Uncontrolled and induced confounding: lessons from simulations

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Uncontrolled and induced confounding lessons
from simulations
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Contents

• Simulations of the 5-variables setting
• Investigating the direction and extent of the
  bias
• A special case the instrumental variables
  approach

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The setting determinants of SBP
Maternal education (years)
Maternal CRP (mg/L)
Maternal SBP (mmHg)
SBP (mmHg)
Birth weight (kg)

• To run the simulations
• -generate each variable in sequence
• Each with independent normally distributed errors
• Relations defined by regression coefficients
  corresponding to the arrows

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N(2,1)
Maternal CRP (mg/L)
Maternal education (years)
N(8,2)
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N(2,1)
Maternal CRP (mg/L)
Maternal education (years)
N(8,2)
1
-2
Maternal SBP (mmHg)
N(100,5.8)
Mat_SBP 100 1 Mat_CRP 2 Mat_Educ e
eN(0,4)
Var(Mat_SBP) 11 44 16 33 and
sd(Mat_SBP)5.8
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N(2,1)
Maternal CRP (mg/L)
Maternal education (years)
N(8,2)
1
-2
Maternal SBP (mmHg)
N(100,5.8)
0.2
-0.05
Birth weight (kg)
N(3.5, 0.7)
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N(2,1)
Maternal CRP (mg/L)
Maternal education (years)
N(8,2)
1
-2
0.25
Maternal SBP (mmHg)
N(100,5.8)
0.2
-0.05
0.02
-2
SBP (mmHg)
Birth weight (kg)
N(3.5, 0.7)
N(90,4.3)
5 models fitted to estimate the effect of birth
weight on SBP - crude - adjusted for Mat
SBP - adjusted for Mat SBP Mat_CRP -
adjusted for Mat SBP Mat_educ - adjusted for
all three
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Estimates of the effect of birth weight (true
value is -2)
After 10,000 simulations
Sample size 100
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Estimates of the effect of birth weight (true
value is -2)
After 10,000 simulations
Sample size 1000

-ve bias
ve bias
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What determines the magnitude and direction of
the bias?

• the backdoor paths.
• Consider
• A- omitted confounders crude estimate
• B- wrong confounder controlled for Mat_SBP

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A- Omitted confounder
TRUE MODEL x3 ß31 x1 ß32 x2 e3
Bias
FITTED MODEL x3 ß31 x1 e3
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A- Omitted confounder
x2
ß12
ß32
X3
X1
ß31
Bias -increases with the variance of X2 and
conditional variance of X3 -increases with the
correlations along the indirect paths -its
direction is determined by the signs of these
correlations
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In a more complex setting
ß32
ß31
ß42
ß51
ß43
ß53
ß54
Crude estimate based on x5 ß54 x4 e5
Bias is affected as before by signs of the
correlations / regression coefficients, and the
variances along the indirect paths
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-
ß32
ß31

ß42
?51

-
ß43
ß53

ß54
In our example, bias depends on ß34
ß53 ß24 ß32 ß53
ß24 ß32 ß51 ß14 ß51

• sum of biases is negative
• crude estimate lt ß54

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B- wrong confounder
X2
X1
?12
X3
X5
X4
Controlling for X3 induces a correlation between
X1 and X2

• It turns out that ?12
• has the opposite sign of (ß32 ß13 )
• increases with var(X134) and var(X214)
• but decreases with var(X34) and var(X4)

• In our example
• ?12 is positive
• adj estimate gt ß54

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When ß32 (Mat Educ to mat SBP) is negative
bias increases with var(Mat CRP) bias
decreases with var(Mat SBP)

ß543

Increasing Var(Mat_CRP)
True value
Increasing Var(Mat SBP)
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When ß32 (Mat Educ to mat SBP) is positive
bias increases with var(Mat CRP) bias
decreases with var(Mat SBP)
True value

ß543

Increasing Var(Mat_CRP)
Increasing Var(Mat SBP)
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A special example the IV approach
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Instrumental variables
What if this is not zero?
Maternal CRP (mg/L)
Maternal SBP (mmHg)
2
ß53
SBP (mmHg)
U
ß53
If U is unmeasured, we can obtain unbiased
estimates of using the regression of SBP on Mat
CRP and of SBP on Mat CRP
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True value
IV estimate
Increasing strength of the instrument
Regression coefficient relating the instrument
with the unmeasured confounder