26 Loni Philip Tabb et al.

8 Online Resource 1

Marginalized zero-altered models for longitudinal count data 27

Table 5 Detailed. Simulation 1. Average estimates (standard deviation, standard error;β95% CI coverage) for generalized estimating equations (GEE), Poisson mixed effect (PME),

negative binomial mixed effect (NBME), and marginalized zero-altered Poisson (MZAP)models. 500 replications.

γ2 γ1 Model βtime = −0.87 βexp = 0.29 βtime×exp = 1.10 σ2 = 0.10

1 0 GEE -0.87 (0.13, 0.13; 0.94) 0.33 (0.08, 0.11; 0.98) 1.10 (0.15, 0.15; 0.95)PME -0.87 (0.13, 0.13; 0.94) 0.32 (0.08, 0.11; 0.98) 1.10 (0.15, 0.15; 0.95) 0.10 (0.02, 0.02)

NBME -0.87 (0.13, 0.13; 0.94) 0.32 (0.08, 0.11; 0.98) 1.10 (0.15, 0.15; 0.96) 0.10 (0.02, 0.02)MZAP -0.87 (0.13, 0.13; 0.95) 0.32 (0.08, 0.11; 0.98) 1.09 (0.15, 0.15; 0.95) 0.10 (0.02, 0.02)

-0.25 GEE -0.87 (0.14, 0.14; 0.95) 0.33 (0.09, 0.11; 0.97) 1.10 (0.16, 0.16; 0.94)PME -0.87 (0.14, 0.14; 0.95) 0.33 (0.09, 0.11; 0.97) 1.10 (0.16, 0.16; 0.95) 0.12 (0.02, 0.03)

NBME -0.87 (0.14, 0.14; 0.95) 0.33 (0.09, 0.11; 0.97) 1.10 (0.16, 0.16; 0.95) 0.11 (0.02, 0.03)MZAP -0.87 (0.14, 0.14; 0.95) 0.33 (0.09, 0.11; 0.98) 1.10 (0.16, 0.16; 0.95) 0.10 (0.02, 0.02)

-0.50 GEE -0.87 (0.15, 0.15; 0.93) 0.34 (0.10, 0.12; 0.96) 1.09 (0.18, 0.17; 0.93)PME -0.87 (0.15, 0.15; 0.93) 0.34 (0.10, 0.12; 0.96) 1.09 (0.18, 0.17; 0.93) 0.15 (0.03, 0.03)

NBME -0.87 (0.15, 0.15; 0.93) 0.34 (0.10, 0.12; 0.96) 1.10 (0.19, 0.17; 0.94) 0.13 (0.03, 0.03)MZAP -0.87 (0.15, 0.15; 0.96) 0.33 (0.10, 0.12; 0.97) 1.09 (0.18, 0.17; 0.94) 0.10 (0.02, 0.02)

-0.75 GEE -0.86 (0.16, 0.16; 0.95) 0.34 (0.11, 0.13; 0.96) 1.09 (0.19, 0.19; 0.94)PME -0.86 (0.16, 0.16; 0.95) 0.35 (0.11, 0.13; 0.97) 1.10 (0.19, 0.19; 0.94) 0.18 (0.03, 0.04)

NBME -0.87 (0.16, 0.16; 0.95) 0.34 (0.11, 0.13; 0.97) 1.10 (0.20, 0.19; 0.95) 0.15 (0.03, 0.04)MZAP -0.86 (0.16, 0.16; 0.95) 0.34 (0.11, 0.13; 0.97) 1.09 (0.19, 0.18; 0.95) 0.10 (0.02, 0.02)

0.50 0 GEE -0.86 (0.15, 0.14; 0.95) 0.34 (0.11, 0.12; 0.96) 1.08 (0.18, 0.17; 0.95)PME -0.86 (0.15, 0.14; 0.95) 0.33 (0.11, 0.12; 0.97) 1.08 (0.18, 0.17; 0.96) 0.13 (0.02, 0.03)

NBME -0.86 (0.15, 0.14; 0.95) 0.33 (0.11, 0.12; 0.97) 1.08 (0.19, 0.18; 0.95) 0.10 (0.03, 0.03)MZAP -0.86 (0.14, 0.14; 0.95) 0.34 (0.10, 0.12; 0.96) 1.08 (0.17, 0.16; 0.94) 0.10 (0.02, 0.02)

-0.25 GEE -0.86 (0.16, 0.16; 0.93) 0.34 (0.12, 0.13; 0.95) 1.08 (0.20, 0.19; 0.94)PME -0.86 (0.16, 0.16; 0.93) 0.33 (0.12, 0.13; 0.96) 1.08 (0.20, 0.19; 0.94) 0.17 (0.03, 0.03)

NBME -0.86 (0.17, 0.16; 0.93) 0.33 (0.12, 0.14; 0.96) 1.09 (0.21, 0.20; 0.94) 0.09 (0.03, 0.04)MZAP -0.85 (0.15, 0.15; 0.95) 0.34 (0.10, 0.12; 0.97) 1.08 (0.18, 0.18; 0.94) 0.10 (0.02, 0.02)

-0.50 GEE -0.86 (0.18, 0.18; 0.95) 0.34 (0.13, 0.15; 0.97) 1.09 (0.22, 0.22; 0.96)PME -0.86 (0.18, 0.18; 0.95) 0.34 (0.13, 0.15; 0.97) 1.09 (0.22, 0.22; 0.96) 0.21 (0.04, 0.04)

NBME -0.87 (0.18, 0.18; 0.95) 0.34 (0.13, 0.15; 0.98) 1.10 (0.22, 0.22; 0.95) 0.06 (0.04, 0.05)MZAP -0.87 (0.15, 0.16; 0.96) 0.34 (0.11, 0.13; 0.97) 1.09 (0.19, 0.19; 0.95) 0.10 (0.02, 0.02)

-0.75 GEE -0.86 (0.20, 0.19; 0.94) 0.28 (0.14, 0.16; 0.97) 1.33 (0.24, 0.24; 0.83)PME -0.86 (0.20, 0.19; 0.94) 0.28 (0.15, 0.17; 0.97) 1.33 (0.24, 0.24; 0.84) 0.27 (0.05, 0.05)

NBME -0.86 (0.20, 0.20; 0.94) 0.29 (0.14, 0.16; 0.98) 1.32 (0.24, 0.24; 0.85) 0.02 (0.03, 0.06)MZAP -0.86 (0.16, 0.17; 0.96) 0.28 (0.11, 0.14; 0.98) 1.33 (0.19, 0.20; 0.83) 0.10 (0.02, 0.02)

0.25 0 GEE -0.86 (0.15, 0.16; 0.95) 0.33 (0.12, 0.14; 0.97) 1.09 (0.19, 0.20; 0.95)PME -0.86 (0.15, 0.16; 0.95) 0.32 (0.12, 0.14; 0.98) 1.09 (0.19, 0.20; 0.96) 0.16 (0.03, 0.03)

NBME -0.86 (0.15, 0.16; 0.95) 0.33 (0.12, 0.14; 0.97) 1.09 (0.20, 0.20; 0.95) 0.05 (0.03, 0.04)MZAP -0.86 (0.13, 0.14; 0.95) 0.33 (0.10, 0.12; 0.97) 1.08 (0.16, 0.17; 0.96) 0.10 (0.02, 0.02)

-0.25 GEE -0.87 (0.18, 0.18; 0.93) 0.33 (0.13, 0.15; 0.97) 1.10 (0.24, 0.23; 0.93)PME -0.87 (0.18, 0.18; 0.93) 0.32 (0.14, 0.15; 0.97) 1.10 (0.24, 0.23; 0.94) 0.21 (0.04, 0.04)

NBME -0.87 (0.18, 0.18; 0.94) 0.33 (0.14, 0.15; 0.97) 1.10 (0.24, 0.23; 0.94) 0.02 (0.03, 0.05)MZAP -0.87 (0.14, 0.15; 0.96) 0.32 (0.10, 0.12; 0.97) 1.10 (0.18, 0.18; 0.93) 0.10 (0.02, 0.02)

9 Online Resource 2

28

LoniP

hilip

Tabb

etal.

Table 6 Detailed. Model estimates, standard errors (SE), and p-values for the frequency of arrhythmias, with -2 log likelihood and Akaike informationcriterion (AIC) values for the marginalized zero-altered negative binomial (MZANB), marginalized zero-altered Poisson (MZAP), negative binomialmixed effect (NBME), Poisson mixed effect (PME), generalized estimating equations (GEE), and Hurdle models.

MZANB MZAP NBME PME GEECoefficient Estimate (SE) p Estimate (SE) p Estimate (SE) p Estimate (SE) p Estimate (SE) p

β0 0.284 (0.359) 0.435 0.422 (0.320) 0.197 0.552 (0.472) 0.252 0.734 (0.440) 0.106 0.399 (0.333) 0.231βE 0.705 (0.266) 0.013 0.667 (0.300) 0.034 0.327 (0.490) 0.511 0.304 (0.503) 0.550 0.657 (0.326) 0.044βh1 0.734 (0.320) 0.030 0.485 (0.213) 0.031 0.620 (0.352) 0.090 0.399 (0.196) 0.051 0.432 (0.374) 0.248βh2 0.469 (0.343) 0.183 0.364 (0.214) 0.100 0.632 (0.346) 0.078 0.410 (0.193) 0.043 0.437 (0.378) 0.247βh3 0.237 (0.325) 0.472 0.126 (0.222) 0.575 0.329 (0.363) 0.372 0.167 (0.208) 0.428 0.187 (0.344) 0.587βh4 0.518 (0.369) 0.171 0.588 (0.224) 0.014 0.749 (0.354) 0.043 0.681 (0.192) 0.001 0.672 (0.313) 0.032σ2 0.187 (0.173) 0.289 0.409 (0.208) 0.059 1.290 (0.577) 0.033 1.561 (0.583) 0.012a 0.433 (0.187) 0.028 0.636 (0.192) 0.003

γ1 0.102 (0.731) 0.890 -0.373 (0.516) 0.476γ2 0.215 (0.553) 0.700 0.343 (0.427) 0.428γ3 3.501 (1.982) 0.088 1.512 (0.543) 0.010−2lnL 486.9 512.8 502.2 551.8AIC 508.9 532.8 518.2 565.8

Hurdle - Zero Hurdle - CountCoefficient Estimate (SE) p Estimate (SE) p

β0 0.304 (0.869) 0.730 0.603 (0.275) 0.037βE -0.324 (0.989) 0.746 0.663 (0.295) 0.033βh1 0.180 (0.795) 0.822 0.408 (0.212) 0.064βh2 1.230 (0.832) 0.151 0.279 (0.206) 0.188βh3 0.721 (0.831) 0.393 0.054 (0.224) 0.811βh4 1.456 (0.893) 0.115 0.524 (0.205) 0.017σ2 2.147 (0.646) 0.003 0.613 (0.152) 0.000−2lnL 503.0AIC 533.0

Marginalized zero-altered models for longitudinal count data 29

10 Online Resource 3

Table 7 Detailed. Model estimates (standard errors) for exposure and % change for themarginalized zero-altered negative binomial (MZANB), marginalized zero-altered Poisson(MZAP), negative binomial mixed effect (NBME), Poisson mixed effect (PME), and gener-alized estimating equations (GEE) models with and without an outlier.

βE βE

w/outlier w/o outlier % Change

MZANB 0.588 (0.283) 0.705 (0.266) 16.596MZAP 0.658 (0.354) 0.667 (0.300) 1.349NBME 0.233 (0.523) 0.327 (0.490) 28.746PME 0.139 (0.546) 0.304 (0.503) 54.276GEE 0.214 (0.446) 0.657 (0.326) 67.428Hurdle - Zero -0.426 (0.949) -0.324 (0.989) 31.485Hurdle - Count 0.597 (0.350) 0.663 (0.295) 9.955