# [R-sig-ME] Logistic regression with spatial autocorrelation structure

Douglas Bates bates at stat.wisc.edu
Mon Feb 7 19:56:51 CET 2011

```On Mon, Feb 7, 2011 at 10:58 AM,  <Dale.W.Steele at gmail.com> wrote:
> Dear Prof. Bates - Thanks for your response. How would I calculate a 95%
> confidence interval for
> the probability of appendicitis in a new patient given a particular set of
> covariates for

> 1) patients enrolled by a specific clinician (eg. doc == "18")?
> 2) some future clinician?

Didn't I answer 2 in my previous message in the part about

> fixef(m2)
(Intercept)   priorprob  genderMale         wbc
-8.2886183   0.0502540   1.1870539   0.3412998
> eta <- crossprod(c(1, 55, 0, 9.45), fixef(m2))
> eta
[,1]
[1,] -2.299365
> binomial()\$linkinv(eta)
[,1]
[1,] 0.09117557

It is certainly possible to automate the process a bit more with a
predict method but, for the time being, I think that should be the way
to go about it.

If you have a simple random effect for doc, as in model m2, then to
get a prediction for a specific physician you add the random effect
for, say, doc == "18", to the value of eta before transforming it.
You can get the conditional modes of the random effects with the ranef
extractor.

> Thanks. --Dale
>
> On Feb 1, 2011 1:08pm, Douglas Bates <bates at stat.wisc.edu> wrote:
>> On Tue, Feb 1, 2011 at 12:05 PM, Douglas Bates bates at stat.wisc.edu> wrote:
>>
>> > On Mon, Jan 31, 2011 at 11:45 AM, Dale Steele dale.w.steele at gmail.com>
>> > wrote:
>>
>> >> Dear mixed-modeling  experts,
>>
>> >>
>>
>> >> I'm interested in modeling the probability of appendicitis in patients
>>
>> >> with abdominal pain.
>>
>> >>
>>
>> >> The R binary data file 'http://www.ped-em.org/appy.rda' contains
>>
>> >> the following variables from a pilot study of 138 children with
>>
>> >> abdominal pain.
>>
>> >
>>
>> > Thank you for providing the data.
>>
>> >
>>
>> >> 'dx'        eventual diagnosis:   0=no appendicitis, 1=appendicitis
>>
>> >> 'gender'       Male/Female
>>
>> >> 'wbc'       total white blood cell count
>>
>> >> 'priorprob'    Clinical predicted probability of appendicitis
>>
>> >> 'doc'       doctor who assigned 'priorprob'
>>
>> >>
>>
>> >> After taking a history and performing a physical examination, the ER
>> >> doctor
>>
>> >> was asked to make a vertical mark on a 100 mm horizontal line to
>> >> represent
>>
>> >> her estimate of the (percent) probability that the patient had
>> >> appendicitis.
>>
>> >>
>>
>> >> My initial thought was to fit a multiple logistic regression model:
>>
>> >>
>>
>> >> m1
>> >>
>>
>> >> However, it seems likely that each doctor interpreted the probability
>> >> scale
>>
>> >> differently.  The 23 doctors evaluated from 1 to 17 patients each. I'm
>>
>> >> not primarily interest in predictions by a specific clinician.  Thus,
>>
>> >> it seems to make sense to fit a generalized linear mixed model.
>>
>> >>
>>
>> >> At this point I get muddled. Have I correctly specified a random
>>
>> >> intercept model (m2) and a random intercept/random slope model (m3)?
>>
>> >> Are there other sensible models?
>>
>> >
>>
>> > Yes, you have correctly specified them.  However, if you check the
>>
>> > output from the fits
>>
>> >> (m2
>> > Generalized linear mixed model fit by the Laplace approximation
>>
>> > Formula: dx ~ priorprob + gender + wbc + (1 | doc)
>>
>> >   Data: appy
>>
>> >   AIC   BIC logLik deviance
>>
>> >  108.1 122.7 -49.03    98.07
>>
>> > Random effects:
>>
>> >  Groups Name        Variance Std.Dev.
>>
>> >  doc    (Intercept) 0.82922  0.91062
>>
>> > Number of obs: 138, groups: doc, 23
>>
>> >
>>
>> > Fixed effects:
>>
>> >            Estimate Std. Error z value Pr(>|z|)
>>
>> > (Intercept) -8.28862    1.45666  -5.690 1.27e-08
>>
>> > priorprob    0.05025    0.01291   3.892 9.96e-05
>>
>> > genderMale   1.18705    0.56752   2.092   0.0365
>>
>> > wbc          0.34130    0.07090   4.814 1.48e-06
>>
>> >
>>
>> > Correlation of Fixed Effects:
>>
>> >           (Intr) prrprb gndrMl
>>
>> > priorprob  -0.775
>>
>> > genderMale -0.293  0.130
>>
>> > wbc        -0.790  0.353  0.040
>>
>> >> (m3
>> > Generalized linear mixed model fit by the Laplace approximation
>>
>> > Formula: dx ~ priorprob + gender + wbc + (priorprob | doc)
>>
>> >   Data: appy
>>
>> >   AIC BIC logLik deviance
>>
>> >  111.5 132 -48.76    97.52
>>
>> > Random effects:
>>
>> >  Groups Name        Variance   Std.Dev. Corr
>>
>> >  doc    (Intercept) 0.02897010 0.170206
>>
>> >        priorprob   0.00016799 0.012961 1.000
>>
>> > Number of obs: 138, groups: doc, 23
>>
>> >
>>
>> > Fixed effects:
>>
>> >            Estimate Std. Error z value Pr(>|z|)
>>
>> > (Intercept) -8.23977    1.44408  -5.706 1.16e-08
>>
>> > priorprob    0.04957    0.01328   3.733 0.000189
>>
>> > genderMale   1.16593    0.57481   2.028 0.042521
>>
>> > wbc          0.34485    0.07056   4.887 1.02e-06
>>
>> >
>>
>> > Correlation of Fixed Effects:
>>
>> >           (Intr) prrprb gndrMl
>>
>> > priorprob  -0.757
>>
>> > genderMale -0.299  0.141
>>
>> > wbc        -0.800  0.350  0.030
>>
>> >> anova(m2, m3)
>>
>> > Data: appy
>>
>> > Models:
>>
>> > m2: dx ~ priorprob + gender + wbc + (1 | doc)
>>
>> > m3: dx ~ priorprob + gender + wbc + (priorprob | doc)
>>
>> >   Df    AIC    BIC  logLik  Chisq Chi Df Pr(>Chisq)
>>
>> > m2  5 108.07 122.70 -49.034
>>
>> > m3  7 111.52 132.01 -48.759 0.5501      2     0.7595
>>
>> >> (m4
>> > Generalized linear mixed model fit by the Laplace approximation
>>
>> > Formula: dx ~ priorprob + wbc + (1 | doc)
>>
>> >   Data: appy
>>
>> >   AIC   BIC logLik deviance
>>
>> >  110.5 122.2 -51.24    102.5
>>
>> > Random effects:
>>
>> >  Groups Name        Variance Std.Dev.
>>
>> >  doc    (Intercept) 0.63458  0.7966
>>
>> > Number of obs: 138, groups: doc, 23
>>
>> >
>>
>> > Fixed effects:
>>
>> >            Estimate Std. Error z value Pr(>|z|)
>>
>> > (Intercept) -7.62797    1.33760  -5.703 1.18e-08
>>
>> > priorprob    0.04834    0.01229   3.934 8.35e-05
>>
>> > wbc          0.34395    0.06771   5.080 3.77e-07
>>
>> >
>>
>> > Correlation of Fixed Effects:
>>
>> >          (Intr) prrprb
>>
>> > priorprob -0.786
>>
>> > wbc       -0.813  0.357
>>
>> >> anova(m4, m3)
>>
>> > Data: appy
>>
>> > Models:
>>
>> > m4: dx ~ priorprob + wbc + (1 | doc)
>>
>> > m3: dx ~ priorprob + gender + wbc + (priorprob | doc)
>>
>> >   Df    AIC    BIC  logLik  Chisq Chi Df Pr(>Chisq)
>>
>> > m4  4 110.48 122.19 -51.240
>>
>> > m3  7 111.52 132.01 -48.759 4.9639      3     0.1745
>>
>>
>>
>> Sorry, that test should have been
>>
>>
>>
>> > anova(m4, m2)
>>
>> Data: appy
>>
>> Models:
>>
>> m4: dx ~ priorprob + wbc + (1 | doc)
>>
>> m2: dx ~ priorprob + gender + wbc + (1 | doc)
>>
>>   Df    AIC    BIC  logLik  Chisq Chi Df Pr(>Chisq)
>>
>> m4  4 110.48 122.19 -51.240
>>
>> m2  5 108.07 122.70 -49.034 4.4138      1    0.03565
>>
>>
>>
>> > you will see that the random intercept/random slope model produces a
>>
>> > degenerate fit (estimated correlation of the within-doctor random
>>
>> > effects is -1.000), which is not significantly better than the random
>>
>> > intercept fit.  I would therefore use m2. (Because the fixed-effect
>>
>> > for gender had a z-value close to 2 I performed the more reliable
>>
>> > likelihood-ratio test, just to check.)
>>
>> >
>>
>> >
>>
>> >> library(lme4)
>>
>> >> m2
>> >>            family=binomial, data=appy)
>>
>> >>
>>
>> >> m3
>> >>            family=binomial, data=appy)
>>
>> >>
>>
>> >> My ultimate goal is to estimate the probability of appendicitis
>>
>> >> (and a prediction interval), given a specific 'gender', 'wbc' and
>>
>> >> 'priorprob' assigned by a doctor with similar diagnostic ability to
>>
>> >> those who participated in our pilot study. I'm stuck on how to code
>> >> this
>>
>> >> prediction.
>>
>> >
>>
>> > The prediction will be based on the fixed-effects only.  Because it
>>
>> > applies to a doctor not in this study, we assign the random effect for
>>
>> > that doctor to be zero, which is the expected value in the absence of
>>
>> > any information on that doctor.
>>
>> >
>>
>> > The fixef extractor returns the estimates of the fixed-effects
>>
>> > parameters.  For a female patient with median prior probability (55%)
>>
>> > and median white blood cell count (9.45) the estimated linear
>>
>> > predictor (eta = -2.299) corresponds to a probability of 9.1%
>>
>> >
>>
>> >
>>
>> >> fixef(m2)
>>
>> > (Intercept)   priorprob  genderMale         wbc
>>
>> >  -8.2886183   0.0502540   1.1870539   0.3412998
>>
>> >> eta
>> >> eta
>>
>> >          [,1]
>>
>> > [1,] -2.299365
>>
>> >> binomial()\$linkinv(eta)
>>
>> >           [,1]
>>
>> > [1,] 0.09117557
>>
>> >
>>

```

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