Figures and data in Misstatements, misperceptions, and mistakes in controlling for covariates in observational research

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Tables

7 figures and 5 tables

Figures

Figure 1

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Agree (a) vs. disagree (b) with the interpretation of Misperception 5a.

Demonstrates a nonlinear and non-monotonic association between body mass index (BMI) and mortality among U.S. adults aged 18–85 years old. This figure suggests that BMI ranging between 23–26 kg/m² formed the nadir of the curve with the best outcome while persons with BMI levels below or above the nadir of the curve experienced increased mortality on average. Source: (Fontaine et al., 2003).

Figure 2

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Association between body mass index and hazard ratio for death among U.S. adults aged 18–85 years old.

Figure 3

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Causal relationships of health outcome, dietary fat consumption, and the belief that consumption of dietary fat is not dangerous.

Direction of arrows represents causal directions and *λ_A, λ_B, β_A*, and *β_B* are structural coefficients.

Figure 4

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Causal relationships of outcome, covariate, and potentially biasing covariate (PBC).

Direction of arrows represents causal directions and *λ_z,* α*_z,* α*_x, β_z*, and βx are structural coefficients. The error terms e1 and e2 have variances chosen so Y1 and Y2 have variances 1 (see the Appendices for more details).

Appendix 1—figure 1

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Appendix 2—figure 1

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Causal relationships of outcome, covariate, and confounding.

Direction of arrows represents causal directions and 𝜆_z, 𝛼_z, 𝛼_x, 𝛽_z, and 𝛽_x are structural coefficients.

Appendix 2—figure 2

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Possible values of $λ_{z}$ based on each choice of the pairs of a, b.

The area shaded in green denotes the area for which a $λ_{z}$ value has a value τ that makes Equation 13 equal zero.

Tables

Appendix 1—table 1

Parameters used to generate simulated data for the simulation studies under Misperception 9.

Scenario	β_A	β_B	λ_A	λ_B	$σ_{ϵ}^{2}$	$σ_{η}^{2}$	$σ_{γ}^{2}$
I	–0.4	0.3	$\sqrt{3} / 2$	$\sqrt{3} / 2$	0.93	0.25	0.25
II	0.4	–0.3	$\sqrt{3} / 2$	$\sqrt{3} / 2$	0.93	0.25	0.25
III	–0.5	0.24	0.8	0.6	0.8076	0.36	0.64
IV	0.5	–0.24	0.8	0.6	0.8076	0.36	0.64

Appendix 1—table 2

Summary of bias when fitting the full model (𝑀_𝐹) and the reduced model (M_R).

The bias is defined as $\hat{β} - β_{A}$ , where $\hat{β}$ is the least-squares estimate under the corresponding model.

Scenario	n=500		n=1000		n=2000
	M_F	M_R	M_F	M_R	M_F	M_R
I	–0.0007	0.2248	0.0001	0.2251	–0.0001	0.2249
II	0.0005	–0.2249	0.0003	–0.2249	0.0002	–0.2248
III	–0.0001	0.24	0.0003	0.2405	–0.0003	0.2399
IV	0.0004	–0.2396	–0.0002	–0.24	–0.0005	–0.2402

Appendix 2—table 1

The correlation matrix among Z, X, Y₂, and Y₁ without selecting on Y₁.

$Σ$	Z	X	Y₂	Y₁
Z	1	$λ_{z}$	$α_{z} + α_{x} λ_{z}$	$β_{z} + β_{x} λ_{z}$
X	$λ_{z}$	1	$α_{x} + α_{z} λ_{z}$	$β_{x} + β_{z} λ_{z}$
Y₂	$α_{z} + α_{x} λ_{z}$	$α_{x} + α_{z} λ_{z}$	1	$(α_{z} + α_{x} λ_{z}) β_{z} + β_{x} (α_{x} + α_{z} λ_{z})$
Y₁	$β_{z} + β_{x} λ_{z}$	$β_{x} + β_{z} λ_{z}$	$(α_{z} + α_{x} λ_{z}) β_{z} + β_{x} (α_{x} + α_{z} λ_{z})$	1

Appendix 2—table 2

The squared correlation and slope of regression.

Quantity of interest	Without selection on Y₁	With selection on Y₁
Squared (zero-order) correlation of X and Y₂	${(λ_{z} α_{z} + α_{x})}^{2}$	${(\frac{{\tilde{σ}}_{23}}{\sqrt{{\tilde{σ}}_{22}} \sqrt{{\tilde{σ}}_{33}}})}^{2}$
Squared (partial) correlation of X and Y₂, controlling for Z	$\frac{{(- {\tilde{σ}}_{23})}^{2}}{{\tilde{σ}}_{22} {\tilde{σ}}_{33}}$	${(\frac{{\tilde{σ}}_{23}^{}}{\sqrt{{\tilde{σ}}_{22}^{}} \sqrt{{\tilde{σ}}_{33}^{*}}})}^{2}$
Slope of univariable regression of Y₂ on X	$(λ_{z} α_{z} + α_{x})$	$\frac{{\tilde{σ}}_{23}}{{\tilde{σ}}_{22}}$
Partial slope of regression of Y₂ on X, controlling for Z	$\frac{α_{x} + α_{x} λ_{z}^{2}}{1 - λ_{z}^{2}}$	$(\frac{\sqrt{{\tilde{σ}}_{33}}}{\sqrt{{\tilde{σ}}_{22}}}) (\frac{{\tilde{ρ}}_{23} - {\tilde{ρ}}_{13} {\tilde{ρ}}_{12}}{1 - {\tilde{ρ}}_{12}^{2}})$

Appendix 2—table 3

Estimated Average Bias of $α_{x}$ Under Various Scenarios.

Where $τ, β_{x}, β_{z}$ are selected to Induce a Zero Correlation Between $X$ and $Z$ After Selecting on $Y_{1}$ . Results are based on sample size of n=50,000 and 1000 samples obtained from the data-generating model described above.

$a$	$b$	$λ_{z} (m a x)$	$β_{x}$	$β_{z}$	$λ_{z}$	$τ$	All data		Select on $Y_{1} > τ$
$a$	$b$	$λ_{z} (m a x)$	$β_{x}$	$β_{z}$	$λ_{z}$	$τ$	$α_{x . z}$	$α_{x}$	$α_{x . z}$	$α_{x}$
0.2	0.2	0.0422	0.2000	0.1952	0.0200	–0.5774	–0.0000	0.0118	–0.0002	–0.0002
0.2	0.2	0.0422	0.1999	0.1946	0.0350	1.3809	–0.0001	0.0208	–0.0004	–0.0008
0.2	0.9	0.5519	0.1931	0.8361	0.2700	0.4647	–0.0001	0.1620	–0.0002	–0.0001
0.2	0.9	0.5519	0.1857	0.8175	0.4000	1.8276	–0.0001	0.2398	–0.0002	–0.0003
0.9	0.2	0.2277	0.8936	0.0683	0.1200	0.6717	0.0001	0.0721	0.0005	0.0009
0.9	0.2	0.2277	0.8842	0.0598	0.1900	3.0584	0.0001	0.1140	–0.0080	–0.0118
0.9	0.9	0.5156	0.8710	0.2383	0.2600	–0.1627	–0.0002	0.1556	–0.0001	–0.0003
0.9	0.9	0.5156	0.8207	0.1818	0.4500	2.3590	–0.0002	0.2699	0.0011	–0.0005

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Xiaoxin Yu
Roger S Zoh
David A Fluharty
Luis M Mestre
Danny Valdez
Carmen D Tekwe
Colby J Vorland
Yasaman Jamshidi-Naeini
Sy Han Chiou
Stella T Lartey
David B Allison

(2024)

Misstatements, misperceptions, and mistakes in controlling for covariates in observational research

eLife 13:e82268.

https://doi.org/10.7554/eLife.82268

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Agree (a) vs. disagree (b) with the interpretation of Misperception 5a.

Association between body mass index and hazard ratio for death among U.S. adults aged 18–85 years old.

Causal relationships of health outcome, dietary fat consumption, and the belief that consumption of dietary fat is not dangerous.

Causal relationships of outcome, covariate, and potentially biasing covariate (PBC).

Causal relationships of health outcome, dietary fat consumption, and the belief that consumption of dietary fat is not dangerous.

Causal relationships of outcome, covariate, and confounding.

Possible values of λz based on each choice of the pairs of a, b.

Parameters used to generate simulated data for the simulation studies under Misperception 9.

Summary of bias when fitting the full model (𝑀𝐹) and the reduced model (MR).

The correlation matrix among Z, X, Y2, and Y1 without selecting on Y1.

The squared correlation and slope of regression.

Estimated Average Bias of αx Under Various Scenarios.

Downloads (link to download the article as PDF)

Open citations (links to open the citations from this article in various online reference manager services)

Cite this article (links to download the citations from this article in formats compatible with various reference manager tools)

Possible values of $λ_{z}$ based on each choice of the pairs of a, b.

Summary of bias when fitting the full model (𝑀_𝐹) and the reduced model (M_R).

The correlation matrix among Z, X, Y₂, and Y₁ without selecting on Y₁.

Estimated Average Bias of $α_{x}$ Under Various Scenarios.