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Regression analysis of unmeasured confounding

  • Brian Knaeble , Braxton Osting and Mark A. Abramson EMAIL logo
Published/Copyright: May 4, 2020
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Epidemiologic Methods
From the journal Epidemiologic Methods Volume 9 Issue 1

Abstract

When studying the causal effect of x on y, researchers may conduct regression and report a confidence interval for the slope coefficient βx. This common confidence interval provides an assessment of uncertainty from sampling error, but it does not assess uncertainty from confounding. An intervention on x may produce a response in y that is unexpected, and our misinterpretation of the slope happens when there are confounding factors w. When w are measured we may conduct multiple regression, but when w are unmeasured it is common practice to include a precautionary statement when reporting the confidence interval, warning against unwarranted causal interpretation. If the goal is robust causal interpretation then we can do something more informative. Uncertainty, in the specification of three confounding parameters can be propagated through an equation to produce a confounding interval. Here, we develop supporting mathematical theory and describe an example application. Our proposed methodology applies well to studies of a continuous response or rare outcome. It is a general method for quantifying error from model uncertainty. Whereas, confidence intervals are used to assess uncertainty from unmeasured individuals, confounding intervals can be used to assess uncertainty from unmeasured attributes.

Keywords: correlation; model uncertainty; optimization; propensity
Corresponding author: Mark A. Abramson,Department of Mathematics, Utah Valley University, Orem, UT, USA, E-mail:

A Proof of Proposition 1.1

Let e∈ℝn denote the ones vector, W=[e|w1|⋯|wp],

P=W(WtW)−1Wt,  and  E=1neet.

Note that P and E are projection matrices, i. e., P2=P and E2=E. Furthermore, since W includes e as a column, we have that EP=PE=E, (I−P)(I−E)=(I−E)(I−P)=I−P, and P−E is also a projection matrix.

We use P and E to write x^=Px, y^=Py,

σx2=1n|(I−E)x|2, σy2=1n|(I−E)y|2,σx^2=1n|(I−E)Px|2, σy^2=1n|(I−E)Py|2,Rwx2=σx^2σx2=|(I−E)Px|2|(I−E)x|2, Rwy2=σy^2σy2=|(I−E)Py|2|(I−E)y|2,ρxy=〈(I−E)x, (I−E)y〉nσxσy, ρx^y^=〈(I−E)Px, (I−E)Py〉nσx^σy^,

and

ρ(x−x^)(y−y^)=〈(I−P)x, (I−P)y〉|(I−P)x| |(I−P)y|.

Using the above expressions, we compute

nσxσyρxy=〈(I−E)x, (I−E)y〉=〈(I−E)Px, (I−E)Py〉+〈(I−P)x, (I−P)y〉=nσx^σy^ρx^y^+nσxσy1−Rwx21−Rwy2ρ(x−x^)(y−y^).

Dividing both sides by nσxσy, we obtain

(8)ρxy=RwxRwyρx^y^+1−Rwx21−Rwy2ρ(x−x^)(y−y^).

Solving for ρ(x−x^)(y−y^), we obtain an expression for the partial correlation,

(9)ρ(x−x^)(y−y^)=ρxy−RwxRwyρx^y^1−Rwx21−Rwy2.

We now consider our model from (1) rewritten as

y=βx|wx+Wβ+ε,

where β=(β0|w, β1, ⋯, βp). Write X=[x|W] and Q=X(XtX)−1Xt. Note that Q is a projection matrix, and since range(W)⊂range(X), we have that PQ=QP=P and (I−P)(I−Q)=(I−Q)(I−P)=(I−Q). In this notation, the fitted values are given by

Qy=βx|wx+Wβ.

We now add and subtract terms as follows:

Q(y−Py+Py)=βx|w(x−Px+Px)+Wβ

and rearrange to obtain:

Q(I−P)y=βx|w(I−P)x+P(Wβ+βx|wx−y).

We now apply I−P to both sides, take the inner product with x on both sides, use Qx=x, and rearrange to obtain

(10a)βx|w=〈x, (I−P)y〉〈x, (I−P)x〉
(10b)=|y−y^||x−x^|ρ(x−x^)(y−y^)
(10c)=σyσx1−Rwy21−Rwx2ρ(x−x^)(y−y^).

Combining (9) and (10), we obtain the desired result.

B Proof of Proposition 2.1

In the setting of the proof of Proposition 1.1, consider (8). Since ρ(y−y^)(x−x^)∈[−1,1], we obtain

(11)ρxy∈[ξ−, ξ+], where ξ±=RwxRwyρx^y^±1−Rwx21−Rwy2.

Manipulating (11) to isolate ρx^y^ gives (4).

Conversely, if n>p+2, then there are sufficient degrees of freedom for ρ(y−y^)(x−x^) to take any desired value in [−1, 1].

C Proof of Proposition 2.2

We assume that the feasible set Ω is non-empty and is defined by the upper and lower inequality constraints in (3a), (3b), (3c), and (4), for eight inequality constraints in total. We do not allow α−=α+ in (4), but we do allow lx=ux, ly=uy, and or lx^y^=ux^y^. We say an inequality constraint is active, when it holds with equality. In what follows for k=1,2,3 we say exactly k constraints are active when exactly k constraints are active and each active constraint is from a different line of (3a), (3b), (3c), or (4), e. g., lx=Rwx2=ux counts as one constraint not two constraints. There is no ambiguity when zero constraints are active. In three dimensions it is impossible or redundant to have four active constraints.

Recalling our notation l=minΩ(βx|w) and u=maxΩ(βx|w), we say a point (Rwx2, Rwy2, ρx^y^) is optimal if it satisfies either

βx|w(Rwx2, Rwy2, ρx^y^)=l or βx|w(Rwx2, Rwy2, ρx^y^)=u.

To prove Proposition 2.2, we show that any optimal points in Ω must also be within the finite subset of points S. The sets Ω and S depend on the values of the parameters {ρxy, σy/σx;lx, ux, ly, uy, lx^y^, ux^y^}, as described in Section 2. Recall, βx|w=σyσxρxy−RwxRwyρx^y^1−Rwx2. Note that

∂βx|w∂Rwx2=0⇔∂βx|w∂Rwx=0 and ∂βx|w∂Rwy2=0⇔∂βx|w∂Rwx=0

wherever Rwx≠0 and Rwy≠0. To simplify algebra in what follows we compute ∂βx|w/∂Rwx in place of ∂βx|w/∂Rwx2 and ∂βx|w/∂Rwy in place of ∂βx|w/∂Rwy2. Likewise, we infer the constancy of βx|w in Rwx2 or Rwy2 from its constancy in Rwx or Rwy, respectfully.

Wherever no constraints are active ∂βx|w∂βx^y^=σyσx−RwxRwy1−Rwx2≠0 and no optimal points exist. If exactly one constraint is active, we have the following cases. If the constraint is from either (3a) or (3b), then ∂βx|w∂ρx^y^=σyσx−RwxRwy1−Rwx2≠0. If that constraint is (3c), then either ∂βx|w∂Rwy=σyσx−Rwxρx^y^1−Rwx2≠0 or, if ρx^y^=0, βx|w is constant in Rwy2. Finally, if that one constraint is (4), then

(12)βx|w=±σyσx1−Rwy2/1−Rwx2,

which is strictly monotonic in both Rwx2 and Rwy2 on both surfaces

(13)ρx^y^=ρxy±1−Rwx21−Rwy2RwxRwy.

We have thus far shown that two or more constraints must be active in order for a point to be optimal.

Suppose exactly two constraints are active. If those two constraints are (3a) and (3b) then βx|w=σyσxρxy−bxbyρx^y^1−bx2 and either ∂βx|w∂ρx^y^≠0 or βx|w is constant in ρx^y^. If those two constraints are (3a) and (3c) then βx|w=σyσxρxy−bxRwybx^y^1−bx2 and either ∂βx|w∂Rwy≠0 or βx|w is constant in Rwy2. If those two constraints are (3b) and (3c) then βx|w=σyσxρxy−Rwxbybx^y^1−Rwx2 and because ∂βx|w∂Rwx=σyσx−bybx^y^Rwx2+2ρxyRwx2−bybx^y^(1−Rwx2)2 optimal points require

Rwx2=q±2(−bybx^y^, 2ρxy, −bybx^y^)

producing the options in line (7a) of Proposition 2.2.

We now consider exactly two active constraints and require one of them to be (4). If those two constraints are (3a) and (4) then as in (12) we have βx|w strictly monotonic in Rwy2. If those two constraints are (3b) and (4) then as in (12) we have βx|w strictly monotonic in Rwx2. If those two constraints are (3c) and (4) then bx^y^=ρxy±1−Rwx21−Rwy2RwxRwy, or written differently

(14)g(Rwx, Rwy):=bx^y^RwxRwy±1−Rwx21−Rwy2=ρxy.

Along a curve g(Rwx, Rwy)=ρxy, via (12),

h(Rwx, Rwy):=βx|w=±σyσx1−Rwy2/1−Rwx2.

At an optimal point, for some real λ, we must have ∇h=λ∇g (Nocedal and Wright 2006, p. 321), implying equality between

∂g∂Rwx/∂g∂Rwy=bx^y^Rwy±2Rwx(1+Rwy2)1/2(1−Rwx2)−1/2bx^y^Rwx±2Rwy(1+Rwx2)1/2(1−Rwy2)−1/2

and

∂h∂Rwx/∂h∂Rwy=Rwx(1−Rwy2)1/2(1−Rwx2)−3/2−Rwy(1−Rwx2)−1/2(1−Rwy)−1/2=Rwx(1−Rwy2)−Rwy(1−Rwx2),

which in turn implies Rwx=Rwy. Solving for R2=Rwx2=Rwy2 within (14) produces R2=ρxy+1bx^y^+1 or R2=ρxy−1bx^y^−1 resulting in lines (7b) and (7c) of Proposition 2.2.

Suppose exactly three constraints are active. If those three constraints are (3a), (3b), and (3c) then any candidate point is of the form (bx2, by2, bx^y^) as in line (7d) of Proposition 2.2. If those three constraints are (3a), (3b), and (4) then via (13) we have ρx^y^=ρxy±1−bx21−by2bxby resulting in line (7e) of Proposition 2.2. If those three constraints are (3a), (3c), and (4) then via (13) we have bx^y^=ρxy±1−bx21−Rwy2bxRwy, which after rearrangement and squaring gives

(15)(bx2bx^y^2+1−bx2)Rwy2−2bxbx^y^ρxyRwy+(bx2−1+ρxy2)=0.

Using the quadratic formula on (15) results in line (7f) of Propositon 2.2. If those three constraints are (3b), (3c), and (4), then likewise, via (13), we have bx^y^=ρxy±1−Rwx21−by2Rwxby, and with analogous rearrangement, squaring, and use of the quadratic formula, we solve for Rwx and derive line (7g) of Proposition 2.2.

  1. Research funding: None declared.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Competing interests: Authors state no conflict of interest.

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Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/em-2019-0028).

Received: 2019-08-29
Accepted: 2020-03-23
Published Online: 2020-05-04

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/em-2019-0028
Keywords for this article
correlation; model uncertainty; optimization; propensity

Articles in the same Issue

  1. Editorial
  2. The mean prevalence
  3. Research Articles
  4. Heterogeneous indirect effects for multiple mediators using interventional effect models
  5. Sleep habits and their association with daytime sleepiness among medical students of Tanta University, Egypt
  6. Population attributable fractions for continuously distributed exposures
  7. A real-time search strategy for finding urban disease vector infestations
  8. Disease mapping models for data with weak spatial dependence or spatial discontinuities
  9. A comparison of cause-specific and competing risk models to assess risk factors for dementia
  10. A simple index of prediction accuracy in multiple regression analysis
  11. A comparison of approaches for estimating combined population attributable risks (PARs) for multiple risk factors
  12. Posterior predictive treatment assignment methods for causal inference in the context of time-varying treatments
  13. Random effects tumour growth models for identifying image markers of mammography screening sensitivity
  14. Extrapolating sparse gold standard cause of death designations to characterize broader catchment areas
  15. Extending balance assessment for the generalized propensity score under multiple imputation
  16. Regression analysis of unmeasured confounding
  17. The Use of Logic Regression in Epidemiologic Studies to Investigate Multiple Binary Exposures: An Example of Occupation History and Amyotrophic Lateral Sclerosis
  18. Meeting the Assumptions of Inverse-Intensity Weighting for Longitudinal Data Subject to Irregular Follow-Up: Suggestions for the Design and Analysis of Clinic-Based Cohort Studies
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