Supplementary Tables and Figures. Web Material.

Consequences of kriging and land use regression for PM2.5 predictions in epidemiologic analyses: Insights into spatial variability using high-resolution satellite data

Simulation Results Under the Null

Supplementary Table 1. Size of test for linear regression health effect model with chronic exposure to air pollution under the null.

Exposure Scenario / κ / m / Effect estimate,
β / Empirical S.E. / Model S.E. / Mean Square Error / Size
Chronic, True exposure / -0.001 / 0.052 / 0.054 / 1.006 / 0.048
A1. Chronic, Kriging / 0.5 / 100 / -0.010 / 0.151 / 0.112 / 1.044 / 0.048
A1. Chronic, Kriging / 1 / 100 / -0.007 / 0.194 / 0.093 / 1.052 / 0.048
A1. Chronic, Kriging / 2 / 100 / 0.000 / 0.116 / 0.066 / 1.014 / 0.052
A1. Chronic, Kriging / 3 / 100 / -0.001 / 0.067 / 0.049 / 1.007 / 0.046
B1. Chronic, LUR / 0.5 / 100 / 0.002 / 0.067 / 0.067 / 1.000 / 0.048
B1. Chronic, LUR / 1 / 100 / 0.001 / 0.067 / 0.067 / 1.002 / 0.050
B1. Chronic, LUR / 2 / 100 / 0.002 / 0.067 / 0.067 / 1.001 / 0.054
B1. Chronic, LUR / 3 / 100 / 0.002 / 0.067 / 0.067 / 1.001 / 0.053

Supplementary Table 2. Size of test for logistic regression health effect model with chronic exposure to air pollution under the null.

Exposure Scenario / κ / m / Odds Ratio / Empirical S.E. / Model S.E. / Mean Square Error / Size
Chronic, True exposure / 1.006 / 0.131 / 0.130 / 0.017 / 0.048
A1. Chronic, Kriging / 0.5 / 100 / 0.977 / 0.521 / 0.292 / 0.272 / 0.029
A1. Chronic, Kriging / 1 / 100 / 1.002 / 0.337 / 0.217 / 0.114 / 0.038
A1. Chronic, Kriging / 2 / 100 / 1.001 / 0.198 / 0.159 / 0.039 / 0.037
A1. Chronic, Kriging / 3 / 100 / 1.001 / 0.150 / 0.118 / 0.023 / 0.045
B1. Chronic, LUR / 0.5 / 100 / 1.013 / 0.167 / 0.163 / 0.028 / 0.052
B1. Chronic, LUR / 1 / 100 / 1.013 / 0.167 / 0.163 / 0.028 / 0.049
B1. Chronic, LUR / 2 / 100 / 1.014 / 0.167 / 0.163 / 0.028 / 0.052
B1. Chronic, LUR / 3 / 100 / 1.013 / 0.166 / 0.163 / 0.028 / 0.050

Supplementary Table 3. Size of test for linear regression health effect model with acute exposure to air pollution under the null.

Exposure Scenario / κ / m / Effect estimate,
β / Empirical S.E. / Model S.E. / Mean Square Error / Size
Acute, True exposure / 0.000 / 0.034 / 0.034 / 1.001 / 0.044
C1. Acute, Kriging / 0.5 / 100 / -0.001 / 0.037 / 0.037 / 1.003 / 0.044
C1. Acute, Kriging / 1 / 100 / -0.002 / 0.039 / 0.037 / 1.006 / 0.060
C1. Acute, Kriging / 2 / 100 / 0.000 / 0.036 / 0.037 / 1.002 / 0.044
C1. Acute, Kriging / 3 / 100 / 0.000 / 0.038 / 0.037 / 1.002 / 0.054
D1. Acute, LUR / 0.5 / 100 / -0.001 / 0.028 / 0.028 / 1.003 / 0.060
D1. Acute, LUR / 1 / 100 / -0.001 / 0.028 / 0.027 / 1.003 / 0.058
D1. Acute, LUR / 2 / 100 / -0.001 / 0.027 / 0.026 / 1.003 / 0.054
D1. Acute, LUR / 3 / 100 / -0.001 / 0.027 / 0.026 / 1.002 / 0.058

Supplementary Table 4. Size of test for logistic regression health effect model with acute exposure to air pollution under the null.

Exposure Scenario / κ / m / Odds Ratio / Empirical S.E. / Model
S.E. / Mean Square Error / Size
Acute, True exposure / 0.999 / 0.020 / 0.019 / 1.002 / 0.052
C1. Acute, Kriging / 0.5 / 100 / 0.999 / 0.022 / 0.020 / 1.002 / 0.060
C1. Acute, Kriging / 1 / 100 / 0.999 / 0.022 / 0.021 / 1.002 / 0.060
C1. Acute, Kriging / 2 / 100 / 0.998 / 0.023 / 0.021 / 1.005 / 0.060
C1. Acute, Kriging / 3 / 100 / 0.999 / 0.020 / 0.021 / 1.002 / 0.050
D1. Acute, LUR / 0.5 / 100 / 0.999 / 0.015 / 0.015 / 1.001 / 0.060
D1. Acute, LUR / 1 / 100 / 0.999 / 0.015 / 0.015 / 1.002 / 0.062
D1. Acute, LUR / 2 / 100 / 0.999 / 0.014 / 0.015 / 1.002 / 0.056
D1. Acute, LUR / 3 / 100 / 0.999 / 0.014 / 0.014 / 1.002 / 0.056

Results for Berkson Error, Bias and R2 of Exposure Surfaces

Although the out-of-sample R2 is a helpful tool for determining how correlated the predicted exposures are to the true exposures for some sample, it does not tell you whether your model is correctly specified. If an exposure model is correctly specified by a spatial component and a non-spatial i.i.d. Normal error component, that i.i.d. Normal error (often called the nugget) is classified as Berkson measurement error which will not yield bias in the health effect estimates. Here in Supplementary Table 5we showresults from a simulation where the spatial exposure surfaces have different proportions of Berkson error. The resulting predicted exposures have widely varying R2 values, yet all of those predicted exposures yield no bias in the health effect estimate.

Supplementary Table 5. Bias and coverage for linear regression health effects with chronic exposure to air pollution.

Exposure Scenario / Berkson variance / m / Predicted exposure R2 / Effect estimate,
β / Empirical S.E. / Model S.E. / Mean Square Error / Naïve 95% CI
coverage
Chronic, True exposure / 0.1 / 0.999 / 0.031 / 0.033 / 0.001 / 96.4
Chronic, LUR / 0.1 / 100 / 0.87 / 0.999 / 0.051 / 0.039 / 0.003 / 87.8
Chronic, True exposure / 0.2 / 0.999 / 0.029 / 0.031 / 0.001 / 96.0
Chronic, LUR / 0.2 / 100 / 0.78 / 0.998 / 0.066 / 0.042 / 0.004 / 81.5
Chronic, True exposure / 0.5 / 0.999 / 0.026 / 0.027 / 0.001 / 96.3
Chronic, LUR / 0.5 / 100 / 0.59 / 0.994 / 0.097 / 0.052 / 0.009 / 69.9
Chronic, True exposure / 1.0 / 0.999 / 0.022 / 0.023 / 0.001 / 96.1
Chronic, LUR / 1.0 / 100 / 0.43 / 0.981 / 0.132 / 0.064 / 0.018 / 63.7

Results for Simulated Exposure Surfaces

The simulation results in the main text using the satellite-basedPM surface illustrate that the standard practice of simulated exposure surfaces using spatial models is insufficient to emulate the measurement error consequences of true exposure surfaces. Thus, a natural follow-up question is whether the standard simulated exposure surfaces could be modified in some way to produce results more similar to the results we presented using thesatellite-based PM surface. To address this question, we have added a simulation setting that uses a simulated exposure surface generated using spatial covariates andfit with a slightly misspecified exposure model. Specifically, we generated an exposure with two spatial covariates and Matern residuals. Results of this simulation are shown in Supplementary Table 6. This suggests one possible simulation setup which illustrates the measurement error consequences under model misspecification that may be more realistic to true exposure settings.

Supplementary Table6. Bias and coverage for linear regression health effects with chronic exposure to air pollution for a simulated exposure surface and a misspecified exposure model.

Exposure Scenario / κ / m / Predicted exposure R2 / Effect estimate, β / Empirical S.E. / Model S.E. / Mean Square Error / Naïve
95% CI
coverage
Chronic, True exposure / 0.999 / 0.024 / 0.024 / 0.001 / 94.8
A1. Chronic, Kriging / 0.3 / 50 / 0.30 / 1.164 / 0.288 / 0.100 / 0.107 / 45.9
A1. Chronic, Kriging / 0.5 / 50 / 0.30 / 1.089 / 0.262 / 0.094 / 0.076 / 51.8
A1. Chronic, Kriging / 2 / 50 / 0.29 / 1.396 / 0.392 / 0.133 / 0.224 / 49.1
A1. Chronic, Kriging / 0.3 / 100 / 0.31 / 1.040 / 0.213 / 0.087 / 0.047 / 54.6
A1. Chronic, Kriging / 0.5 / 100 / 0.29 / 1.108 / 0.437 / 0.115 / 0.201 / 42.8
A1. Chronic, Kriging / 2 / 100 / 0.23 / 1.112 / 0.333 / 0.157 / 0.123 / 59.5