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Quantifying the extent of visit irregularity in longitudinal data

  • Armend Lokku EMAIL logo , Catherine S Birken , Jonathon L Maguire , Eleanor M Pullenayegum and on behalf of the TARGet Kids! Collaboration
Published/Copyright: August 16, 2021
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The International Journal of Biostatistics
From the journal The International Journal of Biostatistics Volume 18 Issue 2

Abstract

The timings of visits in observational longitudinal data may depend on the study outcome, and this can result in bias if ignored. Assessing the extent of visit irregularity is important because it can help determine whether visits can be treated as repeated measures or as irregular data. We propose plotting the mean proportions of individuals with 0 visits per bin against the mean proportions of individuals with >1 visit per bin as bin width is varied and using the area under the curve (AUC) to assess the extent of irregularity. The AUC is a single score which can be used to quantify the extent of irregularity and assess how closely visits resemble repeated measures. Simulation results confirm that the AUC increases with increasing irregularity while being invariant to sample size and the number of scheduled measurement occasions. A demonstration of the AUC was performed on the TARGet Kids! study which enrolls healthy children aged 0–5 years with the aim of investigating the relationship between early life exposures and later health problems. The quality of statistical analyses can be improved by using the AUC as a guide to select the appropriate analytic outcome approach and minimize the potential for biased results.

Keywords: irregular visits; longitudinal data; summary measures; visit intensity; visit process
Corresponding author: Armend Lokku, Child Health Evaluative Sciences, Hospital for Sick Children, Toronto, ON, Canada; and Dalla Lana School of Public Health, University of Toronto, 155 College Street, Toronto, ON, M5T 3M7, Canada, E-mail:

Funding source: Natural Sciences and Engineering Research Council of Canada 10.13039/501100000038

Acknowledgements

We thank all of the participating families for their time and involvement in TARGet Kids! and are grateful to all practitioners who are currently involved in the TARGet Kids! practice-based research network.

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: This research received funding from the Natural Sciences and Engineering Research Council of Canada.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix

See Tables 4 –18 and Figures 16–19.

Table 4:

The mean observed AUCs (AUCOBS) for Log-normal gap times across sample size (n) and study duration (τ).

τ n Gap time variance Mean AUCOBS Standard error
15 30 1.683 0.073 0.014
3.542 0.104 0.017
5.598 0.132 0.018
7.869 0.154 0.020
10.380 0.174 0.020
100 1.683 0.072 0.008
3.542 0.102 0.009
5.598 0.132 0.010
7.869 0.156 0.011
10.380 0.175 0.010
30 30 1.683 0.062 0.008
3.542 0.103 0.011
5.598 0.139 0.013
7.869 0.169 0.015
10.380 0.193 0.016
100 1.683 0.061 0.004
3.542 0.102 0.006
5.598 0.139 0.008
7.869 0.168 0.008
10.380 0.193 0.008
Table 5:

The mean observed AUCs (AUCOBS) for Gamma gap times across sample size (n) and study duration (τ).

τ n Gap time variance Mean AUCOBS Standard error
15 30 4.000 0.131 0.019
5.000 0.144 0.019
6.667 0.164 0.020
10.000 0.199 0.020
20.000 0.259 0.020
100 4.000 0.132 0.010
5.000 0.144 0.011
6.667 0.165 0.011
10.000 0.198 0.011
20.000 0.259 0.011
30 30 4.000 0.122 0.011
5.000 0.137 0.011
6.667 0.160 0.013
10.000 0.200 0.014
20.000 0.273 0.016
100 4.000 0.122 0.006
5.000 0.137 0.007
6.667 0.160 0.007
10.000 0.200 0.008
20.000 0.272 0.009
Table 6:

The mean observed AUCs (AUCOBS) and likelihood-based AUCs (AUCMLE and AUC0) across the level of missingness (π), rate of unscheduled visits (λ), and the standard deviation of scheduled visit timings (σ) for three scheduled measurement occasions and sample size (n) 30.

σ π λ Mean AUCOBS Standard error Mean AUCMLE Standard error Mean AUC0 Standard error
0.1 0.1 0.10 0.023 0.008 0.023 0.006 0.011 0.003
0.15 0.033 0.011 0.032 0.008 0.015 0.003
0.20 0.041 0.012 0.041 0.009 0.020 0.004
0.25 0.049 0.013 0.049 0.009 0.025 0.004
0.30 0.055 0.014 0.055 0.010 0.029 0.005
0.2 0.10 0.032 0.010 0.032 0.008 0.010 0.003
0.15 0.046 0.012 0.046 0.010 0.015 0.003
0.20 0.057 0.014 0.057 0.011 0.020 0.004
0.25 0.068 0.015 0.067 0.011 0.025 0.004
0.30 0.076 0.016 0.076 0.012 0.029 0.005
0.3 0.10 0.041 0.012 0.040 0.009 0.011 0.003
0.15 0.057 0.014 0.056 0.011 0.015 0.003
0.20 0.070 0.015 0.070 0.012 0.020 0.004
0.25 0.083 0.017 0.082 0.013 0.025 0.005
0.30 0.094 0.017 0.093 0.013 0.030 0.005
0.3 0.1 0.10 0.039 0.013 0.039 0.010 0.031 0.007
0.15 0.056 0.014 0.056 0.011 0.044 0.008
0.20 0.070 0.015 0.070 0.012 0.056 0.009
0.25 0.083 0.017 0.083 0.012 0.067 0.010
0.30 0.095 0.018 0.095 0.013 0.078 0.010
0.2 0.10 0.046 0.013 0.046 0.011 0.031 0.007
0.15 0.064 0.016 0.065 0.013 0.044 0.009
0.20 0.079 0.017 0.080 0.014 0.056 0.010
0.25 0.094 0.018 0.095 0.014 0.067 0.010
0.30 0.108 0.019 0.108 0.014 0.078 0.011
0.3 0.10 0.050 0.015 0.050 0.013 0.030 0.008
0.15 0.071 0.017 0.071 0.014 0.044 0.009
0.20 0.088 0.017 0.088 0.015 0.056 0.010
0.25 0.103 0.019 0.104 0.015 0.067 0.011
0.30 0.117 0.019 0.117 0.015 0.077 0.011
Table 7:

The mean observed AUCs (AUCOBS) and likelihood-based AUCs (AUCMLE and AUC0) across the level of missingness (π), rate of unscheduled visits (λ), and the standard deviation of scheduled visit timings (σ) for five scheduled measurement occasions and sample size (n) 30.

σ π λ Mean AUCOBS Standard error Mean AUCMLE Standard error Mean AUC0 Standard error
0.1 0.1 0.10 0.023 0.007 0.023 0.004 0.010 0.002
0.15 0.033 0.008 0.033 0.005 0.016 0.002
0.20 0.041 0.009 0.041 0.006 0.020 0.003
0.25 0.048 0.010 0.049 0.007 0.025 0.003
0.30 0.056 0.011 0.056 0.007 0.030 0.003
0.2 0.10 0.033 0.008 0.033 0.006 0.011 0.002
0.15 0.046 0.010 0.046 0.007 0.015 0.002
0.20 0.058 0.010 0.058 0.008 0.021 0.003
0.25 0.068 0.012 0.069 0.008 0.025 0.003
0.30 0.077 0.013 0.078 0.009 0.030 0.003
0.3 0.10 0.040 0.009 0.040 0.007 0.011 0.002
0.15 0.056 0.011 0.057 0.008 0.015 0.002
0.20 0.071 0.012 0.071 0.008 0.021 0.003
0.25 0.083 0.013 0.084 0.009 0.025 0.003
0.30 0.094 0.014 0.094 0.009 0.030 0.004
0.3 0.1 0.10 0.040 0.010 0.040 0.007 0.031 0.005
0.15 0.056 0.012 0.057 0.008 0.045 0.007
0.20 0.071 0.013 0.071 0.009 0.057 0.007
0.25 0.084 0.013 0.084 0.009 0.068 0.007
0.30 0.095 0.014 0.096 0.009 0.079 0.008
0.2 0.10 0.047 0.011 0.047 0.008 0.031 0.006
0.15 0.064 0.012 0.065 0.009 0.044 0.006
0.20 0.082 0.013 0.082 0.010 0.057 0.007
0.25 0.095 0.014 0.096 0.010 0.068 0.008
0.30 0.109 0.015 0.109 0.011 0.079 0.008
0.3 0.10 0.049 0.011 0.050 0.010 0.030 0.006
0.15 0.071 0.013 0.072 0.011 0.044 0.007
0.20 0.089 0.014 0.090 0.011 0.057 0.007
0.25 0.105 0.015 0.105 0.012 0.068 0.008
0.30 0.118 0.014 0.119 0.011 0.078 0.008
Table 8:

The mean observed AUCs (AUCOBS) and likelihood-based AUCs (AUCMLE and AUC0) across the level of missingness (π), rate of unscheduled visits (λ), and the standard deviation of scheduled visit timings (σ) for three scheduled measurement occasions and sample size (n) 100.

σ π λ Mean AUCOBS Standard error Mean AUCMLE Standard error Mean AUC0 Standard error
0.1 0.1 0.10 0.023 0.005 0.023 0.003 0.010 0.001
0.15 0.033 0.006 0.033 0.004 0.015 0.002
0.20 0.041 0.007 0.041 0.005 0.020 0.002
0.25 0.049 0.007 0.049 0.005 0.025 0.002
0.30 0.056 0.008 0.056 0.006 0.030 0.002
0.2 0.10 0.033 0.006 0.033 0.004 0.010 0.001
0.15 0.046 0.007 0.046 0.005 0.015 0.002
0.20 0.058 0.008 0.058 0.006 0.020 0.002
0.25 0.068 0.008 0.068 0.006 0.025 0.002
0.30 0.077 0.009 0.077 0.007 0.030 0.003
0.3 0.10 0.040 0.007 0.040 0.005 0.010 0.001
0.15 0.056 0.008 0.056 0.006 0.015 0.002
0.20 0.071 0.009 0.070 0.006 0.020 0.002
0.25 0.082 0.009 0.083 0.007 0.025 0.002
0.30 0.094 0.009 0.093 0.007 0.029 0.003
0.3 0.1 0.10 0.040 0.007 0.040 0.005 0.031 0.004
0.15 0.056 0.008 0.056 0.006 0.044 0.005
0.20 0.071 0.009 0.071 0.007 0.056 0.005
0.25 0.083 0.010 0.084 0.007 0.068 0.005
0.30 0.095 0.010 0.095 0.007 0.078 0.006
0.2 0.10 0.046 0.008 0.046 0.006 0.031 0.004
0.15 0.065 0.008 0.065 0.007 0.044 0.005
0.20 0.082 0.010 0.082 0.007 0.057 0.005
0.25 0.096 0.010 0.096 0.008 0.068 0.006
0.30 0.109 0.011 0.109 0.008 0.079 0.006
0.3 0.10 0.050 0.008 0.050 0.007 0.031 0.004
0.15 0.071 0.009 0.071 0.008 0.044 0.005
0.20 0.089 0.010 0.089 0.008 0.057 0.005
0.25 0.104 0.010 0.105 0.008 0.068 0.006
0.30 0.118 0.011 0.119 0.008 0.078 0.006
Table 9:

The mean observed AUCs (AUCOBS) and likelihood-based AUCs (AUCMLE) for dependent regular and irregular visit process across the increase in the probability of missingness due to unscheduled visits (θ), rate of unscheduled visits (λ), and the standard deviation of scheduled visit timings (σ) for three scheduled measurement occasions and sample size (n) 100.

σ θ λ Mean AUCOBS Standard error Correct mean Standard error Mis-specified Standard error
AUCMLE mean AUCMLE
0.05 0.1 0.10 0.018 0.004 0.018 0.003 0.018 0.003
0.30 0.042 0.007 0.043 0.006 0.045 0.006
0.5 0.10 0.016 0.003 0.017 0.003 0.019 0.003
0.30 0.040 0.006 0.042 0.007 0.053 0.006
0.2 0.1 0.10 0.030 0.006 0.034 0.005 0.031 0.004
0.30 0.075 0.009 0.076 0.008 0.076 0.006
0.5 0.10 0.025 0.005 0.031 0.004 0.028 0.004
0.30 0.064 0.008 0.070 0.008 0.071 0.006
Table 10:

The mean AUCs assuming no missingness (AUC0) for dependent regular and irregular visit process across the increase in the probability of missingness due to unscheduled visits (θ), rate of unscheduled visits (λ), and the standard deviation of scheduled visit timings (σ) for three scheduled measurement occasions and sample size (n) 100.

σ θ λ Correct mean AUC0 Standard error Mis-specified mean AUC0 Standard error
0.05 0.1 0.10 0.005 0.001 0.005 0.001
0.30 0.014 0.001 0.014 0.001
0.5 0.10 0.005 0.001 0.005 0.001
0.30 0.014 0.001 0.014 0.001
0.2 0.1 0.10 0.029 0.004 0.021 0.003
0.30 0.059 0.004 0.056 0.004
0.5 0.10 0.028 0.003 0.019 0.003
0.30 0.057 0.004 0.053 0.004
Table 11:

The mean observed AUCs (AUCOBS) and likelihood-based AUCs (AUCMLE and AUC0) for the unobserved process across the probability of the increased rate of unscheduled visits activating and deactivating (P act and P deact), the initial rate of unscheduled visits (λ), the increased rate of unscheduled visits (λ act) for sample size (n) 100.

P deact P act λ λ act Mean AUCOBS Standard error Mean AUCMLE Standard error Mean AUC0 Standard error
0.1 0.3 0.10 0.40 0.049 0.007 0.050 0.006 0.026 0.003
0.60 0.063 0.009 0.065 0.006 0.036 0.003
0.20 0.40 0.059 0.008 0.059 0.006 0.032 0.003
0.60 0.070 0.009 0.071 0.007 0.041 0.003
0.6 0.10 0.40 0.060 0.008 0.061 0.006 0.033 0.003
0.60 0.077 0.010 0.077 0.007 0.046 0.003
0.20 0.40 0.067 0.009 0.066 0.006 0.037 0.003
0.60 0.080 0.010 0.080 0.007 0.049 0.003
0.3 0.3 0.10 0.40 0.042 0.007 0.042 0.005 0.021 0.002
0.60 0.052 0.008 0.053 0.006 0.028 0.002
0.20 0.40 0.054 0.008 0.053 0.005 0.028 0.002
0.60 0.061 0.009 0.062 0.006 0.034 0.003
0.6 0.10 0.40 0.052 0.008 0.052 0.005 0.027 0.002
0.60 0.066 0.009 0.066 0.006 0.037 0.003
0.20 0.40 0.060 0.009 0.060 0.006 0.033 0.003
0.60 0.071 0.009 0.071 0.007 0.041 0.003
Table 12:

The mean observed AUCs (AUCOBS) and mean bias across the number of scheduled measurement occasions (k), baseline rate of unscheduled visits (λ 0), and the level of informativeness of the unscheduled visit process (γ) for sample size (n) 100.

Number of scheduled γ λ 0 Mean AUCOBS Standard error Mean bias Standard error
measurement occasions (k)
2 0 0.10 0.007 0.004 0.001 0.080
0.20 0.017 0.006 0.000 0.081
0.30 0.027 0.008 0.003 0.078
0.40 0.037 0.008 −0.004 0.077
0.5 0.10 0.008 0.004 0.073 0.084
0.20 0.018 0.006 0.118 0.083
0.30 0.029 0.008 0.145 0.083
0.40 0.040 0.009 0.166 0.084
1 0.10 0.011 0.005 0.162 0.095
0.20 0.023 0.007 0.236 0.095
0.30 0.035 0.008 0.279 0.101
0.40 0.045 0.010 0.312 0.104
4 0 0.10 0.004 0.002 0.003 0.075
0.20 0.012 0.004 −0.001 0.079
0.30 0.020 0.005 0.003 0.074
0.40 0.030 0.005 −0.001 0.078
0.5 0.10 0.005 0.002 0.040 0.079
0.20 0.013 0.004 0.077 0.082
0.30 0.023 0.005 0.113 0.080
0.40 0.032 0.006 0.131 0.082
1 0.10 0.007 0.003 0.093 0.085
0.20 0.017 0.005 0.169 0.093
0.30 0.027 0.006 0.226 0.098
0.40 0.038 0.007 0.265 0.103
Figure 16: The mean observed AUCs (AUCOBS) and mean bias across the level of informativeness of the unscheduled visit process (γ) for two scheduled measurement occasions, mean baseline rate of unscheduled visits (λ0) of 0.1, 0.4, 2.0, and a standard deviation of scheduled visit timings (σ) of 0.1, 0.6.
Figure 16:

The mean observed AUCs (AUCOBS) and mean bias across the level of informativeness of the unscheduled visit process (γ) for two scheduled measurement occasions, mean baseline rate of unscheduled visits (λ0) of 0.1, 0.4, 2.0, and a standard deviation of scheduled visit timings (σ) of 0.1, 0.6.

Figure 17: The mean observed AUCs (AUCOBS) and mean bias across the level of informativeness of the unscheduled visit process (γ) for four scheduled measurement occasions, mean baseline rate of unscheduled visits (λ0) of 0.1, 0.4, 2.0, and a standard deviation of scheduled visit timings (σ) of 0.1, 0.6.
Figure 17:

The mean observed AUCs (AUCOBS) and mean bias across the level of informativeness of the unscheduled visit process (γ) for four scheduled measurement occasions, mean baseline rate of unscheduled visits (λ0) of 0.1, 0.4, 2.0, and a standard deviation of scheduled visit timings (σ) of 0.1, 0.6.

Figure 18: The visit timings for random subsets of 30 individuals across the level of informativeness of the unscheduled visit process (γ) for two scheduled measurement occasions, mean baseline rate of unscheduled visits (λ0) of 0.1, 0.4, 2.0, and a standard deviation of scheduled visit timings (σ) of 0.1, 0.6.
Figure 18:

The visit timings for random subsets of 30 individuals across the level of informativeness of the unscheduled visit process (γ) for two scheduled measurement occasions, mean baseline rate of unscheduled visits (λ0) of 0.1, 0.4, 2.0, and a standard deviation of scheduled visit timings (σ) of 0.1, 0.6.

Figure 19: The visit timings for random subsets of 30 individuals across the level of informativeness of the unscheduled visit process (γ) for four scheduled measurement occasions, mean baseline rate of unscheduled visits (λ0) of 0.1, 0.4, 2.0, and a standard deviation of scheduled visit timings (σ) of 0.1, 0.6.
Figure 19:

The visit timings for random subsets of 30 individuals across the level of informativeness of the unscheduled visit process (γ) for four scheduled measurement occasions, mean baseline rate of unscheduled visits (λ0) of 0.1, 0.4, 2.0, and a standard deviation of scheduled visit timings (σ) of 0.1, 0.6.

Table 13:

The mean observed AUCs (AUCOBS) and mean bias across the mean baseline rate of unscheduled visits (λ 0), and the level of informativeness of the unscheduled visit process (γ) for sample size (n) 100, two scheduled measurement occasions, and the standard deviation of scheduled visit timings (σ) of 0.1.

γ λ 0 Mean AUCOBS Standard error Mean bias Standard error
0 0.05 0.003 0.003 0.001 0.079
0.10 0.008 0.004 0.002 0.086
0.25 0.023 0.007 −0.004 0.088
0.40 0.036 0.009 −0.006 0.091
0.60 0.052 0.011 −0.001 0.093
1.10 0.085 0.013 0.004 0.096
1.30 0.096 0.014 0.002 0.095
2.00 0.126 0.015 0.003 0.098
0.5 0.05 0.004 0.003 0.040 0.084
0.10 0.009 0.005 0.074 0.092
0.25 0.024 0.008 0.136 0.097
0.40 0.038 0.009 0.159 0.104
0.60 0.054 0.011 0.186 0.104
1.10 0.087 0.013 0.202 0.105
1.30 0.098 0.014 0.208 0.105
2.00 0.128 0.015 0.216 0.101
1 0.05 0.005 0.003 0.095 0.096
0.10 0.011 0.005 0.159 0.102
0.25 0.028 0.008 0.264 0.121
0.40 0.042 0.010 0.313 0.127
0.60 0.059 0.012 0.350 0.131
1.10 0.092 0.013 0.394 0.135
1.30 0.102 0.015 0.405 0.137
2.00 0.131 0.016 0.432 0.139
Table 14:

The mean observed AUCs (AUCOBS) and mean bias across the mean baseline rate of unscheduled visits (λ 0), and the level of informativeness of the unscheduled visit process (γ) for sample size (n) 100, two scheduled measurement occasions, and the standard deviation of scheduled visit timings (σ) of 0.3.

γ λ 0 Mean AUCOBS Standard error Mean bias Standard error
0 0.05 0.009 0.005 0.001 0.085
0.10 0.022 0.007 0.003 0.084
0.25 0.058 0.011 0.002 0.089
0.40 0.087 0.013 0.001 0.089
0.60 0.116 0.014 −0.001 0.093
1.10 0.164 0.016 0.003 0.096
1.30 0.177 0.016 0.008 0.090
2.00 0.209 0.016 0.006 0.097
0.5 0.05 0.011 0.005 0.038 0.087
0.10 0.024 0.007 0.074 0.088
0.25 0.060 0.012 0.136 0.099
0.40 0.089 0.014 0.153 0.102
0.60 0.117 0.014 0.178 0.100
1.10 0.165 0.016 0.205 0.102
1.30 0.180 0.016 0.205 0.105
2.00 0.211 0.016 0.218 0.105
1 0.05 0.014 0.006 0.095 0.096
0.10 0.029 0.008 0.162 0.105
0.25 0.066 0.013 0.266 0.125
0.40 0.093 0.015 0.311 0.129
0.60 0.122 0.016 0.357 0.137
1.10 0.168 0.017 0.394 0.137
1.30 0.181 0.017 0.412 0.132
2.00 0.214 0.016 0.425 0.141
Table 15:

The mean observed AUCs (AUCOBS) and mean bias across the mean baseline rate of unscheduled visits (λ 0), and the level of informativeness of the unscheduled visit process (γ) for sample size (n) 100, two scheduled measurement occasions, and the standard deviation of scheduled visit timings (σ) of 0.6.

γ λ 0 Mean AUCOBS Standard error Mean bias Standard error
0 0.05 0.018 0.006 0.001 0.081
0.10 0.041 0.010 −0.001 0.085
0.25 0.097 0.014 −0.001 0.086
0.40 0.134 0.015 0.001 0.094
0.60 0.168 0.016 −0.003 0.088
1.10 0.216 0.017 0.001 0.094
1.30 0.229 0.015 0.002 0.098
2.00 0.254 0.014 0.001 0.097
0.5 0.05 0.020 0.007 0.043 0.084
0.10 0.043 0.010 0.074 0.090
0.25 0.099 0.015 0.136 0.097
0.40 0.136 0.016 0.161 0.096
0.60 0.170 0.017 0.179 0.098
1.10 0.218 0.016 0.204 0.106
1.30 0.230 0.016 0.202 0.107
2.00 0.258 0.015 0.221 0.110
1 0.05 0.026 0.008 0.092 0.091
0.10 0.050 0.011 0.160 0.105
0.25 0.104 0.015 0.265 0.120
0.40 0.139 0.016 0.311 0.120
0.60 0.171 0.017 0.344 0.124
1.10 0.220 0.017 0.395 0.128
1.30 0.232 0.016 0.409 0.141
2.00 0.263 0.016 0.421 0.130
Table 16:

The mean observed AUCs (AUCOBS) and mean bias across the mean baseline rate of unscheduled visits (λ 0), and the level of informativeness of the unscheduled visit process (γ) for sample size (n) 100, four scheduled measurement occasions, and the standard deviation of scheduled visit timings (σ) of 0.1.

γ λ 0 Mean AUCOBS Standard error Mean bias Standard error
0 0.05 0.002 0.002 −0.003 0.079
0.10 0.006 0.003 0.004 0.078
0.25 0.019 0.005 −0.001 0.081
0.40 0.032 0.007 0.003 0.085
0.60 0.049 0.008 −0.003 0.086
1.10 0.082 0.010 0.001 0.091
1.30 0.093 0.011 0.003 0.092
2.00 0.125 0.012 −0.003 0.098
0.5 0.05 0.002 0.002 0.019 0.079
0.10 0.006 0.003 0.040 0.082
0.25 0.020 0.005 0.094 0.088
0.40 0.033 0.007 0.131 0.093
0.60 0.049 0.008 0.155 0.093
1.10 0.084 0.010 0.188 0.097
1.30 0.095 0.012 0.189 0.101
2.00 0.126 0.012 0.206 0.105
1 0.05 0.004 0.002 0.046 0.084
0.10 0.008 0.003 0.103 0.094
0.25 0.023 0.006 0.200 0.102
0.40 0.037 0.008 0.257 0.117
0.60 0.054 0.009 0.304 0.120
1.10 0.087 0.012 0.362 0.126
1.30 0.097 0.012 0.371 0.132
2.00 0.129 0.014 0.406 0.132
Table 17:

The mean observed AUCs (AUCOBS) and mean bias across the mean baseline rate of unscheduled visits (λ 0), and the level of informativeness of the unscheduled visit process (γ) for sample size (n) 100, four scheduled measurement occasions, and the standard deviation of scheduled visit timings (σ) of 0.3.

γ λ 0 Mean AUCOBS Standard error Mean bias Standard error
0 0.05 0.006 0.003 −0.005 0.075
0.10 0.016 0.005 0.002 0.080
0.25 0.048 0.008 0.001 0.082
0.40 0.077 0.011 0.004 0.083
0.60 0.108 0.012 0.003 0.085
1.10 0.159 0.013 0.001 0.092
1.30 0.173 0.013 −0.003 0.093
2.00 0.209 0.013 −0.003 0.093
0.5 0.05 0.007 0.003 0.020 0.078
0.10 0.018 0.005 0.037 0.082
0.25 0.050 0.009 0.095 0.087
0.40 0.078 0.011 0.127 0.093
0.60 0.109 0.013 0.156 0.098
1.10 0.160 0.014 0.181 0.101
1.30 0.174 0.014 0.189 0.102
2.00 0.209 0.013 0.209 0.104
1 0.05 0.010 0.004 0.044 0.081
0.10 0.022 0.006 0.093 0.092
0.25 0.055 0.010 0.199 0.108
0.40 0.082 0.012 0.250 0.118
0.60 0.111 0.013 0.303 0.122
1.10 0.160 0.015 0.368 0.130
1.30 0.173 0.014 0.375 0.137
2.00 0.210 0.014 0.406 0.135
Table 18:

The mean observed AUCs (AUCOBS) and mean bias across the mean baseline rate of unscheduled visits (λ 0), and the level of informativeness of the unscheduled visit process (γ) for sample size (n) 100, four scheduled measurement occasions, and the standard deviation of scheduled visit timings (σ) of 0.6.

λ λ 0 Mean AUCOBS Standard error Mean bias Standard error
0 0.05 0.014 0.004 0.002 0.078
0.10 0.031 0.007 0.001 0.076
0.25 0.082 0.011 0.004 0.081
0.40 0.121 0.012 0.001 0.086
0.60 0.158 0.013 −0.003 0.087
1.10 0.211 0.013 −0.001 0.089
1.30 0.225 0.012 −0.004 0.090
2.00 0.254 0.011 −0.007 0.094
0.5 0.05 0.016 0.004 0.019 0.076
0.10 0.034 0.007 0.043 0.081
0.25 0.084 0.011 0.098 0.089
0.40 0.121 0.013 0.122 0.090
0.60 0.158 0.014 0.153 0.094
1.10 0.212 0.013 0.189 0.100
1.30 0.225 0.013 0.196 0.105
2.00 0.257 0.012 0.209 0.109
1 0.05 0.020 0.006 0.052 0.088
0.10 0.040 0.009 0.098 0.091
0.25 0.089 0.013 0.196 0.112
0.40 0.124 0.014 0.259 0.119
0.60 0.158 0.015 0.295 0.122
1.10 0.210 0.015 0.366 0.135
1.30 0.224 0.015 0.378 0.132
2.00 0.257 0.013 0.399 0.131

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Received: 2020-10-28
Revised: 2021-03-01
Accepted: 2021-08-02
Published Online: 2021-08-16

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/ijb-2020-0144
Keywords for this article
irregular visits; longitudinal data; summary measures; visit intensity; visit process

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Doubly robust adaptive LASSO for effect modifier discovery
  4. Increasing the efficiency of randomized trial estimates via linear adjustment for a prognostic score
  5. Review
  6. Review and comparison of treatment effect estimators using propensity and prognostic scores
  7. Research Articles
  8. Error rate control for classification rules in multiclass mixture models
  9. Regression trees and ensembles for cumulative incidence functions
  10. Causal inference under over-simplified longitudinal causal models
  11. Causal inference under interference with prognostic scores for dynamic group therapy studies
  12. Bayesian multi-response nonlinear mixed-effect model: application of two recent HIV infection biomarkers
  13. A Bayesian semiparametric accelerate failure time mixture cure model
  14. Quantifying the extent of visit irregularity in longitudinal data
  15. An improved method for analysis of interrupted time series (ITS) data: accounting for patient heterogeneity using weighted analysis
  16. A robust hazard ratio for general modeling of survival-times
  17. Penalized likelihood estimation of the proportional hazards model for survival data with interval censoring
  18. A parametric approach to relaxing the independence assumption in relative survival analysis
  19. The number of response categories in ordered response models
  20. A comparison of joint dichotomization and single dichotomization of interacting variables to discriminate a disease outcome
  21. Spike detection for calcium activity
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