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I'm stuck and have real problems what to interpret from my current results. Maybe you can help me out? Thanks!

Lets say... I'm investigating on the influence of health factors on dying.

Dependent variable: death yes/no after 10 years time

Independent variables: ml wine per day, cigarettes per day, gramm of fruits or vegetables per day, minutes of excercises per day etc. ...

I'm doing a logistic regression, since I have a binary dependent variable: 

model.binomial <- glm(dv_death ~ 
                            wine +
                            cigarettes +
                            fruits +
                            excercise,
                          data = complete_dataset, family = binomial(link = logit))

I have a questions and I might just lost sight, but..:

If I put in the model all variables (wine, cigarettes, fruits and excercise), all of them are significant. If I only use the independent variable "wine", it is not significant (same goes for all other variables: I have to admit, I also have a correlation between wine + cigarettes of 0.55, but VIFs and Eigenscores are alright). However... when I look at the wine and death data specifically by using: 

ggplot(complete_dataset, aes(x=complete_dataset$wine, y=complete_dataset$death))+ geom_point(size=2, alpha=0.4)+
  stat_smooth(method="loess", colour="blue", size=1.5)+
  xlab("Wine")+
  ylab("Death (yes = 1)")+
  theme_bw()

... I get this kind of plot: see R Code for variable setup

For me this seems to be a u-shape correlation: Too little wine and too much wine reduces your probability of dying, so either be an alcoholic or do not every take a sip...

However, the variable is not significant. Can I test for a u shape in a logistic regression? Or am I on the completely wrong track?

(Don't worry - this is a made up example so pour yourself a drink..)

Update due to the comments:

I added an independent variable squared wine to the model.

Full model without winesquared: wine is not significant.

Full model with wine + winesquared: both are significant - wine (p<0.001), wine squared (p<0.01)

Single model without winesquared: wine is not significant

Single model winesquared only: winesquared is not significant

"Single" model with wine and winesquared: both are significant - both at p<0.1

Update thanks to @Roland: GAM Model: 

model.binomial.gam <- mgcv::gam(dv_death ~ 
                                  s(wine) +
                                  cigarettes +
                                  fruits +
                                  excercise,
                                data = complete_dataset, family = binomial(link = logit), select = TRUE)
summary(model.binomial.gam)
                             Estimate Std. Error z value       Pr(>|z|)    
(Intercept)                -0.9217701  0.3225723  -2.858       0.004269 ** 
cigarettes                 -8.0936235  3.5047369  -2.309       0.020925 *  
fruits                      0.3063182  0.0838298   3.654       0.000258 ***
excercise                   0.1126536  0.0273186   4.124 0.000037284368 ***

Approximate significance of smooth terms
          edf Ref.df Chi.sq p-value    
s(wine) 2.478      9  16.55 0.00014 ***
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  • $\begingroup$ It may help you to read this: How to use boxplots to find the point where values are more likely to come from different conditions? $\endgroup$
    – gung - Reinstate Monica
    Commented Dec 5, 2018 at 17:40
  • 2
    $\begingroup$ Is your death indicator reverse coded (is 1 actually alive and 0 dead?). Usually, the best survival is for intermediate alcohol consumption. $\endgroup$
    – AdamO
    Commented Dec 5, 2018 at 18:02
  • 1
    $\begingroup$ Have you checked whether each independent variable is linear in the log of odds of death? Note that you're assuming that when you run the logistic regression, and by assuming that wine is linear in the log of odds of death you're already assuming a certain non-linear relationship between wine and death. Start by understanding what relationship between wine and death you've assumed to hold and whether that assumption is satisfied. $\endgroup$
    – ColorStatistics
    Commented Dec 5, 2018 at 18:12
  • $\begingroup$ I would fit a GAM to account for possible (and likely) nonlinear relationships on the log-odds scale. Using package mgcv would also combine this with shrinkage and take care of variable selection. $\endgroup$
    – Roland
    Commented Dec 6, 2018 at 7:15
  • $\begingroup$ @ColorStatistics I'm not too sure if I understand you correctly, thank you and sorry. I have reasoning and evidence for both aussumptions (wine & death have a linear vs. a non-linear relationship). With a binary dv I thought I can only run a logistic model. $\endgroup$
    – GreenPirate
    Commented Dec 6, 2018 at 8:57

1 Answer 1

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Yes. Include a quadratic term for wine units consumed. The statistical significance of this term may indicate the presence of an inflection point, at which point the linear trend pivots. It may also indicate an "acceleration" effect, where sequentially higher or lower doses may have escalating trends with the outcome risk. Accompanied by the LOESS smooth which you have already produced, it's compelling evidence in favor of the formerly noted "U-shaped" trend with alcohol consumption.

If one includes an intercept term, a linear term, and a quadratic term, then the resulting model fits a quadratic trend in the log-odds whose apex location and value optimally predict the trend in the data. If one omits the linear term, the quadratic form is constrained to achieve it's extrema at the origin (no wine consumed) which will not reflect the noted reversal of trend in the exposure duration.

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  • 1
    $\begingroup$ If this were a linear regression then the observed u shape between wine and death may justify inclusion of a quadratic term. However, given that this is a logistic regression and the dependent variable is the log of the odd of death, why would a quadratic relationship between wine and death justify the exploration of a quadratic relationship between the log of odds of death and wine? In my mind, there is a disconnect here. $\endgroup$
    – ColorStatistics
    Commented Dec 5, 2018 at 18:32
  • $\begingroup$ @ColorStatistics I think I see your point. I am borrowing on my prior knowledge in this area. You are right that, in general, for GLMs a non-linear trend in the expected response doesn't necessarily mean there's a non-linear trend in the linear predictor. HOWEVER, the inverse-logit is monotonic, and yet the trend reverses direction, so an inflection point is present. $\endgroup$
    – AdamO
    Commented Dec 5, 2018 at 18:45
  • $\begingroup$ I can't be confident that the inclusion of the quadratic term is justified based on the plot of death vs. wine. In my mind, that is not the immediately relevant plot to analyze in order to decide whether or not the inclusion of a quadratic term or splining is justified. On the other hand, if we saw a quadratic relationship between the log of odds and wine, then I'd be confident that we're justified to include a quadratic term. $\endgroup$
    – ColorStatistics
    Commented Dec 5, 2018 at 18:57
  • $\begingroup$ Thank you both. I added a wine squared (so "glm(dv_death ~ wine + winesq + cigarettes + (...)) and both wine and winesq are significant in this model. However, when I use a limited model (either dv_death ~ wine OR dv_death ~ winesq), none of them is significant. $\endgroup$
    – GreenPirate
    Commented Dec 6, 2018 at 9:01
  • $\begingroup$ @GreenPirate think about the model. You know a straight line doesn't fit the trend well. Neither does a quadratic whose apex is constrained to be at 0. Always plot the predicted trend. Practice safe stats. $\endgroup$
    – AdamO
    Commented Dec 6, 2018 at 17:15

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