Regression discontinuity model for TV series

In September, we are usually happy to see our favorite TV series back on air… Or not? Because, admit it, if we are happy to see those characters back, most of the time, we are disappointed, too. So why not look at the data, to confirm this feeling? Nazareno Andrade shared some nice codes to get IMDB ratings in a nice csv file (you can either use the large csv file, or run your own codes)

download.file("https://github.com/nazareno/imdb-series/raw/master/data/series_from_imdb.csv",
destfile="series_from_imdb.csv")
base = read.csv("series_from_imdb.csv")

It is a large dataset, with more than 64,000 episodes of almost 890 TV series,

str(base)
'data.frame':	64018 obs. of  18 variables:
 $ series_name: Factor w/ 889 levels "'Allo 'Allo!",..: 137 137 137 137 137 137 137 137 137 137 ...
 $ episode    : Factor w/ 54090 levels "-30-","¡Viva los muertos!",..: 32314 7446 16 7176 17748 9562 1379 36218 17845 5553 ...
 $ series_ep  : int  1 2 3 4 5 6 7 8 9 10 ...
 $ season     : int  1 1 1 1 1 1 1 2 2 2 ...
 $ season_ep  : int  1 2 3 4 5 6 7 1 2 3 ...
 $ user_rating: num  8.9 8.7 8.7 8.2 8.3 9.2 8.8 8.7 9.2 8.3 ...

Just pick a TV series, for instance Dan Harmon’s Community,

sbase = base[base$series_name=="Community",]

We can plot the evolution of the rating over the 110 episodes.

sbase=sbase[!duplicated(sbase[,c(1,2,4,5)]),]
sbase$series_ep=1:nrow(sbase)

()since there could be some problem with the data (such as duplicates, let us clean it quickly)

plot(sbase$series_ep,sbase$UserRating,xlab=sbase$series_name[1])
idx=c(0,which(diff(sbase$season)!=0),nrow(sbase))
abline(v=idx+.5,lty=2,col=colr[2])
a = unique(sbase$season)
for(u in a){
  ssbase = sbase[sbase$season==u,]
  reg = lm(UserRating~series_ep,data=ssbase)
  lines(ssbase$series_ep,predict(reg),col=colr[3],lwd=2)
}

The vertical lines are here to visualize the seasons. On issue is that the lenght can vary with time. Consider Linwood Boomer’s Malcom in The Middle,

sbase = base[base$series_name=="Malcolm in the Middle",]

or Craig Thomas and Carter Bays’s How I Met Your Mother,

sbase = base[base$series_name=="How I Met Your Mother",]

On those two, the evolution is rather stable. Look at AMC’s The Walking Dead,

sbase = base[base$series_name=="The Walking Dead",]

Now, look at Howard Gordon and Alex Gansa’s Homeland,

sbase = base[base$series_name=="Homeland",]

There is an issue here with the last episode of season4, “Long Time Coming“, that has a very poor rating. If we remove that point, we get the thin line. Note that the regression line is always increasing. For Michael Hirst’s Vickings, we have

sbase = base[base$series_name=="Vicking",]

If we look more carefully on the previous graph, for five seasons (out of six), we have a positive slope. Well, to be honest, it is not significantly positive most of the time, but still. Out of 80 shows, and a total of 583 seasons, the slope is postive 75% of the time (433) and negative 25% of the time (150).

BASE = NULL
L80 = unique(base$series_name)
for(j in 1:length(L)){
sbase=base[base$series_name==L[j],]
sbase=sbase[!duplicated(sbase[,c(1,2,4,5)]),]
sbase=sbase[sbase$season>0,]
sbase$series_ep=1:nrow(sbase)
a=unique(sbase$season)
a=a[!is.na(a)]
for(u in a){
  ssbase=sbase[sbase$season==u,]
  reg=lm(UserRating~series_ep,data=ssbase)
  pente = NA
  if((!is.na(coefficients(reg)[2]))&(!is.na((summary(reg)$coefficients[2,4])))){
  if((summary(reg)$coefficients[2,4]<.05)&(coefficients(reg)[2]>0)) pente="positive"
  if((summary(reg)$coefficients[2,4]<.05)&(coefficients(reg)[2]<0)) pente="negative" sdf=data.frame(nom=sbase$series_name[1],season=u,slope=coefficients(reg)[2],inf=confint(reg)[2,1],sup=confint(reg)[2,2],signe=pente) BASE=rbind(BASE,sdf)} }} str(BASE) 'data.frame': 583 obs. of 6 variables: $ nom : Factor w/ 80 levels "Friends","Game of Thrones",..: 1 1 1 1 1 1 1 1 1 1 ... 
 
mean(BASE$slope>0)
[1] 0.7427101
table(BASE$signe)
negative positive 
      15      144

Most of the time, the slope is not significant. To be more specific, 72% of the time, the slope is not significant. But when it is, 90% of the time, it is positive (144 seasons). Let us look at other TV series, for instance Joel Surnow and Robert Cochran’s 24,

sbase = base[base$series_name=="24",]

Álex Pina’s La Casa de Papel,

sbase = base[base$series_name=="La Casa de Papel",]

Steven Knight’s Peaky Blinders,

sbase = base[base$series_name=="Peaky Blinders",]

or David Simon’s The Wire,

sbase = base[base$series_name=="The Wire",]

The slope is increasing over almost all seasons. But a major drawback is that when we get back to our show, for a new season, we usually get disapointed. More specifically, we can quantify the difference in red below

that can be estimated using

sbase12 = sbase[sbase$season%in%c(a[ij],a[ij+1]),]
seuil = sbase12$series_ep[which(diff(sbase12$season)!=0)]+.5
s = function(x) (x-seuil)*(x>seuil)
reg = lm(UserRating~series_ep+s(series_ep)+I(series_ep>seuil),data=sbase12)

Here we have

summary(reg)
Coefficients:
                         Estimate Std. Error t value  Pr(|t|)    
(Intercept)               8.45000    0.16338  51.719    2e-16 ***
series_ep                 0.10000    0.03235   3.091 0.008598 ** 
s(series_ep)              0.02000    0.04218   0.474 0.643291    
I(series_ep)TRUE.        -1.01778    0.20486  -4.968 0.000257 ***

so the drop of 1 point (out of 10) cannot be claimed as being significant. That is the idea of regression discontinuity.

If we loop again over all our series, we have 485 pairs of consecutive seasons. As expected, in 75% of the casse, from season t-1 to season t, we observe a negative rupture. As previously, in 70% of the cases, it is not significat (with linear models before and after), and when it is significant, it is negative in 96% of the cases ! But an alternative can be to use nonparametric models, on both sides.

To illustrate, consider David Benioff and D. B. Weiss’s Game of Thrones,

sbase = base[base$series_name=="Game of Thrones",]

but let us remove the last season (no spoiler here, but clearly not worst watching)

Consider for instance the drop between season 1 and season 2,

library(rdd)
sbase12=sbase[sbase$season%in%c(1,2),]
lmr=RDestimate(UserRating~series_ep,data=sbase12,cutpoint=mean(range(sbase12$series_ep)))
plot(lmr)

This is very consistent with what we observed with our linear regressions actually,

seuil=10.5
s = function(x) (x-seuil)*(x>seuil)
reg = lm(UserRating~series_ep+s(series_ep)+I(series_ep>seuil),data=sbase12)
summary(reg)
 
Coefficients:
                         Estimate Std. Error t value  Pr(|t|)    
(Intercept)               8.70000    0.15458  56.281    2e-16 ***
series_ep                 0.07273    0.02491   2.919  0.01003 *  
s(series_ep)              0.01455    0.03523   0.413  0.68520    
I(series_ep)TRUE         -0.94000    0.20316  -4.627  0.00028 ***

Here, the drop of one point is significant…

So, your favorite show had an outstanding finale ? and you can’t wait to watch the new season… Well, statistically, it’s very likely that you will be disapointed by the first episode of the forthcoming season…


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