continuous vs categorical logistic regression for marks and admission

Question

I have a list of marks scored by students in Science (X, between 0 to 100%) and whether they went to college to or not (Y).
High marks in science showed a higher concentration of college admits and low marks had the second best hit rate (students went for Arts degree, etc). Scores in the intermediate range had a lower hit rate. Most students have lower scores in Science.

I divided my sample into

5 bins: 0-10, 10-20, 20-80, 80-90, 90-100.

Found Chi-Sq to be significant.

I transformed these bins into dummy categorical variables and then calculated LR coeff which were significant for 80-90, 90-100(+ives), and 0-10(-ive). I concluded that when scores are higher, the odds of getting admissions are high. When scores are low, it is unlikely that the student would go to college.

Q1. Should I instead use marks as continuous variable? My main hypothesis is: highest marks result in college admissions, while the lowest marks imply no college admission.

Q2. The data was linear before. However, what if the data was shown in the chart below, do you think categorizing it made more sense? Event hit rates are higher for a particular range and then trail off. Categorizing it in bins helps as I just to have to analyze the performance of that bin (main focus of the study). May be using dummy variables removes the affect of non-linear relationships?

Answer 1

Yes, you should always use a continuous variable as a continuous variable. There is little reason to ever categorize a continuous variable into bins. If you like, you may want to read my answer here: How to choose between ANOVA and ANCOVA in a designed experiment, which discusses this in greater detail.

Answer 2

Here's a demonstration in r of how @gung's answer applies in logistic regression.

First, a function for simulating data and printing logistic regression summaries using continuous and binned versions of the predictor. Outcome probability is a function of the uniformly distributed predictor. Change n to suit your sample size and runif to suit your theoretical distribution if you like.

thesims=function(n=30,seed=1){set.seed(seed);marks=runif(n,0,100)
admitted=rbinom(n,1,marks/100);binned=c();for(i in 1:n)
{if(marks[i]<10){binned[i]=1}else if(marks[i]<20){binned[i]=2}else
if(marks[i]<80){binned[i]=3}else if(marks[i]<90){binned[i]=4}else{binned[i]=5}}
print(summary(glm(admitted~marks,family='binomial'))$coef[2,3:4])
summary(glm(admitted~binned,family='binomial'))$coef[2,3:4]}

With the default $n=30$ and set.seed(1), the continuous predictor is clearly significant $(z=2.77,p=.006)$, but the binned predictor is very far from it $(z=.01,p=.994)$. This is a pretty extreme difference, but all other seeds will still produce a weaker result for the binned predictor.

Here's a way of visualizing the difference in the same simulated data as above. With continuous data:

require(ggplot2);ggplot(data.frame(marks,admitted),aes(x=marks,y=admitted))+
scale_x_continuous('Marks')+scale_y_continuous('Probability of admission',lim=0:1)+
geom_point(position=position_jitter(width=0,height=.04))+
stat_smooth(method='glm',family='binomial')

Things look as they should...but with binned data:

ggplot(data.frame(admitted,binned),aes(x=binned,y=admitted))+
scale_x_continuous('Binned marks')+scale_y_continuous('Probability of admission',lim=0:1)+
geom_point(position=position_jitter(width=0,height=.04))+
stat_smooth(method='glm',family='binomial')

Fitted probabilities of zero or one are bad news. Again, this is an extreme example, but you will generally find wider confidence bands with binned data. This is to be expected because you're effectively throwing away useful information by binning.