Risk Ratio 和 Odds Ratio

Data Science Day 15

Risk Ratio

Last time, e give a SAS example of Risk Difference to test if to groups are experiencing the same proportion of a certain event. In order to understand the ic better, e ill go over Risk Ratio.

Definition:

Risk Ratio or Relative Risk (RR) is the probability that an event ours in a group 1 relative to the probability that the same event ours in group 2.
Note: Without Loss of Generality, In clinical trials, e take a group to be actual treatment group and the other group is the placebo group.

Formula:

Risk Ratio (Relative Risk)= P(event in group1)/P(event in group 2)

Interpretation:
RR ≈ 1 ⇒ The proportion of events are similar in group 1 and group 2.
RR » 1 ⇒ Increased probability of events among those in group 1 pared to group 2.
RR « 1 ⇒ Decreased probability of events among those in group 1 pared to group 2.
Note: This is a general interpretation, e can explain in different scenarios.

Example:

#import Library
import numpy as np
import pandas as pd
import statsmodels.api as sm
import pylab as pl
import matplotlib.pyplot as plt
import seaborn as sns

#Read in Arthritis data
df = sm.datasets.get_rdataset("Arthritis", "vcd").data
#Create Ne variable Censor:
#Censor=0 means no-improvement
#Censor=1 means some or big improvement 
df["Censor"]=np.here(df["Improved"]=="None", "0", "1")
#table for Treatment and Censor
tab=pd.crosstab(df["Treatment"], df["Censor"])
tab1=sm.stats.Table(tab)
print(tab1.table_orig)
table= np.asarray([[29,14],[13,28]])
t22=sm.stats.Table2x2(table)
print(t22.summary())

#Output
Censor   0   1
Treatment 
Placebo 29  14
Treated 13  28


 Estimate   SE   LCB   UCB   p-value
--------------------------------------------------
Odds ratio 4.4621.785 11.154   0.001
Log odds ratio 1.495 0.467 0.579  2.412   0.001
Risk ratio 2.1270.579  2.412   0.003
Log risk ratio 0.755 0.253 0.260  1.250   0.003
--------------------------------------------------

#Visulatization

%matplotlib inline
pd.crosstab(df.Treatment, df.Censor).plot(kind="bar")
plt.title("Treatment vs Placebo")
plt.xlabel("Treatment Type for Arthritis")
plt.ylabel("Censor Status")

Summary
Censor=0, Arthritis problem remains
Censor=1, Arthritis problem improved
Odds Ratio
Odds Ratio= 4.462, this implies in Placebo Group has 4.5 times more likely to remain the Arthritis problem pared to the Treatment Group.

Risk Ratio
Risk Ratio=2.127 means in Placebo Group the probability of patients’ Arthritis problem remains increased pared to the Patients in Treatment Group.

Connections:

It is very easy to mix the concept for Odds Ratio and Risk Ratio.
Odds Ratio represents the odds of an event in group 1 pared to the event of odds in group 2, here odds means the event over non-event.

$OddsinPlacebo=\frac{29}{14}$Odds in Placebo=1429​

$OddsinTreatment=\frac{13}{28}$Odds in Treatment=2813​

$RiskRatio=\frac{2\mathrm{.}0714}{0\mathrm{.}464}=4\mathrm{.}46$Risk Ratio=0.4642.0714​=4.46

Risk Ratio means the probability of an event ourring in group 1 pared to the probability of the same event ourring in group 2.

$P(censor=0inPlacebo)=\frac{29}{43}$P(censor=0 in Placebo)=4329​

$P(censor=0inTreatment)=\frac{13}{41}$P(censor=0 in Treatment)=4113​

$RiskRatio=\frac{0\mathrm{.}674}{0\mathrm{.}317}=2\mathrm{.}127$Risk Ratio=0.3170.674​=2.127

Conclusion:
Risk ratio pares the probability of the ourrence of the same event in to groups. In addition, e can use the Risk Difference to check if the to groups have the same Risk Ratio.

Happy Studying!