Question

In: Statistics and Probability

Refer to Table 2.9. Construct and interpret a 95% confidence interval for the population (a) odds ratio

 Refer to Table 2.9. Construct and interpret a 95% confidence interval for the population (a) odds ratio, (b) difference of proportions, and (c) relative risk between seat-belt use and type of injury.

TABLE 2.9 Data for Problem 2.3 Injury Safety Equipment in Use Fatal 1601 510 Nonfatal None Seat belt 162,527 412,368 Source: Florida Department of Highway Safety and Motor Vehi- cles. PROBLEMS Applications Refer to Table 2.9. Construct and interpret a 95% confidence interval for the population (a) odds ratio, (b) difference of proportions, and (c) relative risk between seat-belt use and type of injury. 3.1


Solutions

Expert Solution

Answer (a) (b) and (c) Reproduce the table in R and display it.

MI <- matrix(c(1601, 510, 162527, 412368), nrow = 2)

dimnames(MI) <- list("Group" = c("None","Seat belt"), "MI" = c("Fatal","Non Fatal"))

MI

MI

Group Fatal Non Fatal

None 1601 162527

Seat belt 510 412368

We can calculate proportions in R using the prop.table function:

prop.table(MI, margin = 1)

MI

Group Fatal Non Fatal

None 0.009754582 0.9902454

Seat belt 0.001235232 0.9987648

Difference of Proportions:The easiest way is to use the prop.test function

prop.test(MI)

2-sample test for equality of proportions with continuity correction

data: MI

X-squared = 2336.1, df = 1, p-value < 2.2e-16

alternative hypothesis: two.sided

95 percent confidence interval:

0.008027691 0.009011009

sample estimates:

prop 1 prop 2

0.009754582 0.001235232

You can find the details of getting answers at following link:

https://data.library.virginia.edu/comparing-proportions-with-relative-risk-and-odds-ratios/

The 95 percent confidence interval is reported to be about (0.008, 0.009).

The actual estimated difference is not provided but we can calculate it easily enough as follows:

p.out <- prop.test(MI)

> p.out$estimate[1] - p.out$estimate[2]

prop 1

0.00851935

Relative Risk: Relative risk is usually defined as the ratio of two “success” proportions. In our case, that’s the “Fatal” group. We can calculate the relative risk in R “by hand” doing something like this:

prop.out <- prop.table(MI, margin = 1)

> prop.out[1,1]/prop.out[1,2]

[1] 0.009850671

Like the difference in proportions, relative risk is just an estimate when working with a sample. Its a good idea to calculate a confidence interval for the relative risk to determine a range of plausible values. Once again we’ll let R do this for us, but this time we’ll use the epitools package, which provides functions for analysis in epidemiology. The function we want is riskratio.

rr.out <- riskratio(MI)

> rr.out$measure

risk ratio with 95% C.I.

Group estimate lower upper

None 1.000000 NA NA

Seat belt 1.008603 1.008107 1.0091

> rr.out <- riskratio(MI, rev="c")

> rr.out$measure

risk ratio with 95% C.I.

Group estimate lower upper

None 1.0000000 NA NA

Seat belt 0.1266309 0.1146384 0.139878

> rr.out <- riskratio(MI, rev="b")

> rr.out$measure

risk ratio with 95% C.I.

Group estimate lower upper

Seat belt 1.000000 NA NA

None 7.896965 7.149089 8.723078

Odds Ratios:

> prop.out[1,1]/prop.out[1,2]

[1] 0.009850671

> prop.out[2,1]/prop.out[2,2]

[1] 0.001236759


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