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Table 1 Regression parameter estimates (Means and 95% CrI or 95% CI) by estimation method, contaminated and uncontaminated dataa

From: Quantile regression for overdispersed count data: a hierarchical method

 

No contamination (C = 0)

X1

X2

Generating Regression Parameters

0.8

−0.4

Predictor Effects (Mean, 95% CRI or CI), Estimation via:

 Negative Binomial Regression

0.793 (0.769, 0.817)

−0.394 (−0.443,–0.346)

 Robust NB M-Estimation (Aeberhard et al.)

0.791 (0.766, 0.817)

−0.393 (−0.441,–0.344)

 Robust NB M-Estimation (Chambers et al.)

0.755 (−, −)

−0.375 (−, −)

 Count Jittering (Machados & Santos Silva)

0.898 (0.863, 0.932)

−0.435 (−0.506,–0.364)

 Poisson Log-Normal Regression

0.79 (0.763, 0.816)

−0.396 (−0.444,–0.349)

 Hierarchical Median Regression, PLN, Gamma Prior

0.795 (0.77, 0.818)

−0.4 (−0.455,–0.351)

 Hierarchical Median Regression, PLN, Uniform Prior

0.793 (0.768, 0.819)

−0.402 (−0.456,–0.354)

 Hierarchical Median Regression, PLN, Exponential Mean Prior

0.794 (0.77, 0.817)

−0.397 (−0.438,–0.352)

 

C = 5

X1

X2

Generating Regression Parameters

0.8

−0.4

Predictor Effects (Mean, 95% CRI or CI), Estimation via:

 Negative Binomial Regression

0.67 (0.646, 0.695)

−0.352 (−0.398,–0.304)

 Robust NB M-Estimation (Aeberhard et al.)

0.782 (0.754, 0.811)

−0.382 (−0.436,–0.327)

 Robust NB M-Estimation (Chambers et al.)

0.675 (−, −)

−0.35 (−, −)

 Count Jittering (Machados & Santos Silva)

0.858 (0.823, 0.892)

−0.436 (−0.508,–0.363)

 Poisson Log-Normal Regression

0.708 (0.681, 0.734)

−0.371 (−0.422,–0.319)

 Hierarchical Median Regression, PLN, Gamma Prior

0.746 (0.72, 0.773)

−0.384 (−0.432,–0.331)

 Hierarchical Median Regression, PLN, Uniform Prior

0.75 (0.722, 0.776)

−0.384 (−0.434,–0.329)

 Hierarchical Median Regression, PLN, Exponential Mean Prior

0.748 (0.719, 0.774)

−0.386 (−0.436,–0.338)

 

C = 10

X1

X2

Generating Regression Parameters

0.8

−0.4

Predictor Effects (Mean, 95% CRI or CI), Estimation via:

 Negative Binomial Regression

0.581 (0.555, 0.609)

−0.306 (−0.361,–0.249)

 Robust NB M-Estimation (Aeberhard et al.)

0.816 (0.787, 0.845)

−0.418 (−0.473,–0.363)

 Robust NB M-Estimation (Chambers et al.)

0.677 (−, −)

−0.355 (−, −)

 Count Jittering (Machados & Santos Silva)

0.866 (0.832, 0.901)

−0.433 (−0.505,–0.361)

 Poisson Log–Normal Regression

0.711 (0.683, 0.741)

−0.374 (−0.431,–0.318)

 Hierarchical Median Regression, PLN, Gamma Prior

0.777 (0.749, 0.804)

−0.401 (−0.452,–0.346)

 Hierarchical Median Regression, PLN, Uniform Prior

0.776 (0.747, 0.805)

−0.401 (−0.459,–0.349)

 Hierarchical Median Regression, PLN, Exponential Mean Prior

0.774 (0.744, 0.801)

−0.402 (−0.458,–0.344)

 

C = 15

X1

X2

Generating Regression Parameters

0.8

−0.4

Predictor Effects (Mean, 95% CRI or CI), Estimation via:

 Negative Binomial Regression

0.512 (0.484, 0.539)

−0.304 (−0.359,–0.248)

 Robust NB M-Estimation (Aeberhard et al.)

0.805 (0.776, 0.833)

−0.4 (−0.455,–0.346)

 Robust NB M-Estimation (Chambers et al.)

0.677 (−, −)

−0.36 (−, −)

 Count Jittering (Machados & Santos Silva)

0.861 (0.826, 0.897)

−0.436 (−0.508,–0.364)

 Poisson Log-Normal Regression

0.716 (0.686, 0.747)

−0.387 (−0.447,–0.325)

 Hierarchical Median Regression, PLN, Gamma Prior

0.785 (0.758, 0.813)

−0.405 (−0.463,–0.352)

 Hierarchical Median Regression, PLN, Uniform Prior

0.785 (0.757, 0.814)

−0.403 (−0.456,–0.346)

 Hierarchical Median Regression, PLN, Exponential Mean Prior

0.786 (0.755, 0.817)

−0.408 (−0.461,–0.354)

 

C = 20

X1

X2

Generating Regression Parameters

0.8

−0.4

Predictor Effects (Mean, 95% CRI or CI), Estimation via:

 Negative Binomial Regression

0.461 (0.43, 0.49)

−0.285 (−0.347,–0.223)

 Robust NB M-Estimation (Aeberhard et al.)

0.794 (0.765, 0.822)

−0.395 (−0.448,–0.341)

 Robust NB M-Estimation (Chambers et al.)

0.69 (−, −)

−0.368 (−, −)

 Count Jittering (Machados & Santos Silva)

0.861 (0.826, 0.897)

−0.436 (−0.508,–0.364)

 Poisson Log-Normal Regression

0.727 (0.695, 0.761)

−0.396 (−0.46,–0.329)

 Hierarchical Median Regression, PLN, Gamma Prior

0.789 (0.757, 0.82)

−0.408 (−0.465,–0.349)

 Hierarchical Median Regression, PLN, Uniform Prior

0.791 (0.759, 0.821)

−0.407 (−0.468,–0.345)

 Hierarchical Median Regression, PLN, Exponential Mean Prior

0.792 (0.762, 0.822)

−0.409 (−0.466,–0.348)

 

C = 25

X1

X2

Generating Regression Parameters

0.8

−0.4

Predictor Effects (Mean, 95% CRI or CI), Estimation via:

 Negative Binomial Regression

0.421 (0.39, 0.452)

−0.27 (−0.329,–0.205)

 Robust NB M-Estimation (Aeberhard et al.)

0.788 (0.76, 0.816)

−0.394 (−0.448,–0.34)

 Robust NB M-Estimation (Chambers et al.)

0.697 (−, −)

−0.374 (−, −)

 Count Jittering (Machados & Santos Silva)

0.861 (0.826, 0.897)

−0.436 (−0.508,–0.364)

 Poisson Log-Normal Regression

0.736 (0.701, 0.773)

−0.4 (−0.468,–0.333)

 Hierarchical Median Regression, PLN, Gamma Prior

0.797 (0.765, 0.83)

−0.41 (−0.472,–0.355)

 Hierarchical Median Regression, PLN, Uniform Prior

0.796 (0.765, 0.828)

−0.412 (−0.47,–0.357)

 Hierarchical Median Regression, PLN, Exponential Mean Prior

0.797 (0.767, 0.827)

−0.411 (−0.468,–0.356)

 

C = 30

X1

X2

Generating Regression Parameters

0.8

−0.4

Predictor Effects (Mean, 95% CRI or CI), Estimation via:

 Negative Binomial Regression

0.388 (0.356, 0.419)

−0.256 (−0.326,–0.188)

 Robust NB M-Estimation (Aeberhard et al.)

0.786 (0.758, 0.815)

−0.394 (−0.447,–0.34)

 Robust NB M-Estimation (Chambers et al.)

0.703 (−, −)

−0.378 (−, −)

 Count Jittering (Machados & Santos Silva)

0.861 (0.826, 0.897)

−0.436 (−0.508,–0.364)

 Poisson Log-Normal Regression

0.741 (0.705, 0.776)

−0.403 (−0.471,–0.336)

 Hierarchical Median Regression, PLN, Gamma Prior

0.8 (0.769, 0.83)

−0.415 (−0.476,–0.354)

 Hierarchical Median Regression, PLN, Uniform Prior

0.798 (0.765, 0.83)

−0.409 (−0.469,–0.344)

 Hierarchical Median Regression, PLN, Exponential Mean Prior

0.798 (0.768, 0.83)

−0.41 (−0.467,–0.349)

  1. aBayesian estimates except for Robust NB M-estimation, and count jittering