Measurement error is often inescapable when working with real-world data, and has received extensive research in recent decades. Despite this, it is often unaccounted for in practice, particularly in the context of health data. We consider measurement error in the context of precision medicine, an area which has also seen recent growth in the biostatistical literature. We focus on doubly robust estimation methods, which offer consistent parameter estimators as long as at least one of two component models is correctly specified. Measurement error presents a novel challenge in this setting, as its effects become both difficult to predict and more complicated to correct within these multi-model environments. We discuss some broad principles relating to this problem, illustrate the effects of measurement error in some simulated settings, and demonstrate the use (and potential pitfalls) of some common correction methods in this context.

1. Measurement error and precision medicine
Michael Wallace
Assistant Professor, Biostatistics
Department of Statistics and Actuarial Science
University of Waterloo
michael.wallace@uwaterloo.ca
August 14, 2018

2. Outline
1. Crash course 1: personalized medicine
‘’
2. Crash course 2: measurement error
‘’
3. Worlds collide/some specific problems
‘’
4. Application: STAR*D

3. Treating the patient, not the diagnosis
Heterogeneity between patients:
0 months
Patient 1
Patient 2
3 months
Drug A
Drug B
Heterogeneity within patients:
0 months 6 months
Patient 3
3 months
Drug A
0 months 6 months
Patient 3
3 months
Drug A Drug B
Drug A

4. Dynamic treatment regimes
Dynamic treatment regimes (DTRs) ‘formalize’ the process of
personalized medicine:
“If patient BMI over 30 prescribe therapy A, otherwise provide
therapy B.”
DTR
Treatment
recommendation
Patient
information
DTRs can lead to improved results over standard ‘one size fits
all’ approaches.

5. Notation
X A Y
Pre-treatment
covariates
Treatment
received
Outcome
BMI
'Healthiness
metric'
Drug A or
Drug B?
DTR: treatment Aopt that maximizes E[Y |X, Aopt]

6. Identifying the best treatment regime: multi-stage
X
A
Y1
1
X
A
2
2
Stage 1 Stage 2
More often,
we work
in stages.

7. Identifying the best treatment regime: multi-stage
Y
A
2
2
Stage 2
H
H2
X 1
A1
X2
( )= , ,

8. Identifying the best treatment regime: multi-stage
X
A
1
1
Y2
Stage 1 Stage 2
opt
Expected
outcome
if stage 2
treatment
optimal.

9. Lots of methods available:
Q-learning
G-estimation
IPTW
A-learning
dWOLS
MSMs
OWL
etc...

10. Lots of methods available:
Q-learning
G-estimation
IPTW
A-learning
dWOLS
MSMs
OWL
etc...

11. Identifying the best treatment regime
If only one treatment decision:
E[Y |X, A]
Expected outcome
(to be maximized)
We might propose the following model
E[Y |X, A; β, ψ] = β0 + β1BMI + A(ψ0 + ψ1BMI)

12. Identifying the best treatment regime
If only one treatment decision:
E[Y |X, A]
Expected outcome
(to be maximized)
We might propose the following model
E[Y |X, A; β, ψ] = β0 + β1BMI + A(ψ0 + ψ1BMI)
More generally, split outcome into two components:
E[Y |X, A; β, ψ]
Expected outcome
(to be maximized)
=
Impact of patient history
in the absence of treatment
G(X; β) + γ(X, A; ψ)
Impact of treatment
on outcome
Simplifies focus: find Aopt that maximizes γ(X, A; ψ).

13. Identifying the best treatment regime
Suppose the true outcome model is:
E[Y |X, A; β, ψ] = β0 + β1BMI + β2BMI2
+ A(ψ0 + ψ1BMI)
But we propose:
E[Y |X, A; β, ψ] = β0 + β1BMI + A(ψ0 + ψ1BMI)
Problem: what if A depends on BMI?

14. Dynamic WOLS (dWOLS)
E[Y |X, A; β, ψ] = G(X; β) + γ(X, A; ψ)
Three models to specify:
1. Blip model: γ(X, A; ψ).
2. Treatment-free model: G(X; β).
3. Treatment model: P(A = 1|X; α).
Estimate ψ via WOLS of Y on covariates in blip and
treatment-free models, with weights
w = |A − P(A = 1|X; ˆα)|.

15. Dynamic WOLS (dWOLS)
E[Y |X, A; β, ψ] = G(X; β) + γ(X, A; ψ)
Three models to specify:
1. Blip model: γ(X, A; ψ).
2. Treatment-free model: G(X; β).
3. Treatment model: P(A = 1|X; α).
Estimate ψ via WOLS of Y on covariates in blip and
treatment-free models, with weights
w = |A − P(A = 1|X; ˆα)|.
Estimates are doubly-robust: consistent if either G(X; β) or
P(A = 1|X; α) correctly specified.

16. Multi-stage recursion
In the multi-stage setting we conduct a single-stage analysis at
each stage by forming pesudo-outcomes:
˜Yj = Y +
J
k=j+1
[γk(Xk, Aopt
k ; ˆψk) − γk(Xk, Ak; ˆψk)]
˜Yj is the expected outcome assuming optimal treatment from
stage j + 1 onwards.
We plug ˜Yj into our dWOLS procedure and proceed similarly.

17. Summary
Our dWOLS analysis can be broken down into three separate
components:
Find blip parameter estimates ˆψ via dWOLS.

18. Summary
Our dWOLS analysis can be broken down into three separate
components:
Find blip parameter estimates ˆψ via dWOLS.
Implement the rule “treat if ˆψX > 0”.

19. Summary
Our dWOLS analysis can be broken down into three separate
components:
Find blip parameter estimates ˆψ via dWOLS.
Implement the rule “treat if ˆψX > 0”.
(If multi-stage) form pseudo outcomes
˜Yj = Y +
J
k=j+1
[γk(Xk, Aopt
k ; ˆψk) − γk(Xk, Ak; ˆψk)].

20. Summary
Our dWOLS analysis can be broken down into three separate
components:
Find blip parameter estimates ˆψ via dWOLS.
Implement the rule “treat if ˆψX > 0”.
(If multi-stage) form pseudo outcomes
˜Yj = Y +
J
k=j+1
[γk(Xk, Aopt
k ; ˆψk) − γk(Xk, Ak; ˆψk)].
Easy!

21. BUT

22. Measurement Error
True history
Treatment
Outcome

23. Measurement Error
True history
Treatment
Outcome
Reported history

24. Measurement Error
X
A
Y
W

25. Measurement Error
Classical additive measurement error:
Observed = True + Error
W = X + U
Key assumptions:
U ∼ N(0, σ2
u)
Non-differential: Y ⊥ W |X

26. Correction Methods
To account for measurement error we need to know about U.
i W X
1 5 -
2 8 -
3 7 -
4 3 -
5 4 -
... ... ...

27. Correction Methods
We’ll assume replicate data are available:
i W1 W2
1 5 4
2 8 6
3 7 -
4 3 -
5 4 4
... ... ...
We then have numerous error correction methods to choose from.

28. Regression Calibration
Simple correction method: Regression Calibration.
Principle:
1. Use additional data to estimate E[X|W , A] = Xrc.
2. Replace X with Xrc and carry out a standard analysis.
3. Adjust the resulting standard errors to account for the
estimation in step 1.

29. Identifying the best treatment regime
Suppose the true outcome model is:
E[Y |·] = β0 + β1X + β2X2 + A(ψ0 + ψ1X)

30. Identifying the best treatment regime
Suppose the true outcome model is:
E[Y |·] = β0 + β1X + β2X2 + A(ψ0 + ψ1X)
If we have RC estimates Xrc then we could fit
E[Y |·] = β0 + β1Xrc + β2X2
rc + A(ψ0 + ψ1Xrc)

31. Identifying the best treatment regime
Suppose the true outcome model is:
E[Y |·] = β0 + β1X + β2X2 + A(ψ0 + ψ1X)
If we have RC estimates Xrc then we could fit
E[Y |·] = β0 + β1Xrc + β2X2
rc + A(ψ0 + ψ1Xrc)
But we might mis-specify the model as
E[Y |·] = β0 + β1Xrc + A(ψ0 + ψ1Xrc)

32. Identifying the best treatment regime
Suppose the true outcome model is:
E[Y |·] = β0 + β1X + β2X2 + A(ψ0 + ψ1X)
If we have RC estimates Xrc then we could fit
E[Y |·] = β0 + β1Xrc + β2X2
rc + A(ψ0 + ψ1Xrc)
But we might mis-specify the model as
E[Y |·] = β0 + β1Xrc + A(ψ0 + ψ1Xrc)
A depends on W . Establish (approximate) covariate balance
in Xrc by regressing A on Xrc.

33. Simulation Studies
Measurement error: Wr = X + U; U ∼ N(0, 1); r = 1, 2
Outcome: E[Y |X, A; β, ψ] = β0 + β1X + β2X2 + A(ψ0 + ψ1X)
Treatment: P(A = 1|W1) = 1 + exp(−(0.5W1 + 0.5W 2
1 ))
−1
We set ψ0 = −1; ψ1 = 1 ← our target

34. Simulation studies: truth (X)
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Neither Treatment Treatment−free Both
−2−1012
Correct models
Blipparameterestimates
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37. Simulation studies: naive (W )
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38. Truth: “treat if X > 1”; Estimated: “treat if X > 2”
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39. Simulation studies: corrected (Xrc)
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40. Simulation studies
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−2−1012
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ψ0
ψ1

41. Recursion
Recall the multi-stage case requires the computation of
pseudo-outcomes:
˜Yj = Y +
J
k=j+1
[γk(Xk, Aopt
k ; ˆψk) − γk(Xk, Ak; ˆψk)].
Question: What happens if we use Wk instead of Xk? And what
should we do about it?

42. Recursion
Recall the multi-stage case requires the computation of
pseudo-outcomes:
˜Yj = Y +
J
k=j+1
[γk(Xk, Aopt
k ; ˆψk) − γk(Xk, Ak; ˆψk)].
Question: What happens if we use Wk instead of Xk? And what
should we do about it?
Answer: We’re not sure (yet).

43. Future treatment
After estimating ψ, have “Aopt = 1 if ˆψ0 + ˆψ1X > 0”
Suppose “Aopt = 1 if X > τ” (or vice-versa) for ‘threshold’ τ
We observe W = X + U
Questions of practical interest:
P(X < τ|W = w > τ) P(X > τ|W = w < τ)
(e.g., if observed BMI = 31, the probability true BMI < 30)

44. Future treatment
If X ∼ N(µx , σ2
x ), U ∼ N(0, σ2
u), W = X + U then
X|W ∼ N
σ2
x w + σ2
uµx
σ2
x + σ2
u
,
σ2
x σ2
u
σ2
x + σ2
u
Can easily estimate
P(X < τ|W = w > τ) P(X > τ|W = w < τ)
for any given patient.

45. Future treatment
In some settings, results fairly intuitive:
−3 −2 −1 0 1 2 3
0.00.10.20.30.40.5
W
Probabilityofmis−treatment
X ~ N(0,1)
U ~ N(0,1)
τ = 0

46. Future treatment
In others, perhaps more of a surprise (to some):
−3 −2 −1 0 1 2 3
0.00.20.40.6
W
Probabilityofmis−treatment
X ~ N(0,1)
U ~ N(0,1)
τ = 1

47. STAR*D Study
Sequenced Treatment Alternatives to Relieve Depression
Multi-stage: treatment ‘switching’ vs. ‘augmentation’
Outcome: Quick Inventory of Depressive Symptomatology
(QIDS) score (integers from 0-27).
Key variable: patient preference for different types of therapy.

48. STAR*D
QIDS score: clinician- and patient-rated at each stage
=⇒ replicates?
0 5 10 15 20 25
0510152025
Stage 1 Symptom Scores
Clinician−rated
Patient−rated
5 10 15 20 25
510152025
Stage 2 Symptom Scores
Clinician−rated
Patient−rated
(Clinician ratings 0.60 higher at stage 1, 0.25 higher at stage 2.)

49. STAR*D
Treatment: SSRI∗ (A = 1) vs. non-SSRI (A = 0) therapies.
∗
selective serotonin reuptake inhibitor
Variables we’ll consider:
q1, q2: QIDS score at start of stage 1, 2
s1, s2: QIDS ‘slope’ for the previous stage (rate of change
of q1, q2)
p1: patient treatment preference for stage 1
X
A
Y0
0
X
A
1
1
X
A
2
2
Stage 0 Stage 2Stage 1

50. STAR*D
Treatment: SSRI∗ (A = 1) vs. non-SSRI (A = 0) therapies.
∗
selective serotonin reuptake inhibitor
Variables we’ll consider:
q1, q2: QIDS score at start of stage 1, 2
s1, s2: QIDS ‘slope’ for the previous stage (rate of change
of q1, q2)
p1: patient treatment preference for stage 1
X
A
y0
0
q
a1
q
a2
Stage 0 Stage 2Stage 1
(n = 1027) (n = 273)
s
p
s1 1
1
2 2 2
y1

51. STAR*D
Consider ‘full’ blip models at each stage:
stage 1: γ1(hψ1, a1; ψ1) = a1(ψ10 + q1ψ11 + s1ψ12 + p1ψ13)
stage 2: γ2(hψ2, a2; ψ2) = a2(ψ20 + q2ψ21 + s2ψ22)
Model selection: Wald-type or (new!) quasilikelihood information
criterion.

52. STAR*D
stage 1: γ1(hψ1, a1; ψ1) = a1(ψ10 + q1ψ11 + s1ψ12 + p1ψ13)
stage 2: γ2(hψ2, a2; ψ2) = a2(ψ20 + q2ψ21 + s2ψ22)
Stage 1 Stage 2
Variable ˆψ (Naive) ˆψ (RC) ˆψ (Naive) ˆψ (RC)
Intercept -1.498 -2.231
QIDS score (q) 0.117 0.263
QIDS slope (s) -0.512 -0.020
Preference (p) -2.040 - -
Bold: recommended variables in final model.

53. STAR*D
stage 1: γ1(hψ1, a1; ψ1) = a1(ψ10 + q1ψ11 + s1ψ12 + p1ψ13)
stage 2: γ2(hψ2, a2; ψ2) = a2(ψ20 + q2ψ21 + s2ψ22)
Stage 1 Stage 2
Variable ˆψ (Naive) ˆψ (RC) ˆψ (Naive) ˆψ (RC)
Intercept -1.498 -0.792 -2.231 -5.073
QIDS score (q) 0.117 0.066 0.263 0.491
QIDS slope (s) -0.512 -0.031 -0.020 -1.393
Preference (p) -2.040 -1.726 - -
Bold: recommended variables in final model.

54. STAR*D
stage 1: γ1(hψ1, a1; ψ1) = a1(ψ10 + q1ψ11 + s1ψ12 + p1ψ13)
stage 2: γ2(hψ2, a2; ψ2) = a2(ψ20 + q2ψ21 + s2ψ22)
Stage 1 Stage 2
Variable ˆψ (Naive) ˆψ (RC) ˆψ (Naive) ˆψ (RC)
Intercept -1.498 -0.792 -2.231 -5.073
QIDS score (q) 0.117 0.066 0.263 0.491
QIDS slope (s) -0.512 -0.031 -0.020 -1.393
Preference (p) -2.040 -1.726 - -
Bold: recommended variables in final model.

55. STAR*D
stage 1: γ1(hψ1, a1; ψ1) = a1(ψ10 + q1ψ11 + s1ψ12 + p1ψ13)
stage 2: γ2(hψ2, a2; ψ2) = a2(ψ20 + q2ψ21 + s2ψ22)
Stage 1 Stage 2
Variable ˆψ (Naive) ˆψ (RC) ˆψ (Naive) ˆψ (RC)
Intercept -1.498 -0.792 -2.231 -5.073
QIDS score (q) 0.117 0.066 0.263 0.491*
QIDS slope (s) -0.512 -0.031 -0.020 -1.393
Preference (p) -2.040 -1.726 - -
Bold: recommended variables in final model.
*May not survive contact with the bootstrap...

56. Summary/Future Work
So far:
Measurement error poses unique challenges in the
personalized medicine setting.
Biased/incorrect treatment rules.
Theoretical issues (double robustness, recursion).
Consequences for future treatments tailored on error-prone
observations.

57. Summary/Future Work
So far:
Measurement error poses unique challenges in the
personalized medicine setting.
Biased/incorrect treatment rules.
Theoretical issues (double robustness, recursion).
Consequences for future treatments tailored on error-prone
observations.
Moving forward:
Other correction methods (SIMEX, conditional score, etc.).
New methodological work specific to DTR/personalized
framework.
Diagnostics for extant analyses/datasets.

58. References/Acknowledgments
A comprehensive guide to DTRs: B. Chakraborty and E. E.
M. Moodie (2013). Statistical Methods for Dynamic
Treatment Regimens. Springer: New York.
dWOLS: M. P. Wallace and E. E. M. Moodie (2015).
Doubly-robust dynamic treatment regimen estimation via
weighted least squares. Biometrics 71(3) 636-644.
michael.wallace@uwaterloo.ca
@statacake

59. 15 20 25 30 35 40 45
0.000.050.100.150.20
X or W
True

60. 15 20 25 30 35 40 45
0.000.050.100.150.20
X or W
True
Observed

61. Choosing replicates
0.51.01.52.0
Observed value (W)
Errorvarianceestimate
27 28 29 30 31 32 33
But: non-random re-sampling can lead to poor error estimates.

62. Measurement Error Effects
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63. Measurement Error Effects
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64. Measurement Error Effects
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