Statsbook

Risk Example

Deep sea diving and osteonecrosis of the femoral head

In this fictional example, a study-group of 450 subjects over the age of 50 were identified. All subjects were asked if they had been deep-sea diving below a level of 10 metres for at least once in their lives. Furthermore, all subjects had a pelvic radiograph taken that has been scrutinised by a radiologist for signs of osteonecrosis (example of a cross-sectional study).

In our example, we assume that 150 subjects have been deep-sea diving and 300 have not. Of the 150 subjects who have been deep-sea diving, 15 had signs of osteonecrosis on the radiographs whilst 135 did not. In the group of 300 non deep-sea divers, only 2 people had signs of osteonecrosis on the radiographs.

We can now construct a two by two table as follows:

OsteonecrosisNo Osteonecrosis
Deep Sea Diving15135
No Deep Sea Diving2298

Absolute Risk Deep Sea Diving

AR(exposure) = \(\frac{15}{15+135}\) = 0.1

Absolute Risk No Deep Sea Diving

AR(no exposure) = \(\frac{2}{2+298} \approx 0.0067\)

Relative Risk

Risk of exposed individuals getting the disease as compared to non exposed individuals:

RR = \(\frac{15 \over (15+135)}{2 \over (2+298)}\) = 15

Odds Ratio

The odds ratio is the ratio of the odds of a disease occurring among exposed individuals to that of it occurring in unexposed individuals:

OR = \(\frac{a \cdot d}{b \cdot c}\) = \(\frac{15 \cdot 298}{2 \cdot 135} \approx 16.6 \)

Or in the R console:

Use the epiR1 library:

library(epiR)

Create a matrix for use by epiR:

mat<-matrix(c(15,2,135,298),ncol=2)

Perform the calculations:

epi.2by2(mat)
             Outcome +    Outcome -   Total                Inc risk *
Exposed +           15          135     150      10.00 (5.71 to 15.96)
Exposed -            2          298     300       0.67 (0.08 to 2.39)
Total               17          433     450       3.78 (2.22 to 5.98)
Point estimates and 95% CIs:
-------------------------------------------------------------------
Inc risk ratio                                 15.00 (3.48, 64.74)
Inc odds ratio                                 16.56 (3.73, 73.41)
Attrib risk in the exposed *                   9.33 (4.44, 14.22)
Attrib fraction in the exposed (%)            93.33 (71.23, 98.46)
Attrib risk in the population *                3.11 (1.12, 5.10)
Attrib fraction in the population (%)         82.35 (35.59, 95.17)
-------------------------------------------------------------------
Uncorrected chi2 test that OR = 1: chi2(1) = 23.964 Pr>chi2 = <0.001
Fisher exact test that OR = 1: Pr>chi2 = <0.001
 Wald confidence limits
 CI: confidence interval
 * Outcomes per 100 population units

The package epiR2 should be installed.

> summary(epi.tests(mat))
   statistic          est        lower      upper
1         ap  0.333333333 0.2898878816  0.3789871
2         tp  0.037777778 0.0221582162  0.0597982
3         se  0.882352941 0.6355908379  0.9854207
4         sp  0.688221709 0.6422577774  0.7315911
5    diag.ac  0.695555556 0.6507412520  0.7377768
6    diag.or 16.555555556 3.7335839181 73.4110779
7       nndx  1.752619048 1.3946772799  3.5990822
8     youden  0.570574650 0.2778486153  0.7170118
9     pv.pos  0.100000000 0.0570574348  0.1595679
10    pv.neg  0.993333333 0.9761265024  0.9991916
11    lr.pos  2.830065359 2.2644605406  3.5369439
12    lr.neg  0.170943545 0.0464306911  0.6293616
13    p.rout  0.666666667 0.6210129037  0.7101121
14     p.rin  0.333333333 0.2898878816  0.3789871
15    p.tpdn  0.311778291 0.2684089311  0.3577422
16    p.tndp  0.117647059 0.0145793168  0.3644092
17    p.dntp  0.900000000 0.8404321149  0.9429426
18    p.dptn  0.006666667 0.0008083864  0.0238735