Directly adjusted incidence rate estimates
Compute directly adjusted incidence rates.
epi.directadj(obs, tar, std, units = 1, conf.level = 0.95)
obs |
a matrix representing the observed number of events. Rows represent strata (e.g. region); columns represent the covariates to be adjusted for (e.g. age class, gender). The sum of each row will equal the total number of events for each stratum. The rows of the |
tar |
a matrix representing population time at risk. Rows represent strata (e.g. region); columns represent the covariates to be adjusted for (e.g. age class, gender). The sum of each row will equal the total population time at risk within each stratum. The rows of the |
std |
a matrix representing the standard population size for the different levels of the covariate to be adjusted for. The columns of |
units |
multiplier for the incidence rate estimates. |
conf.level |
magnitude of the returned confidence interval. Must be a single number between 0 and 1. |
This function returns unadjusted (crude) and directly adjusted incidence rate estimates for each of the specified population strata. The term ‘covariate’ is used here to refer to the factors we want to control (i.e. adjust) for when calculating the directly adjusted incidence rate estimates.
When the outcome of interest is rare, the confidence intervals for the adjusted incidence rates returned by this function (based on Fay and Feuer, 1997) will be appropriate for incidence risk data. In this situation the argument tar is assumed to represent the size of the population at risk (instead of population time at risk). Example 3 (below) provides an approach if you are working with incidence risk data and the outcome of interest is not rare.
A list containing the following:
crude |
the crude incidence rate estimates for each stratum-covariate combination. |
crude.strata |
the crude incidence rate estimates for each stratum. |
adj.strata |
the directly adjusted incidence rate estimates for each stratum. |
Thanks to Karl Ove Hufthammer for helpful suggestions to improve the execution and documentation of this function.
Fay M, Feuer E (1997). Confidence intervals for directly standardized rates: A method based on the gamma distribution. Statistics in Medicine 16: 791 - 801.
Fleiss JL (1981). Statistical Methods for Rates and Proportions, Wiley, New York, USA, pp. 240.
Frome E, Checkoway H (1985). Use of Poisson regression models in estimating incidence rates and ratios. American Journal of Epidemiology 121: 309 - 323.
Greenland S, Rothman KJ. Introduction to stratified analysis. In: Rothman KJ, Greenland S (1998). Modern Epidemiology. Lippincott Williams, & Wilkins, Philadelphia, pp. 260 - 265.
Thrusfield M (2007). Veterinary Epidemiology, Blackwell Publishing, London, UK, pp. 63 - 64.
Wilcosky T, Chambless L (1985). A comparison of direct adjustment and regression adjustment of epidemiologic measures. Journal of Chronic Diseases 38: 849 - 956.
## EXAMPLE 1 (from Thrusfield 2007 pp. 63 - 64):
## A study was conducted to estimate the seroprevalence of leptospirosis
## in dogs in Glasgow and Edinburgh, Scotland. For the matrix titled pop
## the numbers represent dog-years at risk. The following data were
## obtained for male and female dogs:
dat <- data.frame(obs = c(15,46,53,16), tar = c(48,212,180,71),
sex = c("M","F","M","F"), city = c("ED","ED","GL","GL"))
obs <- matrix(dat$obs, nrow = 2, byrow = TRUE,
dimnames = list(c("ED","GL"), c("M","F")))
tar <- matrix(dat$tar, nrow = 2, byrow = TRUE,
dimnames = list(c("ED","GL"), c("M","F")))
std <- matrix(data = c(250,250), nrow = 1, byrow = TRUE,
dimnames = list("", c("M","F")))
## Compute directly adjusted seroprevalence estimates, using a standard
## population with equal numbers of male and female dogs:
std <- matrix(data = c(250,250), nrow = 1, byrow = TRUE,
dimnames = list("", c("M","F")))
epi.directadj(obs, tar, std, units = 1, conf.level = 0.95)
## > $crude
## > strata cov est lower upper
## > 1 ED M 0.3125000 0.1749039 0.5154212
## > 2 GL M 0.2944444 0.2205591 0.3851406
## > 3 ED F 0.2169811 0.1588575 0.2894224
## > 4 GL F 0.2253521 0.1288082 0.3659577
## > $crude.strata
## > strata est lower upper
## > 1 ED 0.2346154 0.1794622 0.3013733
## > 2 GL 0.2749004 0.2138889 0.3479040
## > $adj.strata
## > strata est lower upper
## > 1 ED 0.2647406 0.1866047 0.3692766
## > 2 GL 0.2598983 0.1964162 0.3406224
## The confounding effect of gender has been removed by the adjusted
## incidence rate estimates.
## The adjusted incidence rate of leptospirosis in Glasgow dogs is 26 (95%
## CI 20 to 34) cases per 100 dog-years at risk.
## EXAMPLE 2 --- A more flexible approach for calculating adjusted incidence
## rate estimates using Poisson regression. See Frome and Checkoway (1985) for
## details.
dat.glm01 <- glm(obs ~ city, offset = log(tar), family = poisson, data = dat)
summary(dat.glm01)
## If you want to obtain adjusted incidence rate estimates, use the predict
## method on a new data set with the time at risk (tar) variable set to 1
## (which means log(tar) = 0). This will return the predicted number of
## cases per one unit of individual time, i.e. the incidence rate.
dat.pred01 <- predict(object = dat.glm01, newdata =
data.frame(city = c("ED","GL"), tar = c(1,1)),
type = "link", se = TRUE)
conf.level <- 0.95
critval <- qnorm(p = 1 - ((1 - conf.level) / 2), mean = 0, sd = 1)
est <- dat.glm01$family$linkinv(dat.pred01$fit)
lower <- dat.glm01$family$linkinv(dat.pred01$fit -
(critval * dat.pred01$se.fit))
upper <- dat.glm01$family$linkinv(dat.pred01$fit +
(critval * dat.pred01$se.fit))
round(data.frame(est, lower, upper), 3)
## est lower upper
## 0.235 0.183 0.302
## 0.275 0.217 0.348
## Results identical to the crude incidence rate estimates from
## epi.directadj.
## We now adjust for the effect of gender and city and report the adjusted
## incidence rate estimates for each city:
dat.glm02 <- dat.glm02 <- glm(obs ~ city + sex, offset = log(tar),
family = poisson, data = dat)
dat.pred02 <- predict(object = dat.glm02, newdata =
data.frame(sex = c("F","F"), city = c("ED","GL"), tar = c(1,1)),
type = "link", se.fit = TRUE)
conf.level <- 0.95
critval <- qnorm(p = 1 - ((1 - conf.level) / 2), mean = 0, sd = 1)
est <- dat.glm02$family$linkinv(dat.pred02$fit)
lower <- dat.glm02$family$linkinv(dat.pred02$fit -
(critval * dat.pred02$se.fit))
upper <- dat.glm02$family$linkinv(dat.pred02$fit +
(critval * dat.pred02$se.fit))
round(data.frame(est, lower, upper), 3)
## est lower upper
## 0.220 0.168 0.287
## 0.217 0.146 0.323
## Using Poisson regression the gender adjusted incidence rate of leptospirosis
## in Glasgow dogs was 22 (95% CI 15 to 32) cases per 100 dog-years at risk.
## These results won't be the same as those using direct adjustment because
## for direct adjustment we use a contrived standard population.
## EXAMPLE 3 --- Logistic regression to return adjusted incidence risk
## estimates:
## Say, for argument's sake, that we are now working with incidence risk data.
## Here we'll re-label the variable 'tar' (time at risk) as 'pop'
## (population size). We adjust for the effect of gender and city and
## report the adjusted incidence risk of canine leptospirosis estimates for
## each city:
dat$pop <- dat$tar
dat.glm03 <- glm(cbind(obs, pop - obs) ~ city + sex,
family = "binomial", data = dat)
dat.pred03 <- predict(object = dat.glm03, newdata =
data.frame(sex = c("F","F"), city = c("ED","GL")),
type = "link", se.fit = TRUE)
conf.level <- 0.95
critval <- qnorm(p = 1 - ((1 - conf.level) / 2), mean = 0, sd = 1)
est <- dat.glm03$family$linkinv(dat.pred03$fit)
lower <- dat.glm03$family$linkinv(dat.pred03$fit -
(critval * dat.pred03$se.fit))
upper <- dat.glm03$family$linkinv(dat.pred03$fit +
(critval * dat.pred03$se.fit))
round(data.frame(est, lower, upper), 3)
## est lower upper
## 0.220 0.172 0.276
## 0.217 0.150 0.304
## The adjusted incidence risk of leptospirosis in Glasgow dogs is 22 (95%
## CI 15 to 30) cases per 100 dogs at risk.Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.