Learning Outcomes
By the end of this session, students will be able to:
 Define and calculate prevalence, risk, odds and rates
 Calculate and interpret risk ratios, odds ratios, rate ratios and their 95% confidence intervals
You can download a copy of the slides here: Video C1.1a
Video C1.1a – Definition of Risk, Odds and Rates (5 minutes)
C1.1a PRACTICAL: Stata
Risks
Calculate the overall risk of death by using the command:
tab death, m
Examine the pattern of risk of death in relation to increasing age. We generally list the outcome variable second when using the ‘tab’ command. For example, for age and death, this command would be:
tab age_grp death, row
Note: we cannot use the ‘tab’ command with continuous variables.
Question C1.1a.i: What does your table suggest about the risk of death overall?
Question C1.1a.ii: What does your table suggest about the risk death across levels of age?
C1.1a. Answer
Answer C1.1a.i:
For this cohort the overall risk of death is 1526/4327=0.3527 or 35.3% over the years of follow up.
Answer C1.1a.ii:
The risk of death appears to increase as age increases.
The ‘cs’ command
A good command in Stata to explore risk and odds ratios is the command ‘cs’. Type the following into Stata to read about this command:
help cs
This command is used to produce epidemiological tables with point estimates and CI for risk differences, risk ratios, odds ratios and attributable/prevented fractions.
The syntax for the command is:
cs var_case var_exposed
which means that we place the outcome variable first, followed by the exposure variable. With this command, your exposure can only be binary. If you have a categorical exposure, you will have to recode it into a set of binary variables.
NOTE: Stata often requires variables to be coded such that the baseline category equals 0. That is the case with the cs command. If you try to use the command in future and find, for example, that your variable has been coded as 1 and 2, then you will need to recode it to 0 and 1. Other commands similarly prefer 0/1 coding, so as good practice you should always recode binary variables this way in Stata (other statistical programs may be different).
Using the ‘cs’ command to calculate a risk ratio
We want to obtain the risk ratio of mortality for individuals that are current smokers compared to nonsmokers. We type the command:
cs death currsmoker
Use the help file for the cs command to find out how to obtain an odds ratio.
Question C1.1b.i: What is the risk ratio and 95% CI of death for individuals that are current smokers compared with individuals that are nonsmokers?
Question C1.1b.ii: What is the odds ratio for death in individuals that are current smokers vs nonsmokers? State and interpret the 95% confidence interval.
C1.1b. Answers
Answer C1.1b.i: The risk ratio for death is 1.16 times higher for current smokers compared to nonsmokers. The 95% confidence interval around the risk ratio is 1.041.30.
Answer C1.1b.ii:
The command is: cs death currsmoker, or
The odds ratio for death in current smokers compared to nonsmokers is 1.27 with a 95% confidence interval of 1.051.52. Or, you could say: “The odds of death in current smokers are 1.27 times those of nonsmokers.”
The 95% CI states that there is a 95% chance that the interval containing the true population odds ratio lies between 1.051.52. Since this interval does not cross 1, it indicates that there are significantly higher odds death for smokers compared to nonsmokers.
The ‘ir’ command for rates
Now let’s have a look at the rate ratio for death in smokers vs nonsmokers. This will require the ‘ir’ command. Use Stata’s help file to explore this command:
help ir
The basic set up follows the command ir with the outcome variable (called the ‘var_case’) variable here and then the exposure variable (called ‘var_exposed’). Then you specify the followup time variable, or ‘var_time’. This command only works with binary exposure variables. You will learn about other commands for examining rates in the Module C2: Survival Data.
Type the following command to produce the rate and rate ratio:
ir death currsmoker fu_years
Question C1.1b.iii: What is the rate of death in current smokers compared to nonsmokers (i.e. what is the rate ratio of ‘currsmoker’)? Is the rate of death in current smokers significantly different from that in nonsmokers?
C1.1b.iii. Answer
The rate ratio of death in current vs nonsmokers is 1.18. This means the rate of death in current smokers is 1.18 times that of nonsmokers. Alternatively, you could say the rate of death in smokers was 18% higher than in nonsmokers. The 95% CI is 1.021.37. This CI does not cross the null value so we know the rate of death in current smokers is significantly higher than that of nonsmokers.
C1.1 PRACTICAL: SPSS
Open the Whitehall FoSSA data set in SPSS
When calculating risk ratios, SPSS automatically places the lowest number first (in this case 0 or ‘No’) and calculates how likely people with X characteristic are NOT to have the outcome of interest (e.g. disease/death), and there is no option to change the reference category. You could conduct the analysis and calculate 1/ans in order to get the ratios for how likely people with X characteristic are TO have the outcome of interest, or you can recode the variable so that Yes= 1 and No =2. You learnt how to recode a variable in practical A1.3b. The rest of this practical will show output tables for the recoded variables, so for each variable mentioned, you will need to recode it to Yes= 1 and No=2 before conducting the analysis, or your risk ratios will be in inverse of those shown here.
Firstly, calculate the overall risk of death.
Select
Analyze >> Descriptive Statistics >> Frequencies
Move ‘death’ to the variables box, make sure that ‘Display frequency tables’ is ticked, then press OK.
Now we are going to examine the pattern of risk of death in relation to increasing age.
Select
Analyze >> Descriptive Statistics >> Crosstabs
Move ‘death’ and age group into rows and columns boxes. It does not really matter which variable goes in which box, the analysis will be the same, it just affects how your output table looks.
Click on ‘cells’ and select what you want to display in the table. You at the very least want to show observed counts, and you can also select to display percentages. Click continue, and then OK to run.
Question C1.1a.i: What does your table suggest about the risk of death overall?
Question C1.1a.ii: What does your table suggest about the risk death across levels of age?
Answer
Answer C1.1a.i:
For this cohort the overall risk of death is 1526/4327=0.3527 or 35.3% over the years of follow up.
Answer C1.1a.ii:
The risk of death appears to increase as age increases.
Risk Ratio
We want to obtain the odds ratio of mortality for individuals that are current smokers compared to nonsmokers.
Select
Analyze >> Descriptive Statistics >> Crosstabs
Move ‘current smoker’ and ‘death’ variables into the rows and columns boxes as before, then click on ‘statistics’ on the righthand side, the tick the box next to ‘risk’. (NB: SPSS calculates an odds ratio AND a risk ratio when you select ‘risk’)
Question C1.1b.i: What is the risk ratio and 95% CI of death for individuals that are current smokers compared with individuals that are nonsmokers?
Question C1.1b.ii: What is the odds ratio for death in individuals that are current smokers vs nonsmokers? State and interpret the 95% confidence interval.
Answer
Your output tables will look like the below. You may have more percentage values in the cross tabs table if you have selected to display percentages for rows, columns and overall. Here I have selected just to display percentages for rows.
Answer C1.1b.i:
You read the risk ratio from the second row down in the first column. This is the ratio of the risk of death (where death=Yes) for the group where current smoker = Yes compared to the groups where current smoker = No.
The risk ratio for death is 1.16 times higher for current smokers compared to nonsmokers. The 95% confidence interval around the risk ratio is 1.041.30. (Figures round to 2dp, as is common for reporting).
Answer C1.1b.ii:
The odds ratio for death in current smokers compared to nonsmokers is 1.27 with a 95% confidence interval of 1.051.52. Or, you could say: “The odds of death in current smokers are 1.27 times those of nonsmokers.”
The 95% CI states that there is a 95% chance that the interval containing the true population odds ratio lies between 1.051.52. Since this interval does not cross 1, it indicates that there are significantly higher odds death for smokers compared to nonsmokers.
NB: SPSS does not offer an option to calculate rate ratios.
C1.1 PRACTICAL: R
Risks
Calculate the overall risk of death by using a combination of ‘table()’ and ‘prop.table()’ commands:
table(df$death)
prop.table(table(df$death))
Examine the pattern of risk of death in relation to increasing age. The margins option in the ‘prop.table()’ command needs to be set to “2” because we want to report the proportions per group. For example, for age and death, this command would be:
table(df$age_grp, df$death)
prop.table(table(df$age_grp, df$death), margins=2)
 Question C1.1a.i: What does your table suggest about the risk of death overall?
 Question C1.1a.ii: What does your table suggest about the risk death across levels of age?
Answer
Answer C1.1a.i:
> table(df$death)
0 1
2761 1505
> prop.table(table(d$death))
0 1
0.6472105 0.3527895
For this cohort the overall risk of death is 1505/(2761+1505) = 0.3527 or 35.3% over the years of follow up.
Answer C1.1a.ii:
The risk of death appears to increase as age increases.
> table(df$age_grp, df$death)
0 1
1 698 135
2 1277 498
3 602 464
4 184 408
> prob.table(table(df$age_grp, df$death),2)
0 1
1 0.25280695 0.08970100
2 0.46251358 0.33089701
3 0.21803694 0.30830565
The epitools R package
A good R package to calculate basic epidemiologic analysis is the epitools R package, which includes commands for the design of twoway and multiway contingency tables and epidemiologic measures, such as risk ratios and odds ratios. You can find more details on the commands included in the epitools R package here.
Install and load the epitools R package with the following commands:
install.packages(“epitools”)
library(epitools)
The epitab() command
A good command in R to explore risk and odds ratios is the epitab() command from the epitools R package. This command is used to produce epidemiological tables with point estimates and CI for risk ratios, and odds ratios.
The syntax for the command is:
epitab(x, y, method)
where x is the vector of the exposure, y is the vector of the outcome and method can be either “oddsratio” to calculate an odds ratio or “riskratio” to calculate a risk ratio. This command works with either binary or categorical exposures. For categorical exposures, R uses by default the lowest value of the exposure as the reference category.
Using the epitab() command to calculate a risk ratio
We want to obtain the risk ratio of mortality for individuals that are current smokers compared to nonsmokers. We type the command:
epitab(x = df$currsmoker, y = df$death, method = “riskratio”)
Use the help file for the epitab() command to find out how to obtain an odds ratio.

 Question C1.1b.i: What is the risk ratio and 95% CI of death for individuals that are current smokers compared with individuals that are nonsmokers?

 Question C1.1b.ii: What is the odds ratio for death in individuals that are current smokers vs nonsmokers? State and interpret the 95% confidence interval.
Answer
Answer C1.1b.i:
> epitab(x = df$currsmoker, y = df$death, method = “riskratio”)
$tab
Outcome
Predictor 0 p0 1 p1 riskratio lower upper p.value
0 2473 0.6549258 1303 0.3450742 1.000000 NA NA NA
1 327 0.6000000 218 0.4000000 1.159171 1.036537 1.296314 0.01263365
$measure
[1] “wald”
$conf.level
[1] 0.95
$pvalue
[1] “fisher.exact”
The risk ratio for death is 1.16 times higher for current smokers compared to nonsmokers. The 95% confidence interval around the risk ratio is 1.041.30.
Answer C1.1b.ii:
The command is:
> epitab(x = df$currsmoker, y = df$death, method = “oddsratio”)
$tab
Outcome
Predictor 0 p0 1 p1 oddsratio lower upper p.value
0 2473 0.8832143 1303 0.8566732 1.000000 NA NA NA
1 327 0.1167857 218 0.1433268 1.265285 1.052595 1.520953 0.01263365
$measure
[1] “wald”
$conf.level
[1] 0.95
$pvalue
[1] “fisher.exact”
The odds ratio for death in current smokers compared to nonsmokers is 1.27 with a 95% confidence interval of 1.051.52. Or, you could say: “The odds of death in current smokers are 1.27 times those of nonsmokers.”
The 95% CI states that there is a 95% chance that the interval containing the true population odds ratio lies between 1.051.52. Since this interval does not cross 1, it indicates that there are significantly higher odds death for smokers compared to nonsmokers.
The ‘ir’ command for rates
Now let’s have a look at the rate ratio for death in smokers vs nonsmokers. This will require the ‘rateratio’ method in the ‘epitab()’ command.
Type the following command to produce the rate and rate ratio:
> death_no_smokers < sum(df$death[df$currsmoker==”0″])
> death_no_smokers
[1] 1290
> death_smokers < sum(df$death[df$currsmoker==”1″])
> death_smokers
[1] 215
> fup_no_smoker < sum(df$fu_years[df$currsmoker==”0″])
> fup_no_smoker
[1] 25460.25
> fup_smoker < sum(df$fu_years[df$currsmoker==”1″])
> fup_smoker
[1] 3593.209
> epitab(x=c(death_no_smokers, death_smokers), y= c(fu_no_smoker, fu_smoker),
method = c(“rateratio”))
Outcome
Predictor c(death_no_smokers, death_smokers) c(fup_no_smoker, fup_smoker)
Exposed1 1290 25460.250
Exposed2 215 3593.209
Predictor rateratio lower upper p.value
Exposed1 1.000000 NA NA NA
Exposed2 1.180943 1.022177 1.364369 0.02633476
$measure
[1] “wald”
$conf.level
[1] 0.95
$pvalue
[1] “midp.exact”

 Question C1.1c: What is the rate of death in current smokers compared to nonsmokers (i.e. what is the rate ratio of ‘currsmoker’)? Is the rate of death in current smokers significantly different from that in nonsmokers?
C1.1c. Answer
Answer C1.1c: The rate ratio of death in current vs nonsmokers is 1.18. This means the rate of death in current smokers is 1.18 times that of nonsmokers. Alternatively, you could say the rate of death in smokers was 18% higher than in nonsmokers. The 95% CI is 1.021.36. This CI does not cross the null value so we know the rate of death in current smokers is significantly higher than that of nonsmokers.