Fig 1.
γ0 = 1 for all simulation scenarios. The over-dispersion parameter κ is set to be 1 for all ZINB simulation scenarios. β0 reflects the log odds of zero inflation in the unexposed group, and is equal to {−1.386, 0, 1.386} for the {20%, 50%, 80%} of zero inflation in this group. β1 reflects the change in log odds of zero inflation when changing from unexposed to exposed group. The corresponding values of β1 of {−5%, 0, +5%} changing in the zero inflation are {−0.349, 0, 0.287}, {0.201, 0, 0.201}, and {−0.287, 0, 0.349} for 20%, 50% and 80% of the zero inflations in the unexposed group, repsectively.
Fig 2.
The flowchart for simulation studies.
Table 1.
The type I error rate estimations.
Fig 3.
The power of test for ZIP simulated data.
The X axis is the value of the covariate effect on the count data γ1 and the Y axis is the power of test when the level of significance is 0.05. Three different cases of covariate effect, i.e., the consonant (ϕt = ϕc − 5%), neutral (ϕt = ϕc) and dissonant (ϕt = ϕc + 5%) effect, are presented in panels (A), (B) and (C); (D), (E) and (F); and (G), (H) and (I), respectively. Each column reflects different proportion of zero inflation in the unexposed group: 20% in (A), (D) and (G); 50% in (B), (E) and (H); and 80% in (C), (F) and (I) from the first to the third column.
Fig 4.
The power of test for ZINB simulated data.
The X axis is the value of the covariate effect on the count data γ1 and the Y axis is the power of test when the level of significance is 0.05. Three different cases of covariate effect, i.e., the consonant (ϕt = ϕc − 5%), neutral (ϕt = ϕc) and dissonant (ϕt = ϕc + 5%) effect, are presented in panels (A) and (B); (C) and (D); and (E) and (F), respectively. Each column reflects different proportion of zero inflation in the unexposed group: 20% in (A), (C) and (E); and 50% in (B), (D) and (F) from the left to the right column, respectively.
Fig 5.
The estimate of γ1 and its standard error for data simulated under ZIP with ϕc = 20%.
The figure displays box-plots of estimates and their standard errors for γ1 from 1000 replications in (A) and (B); (C) and (D); and (E) and (F) for the consonant (ϕt = ϕc − 5%), neutral (ϕt = ϕc) and dissonant (ϕt = ϕc + 5%) effect case, respectively. For each box of the boxplots, the center line represents the median, the bottom line represents the 25th percentiles and the top line represents the 75th percentiles. The whiskers of the boxplots show 1.5 interquartile range (IQR) below the 25th percentiles and 1.5 IQR above the 75th percentiles, and outliers are represented by small circles. The horizontal line in (A), (C) and (E) represents the true value of γ1 (= 0.4) and the bias, standard deviation (SD), and root mean square error (RMSE) of the estimations of γ1 are shown above its box-plot for each method. The mean and standard deviation (SD) of the standard error (SE) estimations are shown above the box-plot for each method in panels (B), (D) and (F).
Fig 6.
The estimate of γ1 and its standard error for data simulated under ZINB with ϕc = 20%.
The figure displays box-plots of estimates and their standard errors for γ1 from 1000 replications in (A) and (B); (C) and (D); and (E) and (F) for the consonant (ϕt = ϕc − 5%), neutral (ϕt = ϕc) and dissonant (ϕt = ϕc + 5%) effect case, respectively. For each box of the boxplots, the center line represents the median, the bottom line represents the 25th percentiles and the top line represents the 75th percentiles. The whiskers of the boxplots show 1.5 interquartile range (IQR) below the 25th percentiles and 1.5 IQR above the 75th percentiles, and outliers are represented by small circles. The horizontal line in (A), (C) and (E) represents the true value of γ1 (= 0.4) and the bias, standard deviation (SD), and root mean square error (RMSE) of the estimations of γ1 are shown above its box-plot for each method. The mean and standard deviation (SD) of the standard error (SE) estimations are shown above the box-plot for each method in panels (B), (D) and (F).
Fig 7.
The estimate of β1 (or ) and its standard error for data simulated under ZINB when ϕc = 20% and γ1 = 0.4.
The figure displays box-plots of estimates and their standard errors for the covariate effect on the log-odds of structural zeros for ZIP and ZINB method and on the log-odds of zeros for hurdle models from 1000 replications when γ1 = 0.4. For each box of the boxplots, the center line represents the median, the bottom line represents the 25th percentiles and the top line represents the 75th percentiles. The whiskers of the boxplots show 1.5 interquartile range (IQR) below the 25th percentiles and 1.5 IQR above the 75th percentiles, and outliers are represented by small circles. Panels (A1), (C1) and (E1) show the estimates of β1 for consonant, neutral and dissonant effect case, respectively. The horizontal line in these panels represents the true value of β1, which is −0.349 in (A1), 0 in (C1) and 0.287 in (E1). Panels (A2), (C2) and (E2) show the estimates of for consonant, neutral and dissonant effect case, respectively. The horizontal line in these panels represents the true value of
, which is −0.420 in (A2), −0.240 in (C2) and −0.070 in (E2). The bias, standard deviation (SD), and root mean square error (RMSE) of the estimates are shown above the box-plot for each method. Panel (B1), (D1) and (F1) show the SEs of the estimates for β1, and panel (B2), (D2) and (F2) show the SEs of the estimates for
. The mean and standard deviation (SD) of the standard error (SE) estimations are shown above the box-plot for each method.
Table 2.
The AIC’s of different methods for data simulated under ZINB distribution with ϕc = 20%.
Fig 8.
The empirical probability of choosing a model using AIC criterion for ZIP distributed data.
The X axis is the value of the covariate effect on the count data γ1 and the Y axis is the empirical probability of choosing a model using AIC criterion. Three different cases of covariate effect, i.e., the consonant (ϕt = ϕc − 5%), neutral (ϕt = ϕc) and dissonant (ϕt = ϕc + 5%) effect, are presented in (A), (B) and (C); (D), (E) and (F); and (G), (H) and (I), respectively. Each column reflects different proportion of zero inflation in the unexposed group: 20% in (A), (D) and (G); 50% in (B), (E) and (H); and 80% in (C), (F) and (I) from the first to the third column.
Fig 9.
The empirical probability of choosing a model using AIC criterion for ZINB distributed data.
The X axis is the value of the covariate effect on the count data γ1 and the Y axis is the empirical probability of choosing a model using AIC criterion. Three different cases of covariate effect, i.e., the consonant (ϕt = ϕc − 5%), neutral (ϕt = ϕc) and dissonant (ϕt = ϕc + 5%) effect, are presented in (A) and (B); (C) and (D); and (E) and (F), respectively. Each column reflects different proportion of zero inflation in the unexposed group: 20% in (A), (C) and (E); and 50% in (B), (D) and (F) from the left to the right column, respectively.
Table 3.
The parameter estimate of the gender effect and goodness of fit for bacteria Campylobacter (proportion of zeros: 77%) using different methods.
Female is the reference category for gender.
Fig 10.
The flowchart for microbiome real data analysis.
Fig 11.
The comparison plots of the observed and expected counts of bacteria for Campylobacter, Anaerotruncus and Dehalobacterium for females and males using the best three models judging by AIC criterion.
The X axis is the possible values of the OTUs, the bars are the observed counts, the red line connects the expected counts produced by the model with smallest AIC values, the green line connects the expected counts produced by the model with the second smallest AIC values and the blue line connects the expected counts produced by the model with the third smallest AIC values. The first, second and third row of the plots are for bacteria Campylobacter, Anaerotruncus, and Dehalobacterium, respectively.