>
>>>>>
>>>>> nVar 2 2
>>>>> 1 3 3
>>>>> BIC -376.9082
>>>>> -376.5588 -376.3094 -375.8468 -374.5582
>>>>> post prob 0.104
>>>>> 0.087 0.077 0.061 0.032
>>>>>
>>>>> [Question 1]
>>>>> Is it O.K to calculate odds ratio and its 95% confidence interval from
>>>>> "EV" (posterior distribution mean) and“SD”(posterior distribution
>>>>> standard deviation)?
>>>>> For example, 95%CI of EV of x2 can be calculated as;
>>>>>> exp(3.1707661)
>>>>> [1] 23.82573 -----> odds ratio
>>>>>> exp(3.1707661+1.96*0.892034)
>>>>> [1] 136.8866
>>>>>> exp(3.1707661-1.96*0.892034)
>>>>> [1] 4.146976
>>>>> ------------------> 95%CI (4.1 to 136.9)
>>>>> Is this O.K.?
>>>>>
>>>>> [Question 2]
>>>>> Is it permissible to delete variables with small value of "p!=0" and
>>>>> "EV", such as age (3.3% and 0.0001634) to reduce the number of
>>>>> explanatory variables and reconstruct new model without those
>>>>> variables
>>>>> for new session of BMA?
>>>>>
>>>>> model 2 (reduced model):
>>>>> I used R package, "pvclust", to reduce the model. The result suggested
>>>>> x1, x2 and x4 belonged to the same cluster, so I picked up only x2.
>>>>> Based on the subject knowledge, I made a simple unweighted sum, by
>>>>> counting the number of clinical features. For 9 features (sex, side,
>>>>> HT2, hyperlipidemia, DM, IHD, smoking, symptom, age), the sum ranges
>>>>> from 0 to 9. This score was defined as ClinicalScore. Consequently, I
>>>>> made up new data set (x6.df), which consists of 5 variables (stenosis,
>>>>> x2, x3, procedure, and ClinicalScore) and one binary outcome
>>>>> (poor/good). Then, for alternative BMA session...
>>>>>
>>>>>> BMAx6.glm<- bic.glm(postopDWI_HI ~ ., data=x6.df,
>>>>> glm.family="binomial", OR=20, strict=FALSE)
>>>>>> summary(BMAx6.glm)
>>>>> (The output below has been cut off at the right edge to save space)
>>>>> Call:
>>>>> bic.glm.formula(f = postopDWI_HI ~ ., data = x6.df, glm.family =
>>>>> "binomial", strict = FALSE, OR = 20)
>>>>>
>>>>>
>>>>> 13 models were selected
>>>>> Best 5 models (cumulative posterior probability = 0.7626 ):
>>>>>
>>>>> p!=0 EV SD model 1 model 2
>>>>> Intercept 100 -5.6918362 1.81220 -4.4688 -6.3166
>>>>> stenosis 8.1 -0.0008417 0.00815 . .
>>>>> x2 100.0 3.0606165 0.87765 3.2221 3.1154
>>>>> x3 46.5 -0.3998864 0.52688 -0.9154 .
>>>>> procedure 49.3 0.5747013 0.70164 . 1.1631
>>>>> ClinicalScore 27.1 0.0966633 0.19645 . .
>>>>>
>>>>>
>>>>> nVar 2 2 1
>>>>> 3 3
>>>>> BIC -376.9082 -376.5588
>>>>> -376.3094 -375.8468 -375.5025
>>>>> post prob 0.208 0.175
>>>>> 0.154 0.122 0.103
>>>>>
>>>>> [Question 3]
>>>>> Am I doing it correctly or not?
>>>>> I mean this kind of model reduction is permissible for BMA?
>>>>>
>>>>> [Question 4]
>>>>> I still have 5 variables, which violates the rule of thumb, "EPV> 10".
>>>>> Is it permissible to delete "stenosis" variable because of small value
>>>>> of "EV"? Or is it O.K. because this is BMA?
>>>>>
>>>>> Sorry for long post.
>>>>>
>>>>> I appreciate your help very much in advance.
>>>>>
>>>>> --
>>>>> KH
>>>>>
>>>>> ______________________________________________
>>>>> R-help_at_r-project.org mailing list
>>>>> https://stat.ethz.ch/mailman/listinfo/r-help
>>>>> PLEASE do read the posting guide
>>>>> http://www.R-project.org/posting-guide.html
>>>>> and provide commented, minimal, self-contained, reproducible code.
>>>>>
>>>>
>>>>
>>>> -----
>>>> Frank Harrell
>>>> Department of Biostatistics, Vanderbilt University
>>>> --
>>>> View this message in context:
>>>> http://r.789695.n4.nabble.com/BMA-logistic-regression-odds-ratio-model-reduction-etc-tp3462416p3462919.html
>>>>
>>>> Sent from the R help mailing list archive at Nabble.com.
>>>>

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