In this notebook we introduce Survival Analysis using both R and Python. We will compare programming languages and leverage Plotly's Python and R APIs to convert graphics to interactive Plotly objects.
Plotly is a platform for making interactive graphs with R, Python, MATLAB, and Excel. You can make graphs and analyze data on Plotly’s free public cloud. For collaboration and sensitive data, you can run Plotly on your own servers.
For a more in depth theoretical background in survival analysis, please refer to these sources:
Need help converting Plotly graphs from R or Python?
For this code to run on your machine, you will need several R and Python packages installed.
Running sudo pip install <package_name> from your terminal will install python libraries.
Running install.packages("<library_name>") in your R console will install R packages.
You will also need to register an account with Plotly to receive your API key.
In [1]:
# You can also install packages from within IPython!
# Install Python Packages
!pip install lifelines
!pip install rpy2
!pip install plotly
!pip install pandas
# Install R libraries
%load_ext rpy2.ipython
%R install.packages("devtools")
%R install_github("ropensci/plotly")
%R install.packages("IOsurv")
%R install.packages("ggplot2")
Survival analysis is a set of statistical methods for analyzing the occurrence of event data over time. It is also used to determine the relationship of co-variates to the time-to-events, and accurately compare time-to-event between two or more groups. For example:
The statistical term survival analysis is analogous to reliability theory in engineering, duration analysis in economics, and event history analysis in sociology.
The two key functions in survival analysis are the survival function and the hazard function.
The survival function is conventionally denoted as $S$, the probability that time of death is later than some specified time $t$, defined as:
$$ S(t) = Pr(T>t)$$where $t$ is some duration, $T$ is a random variable denoting the time of death, and $Pr$ is the probability of the event. Generally, $0\leq S(t)\leq1$ and $S(0) = 1$.
The hazard function gives us the probability of "death" in the next instance of time, given we are still "alive".
$$\lambda(t) = \lim_{dt \to 0} \frac{Pr(t \leq T < t + dt}{dt \cdot S(t)}$$The hazard rate describes the relative likelihood of the event occurring at time $t$, and ignores the accumulation of hazard up to time $t$, unlike $S(t)$.
However, we do not actually observe the true survival function of a population; we must use the observed data to estimate it. A popular method to estimate the survival function $S(t)$ is the Kaplan-Meier estimate.
$$S(t)= \prod_{ti < t} \frac{n_i−d_i}{n_i}$$where $d_i$ are the number of death events at time $t$ and $n_i$ is the number of subjects at risk of death at time t.
Censoring is a type of missing data problem common in survival analysis. Other popular comparison methods, such as linear regression and t-tests do not accommodate for censoring. This makes survival analysis attractive for data from randomized clinical studies.
In an ideal scenario, both the birth and death rates of a patient is known, which means the lifetime is known.
Right censoring occurs when the 'death' is unknown, but it is after some known date. e.g. The 'death' occurs after the end of the study, or there was no follow-up with the patient.
Left censoring occurs when the lifetime is known to be less than a certain duration. e.g. Unknown time of initial infection exposure when first meeting with a patient.
In [2]:
# OIserve contains the survival package and sample datasets
%R library(OIsurv)
%R library(devtools)
%R library(plotly)
%R library(ggplot2)
%R library(IRdisplay)
# Authenticate to plotly's api using your account
%R py <- plotly("rmdk", "0sn825k4r8")
# Load python libraries
import numpy as np
import pandas as pd
import lifelines as ll
# Plotting helpers
from IPython.display import HTML
%matplotlib inline
import matplotlib.pyplot as plt
import plotly.plotly as py
import plotly.tools as tls
from plotly.graph_objs import *
from pylab import rcParams
rcParams['figure.figsize']=10, 5
We will be using the tongue dataset from the KMsurv package in R, then convert the data into a pandas dataframe under the same name.
This data frame contains the following columns:
In [3]:
# Load in data
%R data(tongue)
# Pull data into python kernel
%Rpull tongue
# Convert into pandas dataframe
from rpy2.robjects import pandas2ri
tongue = pandas2ri.ri2py_dataframe(tongue)
We can now refer to tongue using both R and python.
In [4]:
%%R
summary(tongue)
In [5]:
tongue.describe()
Out[5]:
We can even operate on R and Python within the same code cell.
In [6]:
%R print(mean(tongue$time))
print tongue['time'].mean()
In R we need to create a Surv object with the Surv() function. Most functions in the survival package apply methods to this object. For right-censored data, we need to pass two arguments to Surv():
In [7]:
%%R
attach(tongue)
tongue.surv <- Surv(time[type==1], delta[type==1])
tongue.surv
The simplest fit estimates a survival object against an intercept. However, the survfit() function has several optional arguments. For example, we can change the confidence interval using conf.int and conf.type.
See help(survfit.formula) for the comprehensive documentation.
In [8]:
%%R
surv.fit <- survfit(tongue.surv~1)
surv.fit
It is often helpful to call the summary() and plot() functions on this object.
In [9]:
%%R
summary(surv.fit)
In [10]:
%%R -h 400
plot(surv.fit, main='Kaplan-Meier estimate with 95% confidence bounds',
xlab='time', ylab='survival function')
Let's convert this plot into an interactive plotly object using plotly and ggplot2.
First, we will use a helper ggplot function written by Edwin Thoen to plot pretty survival distributions in R.
In [11]:
%%R
ggsurv <- function(s, CI = 'def', plot.cens = T, surv.col = 'gg.def',
cens.col = 'red', lty.est = 1, lty.ci = 2,
cens.shape = 3, back.white = F, xlab = 'Time',
ylab = 'Survival', main = ''){
library(ggplot2)
strata <- ifelse(is.null(s$strata) ==T, 1, length(s$strata))
stopifnot(length(surv.col) == 1 | length(surv.col) == strata)
stopifnot(length(lty.est) == 1 | length(lty.est) == strata)
ggsurv.s <- function(s, CI = 'def', plot.cens = T, surv.col = 'gg.def',
cens.col = 'red', lty.est = 1, lty.ci = 2,
cens.shape = 3, back.white = F, xlab = 'Time',
ylab = 'Survival', main = ''){
dat <- data.frame(time = c(0, s$time),
surv = c(1, s$surv),
up = c(1, s$upper),
low = c(1, s$lower),
cens = c(0, s$n.censor))
dat.cens <- subset(dat, cens != 0)
col <- ifelse(surv.col == 'gg.def', 'black', surv.col)
pl <- ggplot(dat, aes(x = time, y = surv)) +
xlab(xlab) + ylab(ylab) + ggtitle(main) +
geom_step(col = col, lty = lty.est)
pl <- if(CI == T | CI == 'def') {
pl + geom_step(aes(y = up), color = col, lty = lty.ci) +
geom_step(aes(y = low), color = col, lty = lty.ci)
} else (pl)
pl <- if(plot.cens == T & length(dat.cens) > 0){
pl + geom_point(data = dat.cens, aes(y = surv), shape = cens.shape,
col = cens.col)
} else if (plot.cens == T & length(dat.cens) == 0){
stop ('There are no censored observations')
} else(pl)
pl <- if(back.white == T) {pl + theme_bw()
} else (pl)
pl
}
ggsurv.m <- function(s, CI = 'def', plot.cens = T, surv.col = 'gg.def',
cens.col = 'red', lty.est = 1, lty.ci = 2,
cens.shape = 3, back.white = F, xlab = 'Time',
ylab = 'Survival', main = '') {
n <- s$strata
groups <- factor(unlist(strsplit(names
(s$strata), '='))[seq(2, 2*strata, by = 2)])
gr.name <- unlist(strsplit(names(s$strata), '='))[1]
gr.df <- vector('list', strata)
ind <- vector('list', strata)
n.ind <- c(0,n); n.ind <- cumsum(n.ind)
for(i in 1:strata) ind[[i]] <- (n.ind[i]+1):n.ind[i+1]
for(i in 1:strata){
gr.df[[i]] <- data.frame(
time = c(0, s$time[ ind[[i]] ]),
surv = c(1, s$surv[ ind[[i]] ]),
up = c(1, s$upper[ ind[[i]] ]),
low = c(1, s$lower[ ind[[i]] ]),
cens = c(0, s$n.censor[ ind[[i]] ]),
group = rep(groups[i], n[i] + 1))
}
dat <- do.call(rbind, gr.df)
dat.cens <- subset(dat, cens != 0)
pl <- ggplot(dat, aes(x = time, y = surv, group = group)) +
xlab(xlab) + ylab(ylab) + ggtitle(main) +
geom_step(aes(col = group, lty = group))
col <- if(length(surv.col == 1)){
scale_colour_manual(name = gr.name, values = rep(surv.col, strata))
} else{
scale_colour_manual(name = gr.name, values = surv.col)
}
pl <- if(surv.col[1] != 'gg.def'){
pl + col
} else {pl + scale_colour_discrete(name = gr.name)}
line <- if(length(lty.est) == 1){
scale_linetype_manual(name = gr.name, values = rep(lty.est, strata))
} else {scale_linetype_manual(name = gr.name, values = lty.est)}
pl <- pl + line
pl <- if(CI == T) {
if(length(surv.col) > 1 && length(lty.est) > 1){
stop('Either surv.col or lty.est should be of length 1 in order
to plot 95% CI with multiple strata')
}else if((length(surv.col) > 1 | surv.col == 'gg.def')[1]){
pl + geom_step(aes(y = up, color = group), lty = lty.ci) +
geom_step(aes(y = low, color = group), lty = lty.ci)
} else{pl + geom_step(aes(y = up, lty = group), col = surv.col) +
geom_step(aes(y = low,lty = group), col = surv.col)}
} else {pl}
pl <- if(plot.cens == T & length(dat.cens) > 0){
pl + geom_point(data = dat.cens, aes(y = surv), shape = cens.shape,
col = cens.col)
} else if (plot.cens == T & length(dat.cens) == 0){
stop ('There are no censored observations')
} else(pl)
pl <- if(back.white == T) {pl + theme_bw()
} else (pl)
pl
}
pl <- if(strata == 1) {ggsurv.s(s, CI , plot.cens, surv.col ,
cens.col, lty.est, lty.ci,
cens.shape, back.white, xlab,
ylab, main)
} else {ggsurv.m(s, CI, plot.cens, surv.col ,
cens.col, lty.est, lty.ci,
cens.shape, back.white, xlab,
ylab, main)}
pl
}
Voila!
In [12]:
%%R -h 400
p <- ggsurv(surv.fit) + theme_bw()
p
We have to use a workaround to render an interactive plotly object by using an iframe in the ipython kernel. This is a bit easier if you are working in an R kernel.
In [13]:
%%R
# Create the iframe HTML
plot.ly <- function(url) {
# Set width and height from options or default square
w <- "750"
h <- "600"
html <- paste("<center><iframe height=\"", h, "\" id=\"igraph\" scrolling=\"no\" seamless=\"seamless\"\n\t\t\t\tsrc=\"",
url, "\" width=\"", w, "\" frameBorder=\"0\"></iframe></center>", sep="")
return(html)
}
In [14]:
%R p <- plot.ly("https://plot.ly/~rmdk/111/survival-vs-time/")
# pass object to python kernel
%R -o p
# Render HTML
HTML(p[0])
Out[14]:
The y axis represents the probability a patient is still alive at time $t$ weeks. We see a steep drop off within the first 100 weeks, and then observe the curve flattening. The dotted lines represent the 95% confidence intervals.
We will now replicate the above steps using python. Above, we have already specified a variable tongues that holds the data in a pandas dataframe.
In [15]:
from lifelines.estimation import KaplanMeierFitter
kmf = KaplanMeierFitter()
The method takes the same parameters as it's R counterpart, a time vector and a vector indicating which observations are observed or censored. The model fitting sequence is similar to the scikit-learn api.
In [16]:
f = tongue.type==1
T = tongue[f]['time']
C = tongue[f]['delta']
kmf.fit(T, event_observed=C)
Out[16]:
To get a plot with the confidence intervals, we simply can call plot() on our kmf object.
In [17]:
kmf.plot(title='Tumor DNA Profile 1')
Out[17]:
Now we can convert this plot to an interactive Plotly object. However, we will have to augment the legend and filled area manually. Once we create a helper function, the process is simple.
Please see the Plotly Python user guide for more insight on how to update plot parameters.
Don't forget you can also easily edit the chart properties using the Plotly GUI interface by clicking the "Play with this data!" link below the chart.
In [19]:
p = kmf.plot(ci_force_lines=True, title='Tumor DNA Profile 1 (95% CI)')
# Collect the plot object
kmf1 = plt.gcf()
def pyplot(fig, ci=True, legend=True):
# Convert mpl fig obj to plotly fig obj, resize to plotly's default
py_fig = tls.mpl_to_plotly(fig, resize=True)
# Add fill property to lower limit line
if ci == True:
style1 = dict(fill='tonexty')
# apply style
py_fig['data'][2].update(style1)
# Change color scheme to black
py_fig['data'].update(dict(line=Line(color='black')))
# change the default line type to 'step'
py_fig['data'].update(dict(line=Line(shape='hv')))
# Delete misplaced legend annotations
py_fig['layout'].pop('annotations', None)
if legend == True:
# Add legend, place it at the top right corner of the plot
py_fig['layout'].update(
showlegend=True,
legend=Legend(
x=1.05,
y=1
)
)
# Send updated figure object to Plotly, show result in notebook
return py.iplot(py_fig)
pyplot(kmf1, legend=False)
Out[19]:
Many times there are different groups contained in a single dataset. These may represent categories such as treatment groups, different species, or different manufacturing techniques. The type variable in the tongues dataset describes a patients DNA profile. Below we define a Kaplan-Meier estimate for each of these groups in R and Python.
In [19]:
%%R
surv.fit2 <- survfit( Surv(time, delta) ~ type)
p <- ggsurv(surv.fit2) +
ggtitle('Lifespans of different tumor DNA profile') + theme_bw()
p
Convert to a Plotly object.
In [20]:
#%R py$ggplotly(plt)
%R p <- plot.ly("https://plot.ly/~rmdk/173/lifespans-of-different-tumor-dna-profile/")
# pass object to python kernel
%R -o p
# Render HTML
HTML(p[0])
Out[20]:
In [21]:
f2 = tongue.type==2
T2 = tongue[f2]['time']
C2 = tongue[f2]['delta']
ax = plt.subplot(111)
kmf.fit(T, event_observed=C, label=['Type 1 DNA'])
kmf.survival_function_.plot(ax=ax)
kmf.fit(T2, event_observed=C2, label=['Type 2 DNA'])
kmf.survival_function_.plot(ax=ax)
plt.title('Lifespans of different tumor DNA profile')
kmf2 = plt.gcf()
Convert to a Plotly object.
In [25]:
pyplot(kmf2, ci=False)
Out[25]:
It looks like DNA Type 2 is potentially more deadly, or more difficult to treat compared to Type 1. However, the difference between these survival curves still does not seem dramatic. It will be useful to perform a statistical test on the different DNA profiles to see if their survival rates are significantly different.
Python's lifelines contains methods in lifelines.statistics, and the R package survival uses a function survdiff(). Both functions return a p-value from a chi-squared distribution.
It turns out these two DNA types do not have significantly different survival rates.
In [31]:
%%R
survdiff(Surv(time, delta) ~ type)
In [32]:
from lifelines.statistics import logrank_test
summary_= logrank_test(T, T2, C, C2, alpha=99)
print summary_
To estimate the hazard function, we compute the cumulative hazard function using the Nelson-Aalen estimator, defined as:
$$\hat{\Lambda} (t) = \sum_{t_i \leq t} \frac{d_i}{n_i}$$where $d_i$ is the number of deaths at time $t_i$ and $n_i$ is the number of susceptible individuals. Both R and Python modules use the same estimator. However, in R we will use the -log of the Fleming and Harrington estimator, which is equivalent to the Nelson-Aalen.
In [33]:
%%R
haz <- Surv(time[type==1], delta[type==1])
haz.fit <- summary(survfit(haz ~ 1), type='fh')
x <- c(haz.fit$time, 250)
y <- c(-log(haz.fit$surv), 1.474)
cum.haz <- data.frame(time=x, cumulative.hazard=y)
p <- ggplot(cum.haz, aes(time, cumulative.hazard)) + geom_step() + theme_bw() +
ggtitle('Nelson-Aalen Estimate')
p
In [23]:
%R p <- plot.ly("https://plot.ly/~rmdk/185/cumulativehazard-vs-time/")
# pass object to python kernel
%R -o p
# Render HTML
HTML(p[0])
Out[23]:
In [26]:
from lifelines.estimation import NelsonAalenFitter
naf = NelsonAalenFitter()
naf.fit(T, event_observed=C)
naf.plot(title='Nelson-Aalen Estimate')
Out[26]:
In [27]:
naf.plot(ci_force_lines=True, title='Nelson-Aalen Estimate')
py_p = plt.gcf()
pyplot(py_p, legend=False)
Out[27]: