Background
Methods
Data-generating mechanisms
Linear predictor used for data generation
Parameters | Value in simulation | Statistical notation in the linear predictor used for simulation of datasetb |
---|---|---|
Varying-parameters
| ||
Number of trials | 6, 10, 16, 20, 26, 30, 40, 50 | – |
Magnitude of the interaction effecta | 0, −0.01 |
beta.inter
|
Between-trial variance of the interaction effecta | 0, 0.005, 0.05, 0.5 |
V
2
|
Fixed-parameters
| ||
Number of patients included in each trial | between 30 and 199 patients in small trials, and 200 and 400 patients in large trials |
j
|
Ratio of small to large trials | 1:1 |
–
|
Number of trial arms | 2 |
treat
|
Ratio of randomization | 1:1 | – |
Magnitude of the treatment effecta | −1 |
beta.treat
|
Between-trial variance of the treatment effecta | 0.063 |
V
1
|
Effect of age on the outcome | 0.05 |
age
|
Random intercept variance | 0.25 |
σ
2
|
Random slope variance | 0.04 |
τ
2
|
Correlation between random intercept and random slope variances | 0.8 |
ρ
|
Age of individual patients | N(mean. age, variance. age) |
age
|
Mean age of patients within trials | N(67,20.25) | mean. age |
Variance of age within trials | N(49, 1) | variance. age |
Varying parameters for data generation
Interaction effect | |||
---|---|---|---|
-0.01 | 0 | ||
Number of trials | Between-trial variance in the interaction effect | Number of trials | Between-trial variance in the interaction effect |
6 | 0 | 6 | 0 |
10 | 0 | 10 | 0 |
16 | 0 | 16 | 0 |
20 | 0 | 20 | 0 |
26 | 0 | 26 | 0 |
30 | 0 | 30 | 0 |
40 | 0 | 40 | 0 |
50 | 0 | 50 | 0 |
6 | 0.005 | 6 | 0.005 |
10 | 0.005 | 10 | 0.005 |
16 | 0.005 | 16 | 0.005 |
20 | 0.005 | 20 | 0.005 |
26 | 0.005 | 26 | 0.005 |
30 | 0.005 | 30 | 0.005 |
40 | 0.005 | 40 | 0.005 |
50 | 0.005 | 50 | 0.005 |
6 | 0.05 | 6 | 0.05 |
10 | 0.05 | 10 | 0.05 |
16 | 0.05 | 16 | 0.05 |
20 | 0.05 | 20 | 0.05 |
26 | 0.05 | 26 | 0.05 |
30 | 0.05 | 30 | 0.05 |
40 | 0.05 | 40 | 0.05 |
50 | 0.05 | 50 | 0.05 |
6 | 0.5 | 6 | 0.5 |
10 | 0.5 | 10 | 0.5 |
16 | 0.5 | 16 | 0.5 |
20 | 0.5 | 20 | 0.5 |
26 | 0.5 | 26 | 0.5 |
30 | 0.5 | 30 | 0.5 |
40 | 0.5 | 40 | 0.5 |
50 | 0.5 | 50 | 0.5 |
Fixed parameters for data generation
Software used to generate simulated data
Code for data generation
Estimand
Models assessed
Model | Description | Difference from model 1 |
---|---|---|
1 | − Model for analysis of individual patient data − Estimation of the between-trial variance of the interaction between treatment effect and age − Estimation of the between-trial variance of the treatment effect − Separation of within- from between-trial interaction effects | – |
2 | − Model for analysis of individual patient data − Estimation of the between-trial variance of the interaction between treatment effect and age − Separation of within- from between-trial interaction effects | − No estimation of the between-trial variance of the treatment effect |
3 | − Model for analysis of individual patient data − Estimation of the between-trial variance of the treatment effect − Separation of within- from between-trial interaction effects | − No estimation of the between-trial variance of the interaction between treatment effect and age |
4 | − Model for analysis of individual patient data − Estimation of the between-trial variance of the interaction between treatment effect and age − Estimation of the between-trial variance of the treatment effect − No separation of within- from between-trial interaction effects | − No separation of within- from between-trial interaction effects |
5 | − Model for analysis of individual patient data − Estimation of the between-trial variance of the interaction between treatment effect and age − No separation of within- from between-trial interaction effects | − No estimation of the between-trial variance of the treatment effect − No separation of within- from between-trial interaction effects |
6 | − Model for analysis of individual patient data − Estimation of the between-trial variance of the treatment effect − No separation of within- from between-trial interaction effects | − No estimation of the between-trial variance of the interaction between treatment effect and age − No separation of within- from between-trial interaction effects |
7 | − Model for analysis of aggregate data − Estimation of the between-trial variance of the treatment effect | − Model for analysis of aggregate data − No estimation of the between-trial variance of the interaction between treatment effect and age − No separation of within- from between-trial interaction effects |
Performance measures
Sample size calculation
Estimate | MCSE |
---|---|
Bias | ≤0.03 |
Coverage | ≤0.05 |
Empirical SE | ≤0.0005 |
Model SE | ≤0.0005 |
Relative error in the model SE | ≤0.018 |
Software used to analyse the performance of the simulated data
Results
Results of simulations assuming a large between-trial variation of 0.05 in the interaction effect
No. of trials | Model 1 | Model 2 | Model 3 | Model 4 | Model 5 | Model 6 | Model 7 |
---|---|---|---|---|---|---|---|
6 | 0.00179 (0.09149) | −0.00006 (0.09595) | 0.00323 (0.10316) | 0.00179 (0.09149) | 0.00038 (0.09622) | 0.00323 (0.10316) | −0.06454 (1.87747) |
10 | −0.00101 (0.07082) | −0.00079 (0.07411) | − 0.00070 (0.08183) | − 0.00101 (0.07082) | − 0.00072 (0.07410) | −0.00070 (0.08183) | 0.01286 (1.22375) |
16 | −0.00013 (0.05543) | −0.00060 (0.05795) | − 0.00094 (0.06589) | −0.00014 (0.05543) | − 0.00050 (0.05800) | −0.00094 (0.06589) | 0.01857 (0.93068) |
20 | −0.00049 (0.05012) | −0.00102 (0.05260) | − 0.00119 (0.05889) | −0.00049 (0.05012) | − 0.00097 (0.05254) | −0.00119 (0.05889) | − 0.02249 (0.81903) |
26 | −0.00051 (0.04289) | − 0.00079 (0.04524) | 0.00012 (0.05107) | − 0.00051 (0.04289) | −0.00076 (0.04517) | 0.00012 (0.05107) | −0.01126 (0.67902) |
30 | −0.00013 (0.04099) | 0.00051 (0.04331) | 0.00018 (0.04889) | −0.00013 (0.04099) | 0.00049 (0.04323) | 0.00018 (0.04889) | 0.00940 (0.62751) |
40 | 0.00034 (0.03480) | 0.00073 (0.03687) | 0.00050 (0.04180) | 0.00034 (0.03480) | 0.00074 (0.03686) | 0.00050 (0.04180) | 0.01953 (0.55119) |
50 | −0.00010 (0.03198) | −0.00005 (0.03364) | − 0.00013 (0.03806) | −0.00010 (0.03198) | − 0.00004 (0.03368) | −0.00013 (0.03806) | 0.00074 (0.48328) |
No. of trials | Model 1 | Model 2 | Model 3 | Model 4 | Model 5 | Model 6 | Model 7 |
---|---|---|---|---|---|---|---|
6 | 0.89400 (0.00688) | 0.88500 (0.00713) | 0.83850 (0.00823) | 0.89450 (0.00687) | 0.88550 (0.00712) | 0.83850 (0.00823) | 0.94350 (0.00516) |
10 | 0.92250 (0.00598) | 0.92100 (0.00603) | 0.88950 (0.00701) | 0.92250 (0.00598) | 0.92150 (0.00601) | 0.88950 (0.00701) | 0.95250 (0.00476) |
16 | 0.92850 (0.00576) | 0.92600 (0.00585) | 0.90800 (0.00646) | 0.92900 (0.00574) | 0.92700 (0.00582) | 0.90800 (0.00646) | 0.94900 (0.00492) |
20 | 0.93350 (0.00557) | 0.93900 (0.00535) | 0.92350 (0.00594) | 0.93400 (0.00555) | 0.93450 (0.00553) | 0.92350 (0.00594) | 0.94050 (0.00529) |
26 | 0.94150 (0.00525) | 0.94300 (0.00518) | 0.93250 (0.00561) | 0.94150 (0.00525) | 0.94250 (0.00521) | 0.93250 (0.00561) | 0.95300 (0.00473) |
30 | 0.93150 (0.00565) | 0.93300 (0.00559) | 0.91750 (0.00615) | 0.93150 (0.00565) | 0.93400 (0.00555) | 0.91750 (0.00615) | 0.94900 (0.00492) |
40 | 0.94650 (0.00503) | 0.93800 (0.00539) | 0.93250 (0.00561) | 0.94650 (0.00503) | 0.93700 (0.00543) | 0.93250 (0.00561) | 0.94450 (0.00512) |
50 | 0.94700 (0.00501) | 0.94950 (0.00490) | 0.93400 (0.00555) | 0.94700 (0.00501) | 0.94900 (0.00492) | 0.93400 (0.00555) | 0.95400 (0.00468) |
No. of trials | Performance measure | Model 1 | Model 2 | Model 3 | Model 4 | Model 5 | Model 6 | Model 7 |
---|---|---|---|---|---|---|---|---|
6 | Empirical SE | 0.09147 (0.00145) | 0.09595 (0.00152) | 0.10311 (0.00163) | 0.09148 (0.00145) | 0.09621 (0.00152) | 0.10311 (0.00163) | 1.87636 (0.02968) |
Model SE | 0.09125 (0.01059) | 0.09486 (0.01456) | 0.09568 (0.01640) | 0.09125 (0.01059) | 0.09528 (0.01454) | 0.09568 (0.01640) | 1.81917 (177.01665) | |
Relative error in model SE | −0.24168 (1.72411) | −1.13917 (1.77644) | −7.20306 (1.70537) | − 0.24417 (1.72406) | − 0.97397 (1.77358) | −7.20303 (1.70537) | −3.04755 (2.09350) | |
10 | Empirical SE | 0.07081 (0.00112) | 0.07410 (0.00117) | 0.08183 (0.00129) | 0.07081 (0.00112) | 0.07410 (0.00117) | 0.08183 (0.00129) | 1.22368 (0.01935) |
Model SE | 0.07072 (0.00378) | 0.07411 (0.00540) | 0.07776 (0.00676) | 0.07072 (0.00378) | 0.07412 (0.00534) | 0.07776 (0.00676) | 1.23611 (36.09575) | |
Relative error in model SE | −0.13873 (1.66686) | 0.00346 (1.71505) | −4.96244 (1.65106) | −0.13683 (1.66689) | 0.03390 (1.71257) | −4.96244 (1.65106) | 1.01558 (1.86656) | |
16 | Empirical SE | 0.05543 (0.00088) | 0.05794 (0.00092) | 0.06588 (0.00104) | 0.05543 (0.00088) | 0.05800 (0.00092) | 0.06588 (0.00104) | 0.93050 (0.01472) |
Model SE | 0.05606 (0.00144) | 0.05883 (0.00200) | 0.06377 (0.00321) | 0.05606 (0.00144) | 0.05867 (0.00196) | 0.06377 (0.00321) | 0.92118 (10.53523) | |
Relative error in model SE | 1.13187 (1.65210) | 1.53029 (1.68156) | −3.21023 (1.64405) | 1.13079 (1.65209) | 1.15699 (1.67359) | −3.21024 (1.64405) | −1.00201 (1.70189) | |
20 | Empirical SE | 0.05012 (0.00079) | 0.05259 (0.00083) | 0.05887 (0.00093) | 0.05012 (0.00079) | 0.05253 (0.00083) | 0.05887 (0.00093) | 0.81872 (0.01295) |
Model SE | 0.05005 (0.00089) | 0.05253 (0.00128) | 0.05719 (0.00207) | 0.05005 (0.00089) | 0.05253 (0.00130) | 0.05719 (0.00207) | 0.80839 (6.13561) | |
Relative error in model SE | −0.14153 (1.61861) | −0.10115 (1.64049) | −2.86836 (1.62743) | −0.14211 (1.61860) | 0.00514 (1.64349) | −2.86838 (1.62743) | −1.26192 (1.66352) | |
26 | Empirical SE | 0.04289 (0.00068) | 0.04523 (0.00072) | 0.05107 (0.00081) | 0.04289 (0.00068) | 0.04517 (0.00071) | 0.05107 (0.00081) | 0.67892 (0.01074) |
Model SE | 0.04404 (0.00054) | 0.04615 (0.00074) | 0.05064 (0.00127) | 0.04404 (0.00054) | 0.04612 (0.00074) | 0.05064 (0.00127) | 0.69302 (3.18236) | |
Relative error in model SE | 2.69132 (1.65593) | 2.02827 (1.65925) | −0.83126 (1.64152) | 2.69167 (1.65594) | 2.10172 (1.65950) | −0.83126 (1.64152) | 2.07716 (1.68652) | |
30 | Empirical SE | 0.04099 (0.00065) | 0.04331 (0.00068) | 0.04889 (0.00077) | 0.04099 (0.00065) | 0.04323 (0.00068) | 0.04889 (0.00077) | 0.62743 (0.00992) |
Model SE | 0.04073 (0.00040) | 0.04282 (0.00058) | 0.04681 (0.00092) | 0.04073 (0.00040) | 0.04278 (0.00058) | 0.04681 (0.00092) | 0.63396 (2.30260) | |
Relative error in model SE | −0.63932 (1.59872) | −1.13154 (1.60574) | −4.27073 (1.57354) | −0.64006 (1.59871) | −1.03999 (1.60736) | −4.27072 (1.57354) | 1.03939 (1.66192) | |
40 | Empirical SE | 0.03480 (0.00055) | 0.03687 (0.00058) | 0.04180 (0.00066) | 0.03480 (0.00055) | 0.03685 (0.00058) | 0.04180 (0.00066) | 0.55085 (0.00871) |
Model SE | 0.03545 (0.00022) | 0.03722 (0.00034) | 0.04099 (0.00053) | 0.03545 (0.00022) | 0.03719 (0.00034) | 0.04099 (0.00053) | 0.54445 (1.23398) | |
Relative error in model SE | 1.87182 (1.63128) | 0.94769 (1.63044) | −1.92503 (1.59642) | 1.87298 (1.63130) | 0.93484 (1.62997) | −1.92503 (1.59642) | −1.16099 (1.60819) | |
50 | Empirical SE | 0.03198 (0.00051) | 0.03364 (0.00053) | 0.03806 (0.00060) | 0.03198 (0.00051) | 0.03368 (0.00053) | 0.03806 (0.00060) | 0.48328 (0.00764) |
Model SE | 0.03165 (0.00014) | 0.03338 (0.00021) | 0.03673 (0.00034) | 0.03165 (0.00014) | 0.03337 (0.00021) | 0.03673 (0.00034) | 0.48219 (0.75386) | |
Relative error in model SE | −1.03205 (1.58095) | −0.79436 (1.59442) | −3.48739 (1.56226) | −1.03213 (1.58094) | −0.91692 (1.59272) | −3.48739 (1.56226) | − 0.22640 (1.61322) |
No. of trials | Model 1 | Model 2 | Model 3 | Model 4 | Model 5 | Model 6 | Model 7 |
---|---|---|---|---|---|---|---|
6 | 0.00837 (0.00027) | 0.00920 (0.00029) | 0.01064 (0.00034) | 0.00837 (0.00027) | 0.00925 (0.00029) | 0.01064 (0.00034) | 3.52311 (0.18382) |
10 | 0.00501 (0.00016) | 0.00549 (0.00017) | 0.00669 (0.00022) | 0.00501 (0.00016) | 0.00549 (0.00017) | 0.00669 (0.00022) | 1.49681 (0.05581) |
16 | 0.00307 (0.00010) | 0.00336 (0.00011) | 0.00434 (0.00014) | 0.00307 (0.00010) | 0.00336 (0.00011) | 0.00434 (0.00014) | 0.86574 (0.03416) |
20 | 0.00251 (0.00008) | 0.00277 (0.00009) | 0.00347 (0.00011) | 0.00251 (0.00008) | 0.00276 (0.00009) | 0.00347 (0.00011) | 0.67048 (0.02303) |
26 | 0.00184 (0.00006) | 0.00205 (0.00007) | 0.00261 (0.00008) | 0.00184 (0.00006) | 0.00204 (0.00007) | 0.00261 (0.00008) | 0.46083 (0.01572) |
30 | 0.00168 (0.00005) | 0.00187 (0.00006) | 0.00239 (0.00008) | 0.00168 (0.00005) | 0.00187 (0.00006) | 0.00239 (0.00008) | 0.39357 (0.01326) |
40 | 0.00121 (0.00004) | 0.00136 (0.00004) | 0.00175 (0.00005) | 0.00121 (0.00004) | 0.00136 (0.00004) | 0.00175 (0.00005) | 0.30366 (0.00986) |
50 | 0.00102 (0.00003) | 0.00113 (0.00004) | 0.00145 (0.00004) | 0.00102 (0.00003) | 0.00113 (0.00004) | 0.00145 (0.00004) | 0.23345 (0.00767) |