Variance of the log risk
variance_of_the_log_risk.RmdIntroduction
It is often necessary to compare estimates for a categorical variable with more than two categories. In order to make these comparisons, it is necessary to select an appropriate reference category; this can often be challenging as, in some cases, the choice of reference category is not obvious and can be rather arbitrary.
When there is a categorical variable with more than two levels in a regression model, the estimates and corresponding standard errors only allow a direct comparison between each specified category relative to the category selected as the reference group. To address this limitation, ‘floating’ standard errors, or standard errors based on quasi-variances, may be used. These are also called floating absolute risks. This method allows for the estimation of standard errors across all levels of the categorical variable, including the reference group, enabling comparisons between any pair of groups.
In the context of floating absolute risks, various algorithms are available which use slightly different methods for estimation of the variance-covariance matrix. The variance-covariance matrix captures the variability and relationship between estimates across levels of a categorical variable. This enables consistent standard error estimation across all groups, thereby facilitating accurate comparisons between any pair of groups by accounting for the relative differences in their standard errors (Easton et al, 1991; Plummer, 2004; Firth and de Menezes, 2004).
In R there are two functions in two packages that implement this:
Epi::float uses the method of Plummer (2004) and
qvcalc::qvcalc the method of Firth and de Menezes
(2004).
Both functions are re-exported in the ckbtools package,
so they can be used after calling library(ckbtools).
float
Usage:
float(object)where object is any type of regression model.
Example
Cox regression of ihd on categories of
bmi_calc.
library(survival)
ckbtools_participant_data$bmi_grp <- cut(ckbtools_participant_data$bmi_calc, c(0, 18.5, 25, 30, 100), right = FALSE)
ckbtools_participant_data$ihd_time <- as.numeric(ckbtools_participant_data$ihd_date - ckbtools_participant_data$study_date) / 365.25
cox_model <- coxph(Surv(ihd_time, ihd) ~ bmi_grp +
age_at_study_date + is_female + as.factor(region_code),
data = ckbtools_participant_data)
fl <- float(cox_model)
data.frame(coef = fl$coef, var = fl$var)# coef var
# [0,18.5) 0.0000000 0.0074317718
# [18.5,25) 0.4370379 0.0007175254
# [25,30) 0.6873568 0.0015079493
# [30,100) 1.2335243 0.0135786368
qvcalc
Usage:
qvcalc(object)where object is any type of regression model.
Example
Cox regression of ihd on categories of
bmi_calc.
qvcalc(cox_model, factorname = "bmi_grp")You can also include the coef.indices argument to
determine which rows and columns of the variance-covariance matrix to
use (i.e. which factors to consider in the calculation). Add a zero to
include the reference level. This will enable the estimation of the
reference group’s ‘floating’ standard error, which allows the
calculation of confidence intervals. This is particularly useful for
creating display items such as shape plots of hazard ratios/risk ratios
etc. across all categories, as it allows for the inclusion of the
reference group estimate and confidence intervals.
References
Plummer, M. Improved estimates of floating absolute risk. Statistics in Medicine 23, 93–104 (2004). (Not open access)
Firth, D. & De Menezes, R. X. Quasi‐variances. Biometrika 91, 65–80 (2004)
Firth, D. Overcoming the Reference Category Problem in the Presentation of Statistical Models. Sociological Methodology 33, 1–18 (2003). (Not open access)