Solved – How to explain smoothing functions in the logistic regression model

logisticsmoothing

I fitted a logistic regression model with some smoothing functions, and the software made beautiful plots for them. Here is one example:

enter image description here

My main concern is that there is no reference level in the continuous variable, so converting the estimates by an exponential function to odds ratio seems not quite right for smoothing functions. Hence, I am wondering how to explain smoothing functions appropriately?

Here is a part of output of the above smoothing function:

age estimate  95% CI
18  0.802419 (0.708936, 0.90305)
19  0.744268 (0.666983, 0.823497)
20  0.688122 (0.625178, 0.755262)
21  0.634746 (0.57693, 0.69252)
22  0.584887 (0.530877, 0.638431)
23  0.538855 (0.485444, 0.589725)
24  0.496467 (0.44487, 0.550205)
25  0.457521 (0.405356, 0.511471)
26  0.421809 (0.370094, 0.47534)
27  0.389193 (0.340622, 0.441149)

Any suggestion will be appreciated!

Best Answer

The reference age is the age that your function cuts the x-axis (around 67, maybe, by eyeballing your graph). Whatever, let's say that it is 67. The odds ratio is the odds of the event (probability of the event divided by one minus the probability of the event), given the person's age divided by the odds of the event given the age=67: \begin{equation} \frac{\frac{P\{E|age\}}{1-P\{E|age\}}}{\frac{P\{E|67\}}{1-P\{E|67\}}}= \frac{exp(f(age))}{exp(f(67))}=exp(f(age)) \end{equation} So, the odds ratio for an 18-year-old relative to a 67-year-old would be $exp(0.8)=2.22$, or the 18-year-old has odds of the event 222% as high as the 67-year-old.

If you don't want the reference age to be 67, then you can make it anything you like via subtraction. If you want the reference age to be 18, then just subtract 0.8 from the value of f(age) for every value of age. Then the odds ratio for a 27-year-old, compared to (now) an 18-year-old is $exp(0.39-0.80)=0.66$, or the 27-year-old has odds 66% as high as the 18-year-old to have the event.

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Solved – How to estimate Relative Risks in Multivariate Binary Logistic Regression Models, instead of Odds Ratios

Here is more or less the replication of the SAS example mentioned in the comments to the question.

library("sas7bdat")
eyestudy =read.sas7bdat("eyestudy.sas7bdat")
sapply(1:ncol(eyestudy),function(z)summary(eyestudy[,z]))
for(i in c(2:3))eyestudy[,i]<-as.factor(eyestudy[,i])

tabfq=with(eyestudy, table(carrot, lenses))
(tabfqm=addmargins(tabfq))
prop.table(tabfq, 1)
#OR =  (32/17)/(21/30)  = 2.69
#RR =   (32/49)/(21/51)  = 1.59
(OR=tabfqm[1,2]/tabfqm[1,1]/(tabfqm[2,2]/tabfqm[2,1]))
(RR=tabfqm[1,2]/tabfqm[1,3]/(tabfqm[2,2]/tabfqm[2,3]))

#logit
(ml1<-glm(lenses~carrot, data=eyestudy,family =binomial(link = "logit")))
summary(ml1)
exp(-coefficients(ml1)) #OR
exp(-cbind(coef(ml1), confint(ml1)))  
(ml10 <- glm(lenses~1, data=eyestudy,family =binomial(link = "logit")))
anova(ml10, ml1, test="Chisq")

#log
(ml2<-glm(lenses~carrot, data=eyestudy,family =binomial(link = "log")))
summary(ml2)
exp(-coefficients(ml2)) #RR
exp(-cbind(coef(ml2), confint(ml2)))  

#poisson
(ml3<-glm(lenses~carrot, data=eyestudy,family =poisson(link = "log")))
summary(ml3)
exp(-coefficients(ml3)) #RR
exp(-cbind(coef(ml3), confint(ml3)))  

#poisson 2
(ml4<-glm(lenses~carrot+gender+latitude, data=eyestudy,family =poisson(link = "log")))
summary(ml4)
exp(-coefficients(ml4)) #RR
exp(-cbind(coef(ml4), confint(ml4)))

Solved – Logistic regression with multi-level categorical predictors

You do seem to have found an error in the source you cite, but you have to be careful in general. Note this sentence from that source:

Previous use has been split into two components (according to whatever contrasts were specified for this variable).

Exactly what is being displayed in the output of a model for a multi-level categorical variable depends on how the contrasts for that variable were coded, which isn't shown in the excerpts you quote in your question. In this case it seems that the R default treatment contrast was used. In that case, you are correct that the coefficient (and statistical tests) for each level's coefficient is relative to the reference level for that variable.

Several other types of contrasts are possible, however, as explained for example on this page. One might be able to devise a contrast that provides, for each of 2 levels, a comparison of that level to the average of the other levels, as the source claims for this example. It just doesn't appear that such a contrast was actually used in this particular case.

Best Answer

Related Solutions

Solved – How to estimate Relative Risks in Multivariate Binary Logistic Regression Models, instead of Odds Ratios

Solved – Logistic regression with multi-level categorical predictors

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