S IMULATION STUDY WITH A BINARY OUTCOME

In this section, the methods laid out for measurement error correction in the presence of a quadratic transformation of the error-prone variable are applied in a simulation study when the outcome is binary and the substantive model is therefore logistic regression.

Simulation study design and evaluation criteria

The latent exposure 𝑋 and the error prone measure 𝑊 were generated exactly as for the previous simulation studies and no accurately observed covariates were included (Box 4.1). Only two quadratic shapes are considered in this simulation study, a J-shaped association and a weak quadratic association (Table 4.5). The binary outcome, 𝑌𝑖, was generated as a Bernuolli random

variable with probability equal to the logit transformation of the linear predictor. The value of each 𝛽₀ was chosen to result in an event rate for the outcome 𝑌 of 20%.

97 Table 4.5 Models for exposure-outcome association shapes with a binary outcome.

Shape Linear predictor Sample size (N)

J-shape -1.8 + 0.2 ∗ (𝑋 − 9)2 2000

Weak quadratic -4.5 + 0.03 ∗ 𝑋2 10,000

The only setting considered was a validation study in 30% of the study population with measurement error variance¼ the variance of 𝑋, i.e. 𝜎𝑈2= 0.25. For each association shape, 100

simulations were performed (2 × 100).

The criteria for evaluating the fit of the quadratic model after application of measurement error correction can be found in Section 4.5.2. The average MISE for each method was estimated using the outcome probabilities, i.e. the logit transformation of the linear predictor, in lieu of a continuous outcome (Equation 4.9).

Implementation of methods

All methods were implemented exactly as they were in Section 4.5.3 except for the following changes.

For the implementation of the fully Bayesian method in MCMC, there was no scaling of the binary outcome. Three chains were run for each simulation with a burn-in period of 300,000 samples with the last 20,000 burn-in samples evaluated for convergence using 𝑅̂. Another 20,000 samples were collected from each chain for inference. ESS was estimated from these samples.

Fitting the latent 𝑿 and naïve models

Only the J-shaped association had a turning point observed in the range of exposure simulated at 𝑋 = 9.0 (Table 4.6). The average MISE for the J-shaped association was 0.027 and for the weak quadratic association was 0.0073.

The addition of measurement error, 𝜎𝑈2= 0.25, in the naïve analysis resulted in the turning point

of the J-shaped association being biased -0.3 away from the mean of 𝑋 (Table 4.6). The bias toward the null observed in 𝛽̂_𝑋2 for the J-shaped association was 37% of 𝛽_𝑋2. This was near to the 36% reduction predicted by Equation 4.3 which was only approximately true for logistic regression. The bias in 𝛽̂𝑋1 was 39% of 𝛽𝑋1. For the weak quadratic association, 𝛽̂𝑋2 is biased

toward the null by 49%, much higher than predicted by Equation 4.3. Similar to what was observed for linear regression, the undercoverage for the J-shaped association (42% and 33% for 𝛽𝑋1 and 𝛽𝑋2, respectively) was much more severe than for the weak quadratic association (92%

98 Table 4.6 Simulations with a validation study and a binary outcome and measurement error variance ¼ the variance of X: Mean bias in the quadratic model regression estimates, coverage of the 95% confidence intervals or credible intervals and the associated Monte Carlo error 95% confidence interval (MCE 95% CI), average mean integrated square error (MISE) over the exposure range 7-13, and bias in the turning point of the mean curve (where applicable) estimated from 200 simulations with a continuous outcome, a validation study performed on 30% of 2000 study participants after application of the designated method to data generated from two shapes of association. *indicates a MCE 95% CI for bias that excludes zero.

Bias Coverage

(MC error 95% CI) Average

MISE Turning point bias Method 𝜷_𝟎 𝜷_𝑿𝟏 𝜷_𝑿𝟐 𝜷_𝑿𝟏 𝜷_𝑿𝟐 J-shaped, N=2000 True values 14.4 -3.6 0.2 9.0 Latent X -0.034 0.010 -0.0007 97% (94 – 100) 97% (94 – 100) 0.027 0 Naïve -6.620* _1.414* _-0.0746* _{42% (32 – 52) 33% (24 – 42) 0.115 -0.3} RC -0.360 0.073 -0.0035 94% (89 – 99) 94% (89 – 99) 0.043 0 MCMC 0.110 -0.025 0.0013 93% (88 – 98) 93% (88 – 98) 0.039 0 MCMC-RC -0.057 0.012 -0.0005 93% (88 – 98) 94% (89 – 99) 0.037 0 INLA-RC -0.100 0.021 -0.0010 93% (88 – 98) 94% (89 – 99) 0.037 0 Weak quadratic, N=10000 True values -4.5 0 0.03 Latent X -0.267 0.048 -0.0021 97% (94 – 100) 96% (92 – 100) 0.0073 Naïve -0.243 0.173* _-0.0146* _{92% (87 – 97) 82% (74 – 90) 0.0506} RC -0.585* _0.120* _-0.0060* _{94% (89 – 99) 95% (91 – 99) 0.0106} MCMC -0.424 0.078 -0.0036 97% (94 – 100) 98% (95 – 100) 0.0096 MCMC-RC -0.441* _0.090* _-0.0044* _{96% (92 – 100) 97% (94 – 100) 0.0086} INLA-RC -0.439* _0.089* _-0.0044* _{96% (92 – 100) 97% (94 – 100) 0.0086}

Results: Model fit after application of measurement error correction methods No bias in the turning point of the J-shaped association was observed after correction by any method (Table 4.6). All correction methods reduced bias and improved coverage over the naïve analysis; however, all RC methods (RC, MCMC-RC, and INLA-RC) had significant bias remaining in the regression coefficients for the weak quadratic association. The Bayesian RC methods, MCMC-RC and INLA-RC, are subject to the same approximation for logistic regression as standard RC.

MCMC had no discernable bias in the mean regression coefficients for either association shape (given that there were only 100 simulations per scenario), but the average MISE was slightly higher than for MCMC-RC and INLA-RC.

No indication of undercoverage, within the limits of the smaller simulation study, was observed. The uncertainty in the estimation of the substantive model parameters was going to be much higher for logistic regression than for linear regression; this has the effect of minimizing the relative impact of not including the uncertainty from the measurement error and exposure models for MCMC-RC and INLA-RC.

99 MCMC convergence diagnostics

All MCMC chains converged to the stationary distribution as determined by 𝑅̂ < 1.1. The mean ESS for the regression coefficients in the full MCMC model for the J-shaped and weak quadratic associations was 3,955 and 3,206, respectively. The mean ESS for the latent 𝑋_𝑖 values in the MCMC-RC model was approximately 18,000 in both settings.

4.7 S

In this chapter, I introduced hybrid Bayesian RC methods using either MCMC or INLA for the Bayesian posterior estimates of 𝐸[𝑋𝑖|𝑾𝒊, 𝒁𝒊] and 𝐸[𝑋𝑖2|𝑾𝒊, 𝒁𝒊]. These were evaluated in

simulation studies alongside previously published extensions of RC and MCMC.

All methods of measurement error correction applied in this chapter improved the fit of the quadratic model over the naïve analysis. That is, the true turning point was recovered and, in most cases, bias in the regression coefficients was minimized. The average MISE was consistently most minimal for MCMC and largest for RC, reflecting the greater variability in the curve fits estimated by this method, with the hybrid methods in between. A more specific summary of the simulation studies can be found in Box 4.2.

Box 4.2 Simulation study summary

 For linear regression, only RC was observed to have significant bias in the corrected curve in the replicate study setting (Figure 4.7). The mean regression coefficients did not appear to have significant bias. For logistic regression, RC and the hybrid methods had significant bias in the mean regression coefficients for the weak quadratic association.

 RC had the greatest variance in the regression coefficients and the highest average MISE across simulations. The 95% CIs for RC were wider than the 95% CI/CrIs for other methods although the nominal coverage was maintained.

 The fully Bayesian method using MCMC consistently resulted in the lowest average MISE across all linear regression settings while maintaining the nominal coverage for the regression estimates.

 The hybrid methods performed similarly to each other and consistently between MCMC and RC. However, INLA-RC performed slightly better than MCMC-RC in a number of scenarios either having a lower average MISE, better coverage, or less variability.

100 MCMC-RC had a much higher ESS (approx. 18,000) while INLA-RC drew only 3,000 samples (equivalent to an ESS of 3,000). This number of samples was sufficient in this setting as evidenced by the performance of INLA-RC but a larger sample size may be necessary to minimize Monte Carlo error in other settings. It is important to note that the INLA method for the classical measurement error model and a normally distributed exposure is exact.

The hybrid methods had below nominal coverage in general (with the exception of the weak quadratic association for a validation study with 𝜎𝑈2= 1). This was anticipated given that the

uncertainty in the estimation of the measurement error and exposure model parameters was not incorporated into the final SEs; only the uncertainty in the estimation of the substantive model parameters were included. Use of bootstrapping, most feasible with INLA-RC, would improve these estimates.

While MI and INLA have promising application to measurement error correction in some settings, for the setting described in this and future chapters, it was felt that each had reached its limit. In the next chapter, I will evaluate whether RC, MCMC, MCMC-RC, and INLA-RC can recover power lost to measurement error in the naïve analysis while maintaining appropriate type I error rates.

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5 MEASUREMENT ERROR CORRECTION FOR SELECTION