Quantitative bias analysis allows to estimate nonrandom errors in epidemiologic studies, assessing the magnitude and direction of biases, and quantifying their uncertainties. Every study has some random error due to its limited sample size, and is susceptible to systematic errors as well, from selection bias to the presence of (un)known confounders or information bias (measurement error, including misclassification). Bias analysis methods were compiled by Lash et al. in their book “Applying Quantitative Bias Analysis to Epidemiologic Data”. This package implements the various bias analyses from that book, which are also available (for some) as a spreadsheet or a SAS macro. This vignette provides some examples on how to use the package.
We will use a case-control study by Stang et al. on the relation between mobile phone use and uveal melanoma. The observed odds ratio for the association between regular mobile phone use vs. no mobile phone use with uveal melanoma incidence is 0.71 [95% CI 0.51-0.97]. But there was a substantial difference in participation rates between cases and controls (94% vs 55%, respectively) and so selection bias could have an impact on the association estimate. The 2X2 table for this study is the following:
| Regular use | No use | |
|---|---|---|
| Cases | 136 | 107 |
| Controls | 297 | 165 |
We use the function selection as shown below.
library(episensr)
stang <- selection(matrix(c(136, 107, 297, 165),
dimnames = list(c("UM+", "UM-"), c("Mobile+", "Mobile-")),
nrow = 2, byrow = TRUE),
bias_parms = c(.94, .85, .64, .25))
stang
## --Observed data--
## Outcome: UM+
## Comparing: Mobile+ vs. Mobile-
##
## Mobile+ Mobile-
## UM+ 136 107
## UM- 297 165
##
## 2.5% 97.5%
## Observed Relative Risk: 0.7984287 0.6518303 0.9779975
## Observed Odds Ratio: 0.7061267 0.5143958 0.9693215
## ---
##
## Selection Bias Corrected Relative Risk: 1.483780
## Selection Bias Corrected Odds Ratio: 1.634608
The various episensr functions return an object which is a list containing the
input and output variables.
You can check it out with str().
The 2X2 table is provided as a matrix and selection probabilities given with the
argument bias_parms, a vector with the 4 probabilities (guided by the
participation rates in cases and controls) in the following order: among cases
exposed, among cases unexposed, among noncases exposed, and among noncases
unexposed.
The output shows the observed 2X2 table and the observed odds ratio (and relative
risk), followed by the corrected ones.
We will use data from a cross-sectional study by Tyndall et al. on the association between male circumcision and the risk of acquiring HIV, which might be confounded by religion. The code to account for unmeasured or unknown confounders is the following, where the 2X2 table is given as a matrix. We choose a risk ratio implementation, provide a vector defining the risk ratio associating the confounder with the disease, and the prevalence of the confounder, religion, among the exposed and the unexposed.
confounders(matrix(c(105, 85, 527, 93),
dimnames = list(c("HIV+", "HIV-"), c("Circ+", "Circ-")),
nrow = 2, byrow = TRUE),
type = "RR",
bias_parms = c(.63, .8, .05))
## --Observed data--
## Outcome: HIV+
## Comparing: Circ+ vs. Circ-
##
## Circ+ Circ-
## HIV+ 105 85
## HIV- 527 93
##
## 2.5% 97.5%
## Crude Relative Risk: 0.3479151 0.2757026 0.4390417
## Relative Risk, Confounder +: 0.4850550
## Relative Risk, Confounder -: 0.4850550
## ---
## Adjusted RR
## Standardized Morbidity Ratio: 0.4850550 0.7172695
## Mantel-Haenszel: 0.4850550 0.7172695
The output gives the crude 2X2 table, the crude relative risk and confounder specific measures of association between exposure and outcome, and the relationship adjusted for the unknown confounder, using a standardized morbidity ratio (SMR) or a Mantel-Haenszel (MH) estimate of the risk ratio.
We use a study on the effect of smoking during pregnancy on breast cancer risk
by Fink & Lash, where we assume
nondifferential misclassification of the exposure, smoking, with probability
density functions for sensitivities (Se) and specificities (Sp) among cases and
noncases equal to uniform distributions with a minimum of 0.7 and a maximum of
0.95 for sensitivities (0.9 and 0.99 respectively for specificities).
We choose to correct for exposure misclassification with the argument type =
exposure.
We ask for 10000 replications (default is 1000).
The Se and Sp for cases (seca, spca) are given as a list with its first
element referring to the type of distribution (choice between constant, uniform,
triangular, trapezoidal, logit-logistic, and logit-normal) and the second
element giving the distribution parameters (min and max for uniform
distribution).
By avoiding to provide information on the noncases (seexp, spexp), we are
referring to a nondifferential misclassification.
set.seed(123)
smoke.nd <- probsens(matrix(c(215, 1449, 668, 4296),
dimnames = list(c("BC+", "BC-"), c("Smoke+", "Smoke-")),
nrow = 2, byrow = TRUE),
type = "exposure",
reps = 10000,
seca.parms = list("uniform", c(.7, .95)),
spca.parms = list("uniform", c(.9, .99)))
smoke.nd
## --Observed data--
## Outcome: BC+
## Comparing: Smoke+ vs. Smoke-
##
## Smoke+ Smoke-
## BC+ 215 1449
## BC- 668 4296
##
## 2.5% 97.5%
## Observed Relative Risk: 0.9653825 0.8523766 1.0933704
## Observed Odds Ratio: 0.9542406 0.8092461 1.1252141
## ---
## Median 2.5th percentile
## Relative Risk -- systematic error: 0.9425515 0.8820009
## Odds Ratio -- systematic error: 0.9245982 0.8482093
## Relative Risk -- systematic and random error: 0.9355316 0.8186992
## Odds Ratio -- systematic and random error: 0.9158426 0.7676660
## 97.5th percentile
## Relative Risk -- systematic error: 0.9612203
## Odds Ratio -- systematic error: 0.9487950
## Relative Risk -- systematic and random error: 1.0688417
## Odds Ratio -- systematic and random error: 1.0919379
The output gives the 2X2 table, the observed measures of association, and the corrected measures of association.
We saved the probsens analysis in a new object smoke.nd.
We can see its elements with the function str():
str(smoke.nd)
## List of 4
## $ obs.data : num [1:2, 1:2] 215 668 1449 4296
## ..- attr(*, "dimnames")=List of 2
## .. ..$ : chr [1:2] "BC+" "BC-"
## .. ..$ : chr [1:2] "Smoke+" "Smoke-"
## $ obs.measures: num [1:2, 1:3] 0.965 0.954 0.852 0.809 1.093 ...
## ..- attr(*, "dimnames")=List of 2
## .. ..$ : chr [1:2] " Observed Relative Risk:" " Observed Odds Ratio:"
## .. ..$ : chr [1:3] " " "2.5%" "97.5%"
## $ adj.measures: num [1:4, 1:3] 0.943 0.925 0.936 0.916 0.882 ...
## ..- attr(*, "dimnames")=List of 2
## .. ..$ : chr [1:4] " Relative Risk -- systematic error:" " Odds Ratio -- systematic error:" "Relative Risk -- systematic and random error:" " Odds Ratio -- systematic and random error:"
## .. ..$ : chr [1:3] "Median" "2.5th percentile" "97.5th percentile"
## $ sim.df :'data.frame': 10000 obs. of 12 variables:
## ..$ seca : num [1:10000] 0.772 0.897 0.802 0.921 0.935 ...
## ..$ seexp : num [1:10000] 0.772 0.897 0.802 0.921 0.935 ...
## ..$ spca : num [1:10000] 0.928 0.929 0.978 0.93 0.911 ...
## ..$ spexp : num [1:10000] 0.928 0.929 0.978 0.93 0.911 ...
## ..$ A1 : num [1:10000] 135.9 117.6 229.2 115 79.7 ...
## ..$ B1 : num [1:10000] 1528 1546 1435 1549 1584 ...
## ..$ C1 : num [1:10000] 443 383 718 374 269 ...
## ..$ D1 : num [1:10000] 4521 4581 4246 4590 4695 ...
## ..$ corr.RR: num [1:10000] 0.929 0.931 0.958 0.931 0.905 ...
## ..$ corr.OR: num [1:10000] 0.907 0.91 0.945 0.91 0.877 ...
## ..$ tot.RR : num [1:10000] 0.799 0.962 0.968 0.992 0.854 ...
## ..$ tot.OR : num [1:10000] 0.743 0.95 0.958 0.989 0.811 ...
## - attr(*, "class")= chr [1:2] "episensr" "list"
smoke.nd is a list of 4 elements where different information on the analysis
done are saved.
We have smoke.nd$obs.data where we have the observed 2X2 table,
smoke.nd$obs.measures (the observed measures of association),
smoke.nd$adj.measures (the adjusted measures of association), and
smoke.nd$sim.df, a data frame with the simulated variables from each
replication, like the Se and Sp, the 4 cells of the adjusted 2X2 table, and the
adjusted measures.
We can plot the Se prior distribution (and not forgetting to discard the draws that
led to negative adjustments).
hist(smoke.nd$sim.df[!is.na(smoke.nd$sim.df$corr.RR), ]$seca,
breaks = seq(0.65, 1, 0.01),
col = "lightgreen",
main = NULL,
xlab = "Sensitivity for Cases")
Selection and misclassification bias models can be bootstrapped in order to get confidence interval on the adjusted association parameter. We can use the ICU dataset from Hosmer and Lemeshow Applied Logistic Regression textbook as an example. The replicates that give negative cells in the adjusted 2X2 table are silently ignored and the number of effective bootstrap replicates is provided in the output.
library(aplore3) # to get ICU data
data(icu)
## First run the model
misclass_eval <- misclassification(icu$sta, icu$inf,
type = "exposure",
bias_parms = c(.75, .85, .9, .95))
misclass_eval
## --Observed data--
## Outcome: Died
## Comparing: Yes vs. No
##
## exposed
## case Yes No
## Died 24 16
## Lived 60 100
##
## 2.5% 97.5%
## Observed Relative Risk: 2.071429 1.175157 3.651272
## Observed Odds Ratio: 2.500000 1.230418 5.079573
## ---
##
## Misclassification Bias Corrected Relative Risk: 3.627845
## Misclassification Bias Corrected Odds Ratio: 4.871795
## Then bootstrap it
set.seed(123)
misclass_boot <- boot.bias(misclass_eval, R = 10000)
misclass_boot
## 95 % confidence interval of the bias adjusted measures:
## RR: 1.012453 11.50197
## OR: 1.140324 18.58507
## ---
## Based on 9595 bootstrap replicates
Bootstrap replicates can also be plotted, with the confidence interval shown as dotted lines.
plot(misclass_boot, "rr")
