Table 3 Simulation results for the different models showing the simulation-based average of the posterior mean (average of the estimated standard errors)(simulation-based standard errors) for the marginal or population-averaged FOI \(\lambda\) and variance of the random intercepts \(\sigma ^2_{b}\), its corresponding bias, root mean square error (RMSE) and mean length of credibly intervals (MLCI). P represents the percentage of symptomatic infections. True values: (1) \(\lambda =0.0027\) and (2) \(\sigma ^2_{b}=0.25\). Scenario 4 uses all data except for positive routine observations following a positive clinical visit, or positive clinical observations following a positive routine visit within a 35 day period. N represents the total number of observations over all individuals averaged over the M datasets

\({\bar{\hat{\lambda }}}\)

0.0027 (8.9e−5) (8.3e−5)

0.0036 (1.1e−4) (1.0e−4)

0.0032 (2.1e−4) (2.7e−4)

0.0027 (7.6e−5) (7.9e−5)

\(\text{ Bias }(\lambda )\)

5.80e−5

9.20e−4

5.60e−4

4.40e−5

\(\text{ RMSE }(\lambda )\)

1.00e−4

9.30e−4

6.20e−4

9.04e−5

MLCI

3.50e−4

4.30e−4

7.80e−4

3.00e−4

\(\bar{\hat{\sigma ^2_{b}}}\)

0.4086(0.0437)(0.0451)

0.4311 (0.0408) (0.0387)

0.2948 (0.0894) (0.0843)

0.3012 (0.0386) (0.0367)

\(\text{ Bias }(\sigma ^2_{b})\)

0.1586

0.1811

0.0448

0.0513

\(\text{ RMSE }(\sigma ^2_{b})\)

0.1650

0.1850

0.0955

0.0631

MLCI

0.1703

0.15947

0.4235

0.1500

\(\bar{\hat{\lambda }}\)

0.0027 (8.9e−5) (8.3e−5)

0.0046 (1.3e−4) (1.2e−4)

0.0031 (1.4e−04) (1.6e−4)

0.0027 (7.2e−5) (7.6e−5)

\(\text{ Bias }(\lambda )\)

5.80e−5

1.90e−3

4.50e−4

4.00e−5

\(\text{ RMSE }(\lambda )\)

1.00e−4

2.00e−3

4.80e−4

8.85e−5

MLCI

3.50e−4

5.20e−4

5.30e−4

2.80e−4

\(\bar{\hat{\sigma ^2_{b}}}\)

0.4087 (0.0447) (0.0450)

0.4179 (0.0369) (0.0380)

0.2892 (0.0594) (0.0582)

0.2883 (0.0330) (0.0309)

\(\text{ Bias }(\sigma ^2_{b})\)

0.1587

0.1679

0.0392

0.0381

\(\text{ RMSE }(\sigma ^2_{b})\)

0.1650

0.1720

0.0702

0.0491

MLCI

0.1738

0.1439

0.2235

0.1276

\(\bar{\hat{\lambda }}\)

0.0027 (8.9e−5) (8.4e−5)

0.0055 (1.6e−4) (1.5e−4)

0.0030 (1.0e−4) (1.1e−4)

0.0027 (6.9e−5) (7.0e−5)

\(\text{ Bias }(\lambda )\)

5.70e−5

2.80e−3

3.20e−4

2.15e−5

\(\text{ RMSE }(\lambda )\)

1.00e−4

2.80e−3

3.40e−4

7.32e−5

MLCI

3.50e−4

6.10e−4

4.10e−4

2.70e−4

\(\bar{\hat{\sigma ^2_{b}}}\)

0.4092 (0.0444) (0.0431)

0.4000 (0.0336) (0.0325)

0.2704 (0.0451) (0.0503)

0.2752 (0.0282) (0.0266)

\(\text{ Bias }(\sigma ^2_{b})\)

0.1592

0.1500

0.0204

0.0252

\(\text{ RMSE }(\sigma ^2_{b})\)

0.1650

0.153

0.0543

0.0366

MLCI

0.1734

0.1302

0.1716

0.1099

\(\bar{\hat{\lambda }}\)

0.0027 (8.9e−5) (8.2e−5)

0.0065 (1.8e−4) (1.7e−4)

0.0029 (9.0e−5) (1.0e−4)

0.0027 (5.0e−5) (5.7e−5)

\(\text{ Bias }(\lambda )\)

5.80e−5

3.80e−3

2.40e−4

1.80e−5

\(\text{ RMSE }(\lambda )\)

1.00e−4

3.80e−3

2.60e−4

5.90e−5

MLCI

3.50e−4

6.80e−4

3.50e−4

1.90e−4

\(\bar{\hat{\sigma ^2_{b}}}\)

0.4094 (0.0436) (0.0425)

0.3829 (0.0307) (0.0312)

0.2624 (0.0375) (0.0395)

0.2469 (0.0233) (0.0235)

\(\text{ Bias }(\sigma ^2_{b})\)

0.1594

0.1329

0.0124

0.0031

\(\text{ RMSE }(\sigma ^2_{b})\)

0.1650

0.1365

0.0414

0.0237

MLCI

0.1698

0.1189

0.1448

0.0905

\(\bar{\hat{\lambda }}\)

0.0027 (8.9e−5) (8.5e−5)

0.0075 (1.9e−4) (1.8e−4)

0.0028 (7.9e−5) (8.0e−5)

0.0027 (5.0e−5) (5.7e−5)

\(\text{ Bias }(\lambda )\)

6.00e−5

4.80e−3

1.40e−4

1.80e−5

\(\text{ RMSE }(\lambda )\)

1.00e−4

4.80e−3

1.60e−4

5.90e−5

MLCI

3.50e−4

7.50e−4

3.10e−4

1.80e−4

\(\bar{\hat{\sigma ^2_{b}}}\)

0.3984 (0.0434) (0.0449)

0.3640 (0.0286) (0.0299)

0.2499 (0.0323) (0.0332)

0.2489 (0.0210) (0.0203)

\(\text{ Bias }(\sigma ^2_{b})\)

0.1484

0.1140

6.10e−5

0.0011

\(\text{ RMSE }(\sigma ^2_{b})\)

0.1650

0.1178

0.0332

0.0203

MLCI

0.1695

0.1114

0.1204

0.0705