[1] http://en.wikipedia.org/wiki/Kaplan%E2%80%93Meier_estimator
---
# http://blogs.wsj.com/venturecapital/2014/11/20/techstars-graduates-success-rates-what-the-numbers-show/
# data:application/octet-stream;charset=utf-8,Year%2C2007%2C2008%2C2009%2C2010%2C2011%2C2012%2C2013%2C2014%0AActive%2C2%2C2%2C7%2C15%2C36%2C68%2C121%2C120%0AFailed%2C3%2C4%2C5%2C11%2C9%2C10%2C5%2C0%0AAcquired%2C5%2C4%2C7%2C5%2C14%2C15%2C4%2C1
rates <- read.csv(stdin(),header=TRUE)
Year,2007,2008,2009,2010,2011,2012,2013,2014
Active,2,2,7,15,36,68,121,120
Failed,3,4,5,11,9,10,5,0
Acquired,5,4,7,5,14,15,4,1
library(reshape2)
rates2 <- melt(rates)
colnames(rates2) <- c("Status", "Year", "Count")
rates2$Year <- as.integer(substring(as.character(rates2$Year), 2))
startups <- NULL
for (i in 1:nrow(rates2)) { startups <- rbind(data.frame(Status = rep(rates2[i,]$Status, rates2[i,]$Count), Year=rep(rates2[i,]$Year, rates2[i,]$Count)), startups) }
library(survival)
# define startups which have been acquired or failed as dead
startups$Alive <- startups$Status == "Active"
# define startups in 2014 as 0 years old, etc
startups$Age <- 2014 - startups$Year
sf <- survfit(Surv(Age, Alive, type="right") ~ 1, data=startups); summary(sf)
# time n.risk n.event survival std.err lower 95% CI upper 95% CI
# 0 473 120 0.74630 0.020007 0.708099 0.78656
# 1 352 121 0.48976 0.023007 0.446680 0.53699
# 2 222 68 0.33974 0.022007 0.299236 0.38573
# 3 129 36 0.24493 0.020778 0.207412 0.28924
# 4 70 15 0.19245 0.020269 0.156552 0.23657
# 5 39 7 0.15790 0.020407 0.122571 0.20342
# 6 20 2 0.14211 0.021202 0.106083 0.19038
# 7 10 2 0.11369 0.024715 0.074248 0.17409
plot(sf)
# https://i.imgur.com/76B7AxN.png
Since there are no covariates or anything in the provided data, we just get a curve. It looks like a pretty steady decline per year, with half of them 'dying' in the first year. The curve flattens out towards the end, which suggests that there might be some sort of time-varying hazard going on (possibly the accelerator has gotten less picky and the earliest startups were best?).
Of course, there's a bigger problem: one might argue that treating 'failed' & 'acquired' the same is painting a misleadingly negative picture - surely acquisitions represent successes? But we can't mark acquired as 'alive' because then they'll never die and then the graph is just of explicit failure... So let's switch to a form of survival analysis which has multiple kinds of deaths, 'competing risks survival analysis' (https://en.wikipedia.org/wiki/Relative_survival); we'll treat acquisition as one form of death, failure another kind, and any startups which are 'active' are considered censored (we don't yet know what their fate will be):
--- library(cmprsk) sc <- cuminc(startups$Age, startups$Status, cencode="Active"); sc # Estimates and Variances: # $est # 1 2 3 4 5 6 # 1 Acquired 0.0134537767 0.0791545678 0.172799416 0.223444079 0.321616811 0.397350061 # 1 Failed 0.0141745147 0.0579750421 0.118175302 0.229593560 0.299716940 0.375450190 # # $var # 1 2 3 4 5 6 # 1 Acquired 3.62406271e-05 0.000301240943 0.000805565675 0.00118663857 0.00205862260 0.00275961348 # 1 Failed 3.97263451e-05 0.000220342717 0.000570181024 0.00140171794 0.00200661959 0.00272841167 plot(sc, lty=1, color=c(3,2)) # https://i.imgur.com/Tafk9B0.png ---
If we only consider 'failure' as a bad outcome, then this is more helpful than the first survival curve, as we can read off the risk with time easily from the graph or table.
This graph is the opposite of before, we're now seeing the cumulative risk over time for each kind of death - eg by 7 years after founding, a startup has roughly 40% chance of having died at some point, roughly 40% chance of having been bought at some point, and just 20% to still be active; given another few years, I think actives would drop to ~0% and it'd be roughly 50/50 - and a half-failure-rate is close to the summary in OP:
> Techstars failure rates, at least so far, are a little lower than the industry average, according to estimates from the National Venture Capital Association, which says that overall about 40% of venture-backed companies fail, 40% produce moderate returns, and 20% produce high returns.