```
library(data.table)
library(magrittr)
library(ggplot2)
```

```
set.seed(4)
<- data.table(id = 1:100000)
d := rbinom(.N, 1, 0.2)]
d[, is_smoker := ifelse(is_smoker==T, 0.05, 0.3)]
d[, probability_of_exercises_frequently := rbinom(.N, 1, probability_of_exercises_frequently)]
d[, exercises_frequently := rbinom(.N, 1, 0.2)]
d[, has_good_genes := rbinom(.N, 1, 0.2)]
d[, wears_hats_frequently
:= 30 - 10 * is_smoker + 5 * exercises_frequently + 8 * has_good_genes + rnorm(.N, mean = 0, sd = 3)]
d[, lung_function
:= 1/.N]
d[, probability_of_selection_uniform
:= ifelse(is_smoker==T, 5, 1)]
d[, probability_of_selection_oversample_smoker := probability_of_selection_oversample_smoker/sum(probability_of_selection_oversample_smoker)]
d[, probability_of_selection_oversample_smoker
# We have a dataset with oversampled smokers
<- d[sample(1:.N, size = 5000, prob = probability_of_selection_oversample_smoker)]
d_oversampled_smokers <- mean(d$is_smoker)/mean(d_oversampled_smokers$is_smoker)) (weight_smoker
```

`[1] 0.3700516`

`<- mean(!d$is_smoker)/mean(!d_oversampled_smokers$is_smoker)) (weight_non_smoker `

`[1] 1.747289`

```
:= ifelse(is_smoker==T, weight_smoker, weight_non_smoker)]
d_oversampled_smokers[, weights
# The real associations:
# is_smoker: -10 (also associated with exercises_frequently!)
# exercises_frequently: +5 (also associated with is_smoker!)
# has_good_genes: +8 (only associated with outcome, not with other exposures)
# wears_hats_frequently: 0 (not associated with outcome nor other exposures)
summary(lm(lung_function ~ is_smoker + exercises_frequently + has_good_genes + wears_hats_frequently, data=d))
```

```
Call:
lm(formula = lung_function ~ is_smoker + exercises_frequently +
has_good_genes + wears_hats_frequently, data = d)
Residuals:
Min 1Q Median 3Q Max
-12.6549 -2.0112 0.0055 2.0045 12.1835
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 30.02288 0.01421 2112.751 <2e-16 ***
is_smoker -10.05071 0.02427 -414.150 <2e-16 ***
exercises_frequently 4.99532 0.02250 221.992 <2e-16 ***
has_good_genes 7.98073 0.02355 338.831 <2e-16 ***
wears_hats_frequently -0.01537 0.02360 -0.651 0.515
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.989 on 99995 degrees of freedom
Multiple R-squared: 0.7975, Adjusted R-squared: 0.7975
F-statistic: 9.847e+04 on 4 and 99995 DF, p-value: < 2.2e-16
```

```
# When we run the model in the full data, excluding is_smoker, we get the following associations:
# exercises_frequently: +7.2 (biased from association with is_smoker)
# has_good_genes: +8 (not biased)
# wears_hats_frequently: 0 (not biased)
summary(lm(lung_function ~ exercises_frequently + has_good_genes + wears_hats_frequently, data=d))
```

```
Call:
lm(formula = lung_function ~ exercises_frequently + has_good_genes +
wears_hats_frequently, data = d)
Residuals:
Min 1Q Median 3Q Max
-21.056 -2.670 0.928 3.445 14.532
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.747e+01 2.109e-02 1302.327 <2e-16 ***
exercises_frequently 7.172e+00 3.605e-02 198.941 <2e-16 ***
has_good_genes 7.958e+00 3.881e-02 205.044 <2e-16 ***
wears_hats_frequently 4.393e-04 3.888e-02 0.011 0.991
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 4.926 on 99996 degrees of freedom
Multiple R-squared: 0.4502, Adjusted R-squared: 0.4502
F-statistic: 2.73e+04 on 3 and 99996 DF, p-value: < 2.2e-16
```

```
# When we run the model in the biased data, with oversampling of smokers (that has also an association with the outcome):
# exercises_frequently: +9.8 (biased from association with is_smoker and the biased sampling)
# has_good_genes: +7.6 (not biased)
# wears_hats_frequently: +0.3 (not biased)
summary(lm(lung_function ~ exercises_frequently + has_good_genes + wears_hats_frequently, data=d_oversampled_smokers))
```

```
Call:
lm(formula = lung_function ~ exercises_frequently + has_good_genes +
wears_hats_frequently, data = d_oversampled_smokers)
Residuals:
Min 1Q Median 3Q Max
-14.4131 -4.3613 -0.6571 4.4586 17.3734
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 23.7436 0.1014 234.191 <2e-16 ***
exercises_frequently 9.8631 0.2140 46.095 <2e-16 ***
has_good_genes 7.6164 0.1967 38.726 <2e-16 ***
wears_hats_frequently 0.3363 0.1905 1.765 0.0776 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 5.501 on 4996 degrees of freedom
Multiple R-squared: 0.4221, Adjusted R-squared: 0.4218
F-statistic: 1217 on 3 and 4996 DF, p-value: < 2.2e-16
```

```
# Run the model in the biased data, with weights:
# exercises_frequently: +7.4 (biased from association with is_smoker)
# has_good_genes: +7.6 (not biased)
# wears_hats_frequently: +0.3 (not biased)
summary(lm(lung_function ~ exercises_frequently + has_good_genes + wears_hats_frequently, data=d_oversampled_smokers, weights = weights))
```

```
Call:
lm(formula = lung_function ~ exercises_frequently + has_good_genes +
wears_hats_frequently, data = d_oversampled_smokers, weights = weights)
Weighted Residuals:
Min 1Q Median 3Q Max
-12.537 -4.885 -2.767 2.077 18.233
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 27.33919 0.09322 293.291 <2e-16 ***
exercises_frequently 7.40799 0.16055 46.140 <2e-16 ***
has_good_genes 7.60104 0.17490 43.459 <2e-16 ***
wears_hats_frequently 0.26672 0.16734 1.594 0.111
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 4.859 on 4996 degrees of freedom
Multiple R-squared: 0.45, Adjusted R-squared: 0.4497
F-statistic: 1363 on 3 and 4996 DF, p-value: < 2.2e-16
```

```
# Run the model in the biased data, with is_smoker:
# is_smoker: -9.9 (not biased)
# exercises_frequently: +5.3 (not biased)
# has_good_genes: +7.8 (not biased)
# wears_hats_frequently: +0.2 (not biased)
summary(lm(lung_function ~ is_smoker + exercises_frequently + has_good_genes + wears_hats_frequently, data=d_oversampled_smokers))
```

```
Call:
lm(formula = lung_function ~ is_smoker + exercises_frequently +
has_good_genes + wears_hats_frequently, data = d_oversampled_smokers)
Residuals:
Min 1Q Median 3Q Max
-10.6202 -1.9903 -0.0699 1.9855 11.1052
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 29.83110 0.07786 383.149 <2e-16 ***
is_smoker -9.88043 0.08978 -110.051 <2e-16 ***
exercises_frequently 5.25052 0.12300 42.688 <2e-16 ***
has_good_genes 7.79706 0.10630 73.350 <2e-16 ***
wears_hats_frequently 0.15569 0.10296 1.512 0.131
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.973 on 4995 degrees of freedom
Multiple R-squared: 0.8313, Adjusted R-squared: 0.8311
F-statistic: 6152 on 4 and 4995 DF, p-value: < 2.2e-16
```

Conclusion: Biased datasets can be corrected for by either:

- Sample weights
- Including the sampling variables as covariates in the regression model