Multiple Bias Modeling

Epidemiologic studies can suffer from more than one bias. Bias functions in episensr can be applied sequentially to quantify bias resulting from multiple biases.

Following the example in Lash et al., we can use the study by Chien et al.. It is a case-control study looking at the association between antidepressant use and the occurrence of breast cancer. The observed OR was 1.2 [0.9–1.6].

chien <- matrix(c(118, 832, 103, 884),
                dimnames = list(c("BC+", "BC-"), c("AD+", "AD-")),
                nrow = 2, byrow = TRUE)

	AD+	AD-
BC+	118	832
BC-	103	884

Records on medication use differed between participants, from pharmacy records and self-reported use, leading to misclassification:

library(episensr)
chien %>%
    misclassification(., type = "exposure", bias_parms = c(.56, .58, .99, .97))

## --Observed data-- 
##          Outcome: BC+ 
##        Comparing: AD+ vs. AD- 
## 
##     AD+ AD-
## BC+ 118 832
## BC- 103 884
## 
##                                        2.5%     97.5%
## Observed Relative Risk: 1.1012443 0.9646019 1.2572431
##    Observed Odds Ratio: 1.2172330 0.9192874 1.6117443
## ---
##                                                              2.5%    97.5%
## Misclassification Bias Corrected Relative Risk: 1.272939                  
##    Misclassification Bias Corrected Odds Ratio: 1.676452 1.150577 2.442679

Controls and cases also enrolled into the study at different rates. We can combine the misclassification bias with a selection bias thanks to the function multiple.bias:

chien %>%
    misclassification(., type = "exposure", bias_parms = c(.56, .58, .99, .97)) %>%
    multiple.bias(., bias_function = "selection", bias_parms = c(.73, .61, .82, .76))

## 
## Multiple bias analysis
## ---
##                                                 
## Selection Bias Corrected Relative Risk: 1.192461
##    Selection Bias Corrected Odds Ratio: 1.512206

The association between antidepressant use and breast cancer was adjusted for various confounders (race/ethnicity, income, etc.). None of these confounders were found to change the association by more than 10%. However, for illustration, we can add the effect of a potential confounder (e.g. physical activity):

chien %>%
    misclassification(., type = "exposure", bias_parms = c(.56, .58, .99, .97)) %>%
    multiple.bias(., bias_function = "selection",
                  bias_parms = c(.73, .61, .82, .76)) %>%
    multiple.bias(., bias_function = "confounders",
                  type = "OR", bias_parms = c(.92, .3, .44))

## 
## Multiple bias analysis
## ---
##                                         OR_conf
## Standardized Morbidity Ratio: 1.494938 1.011609
##              Mantel-Haenszel: 1.494938 1.011609

We can do the same in a probabilistic framework:

Misclassification bias

set.seed(123)
mod1 <- chien %>%
    probsens(., type = "exposure", reps = 100000,
             seca.parms = list("trapezoidal", c(.45, .5, .6, .65)),
             seexp.parms = list("trapezoidal", c(.4, .48, .58, .63)),
             spca.parms = list("trapezoidal", c(.95, .97, .99, 1)),
             spexp.parms = list("trapezoidal", c(.96, .98, .99, 1)),
             corr.se = .8, corr.sp = .8)
mod1

## --Observed data-- 
##          Outcome: BC+ 
##        Comparing: AD+ vs. AD- 
## 
##     AD+ AD-
## BC+ 118 832
## BC- 103 884
## 
##                                         2.5%     97.5%
##  Observed Relative Risk: 1.1012443 0.9646019 1.2572431
##     Observed Odds Ratio: 1.2172330 0.9192874 1.6117443
## ---
##                                                  Median 2.5th percentile
##            Relative Risk -- systematic error: 1.0712296        0.9538245
##               Odds Ratio -- systematic error: 1.1480025        0.9125458
## Relative Risk -- systematic and random error: 1.0704809        0.8993218
##    Odds Ratio -- systematic and random error: 1.1477524        0.8001310
##                                               97.5th percentile
##            Relative Risk -- systematic error:         1.1895215
##               Odds Ratio -- systematic error:         1.4344889
## Relative Risk -- systematic and random error:         1.2667664
##    Odds Ratio -- systematic and random error:         1.6374497

plot(mod1, "or")

Selection bias

set.seed(123)
mod2 <- chien %>%
    probsens.sel(., reps = 100000,
                 case.exp = list("beta", c(8.08, 24.25)),
                 case.nexp = list("trapezoidal", c(.75, .85, .95, 1)),
                 ncase.exp = list("beta", c(12.6, 50.4)),
                 ncase.nexp = list("trapezoidal", c(0.7, 0.8, 0.9, 1)))
mod2

## --Observed data-- 
##          Outcome: BC+ 
##        Comparing: AD+ vs. AD- 
## 
##     AD+ AD-
## BC+ 118 832
## BC- 103 884
## 
##                                     2.5%     97.5%
## Observed Odds Ratio: 1.2172330 0.9192874 1.6117443
## ---
##                                               Median 2.5th percentile
##            Odds Ratio -- systematic error: 1.0232545        0.4594409
## Odds Ratio -- systematic and random error: 1.0225083        0.4365052
##                                            97.5th percentile
##            Odds Ratio -- systematic error:         2.4217989
## Odds Ratio -- systematic and random error:         2.5253069

plot(mod2, "or")

Confounding

set.seed(123)
mod3 <- chien %>%
    probsens.conf(., reps = 100000,
                  prev.exp = list("beta", c(24.9, 58.1)),
                  prev.nexp = list("beta", c(42.9, 54.6)),
                  risk = list("trapezoidal", c(.2, .58, 1.01, 1.24)))
mod3

## --Observed data-- 
##          Outcome: BC+ 
##        Comparing: AD+ vs. AD- 
## 
##     AD+ AD-
## BC+ 118 832
## BC- 103 884
## 
##                                        2.5%     97.5%
## Observed Relative Risk: 1.1012443 0.9646019 1.2572431
##    Observed Odds Ratio: 1.2172330 0.9192874 1.6117443
## ---
##                                             Median 2.5th percentile
##            RR (SMR) -- systematic error: 1.0693786        0.9292659
## RR (SMR) -- systematic and random error: 1.0595660        0.8854500
##            OR (SMR) -- systematic error: 1.1559562        0.8145649
## OR (SMR) -- systematic and random error: 1.1302762        0.7492557
##                                          97.5th percentile
##            RR (SMR) -- systematic error:         1.1259703
## RR (SMR) -- systematic and random error:         1.2329113
##            OR (SMR) -- systematic error:         1.2774908
## OR (SMR) -- systematic and random error:         1.5561690

plot(mod3, "or")

Misclassification then selection

set.seed(123)
chien %>%
    probsens(., type = "exposure", reps = 100000,
             seca.parms = list("trapezoidal", c(.45, .5, .6, .65)),
             seexp.parms = list("trapezoidal", c(.4, .48, .58, .63)),
             spca.parms = list("trapezoidal", c(.95, .97, .99, 1)),
             spexp.parms = list("trapezoidal", c(.96, .98, .99, 1)),
             corr.se = .8, corr.sp = .8) %>%
    multiple.bias(., bias_function = "probsens.sel",
                  case.exp = list("beta", c(8.08, 24.25)),
                  case.nexp = list("trapezoidal", c(.75, .85, .95, 1)),
                  ncase.exp = list("beta", c(12.6, 50.4)),
                  ncase.nexp = list("trapezoidal", c(0.7, 0.8, 0.9, 1)))

## 
## Multiple bias analysis
## ---
##                                               Median 2.5th percentile
##            Odds Ratio -- systematic error: 0.9796433        0.4763065
## Odds Ratio -- systematic and random error: 0.9844851        0.4568008
##                                            97.5th percentile
##            Odds Ratio -- systematic error:         2.1640513
## Odds Ratio -- systematic and random error:         2.2700749

Misclassification, selection, and confounding

set.seed(123)
mod6 <- chien %>%
    probsens(., type = "exposure", reps = 100000,
             seca.parms = list("trapezoidal", c(.45, .5, .6, .65)),
             seexp.parms = list("trapezoidal", c(.4, .48, .58, .63)),
             spca.parms = list("trapezoidal", c(.95, .97, .99, 1)),
             spexp.parms = list("trapezoidal", c(.96, .98, .99, 1)),
             corr.se = .8, corr.sp = .8) %>%
    multiple.bias(., bias_function = "probsens.sel",
                  case.exp = list("beta", c(8.08, 24.25)),
                  case.nexp = list("trapezoidal", c(.75, .85, .95, 1)),
                  ncase.exp = list("beta", c(12.6, 50.4)),
                  ncase.nexp = list("trapezoidal", c(0.7, 0.8, 0.9, 1))) %>%
    multiple.bias(., bias_function = "probsens.conf",
                  prev.exp = list("beta", c(24.9, 58.1)),
                  prev.nexp = list("beta", c(42.9, 54.6)),
                  risk = list("trapezoidal", c(.2, .58, 1.01, 1.24)))
mod6

## 
## Multiple bias analysis
## ---
##                                             Median 2.5th percentile
##            RR (SMR) -- systematic error: 1.0442703        0.9122304
## RR (SMR) -- systematic and random error: 1.0352565        0.8650944
##            OR (SMR) -- systematic error: 1.0944021        0.7603153
## OR (SMR) -- systematic and random error: 1.0723193        0.7091321
##                                          97.5th percentile
##            RR (SMR) -- systematic error:         1.1014633
## RR (SMR) -- systematic and random error:         1.1996913
##            OR (SMR) -- systematic error:         1.2164556
## OR (SMR) -- systematic and random error:         1.4492568

plot(mod6, "or")

plot(mod6, "or_tot")