Statistics review 11: Assessing risk

R code accompanying paper

Key learning points

• Relative Risk
• Odds Ratio
• Measuring the impact of exposure to a risk factor
• Measures of the success of a treatment

suppressPackageStartupMessages(library(tidyverse))

options(repr.plot.width=4, repr.plot.height=3)

Data

That study investigated the association between surfactant protein B and acute respiratory distress syndrome (ARDS). Patients were classified according to their thymine/cytosine (C/T) gene coding, and patients with the C allele present (genotype CC or CT) were compared with those with genotype TT

ARDS <- c(11,1)
NoARDS <- c(208, 182)
df <- data.frame(row.names=c("CC/CT", "TT"), ARDS=ARDS, NoARDS=NoARDS)
df

	ARDS	NoARDS
CC/CT	11	208
TT	1	182

Relative risk

df1 <- df %>% mutate(Total = ARDS+NoARDS) %>% mutate(risk = ARDS/Total)
df1

ARDS	NoARDS	Total	risk
11	208	219	0.050228311
1	182	183	0.005464481

rr <- df1$risk[1]/df1$risk[2]
round(rr, 2)

9.19

rr <- function(tbl) {
    a <- tbl[1,1]
    b <- tbl[1,2]
    c <- tbl[2,1]
    d <- tbl[2,2]
    round((a/(a+b))/(c/(c+d)), 2)
    }

rr_ci <- function(tbl, alpha=0.05) {
    a <- tbl[1,1]
    b <- tbl[1,2]
    c <- tbl[2,1]
    d <- tbl[2,2]
    log.se <- sqrt(1/a - 1/(a+b) + 1/c - 1/(c+d))
    log.rr <- log((a/(a+b))/(c/(c+d)))
    k <- qnorm(1-alpha/2)
    log.ci <- c(log.rr - k*log.se, log.rr + k*log.se)
    round(exp(log.ci), 2)
}

rr(df)

9.19

rr_ci(df)

1. 1.2
2. 70.53

Odds ratio

df2 <- df %>% mutate(odds = ARDS/NoARDS)
df2

ARDS	NoARDS	odds
11	208	0.052884615
1	182	0.005494505

or <- df2$odds[1]/df2$odds[2]
or

9.625

or <- function(tbl) {
    a <- tbl[1,1]
    b <- tbl[1,2]
    c <- tbl[2,1]
    d <- tbl[2,2]
    round((a/b)/(c/d), 2)
}

or_ci <- function(tbl, alpha=0.05) {
    a <- tbl[1,1]
    b <- tbl[1,2]
    c <- tbl[2,1]
    d <- tbl[2,2]
    log.se <- sqrt(1/a + 1/b + 1/c - 1/d)
    log.rr <- log((a/b)/(c/d))
    k <- qnorm(1-alpha/2)
    log.ci <- c(log.rr - k*log.se, log.rr + k*log.se)
    round(exp(log.ci), 2)
}

or(df)

9.62

or_ci(df)

1. 1.24
2. 74.5

Advantages of odds ratio

• Can be estimated in case-control study
• OR is a symmetric ratio in that the OR for the disease given the risk factor is the same as the OR for the risk factor given the disease.
• form part of the output when carrying out logistic regression

Attributable risk

The proportion of cases in a population that could be prevented if the risk factor were to be eliminated. The AR is the difference between the actual number of cases in a sample and the number of cases that would be expected if exposure to the risk factor were eliminated, expressed as a proportion of the former.

ar <- function(tbl) {
    a <- tbl[1,1]
    b <- tbl[1,2]
    c <- tbl[2,1]
    d <- tbl[2,2]
    n <- a + b + c + d
    ar <- ((a+c) - (n*c)/(c+d))/(a+c)
    round(100*ar, 2)
    }

ar_ci <- function(tbl, alpha=0.05) {
    a <- tbl[1,1]
    b <- tbl[1,2]
    c <- tbl[2,1]
    d <- tbl[2,2]
    n <- a + b + c + d
    k <- qnorm(1-alpha/2)
    u <- (k*(a+c)*(c+d))/(a*d-b*c) * sqrt(((a*d*(n-c) + c^2*b)/(n*c*(a+c)*(c+d))))

    hi <- ((a*d - b*c)*exp(u))/(n*c + (a*d-b*c)*exp(u))
    lo <- ((a*d - b*c)*exp(-u))/(n*c + (a*d-b*c)*exp(-u))
    round(100*c(lo, hi), 2)
}

ar(df)

81.69

ar_ci(df)

1. 31.16
2. 97.78

Risk measurements in clinical trials

df3 <- data.frame(survived=c(79, 60), died=c(38, 59),
                  row.names=c("early", "standard"))
df3

	survived	died
early	79	38
standard	60	59

rr(df3)

1.34

or(df3)

2.04

ar(df3)

14.39

Risk difference

arr <- function(tbl) {
    a <- tbl[1,1]
    b <- tbl[1,2]
    c <- tbl[2,1]
    d <- tbl[2,2]

    r <- (d/(c+d) - b/(a+b))
    round(100*r, 2)
    }

arr(df3)

17.1

Number needed to treat

nnt <- function(df) {
    round(100/arr(df), 0)
}

nnt(df3)

6

Exercise

1. Write a function to calculate the confidence intervals of the absolute risk reduction (ARR) given a \(2 \times 2\) table of outcomes. What is the 90% CI for the data given in df3?

2 Write a function to calculate the confidence intervals for the number needed to treat (NNT) 2×2 table of outcomes. What is the 90% CI for the data given in df3?