A common task in epidemiology is to quantify the strength of association between exposures (‘risk factors’) and disease outcomes. In this context the term ‘exposure’ is taken to mean a variable whose association with the outcome is to be estimated.

Exposures can be harmful, beneficial or both harmful and beneficial (e.g., if an immunisable disease is circulating, exposure to immunising agents helps most recipients but may harm those who experience adverse reactions). The term ‘outcome’ is used to describe all the possible results that may arise from exposure to a causal factor or from preventive or therapeutic interventions (Porta, Greenland, and Last 2014). In human and animal health an ‘outcome-positive’ individual is an individual who has experienced a given disease of interest.

In this vignette we outline describe how `epiR`

can be
used to compute the various measures of association used in epidemiology
notably the risk ratio, odds ratio, attributable risk in the exposed,
attributable fraction in the exposed, attributable risk in the
population and attributable fraction in the population. Examples are
provided to demonstrate how the package can be used to deal with
exposure-outcome data presented in various formats.

This vignette has been written assuming the reader routinely formats
their 2 \(\times\) 2 table data with
the outcome status as columns and exposure status as rows. If this is
not the case the argument `outcome = "as.columns"`

(the
default) can be changed to `outcome = "as.rows"`

.

The EpiToolbox app for iPhone
and Android
devices provides access to many of the measures of association functions
in `epiR`

using a smart phone.

Use the interactive 2 by 2 table page to develop an intuitive understanding of the calculation methods for the incidence risk ratio, odds ratio, attributable risk, attributable fraction, population attributable risk and population attributable fraction.

**Measures of association strength**

*The incidence risk ratio*

Consider a study where study subjects are monitored for the presence of disease over a defined follow-up period. At the start of the follow-up period study subjects are classified according to exposure to a hypothesised risk factor. If the exposure and outcome are binary variables (yes or no) we can present the counts of subjects in each of the four exposure-outcome categories in a 2 \(\times\) 2 table.

Outcome + | Outcome - | Total | |
---|---|---|---|

Exp + | a | b | a + b |

Exp - | c | c | c + d |

Total | a + c | b + c | a + b + c + d |

When our data are in this format we can calculate the incidence risk of the outcome in those that were exposed \(R_E+\), the incidence risk in those that were not exposed \(R_{E-}\) and finally the incidence risk in the total study population \(R_{T}\):

Outcome + | Outcome - | Total | Risk | |
---|---|---|---|---|

a | b | a + b | RE+ = a / (a + b) | |

Exp + | c | c | c + d | RE- = c / (c + d) |

Exp - | a + c | b + c | a + b + c + d | RT = (a + c) / (a + b + c + d) |

In Figure 1 we present the outcome incidence risk in the exposed, unexposed and (exposed and unexposed) groups as a bar chart. The outcome incidence risk ratio equals the outcome incidence risk in the exposed divided by the outcome incidence risk in the unexposed:

The outcome incidence risk ratio provides an estimate of how many times more likely exposed individuals are to experience the outcome of interest, compared with non-exposed individuals.

If the outcome incidence risk ratio equals 1, then the risk of the outcome in both the exposed and non-exposed groups are the same. If the outcome incidence risk ratio is greater than 1, then exposure increases the outcome incidence risk with greater departures from 1 indicative of a stronger association If the outcome incidence risk ratio is less than 1, exposure reduces the outcome incidence risk and exposure is said to be protective.

Tip: The interactive 2 by 2 table page and the glossary in the EpiToolbox app contains an animated version of Figure 1.

*The odds ratio — cohort studies*

In a cohort study definition of exposure status (exposure-positive, exposure-negative) comes first. Subjects are then followed over time to determine their outcome status (outcome-positive, outcome-negative). The outcome odds in the exposed and unexposed populations are calculated as follows:

Outcome + | Outcome - | Total | Odds | |
---|---|---|---|---|

Exp + | a | b | a + b | OE+ = a / b |

Exp - | c | d | c + d | OE- = c / d |

Total | a + c | b + d | a + b + c + d | OT = (a + c) / (b + d) |

The outcome odds ratio for a cohort study is then the outcome odds in the exposed divided by the outcome odds in the unexposed.

*The odds ratio — case-control studies*

In a case-control study outcome status (outcome-positive,
outcome-negative) is defined first. The history provided by each study
subject then provides information about exposure status
(exposure-positive, exposure-negative). For case-control studies,
instead of talking about the outcome odds in the exposed and unexposed
groups (as we did when we were working with data from a cohort study) we
talk about the odds of *exposure odds* in the case
(outcome-positive) and control (outcome-negative) groups. To emphasise
this difference we list outcome status (case, control) as rows and
exposure status (exposure-positive, exposure-negative) in our 2 \(\times\) 2 table. A handy way to remember
this is to think about the temporal order of events in a study. In a
case-control study outcome status is defined first: exposure status is
defined second. Because outcome status comes first it is listed as the
first column of your 2 \(\times\) 2
table.

Exp + | Exp - | Total | Odds | |
---|---|---|---|---|

Outcome + (case) | a | b | a + b | OD+ = a / b |

Outcome - (control) | c | d | c + d | OD- = c / d |

Total | a + c | b + d | a + b + c + d | OT = (a + c) / (b + d) |

The odds ratio for a case-control study is defined as exposure odds in the cases (\(O_{D+}\)) divided by the exposure odds in the controls (\(O_{D-}\)).

Note that the numeric estimate of the odds ratio is exactly the same as that calculated for a cohort study. The expression of the result is the only thing that differs. Key points:

In a

**cohort study**we talk about the outcome odds being \(x\) times greater (or less) in**the exposed**, compared with the unexposed.In a

**case-control study**we talk about the exposure odds being \(x\) times greater (or less) in**cases**, compared with controls.

**Measures of effect in the exposed**

*The attributable risk in the exposed*

The attributable risk is defined as the increase or decrease in the incidence risk of the outcome in the exposed that is attributable to exposure (Figure 2). Attributable risk (unlike the incidence risk ratio) provides a measure of the absolute frequency of the outcome that is associated with exposure.

A useful way of expressing attributable risk in a clinical setting is in terms of the number needed to treat, NNT. NNT equals the inverse of the attributable risk. Depending on the outcome of interest we use different labels for NNT.

When dealing with an outcome that is ‘desirable’ (e.g., treatment success) we call NNT the number needed to treat for benefit, NNTB. NNTB equals the number of subjects who would have to be exposed to result in a single (desirable) outcome. When dealing with an outcome that is ‘undesirable’ (e.g., death) we call NNT the number needed to treat for harm, NNTH. NNTH equals the number of subjects who would have to be exposed to result in a single (undesirable) outcome.

*The attributable fraction in the
exposed*

The attributable fraction in the exposed is the proportion of outcome-positive subjects in the exposed group that is due to exposure (Figure 3).

**Measures of effect in the population**

*The attributable risk in the population*

The population attributable risk is the increase or decrease in incidence risk of the outcome in the study population that is attributable to exposure (Figure 4).

*The attributable fraction in the
population*

The population attributable fraction (also known as the aetiologic fraction) is the proportion of outcome-positive subjects in the study population that is due to the exposure (Figure 5).

On the condition that the exposure of interest is a cause of the disease outcome, the population attributable fraction represents the proportional reduction in average disease risk over a specified period of time that would be achieved by eliminating the exposure of interest while the distribution of other risk factors in the population remained unchanged. For this reason, PAFs are particularly useful to guide policy makers when planning public health interventions. If you’re going to use PAFs as a means for informing policy, make sure that: (1) the exposure of interest is causally related to the outcome; and (2) the exposure of interest is something amenable to intervention.

**Theory to practice: Calculating measures of association using
R**

*Direct entry of 2 by 2 table contingency table cell
frequencies*

Firstly, a 2 \(\times\) 2 table can be created by listing the contingency table cell frequencies in vector format. Take the following example.

A cross sectional study investigating the relationship between dry cat food (DCF) and feline lower urinary tract disease (FLUTD) was conducted (Willeberg 1977). Counts of individuals in each group were as follows. DCF-exposed cats (outcome-positive, outcome-negative) 13, 2163. Non DCF-exposed cats (outcome-positive, outcome-negative) 5, 3349. We can enter these data directly into R as a vector of length four. Check that your counts have been entered in the correct order by viewing the data as a matrix.

```
dat.v01 <- c(13,2163,5,3349); dat.v01
#> [1] 13 2163 5 3349
# View the data in the usual 2 by 2 table format:
matrix(dat.v01, nrow = 2, byrow = TRUE)
#> [,1] [,2]
#> [1,] 13 2163
#> [2,] 5 3349
```

Calculate the outcome prevalence ratio, odds ratio, attributable
prevalence in the exposed, attributable fraction in the exposed,
attributable prevalence in the population and the attributable fraction
in the population using function `epi.2by2`

. Note that we use
the term prevalence risk ratio and prevalence odds ratio (instead of
incidence risk ratio and incidence odds ratio) because we’re dealing
with data from a cross-sectional study — the frequency of disease is
expressed as a prevalence, not an incidence.

```
library(epiR)
epi.2by2(dat = dat.v01, method = "cross.sectional", conf.level = 0.95, units = 100,
interpret = FALSE, outcome = "as.columns")
#> Outcome + Outcome - Total Prev risk *
#> Exposed + 13 2163 2176 0.60 (0.32 to 1.02)
#> Exposed - 5 3349 3354 0.15 (0.05 to 0.35)
#> Total 18 5512 5530 0.33 (0.19 to 0.51)
#>
#> Point estimates and 95% CIs:
#> -------------------------------------------------------------------
#> Prev risk ratio 4.01 (1.43, 11.23)
#> Prev odds ratio 4.03 (1.43, 11.31)
#> Attrib prev in the exposed * 0.45 (0.10, 0.80)
#> Attrib fraction in the exposed (%) 75.05 (30.11, 91.09)
#> Attrib prev in the population * 0.18 (-0.02, 0.38)
#> Attrib fraction in the population (%) 54.20 (3.61, 78.24)
#> -------------------------------------------------------------------
#> Uncorrected chi2 test that OR = 1: chi2(1) = 8.177 Pr>chi2 = 0.004
#> Fisher exact test that OR = 1: Pr>chi2 = 0.006
#> Wald confidence limits
#> CI: confidence interval
#> * Outcomes per 100 population units
```

The prevalence of FLUTD in DCF exposed cats was 4.0 (95% CI 1.4 to 11) times greater than the prevalence of FLUTD in non-DCF exposed cats. In DCF exposed cats, 75% of FLUTD was attributable to DCF (95% CI 30% to 91%). Fifty-four percent of FLUTD cases in this cat population were attributable to DCF (95% CI 4% to 78%).

Need a hand to get the correct wording to explain each of the listed
measures of association and measures of effect? Set
`interpret = TRUE`

in `epi.2by2`

:

```
epi.2by2(dat = dat.v01, method = "cross.sectional", conf.level = 0.95, units = 100,
interpret = TRUE, outcome = "as.columns")
#> Outcome + Outcome - Total Prev risk *
#> Exposed + 13 2163 2176 0.60 (0.32 to 1.02)
#> Exposed - 5 3349 3354 0.15 (0.05 to 0.35)
#> Total 18 5512 5530 0.33 (0.19 to 0.51)
#>
#> Point estimates and 95% CIs:
#> -------------------------------------------------------------------
#> Prev risk ratio 4.01 (1.43, 11.23)
#> Prev odds ratio 4.03 (1.43, 11.31)
#> Attrib prev in the exposed * 0.45 (0.10, 0.80)
#> Attrib fraction in the exposed (%) 75.05 (30.11, 91.09)
#> Attrib prev in the population * 0.18 (-0.02, 0.38)
#> Attrib fraction in the population (%) 54.20 (3.61, 78.24)
#> -------------------------------------------------------------------
#> Uncorrected chi2 test that OR = 1: chi2(1) = 8.177 Pr>chi2 = 0.004
#> Fisher exact test that OR = 1: Pr>chi2 = 0.006
#> Wald confidence limits
#> CI: confidence interval
#> * Outcomes per 100 population units
#>
#> Measures of association strength:
#> The outcome prevalence risk among the exposed was 4.01 (95% CI 1.43 to 11.23) times the outcome prevalence among the unexposed: exposure increased outcome prevalence risk among the exposed.
#>
#> The outcome prevalence odds among the exposed was 4.03 (95% CI 1.43 to 11.31) times the outcome prevalence odds among the unexposed: exposure increased outcome prevalence odds among the exposed.
#>
#> Measures of effect in the exposed:
#> Exposure changed the outcome prevalence risk in the exposed by 0.45 (95% CI 0.1 to 0.8) per 100 population units. 75.2% of outcomes in the exposed were attributable to exposure (95% CI 25.7% to 93.1%).
#>
#> Number needed to treat for benefit (NNTB) and harm (NNTH):
#> Exposure increased the outcome incidence risk in the exposed. The number needed to treat (i.e., expose) to increase the outcome frequency by one was 223 (95% CI 125 to 1008).
#>
#> Measures of effect in the population:
#> Exposure changed the outcome prevalence risk in the population by 0.18 (95% CI -0.02 to 0.38) per 100 population units. 54.3% of outcomes in the population were attributable to exposure (95% CI 32.5% to 75%).
```

*Data frame with one row per observation*

Here we provide examples where you have exposure status and outcome
status listed for each member of your study population. There are two
options for contingency table preparation in this situation: (1) using
base R’s table function; or (2) using the `tidyverse`

group
of packages.

For this example we use the low infant birth weight data presented by
Hosmer and Lemeshow (2000)
and available in the `MASS`

package in R. The
`birthwt`

data frame has 189 rows and 10 columns. The data
were collected at Baystate Medical Center, Springfield, Massachusetts
USA during 1986.

**Two by two table preparation using the table
function in base R**

```
library(MASS)
# Load and view the data:
dat.df02 <- birthwt; head(dat.df02)
#> low age lwt race smoke ptl ht ui ftv bwt
#> 85 0 19 182 2 0 0 0 1 0 2523
#> 86 0 33 155 3 0 0 0 0 3 2551
#> 87 0 20 105 1 1 0 0 0 1 2557
#> 88 0 21 108 1 1 0 0 1 2 2594
#> 89 0 18 107 1 1 0 0 1 0 2600
#> 91 0 21 124 3 0 0 0 0 0 2622
```

Each row of this data set represents data for one mother. We’re
interested in the association between `smoke`

(the mother’s
smoking status during pregnancy) and `low`

(delivery of a
baby less than 2.5 kg bodyweight).

Its important that the table you present to `epi.2by2`

is
in the correct format: Outcome positives in the first column, outcome
negatives in the second column, exposure positives in the first row and
exposure negatives in the second row. If we run the `table`

function on the `bwt`

data the output table is in the
incorrect format - outcome negatives are listed first in the columns of
the table and exposure negatives are listed first in the rows:

```
dat.tab02 <- table(dat.df02$smoke, dat.df02$low, dnn = c("Smoke", "Low BW")); dat.tab02
#> Low BW
#> Smoke 0 1
#> 0 86 29
#> 1 44 30
```

There are two ways to fix this problem. The quick fix is to simply ask R to switch the order of the rows and columns in the output table:

```
dat.tab02 <- table(dat.df02$smoke, dat.df02$low, dnn = c("Smoke", "Low BW")); dat.tab02
#> Low BW
#> Smoke 0 1
#> 0 86 29
#> 1 44 30
dat.tab02 <- dat.tab02[2:1,2:1]; dat.tab02
#> Low BW
#> Smoke 1 0
#> 1 30 44
#> 0 29 86
```

The second approach is to set the exposure variable and the outcome
variable as a factor and to define the levels of each factor using
`levels = c(1,0)`

:

```
# Variables low, smoke and race as factors. Put an 'f' in front of the variable names to remind you that they're factors:
dat.df02$flow <- factor(dat.df02$low, levels = c(1,0))
dat.df02$fsmoke <- factor(dat.df02$smoke, levels = c(1,0))
dat.df02$frace <- factor(dat.df02$race, levels = c(1,2,3))
dat.tab02 <- table(dat.df02$fsmoke, dat.df02$flow, dnn = c("Smoke", "Low BW")); dat.tab02
#> Low BW
#> Smoke 1 0
#> 1 30 44
#> 0 29 86
```

Now compute the odds ratio for smoking and delivery of a low birth weight baby:

```
dat.epi02 <- epi.2by2(dat = dat.tab02, method = "cohort.count", conf.level = 0.95,
units = 100, interpret = FALSE, outcome = "as.columns")
dat.epi02
#> Outcome + Outcome - Total Inc risk *
#> Exposed + 30 44 74 40.54 (29.27 to 52.59)
#> Exposed - 29 86 115 25.22 (17.58 to 34.17)
#> Total 59 130 189 31.22 (24.69 to 38.35)
#>
#> Point estimates and 95% CIs:
#> -------------------------------------------------------------------
#> Inc risk ratio 1.61 (1.06, 2.44)
#> Inc odds ratio 2.02 (1.08, 3.78)
#> Attrib risk in the exposed * 15.32 (1.61, 29.04)
#> Attrib fraction in the exposed (%) 37.80 (5.47, 59.07)
#> Attrib risk in the population * 6.00 (-4.33, 16.33)
#> Attrib fraction in the population (%) 19.22 (-0.21, 34.88)
#> -------------------------------------------------------------------
#> Uncorrected chi2 test that OR = 1: chi2(1) = 4.924 Pr>chi2 = 0.026
#> Fisher exact test that OR = 1: Pr>chi2 = 0.036
#> Wald confidence limits
#> CI: confidence interval
#> * Outcomes per 100 population units
```

The odds of having a low birth weight child for smokers is 2.0 (95%
CI 1.1 to 3.8) times greater than the odds of having a low birth weight
child for non-smokers. All of the calculated measures of association
computed by `epi.2by2`

can be listed using:

```
names(dat.epi02$massoc.detail)
#> [1] "RR.strata.wald" "RR.strata.taylor" "RR.strata.score"
#> [4] "OR.strata.wald" "OR.strata.cfield" "OR.strata.score"
#> [7] "OR.strata.mle" "ARisk.strata.wald" "ARisk.strata.score"
#> [10] "NNT.strata.wald" "NNT.strata.score" "AFRisk.strata.wald"
#> [13] "PARisk.strata.wald" "PARisk.strata.piri" "PAFRisk.strata.wald"
#> [16] "chi2.strata.uncor" "chi2.strata.yates" "chi2.strata.fisher"
#> [19] "chi2.correction"
```

Compare the Wald confidence intervals for the odds ratio with the score confidence intervals:

```
dat.epi02$massoc.detail$OR.strata.wald
#> est lower upper
#> 1 2.021944 1.08066 3.783112
# Wald confidence intervals: 2.0 (95% CI 1.1 to 3.8)
dat.epi02$massoc.detail$OR.strata.score
#> est lower upper
#> 1 2.021944 1.084385 3.773885
# Score confidence intervals: 2.0 (95% CI 1.1 to 3.8)
```

**Two by two table preparation using
tidyverse**

The `dplyr`

and `tidyr`

packages (available
within the `tidyverse`

group of packages) can also be used to
prepare your data in the required format:

```
library(dplyr); library(tidyr)
dat.df03 <- birthwt; head(dat.df03)
#> low age lwt race smoke ptl ht ui ftv bwt
#> 85 0 19 182 2 0 0 0 1 0 2523
#> 86 0 33 155 3 0 0 0 0 3 2551
#> 87 0 20 105 1 1 0 0 0 1 2557
#> 88 0 21 108 1 1 0 0 1 2 2594
#> 89 0 18 107 1 1 0 0 1 0 2600
#> 91 0 21 124 3 0 0 0 0 0 2622
# Here we set the factor levels and tabulate the data in a single call using pipe operators:
dat.tab03 <- dat.df03 %>%
mutate(flow = factor(low, levels = c(1,0), labels = c("yes","no"))) %>%
mutate(fsmoke = factor(smoke, levels = c(1,0), labels = c("yes","no"))) %>%
group_by(fsmoke, flow) %>%
summarise(n = n())
# View the data:
dat.tab03
#> # A tibble: 4 × 3
#> # Groups: fsmoke [2]
#> fsmoke flow n
#> <fct> <fct> <int>
#> 1 yes yes 30
#> 2 yes no 44
#> 3 no yes 29
#> 4 no no 86
# View the data in conventional 2 by 2 table format:
pivot_wider(dat.tab03, id_cols = c(fsmoke),
names_from = flow, values_from = n)
#> # A tibble: 2 × 3
#> # Groups: fsmoke [2]
#> fsmoke yes no
#> <fct> <int> <int>
#> 1 yes 30 44
#> 2 no 29 86
```

As before, compute the odds ratio for smoking and delivery of a low birth weight baby:

```
dat.epi03 <- epi.2by2(dat = dat.tab03, method = "cohort.count",
conf.level = 0.95, units = 100, interpret = FALSE, outcome = "as.columns")
dat.epi03
#> Outcome + Outcome - Total Inc risk *
#> Exposed + 30 44 74 40.54 (29.27 to 52.59)
#> Exposed - 29 86 115 25.22 (17.58 to 34.17)
#> Total 59 130 189 31.22 (24.69 to 38.35)
#>
#> Point estimates and 95% CIs:
#> -------------------------------------------------------------------
#> Inc risk ratio 1.61 (1.06, 2.44)
#> Inc odds ratio 2.02 (1.08, 3.78)
#> Attrib risk in the exposed * 15.32 (1.61, 29.04)
#> Attrib fraction in the exposed (%) 37.80 (5.47, 59.07)
#> Attrib risk in the population * 6.00 (-4.33, 16.33)
#> Attrib fraction in the population (%) 19.22 (-0.21, 34.88)
#> -------------------------------------------------------------------
#> Uncorrected chi2 test that OR = 1: chi2(1) = 4.924 Pr>chi2 = 0.026
#> Fisher exact test that OR = 1: Pr>chi2 = 0.036
#> Wald confidence limits
#> CI: confidence interval
#> * Outcomes per 100 population units
```

The odds of having a low birth weight child for smokers is 2.0 (95% CI 1.1 to 3.8) times greater than the odds of having a low birth weight child for non-smokers.

**Measures of association for a series of candidate risk
factors**

The code that follows provides and approach for calculating measures of association for a series of candidate risk factors, writing the results to a data frame. Here we set up a loop to calculate the odds ratio for age, smoke and race:

```
dat.df04 <- birthwt; head(dat.df04)
#> low age lwt race smoke ptl ht ui ftv bwt
#> 85 0 19 182 2 0 0 0 1 0 2523
#> 86 0 33 155 3 0 0 0 0 3 2551
#> 87 0 20 105 1 1 0 0 0 1 2557
#> 88 0 21 108 1 1 0 0 1 2 2594
#> 89 0 18 107 1 1 0 0 1 0 2600
#> 91 0 21 124 3 0 0 0 0 0 2622
dat.df04$flow <- factor(dat.df04$low, levels = c(1,0))
dat.df04$fage <- ifelse(dat.df04$age > 23, 0,1)
dat.df04$fage <- factor(dat.df04$fage, levels = c(1,0))
dat.df04$fsmoke <- factor(dat.df04$smoke, levels = c(1,0))
# Race is coded 1 = white, 2 = black and 3 = other. Set white as the reference level:
dat.df04$frace <- ifelse(dat.df04$race == 1, 0, 1)
dat.df04$frace <- factor(dat.df04$frace, levels = c(1,0))
# Empty vectors to collect results:
rfactor <- ref <- or.p <- or.l <- or.u <- c()
# The candidate risk factors are in columns 12 to 14 of data frame dat.df04:
for(i in 12:14){
tdat.tab04 <- table(dat.df04[,i], dat.df04$flow)
tdat.epi04 <- epi.2by2(dat = tdat.tab04, method = "cohort.count",
conf.level = 0.95, units = 100, interpret = FALSE, outcome = "as.columns")
trfactor <- as.character(names(dat.df04)[i])
rfactor <- c(rfactor, trfactor)
tref <- as.character(paste("Reference: ", trfactor, " - ", levels(dat.df04[,i])[2], sep = ""))
ref <- c(ref, tref)
tor.p <- as.numeric(tdat.epi04$massoc.detail$OR.strata.wald[1])
or.p <- c(or.p, tor.p)
tor.l <- as.numeric(tdat.epi04$massoc.detail$OR.strata.wald[2])
or.l <- c(or.l, tor.l)
tor.u <- as.numeric(tdat.epi04$massoc.detail$OR.strata.wald[3])
or.u <- c(or.u, tor.u)
}
gdat.df04 <- data.frame(yat = 1:3, ylab = rfactor, ref, or.p, or.l, or.u)
gdat.df04
#> yat ylab ref or.p or.l or.u
#> 1 1 fage Reference: fage - 0 1.174769 0.6294241 2.192609
#> 2 2 fsmoke Reference: fsmoke - 0 2.021944 1.0806596 3.783112
#> 3 3 frace Reference: frace - 0 2.004577 1.0703040 3.754380
```

Plot the point estimate of the odds ratio for each candidate risk factor and its 95% confidence interval. Annotate the plot with the reference category for each risk factor:

```
library(ggplot2); library(scales)
x.at <- c(0.25,0.5,1,2,4,8,16,32)
ggplot(data = gdat.df04, aes(x = or.p, y = yat)) +
theme_bw() +
geom_point() +
geom_errorbarh(aes(xmax = or.l, xmin = or.u, height = 0.2)) +
scale_x_continuous(trans = log2_trans(), breaks = x.at, limits = c(0.25,8),
name = "Odds ratio") +
scale_y_continuous(breaks = gdat.df04$yat, labels = gdat.df04$ylab,
name = "Risk factor") +
geom_vline(xintercept = 1, lwd = 1) +
annotate("text", x = 0.25, y = gdat.df04$yat, label = gdat.df04$ref, hjust = 0, size = 3) +
coord_fixed(ratio = 0.75 / 1) +
theme(axis.title.y = element_text(vjust = 0))
```

**Confounding**

We’re concerned that the mother’s race may confound the association
between low birth weight and delivery of a low birth weight baby so
we’ll stratify the data by race and compute the Mantel-Haenszel adjusted
odds ratio. As before, our tables can be prepared using either base R or
`tidyverse`

.

*Stratified two by two table preparation using the table
function in base R*

```
dat.df05 <- birthwt; head(dat.df05)
#> low age lwt race smoke ptl ht ui ftv bwt
#> 85 0 19 182 2 0 0 0 1 0 2523
#> 86 0 33 155 3 0 0 0 0 3 2551
#> 87 0 20 105 1 1 0 0 0 1 2557
#> 88 0 21 108 1 1 0 0 1 2 2594
#> 89 0 18 107 1 1 0 0 1 0 2600
#> 91 0 21 124 3 0 0 0 0 0 2622
dat.df05$flow <- factor(dat.df05$low, levels = c(1,0))
dat.df05$fsmoke <- factor(dat.df05$smoke, levels = c(1,0))
dat.df05$frace <- factor(dat.df05$race, levels = c(1,2,3))
dat.tab05 <- table(dat.df05$fsmoke, dat.df05$flow, dat.df05$frace,
dnn = c("Smoke", "Low BW", "Race")); dat.tab05
#> , , Race = 1
#>
#> Low BW
#> Smoke 1 0
#> 1 19 33
#> 0 4 40
#>
#> , , Race = 2
#>
#> Low BW
#> Smoke 1 0
#> 1 6 4
#> 0 5 11
#>
#> , , Race = 3
#>
#> Low BW
#> Smoke 1 0
#> 1 5 7
#> 0 20 35
```

Compute the Mantel-Haenszel adjusted odds ratio for smoking and
delivery of a low birth weight baby, adjusting for the effect of race.
Function `epi.2by2`

automatically calculates the
Mantel-Haenszel odds ratio and risk ratio when presented with stratified
contingency tables.

```
dat.epi05 <- epi.2by2(dat = dat.tab05, method = "cohort.count",
conf.level = 0.95, units = 100, interpret = FALSE, outcome = "as.columns")
dat.epi05
#> Outcome + Outcome - Total Inc risk *
#> Exposed + 30 44 74 40.54 (29.27 to 52.59)
#> Exposed - 29 86 115 25.22 (17.58 to 34.17)
#> Total 59 130 189 31.22 (24.69 to 38.35)
#>
#>
#> Point estimates and 95% CIs:
#> -------------------------------------------------------------------
#> Inc risk ratio (crude) 1.61 (1.06, 2.44)
#> Inc risk ratio (M-H) 2.15 (1.29, 3.58)
#> Inc risk ratio (crude:M-H) 0.75
#> Inc odds ratio (crude) 2.02 (1.08, 3.78)
#> Inc odds ratio (M-H) 3.09 (1.49, 6.39)
#> Inc odds ratio (crude:M-H) 0.66
#> Attrib risk in the exposed (crude) * 15.32 (1.61, 29.04)
#> Attrib risk in the exposed (M-H) * 22.17 (1.41, 42.94)
#> Attrib risk (crude:M-H) 0.69
#> -------------------------------------------------------------------
#> M-H test of homogeneity of IRRs: chi2(2) = 3.862 Pr>chi2 = 0.145
#> M-H test of homogeneity of ORs: chi2(2) = 2.800 Pr>chi2 = 0.247
#> Test that M-H adjusted OR = 1: chi2(1) = 9.413 Pr>chi2 = 0.001
#> Wald confidence limits
#> M-H: Mantel-Haenszel; CI: confidence interval
#> * Outcomes per 100 population units
```

The Mantel-Haenszel test of homogeneity of the strata odds ratios is not significant (chi square test statistic 2.800; df 2; p-value = 0.25). We accept the null hypothesis and conclude that the odds ratios for each strata of race are the same. Because the stratum specific odds ratios are not statistically significantly different the Mantel-Haenszel adjusted odds ratio provides an appropriate summary of the association between smoking and low birth weight.

After accounting for the confounding effect of race, the odds of having a low birth weight baby for smokers was 3.1 (95% CI 1.5 to 6.4) times that of non-smokers.

*Stratified two by two table preparation using
tidyverse*

```
dat.df06 <- birthwt
dat.tab06 <- dat.df06 %>%
mutate(flow = factor(low, levels = c(1,0), labels = c("yes","no"))) %>%
mutate(fsmoke = factor(smoke, levels = c(1,0), labels = c("yes","no"))) %>%
mutate(frace = factor(race)) %>%
group_by(frace, fsmoke, flow) %>%
summarise(n = n())
dat.tab06
#> # A tibble: 12 × 4
#> # Groups: frace, fsmoke [6]
#> frace fsmoke flow n
#> <fct> <fct> <fct> <int>
#> 1 1 yes yes 19
#> 2 1 yes no 33
#> 3 1 no yes 4
#> 4 1 no no 40
#> # ℹ 8 more rows
# View the data in conventional 2 by 2 table format:
pivot_wider(dat.tab06, id_cols = c(frace, fsmoke),
names_from = flow, values_from = n)
#> # A tibble: 6 × 4
#> # Groups: frace, fsmoke [6]
#> frace fsmoke yes no
#> <fct> <fct> <int> <int>
#> 1 1 yes 19 33
#> 2 1 no 4 40
#> 3 2 yes 6 4
#> 4 2 no 5 11
#> # ℹ 2 more rows
```

Compute the Mantel-Haenszel adjusted odds ratio for smoking and delivery of a low birth weight baby, adjusting for the effect of race:

```
dat.epi06 <- epi.2by2(dat = dat.tab06, method = "cohort.count",
conf.level = 0.95, units = 100, interpret = FALSE, outcome = "as.columns")
dat.epi06
#> Outcome + Outcome - Total Inc risk *
#> Exposed + 30 44 74 40.54 (29.27 to 52.59)
#> Exposed - 29 86 115 25.22 (17.58 to 34.17)
#> Total 59 130 189 31.22 (24.69 to 38.35)
#>
#>
#> Point estimates and 95% CIs:
#> -------------------------------------------------------------------
#> Inc risk ratio (crude) 1.61 (1.06, 2.44)
#> Inc risk ratio (M-H) 2.15 (1.29, 3.58)
#> Inc risk ratio (crude:M-H) 0.75
#> Inc odds ratio (crude) 2.02 (1.08, 3.78)
#> Inc odds ratio (M-H) 3.09 (1.49, 6.39)
#> Inc odds ratio (crude:M-H) 0.66
#> Attrib risk in the exposed (crude) * 15.32 (1.61, 29.04)
#> Attrib risk in the exposed (M-H) * 22.17 (1.41, 42.94)
#> Attrib risk (crude:M-H) 0.69
#> -------------------------------------------------------------------
#> M-H test of homogeneity of IRRs: chi2(2) = 3.862 Pr>chi2 = 0.145
#> M-H test of homogeneity of ORs: chi2(2) = 2.800 Pr>chi2 = 0.247
#> Test that M-H adjusted OR = 1: chi2(1) = 9.413 Pr>chi2 = 0.001
#> Wald confidence limits
#> M-H: Mantel-Haenszel; CI: confidence interval
#> * Outcomes per 100 population units
```

Plot the individual strata odds ratios and the Mantel-Haenszel summary odds ratio as an error bar plot to better understand how the Mantel-Haenszel adjusted odds ratio relates to the individual strata odds ratios:

```
nstrata <- 1:length(unique(dat.tab06$frace))
strata.lab <- paste("Strata ", nstrata, sep = "")
y.at <- c(nstrata, max(nstrata) + 1)
y.lab <- c("M-H", strata.lab)
x.at <- c(0.25,0.5,1,2,4,8,16,32)
or.p <- c(dat.epi06$massoc.detail$OR.mh$est,
dat.epi06$massoc.detail$OR.strata.cfield$est)
or.l <- c(dat.epi06$massoc.detail$OR.mh$lower,
dat.epi05$massoc.detail$OR.strata.cfield$lower)
or.u <- c(dat.epi06$massoc.detail$OR.mh$upper,
dat.epi05$massoc.detail$OR.strata.cfield$upper)
gdat.df06 <- data.frame(y.at, y.lab, or.p, or.l, or.u)
ggplot(data = gdat.df06, aes(x = or.p, y = y.at)) +
theme_bw() +
geom_point() +
geom_errorbarh(aes(xmax = or.l, xmin = or.u, height = 0.2)) +
labs(x = "Odds ratio", y = "Strata") +
scale_x_continuous(trans = log2_trans(), breaks = x.at,
limits = c(0.25,32)) +
scale_y_continuous(breaks = y.at, labels = y.lab) +
geom_vline(xintercept = 1, lwd = 1) +
coord_fixed(ratio = 0.75 / 1) +
theme(axis.title.y = element_text(vjust = 0))
```

Altman, DG. 1998. “Confidence Intervals for the Number Needed to
Treat.” *British Medical Journal* 317: 1309–12.

Grimes, DA, and KF Schulz. 2008. “Making Sense of Odds and Odds
Ratios.” *Obstetrics and Gynecology* 111: 423–26.

Hosmer, DW, and S Lemeshow. 2000. *Applied Logistic
Regression*. London: Jon Wiley; Sons Inc.

Kuritz, SJ, JR Landis, and GG Koch. 1988. “A
general overview of Mantel-Haenszel methods: Applications and recent
developments.” *Annual Reviews in Public Health* 9:
123–60.

Porta, M, S Greenland, and JM Last. 2014. *A
Dictionary of Epidemiology*. London: Oxford University Press.

Prasad, K, R Jaeschke, P Wyer, S Keitz, and G Guyatt. 2007. “Tips for teachers of evidence-based medicine:
Understanding odds ratios and their relationship to risk
ratios.” *Journal of General Internal Medicine* 23
(5): 635–40.

Siegerink, B, and JL Rohmann. 2018. “Impact
of your results: Beyond the relative risk.” *Research
and Practice in Thrombosis and Haemostasis* 2: 653–57.

Willeberg, P. 1977. “Animal disease
information processing: Epidemiologic analyses of the feline urologic
syndrome.” *Acta Veterinaria Scandinavica* 64:
1–48.