Selection bias in clinical sciences
Selection bias in clinical sciences is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
In epidemiology, retrospective analyses have well-known limitations compared to prospective studies.
One such limitation is the occurrence of selection bias in the choice of subjects between treated and untreated groups about whom the data is collected. For example, a treatment may have only been given to persons who were less severely ill, which would bias the results in favor of such subjects appearing to have done better because of the treatment when the biased group is then compared to those who who did not receive the study treatment. Or, in a retrospective study, there may a choice to place subjects in a particular study group using a method which is inadvertently biased by the outcome being measured. Creating a programming example of a simulation of such selection bias in the design of a retrospective study is the topic of this task.
The genuine, historical example (only partially approximated in this task) is of a study done of persons who, over a course of 180 days, may or may not have become infected with Covid-19. Prior to becoming ill, these subjects may or may not have taken an available medication, which was actually taken on a particular schedule not used here, but is approximated by stating the medication was taken in doses of 3, 6, or 9 mg daily. The historical study then divided its subjects into three groups based on their cumulative dosage of the study medication:
- Group UNTREATED were those who did not take the study medication at all before they got Covid-19, including those who exited the study period without Covid-19 and having never taken the study medication.
- Group IRREGULAR is those who took the study medication but whose cumulative dose was less than a certain amount (approximated for our purposes as 100 mg) before they either came down with Covid-19 during the study or the study period ended.
- Group REGULAR is those who took (our approximation is >= 100 mg) of the study medication either before they came down with Covid-19 or took >= 100 mg by the end of the study and never became infected during the study.
- Assumptions for the study simulation programming task
- Daily risk of getting Covid-19 infection for each subject was 0.1% per day, or 18% over the 180 cumulative days of the study.
- The probability of starting treatment medication for anyone not already taking it was 0.5% per day. For those who started medication, the chance of continuing the treatment was increased 50-fold to 25% each day, since most who started the medication continued to take it to some extent.
- Study dose per day is random between the approximation for the simulation of 3, 6 and 9 mg. The daily cumulative dosage is used to determine the group the subject is in, unless a subject develops Covid-19. If a subject was diagnosed with Covid-19, their group at the time of that diagnosis is used in the statistical analysis of that group.
- Task
- Create a simulation of the subjects, keeping track of their medication dosages, group membership, and Covid-19 status during the study.
- Use at least 1000 subjects in the simulation over the 180 days (historically, the study size was 80,000).
- Statistics used are to be the Kruscal statistic for the analysis of multiple groups, with the boolean study outcome variable whether the subject got Covid-19 during the study period, analyzed versus category.
- You should get a statistical result highly favoring the REGULAR group.
- Stretch task
- Show monthly outcomes.
A note regarding outcome: Note that by simulation design all subjects must have an IDENTICAL risk, that is 0.1 per cent or p = 0.001 per day, of developing Covid-19. Because of the design, any statistical differences between the groups CANNOT come from an influence of the treatment on that risk, but must come from some other feature of the study design.
- See also
Julia
using HypothesisTests
@enum TreatmentClass Untreated Irregular Regular
mutable struct Subject
cum_dose::Float64
treatment_class::TreatmentClass
had_covid::Bool
end
function update!(subjects::Vector{Subject}, pcovid = 0.001, pstart = 0.005, pdosing = 0.25, drange = 3:3:9)
for subj in subjects
if subj.had_covid
continue
elseif rand() < pcovid
subj.had_covid = true
elseif subj.cum_dose > 0 && rand() <= pdosing || subj.cum_dose == 0 && rand() <= pstart
subj.cum_dose += rand(drange)
subj.treatment_class =
subj.cum_dose == 0 ? Untreated : subj.cum_dose >= 100 ? Regular : Irregular
end
end
end
function run_study(N = 10_000, duration = 180)
population = [Subject(0.0, Untreated, false) for _ in 1:N]
unt, unt_covid, irr, irr_covid, reg, reg_covid = 0, 0, 0, 0, 0, 0
println("Population size $N, daily infection risk 0.1%")
for day in 1:duration
update!(population)
if day % 30 == 0
println("\nDay $day:")
unt = count(s -> s.treatment_class == Untreated, population)
unt_covid = count(s -> (s.treatment_class == Untreated) && s.had_covid, population)
println("Untreated: N = $unt, with infection = $unt_covid")
irr = count(s -> s.treatment_class == Irregular, population)
irr_covid = count(s -> (s.treatment_class == Irregular) && s.had_covid, population)
println("Irregular Use: N = $irr, with infection = $irr_covid")
reg = count(s -> s.treatment_class == Regular, population)
reg_covid = count(s -> (s.treatment_class == Regular) && s.had_covid, population)
println("Regular Use: N = $reg, with infection = $reg_covid")
end
if day == 90
println("\nAt midpoint, Infection case percentages are:")
println(" Untreated : ", Float16(100 * unt_covid / unt))
println(" Irregulars: ", Float16(100 * irr_covid / irr))
println(" Regulars : ", Float16(100 * reg_covid / reg))
end
end
println("\nAt study end, Infection case percentages are:")
println(" Untreated : ", Float16(100 * unt_covid / unt), " of group size of $unt")
println(" Irregulars: ", Float16(100 * irr_covid / irr), " of group size of $irr")
println(" Regulars : ", Float16(100 * reg_covid / reg), " of group size of $reg")
untreated = [s.had_covid for s in population if s.treatment_class == Untreated]
irregular = [s.had_covid for s in population if s.treatment_class == Irregular]
regular = [s.had_covid for s in population if s.treatment_class == Regular]
println("\n\n Final statistics:\n")
@show KruskalWallisTest(untreated, irregular, regular)
end
run_study()
- Output:
Population size 10000, daily infection risk 0.1%
Day 30:
Untreated: N = 8633, with infection = 288
Irregular Use: N = 1367, with infection = 21
Regular Use: N = 0, with infection = 0
Day 60:
Untreated: N = 7513, with infection = 519
Irregular Use: N = 2325, with infection = 79
Regular Use: N = 162, with infection = 2
Day 90:
Untreated: N = 6559, with infection = 692
Irregular Use: N = 2362, with infection = 159
Regular Use: N = 1079, with infection = 24
At midpoint, Infection case percentages are:
Untreated : 10.55
Irregulars: 6.73
Regulars : 2.225
Day 120:
Untreated: N = 5794, with infection = 845
Irregular Use: N = 2071, with infection = 221
Regular Use: N = 2135, with infection = 72
Day 150:
Untreated: N = 5115, with infection = 987
Irregular Use: N = 1835, with infection = 266
Regular Use: N = 3050, with infection = 156
Day 180:
Untreated: N = 4538, with infection = 1106
Irregular Use: N = 1654, with infection = 302
Regular Use: N = 3808, with infection = 263
At study end, Infection case percentages are:
Untreated : 24.38 of group size of 4538
Irregulars: 18.27 of group size of 1654
Regulars : 6.906 of group size of 3808
Final statistics:
KruskalWallisTest(untreated, irregular, regular) = Kruskal-Wallis rank sum test (chi-square approximation)
-------------------------------------------------------
Population details:
parameter of interest: Location parameters
value under h_0: "all equal"
point estimate: NaN
Test summary:
outcome with 95% confidence: reject h_0
one-sided p-value: <1e-99
Details:
number of observation in each group: [4538, 1654, 3808]
χ²-statistic: 457.179
rank sums: [2.44308e7, 8.39891e6, 1.71753e7]
degrees of freedom: 2
adjustment for ties: 0.417533
Kruskal-Wallis rank sum test (chi-square approximation)
-------------------------------------------------------
Population details:
parameter of interest: Location parameters
value under h_0: "all equal"
point estimate: NaN
Test summary:
outcome with 95% confidence: reject h_0
one-sided p-value: <1e-99
Details:
number of observation in each group: [4538, 1654, 3808]
χ²-statistic: 457.179
rank sums: [2.44308e7, 8.39891e6, 1.71753e7]
degrees of freedom: 2
adjustment for ties: 0.417533
Python
''' Rosetta code rosettacode.org/wiki/Study_Bias_in_Clinical_Sciences '''
from random import randrange
from numpy.random import rand
from scipy.stats import kruskal
UNTREATED = 0
IRREGULAR = 1
REGULAR = 2
DOSE_FOR_REGULAR = 100
class Subject:
''' A subject for the study '''
def __init__(self):
self.cum_dose = 0.0
self.category = UNTREATED
self.had_covid = False
self.update_count = 0
def update(self, p_covid=0.001, p_starting_treatment=0.005, p_redose=0.25, drange=(3, 10, 3)):
''' daily update on the subject to check for infection and randomly dose. '''
if not self.had_covid:
if rand() < p_covid:
self.had_covid = True
elif (self.cum_dose == 0 and rand() < p_starting_treatment) or\
(self.cum_dose > 0 and rand() < p_redose):
self.cum_dose += randrange(*drange)
self.categorize()
self.update_count += 1
def categorize(self):
''' Set treatment category based on cumulative treatment taken. '''
self.category = UNTREATED if self.cum_dose == 0 else REGULAR if\
self.cum_dose >= DOSE_FOR_REGULAR else IRREGULAR
return self.category
def run_study(num_subjects=1000, duration=180, interval=30):
''' Run the study using the population of size `N` for `duration` days. '''
population = [Subject() for _ in range(num_subjects)]
unt, unt_covid, irr, irr_covid, reg, reg_covid = 0, 0, 0, 0, 0, 0
print(f'Total subjects: {num_subjects:,}')
for day in range(duration):
for subj in population:
subj.update()
if (day + 1) % interval == 0:
print(f'\nDay {day + 1}:')
unt = sum(s.category == UNTREATED for s in population)
unt_covid = sum(s.category ==
UNTREATED and s.had_covid for s in population)
print(f'Untreated: N = {unt}, with infection = {unt_covid}')
irr = sum(s.category == IRREGULAR for s in population)
irr_covid = sum(s.category ==
IRREGULAR and s.had_covid for s in population)
print(f'Irregular Use: N = {irr}, with infection = {irr_covid}')
reg = sum(s.category == REGULAR for s in population)
reg_covid = sum(s.category ==
REGULAR and s.had_covid for s in population)
print(f'Regular Use: N = {reg}, with infection = {reg_covid}')
if day == duration // 2 - 1:
print('\nAt midpoint, Infection case percentages are:')
print(' Untreated : ', 100 * unt_covid / unt)
print(' Irregulars: ', 100 * irr_covid / irr)
print(' Regulars : ', 100 * reg_covid / reg)
print('\nAt study end, Infection case percentages are:')
print(f' Untreated : {100 * unt_covid / unt} of group size of {unt}')
print(f' Irregulars: {100 * irr_covid / irr} of group size of {irr}')
print(f' Regulars : {100 * reg_covid / reg} of group size of {reg}')
untreated = [
s.had_covid for s in population if s.category == UNTREATED]
irregular = [
s.had_covid for s in population if s.category == IRREGULAR]
regular = [s.had_covid for s in population if s.category == REGULAR]
print('\nFinal statistics: ', kruskal(untreated, irregular, regular))
run_study()
- Output:
Total subjects: 1,000 Day 30: Untreated: N = 872, with infection = 25 Irregular Use: N = 128, with infection = 2 Regular Use: N = 0, with infection = 0 Day 60: Untreated: N = 755, with infection = 55 Irregular Use: N = 222, with infection = 8 Regular Use: N = 23, with infection = 1 Day 90: Untreated: N = 671, with infection = 70 Irregular Use: N = 219, with infection = 13 Regular Use: N = 110, with infection = 4 At midpoint, Infection case percentages are: Untreated : 10.432190760059612 Irregulars: 5.936073059360731 Regulars : 3.6363636363636362 Day 120: Untreated: N = 600, with infection = 88 Irregular Use: N = 189, with infection = 17 Regular Use: N = 211, with infection = 8 Day 150: Untreated: N = 514, with infection = 108 Irregular Use: N = 194, with infection = 21 Regular Use: N = 292, with infection = 16 Day 180: Untreated: N = 447, with infection = 119 Irregular Use: N = 189, with infection = 26 Regular Use: N = 364, with infection = 26 At study end, Infection case percentages are: Untreated : 26.62192393736018 of group size of 447 Irregulars: 13.756613756613756 of group size of 189 Regulars : 7.142857142857143 of group size of 364 Final statistics: KruskalResult(statistic=55.48204323818349, pvalue=8.95833684545873e-13)