1 Introduction
2 Marginal and partially conditional rate-based models
2.1 Multiplicative models based on marginal rate functions
2.2 Multiplicative models based on partially conditional rate functions
3 Inference regarding treatment effects with omitted covariates
3.1 Asymptotic properties for estimators of treatment effect
3.2 A case-study involving an omitted fixed covariate
4 Empirical studies of finite sample behaviour
4.1 Misspecified rate-based models for Poisson processes
\(\phi \)
|
\(\zeta = 0\)
|
\(\zeta = \log 1.5\)
|
\(\zeta = \log 3.0\)
| |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
BIAS | ESE | ASE\(^{1}\) | ASE\(^2\) | ECP\(^1\) | ECP\(^2\) | BIAS | ESE | ASE\(^{1}\) | ASE\(^2\) | ECP\(^1\) | ECP\(^2\) | BIAS | ESE | ASE\(^{1}\) | ASE\(^2\) | ECP\(^1\) | ECP\(^2\) | |
Marginal model, \(P(Z=1) = 0.25\) | ||||||||||||||||||
0.5 | \(-\) 0.15 | 5.06 | 5.14 | 5.14 | 96.0 | 96.1 | \(-\) 5.59 | 4.97 | 4.85 | 5.01 | 78.3 | 80.2 | \(-\) 17.19 | 5.72 | 4.24 | 5.65 | 6.6 | 14.4 |
1.0 | \(-\) 0.15 | 5.06 | 5.14 | 5.14 | 96.0 | 96.1 | \(-\) 0.47 | 5.08 | 4.85 | 5.00 | 93.4 | 94.2 | \(-\) 0.09 | 6.10 | 4.20 | 5.65 | 82.1 | 92.4 |
2.0 | \(-\) 0.15 | 5.06 | 5.14 | 5.14 | 96.0 | 96.1 | 5.74 | 4.92 | 4.84 | 4.99 | 76.3 | 78.9 | 17.22 | 5.72 | 4.19 | 5.61 | 5.8 | 14.8 |
4.0 | \(-\) 0.15 | 5.06 | 5.14 | 5.14 | 96.0 | 96.1 | 11.04 | 5.24 | 4.83 | 4.97 | 37.0 | 38.8 | 32.51 | 5.58 | 4.21 | 5.53 | 0.0 | 0.0 |
Marginal model, \(P(Z=1) = 0.50\) | ||||||||||||||||||
0.5 | \(-\) 0.15 | 5.06 | 5.14 | 5.14 | 96.0 | 96.1 | \(-\) 6.77 | 4.81 | 4.61 | 4.78 | 68.2 | 70.1 | \(-\) 17.28 | 4.81 | 3.67 | 4.92 | 2.1 | 5.7 |
1.0 | \(-\) 0.15 | 5.06 | 5.14 | 5.14 | 96.0 | 96.1 | \(-\) 0.05 | 4.55 | 4.60 | 4.78 | 94.9 | 95.4 | 0.13 | 5.06 | 3.63 | 4.90 | 83.7 | 95.0 |
2.0 | \(-\) 0.15 | 5.06 | 5.14 | 5.14 | 96.0 | 96.1 | 7.23 | 4.85 | 4.59 | 4.76 | 64.3 | 65.9 | 17.03 | 5.09 | 3.62 | 4.89 | 2.9 | 7.2 |
4.0 | \(-\) 0.15 | 5.06 | 5.14 | 5.14 | 96.0 | 96.1 | 13.32 | 4.75 | 4.59 | 4.75 | 18.2 | 19.7 | 33.51 | 4.75 | 3.64 | 4.87 | 0.0 | 0.0 |
Partially conditional model, \(P(Z=1) = 0.25\) | ||||||||||||||||||
0.5 | \(-\) 0.15 | 5.12 | 5.20 | 5.19 | 96.2 | 96.2 | \(-\) 4.66 | 4.95 | 4.93 | 4.92 | 85.2 | 84.9 | \(-\) 5.15 | 4.37 | 4.37 | 4.39 | 79.2 | 79.3 |
1.0 | \(-\) 0.15 | 5.12 | 5.20 | 5.19 | 96.2 | 96.2 | 0.87 | 4.90 | 4.90 | 4.89 | 94.8 | 94.7 | 7.47 | 4.39 | 4.27 | 4.25 | 57.8 | 57.6 |
2.0 | \(-\) 0.15 | 5.12 | 5.20 | 5.19 | 96.2 | 96.2 | 6.42 | 4.96 | 4.88 | 4.88 | 72.3 | 72.4 | 19.73 | 3.98 | 4.22 | 4.19 | 0.2 | 0.2 |
4.0 | \(-\) 0.15 | 5.12 | 5.20 | 5.19 | 96.2 | 96.2 | 10.96 | 5.11 | 4.87 | 4.87 | 40.7 | 40.6 | 31.33 | 4.26 | 4.24 | 4.23 | 0.0 | 0.0 |
Partially conditional model, \(P(Z=1) = 0.50\) | ||||||||||||||||||
0.5 | \(-\) 0.15 | 5.12 | 5.20 | 5.19 | 96.2 | 96.2 | \(-\) 5.39 | 4.58 | 4.69 | 4.69 | 80.5 | 80.1 | \(-\) 4.96 | 3.90 | 3.79 | 3.82 | 72.6 | 73.2 |
1.0 | \(-\) 0.15 | 5.12 | 5.20 | 5.19 | 96.2 | 96.2 | 1.21 | 4.61 | 4.66 | 4.65 | 94.6 | 94.6 | 7.47 | 3.70 | 3.70 | 3.65 | 48.1 | 47.4 |
2.0 | \(-\) 0.15 | 5.12 | 5.20 | 5.19 | 96.2 | 96.2 | 7.70 | 4.66 | 4.63 | 4.61 | 59.7 | 59.1 | 19.78 | 3.67 | 3.66 | 3.60 | 0.0 | 0.0 |
4.0 | \(-\) 0.15 | 5.12 | 5.20 | 5.19 | 96.2 | 96.2 | 13.84 | 4.32 | 4.62 | 4.60 | 13.6 | 13.3 | 31.77 | 3.74 | 3.68 | 3.69 | 0.0 | 0.0 |
4.2 Misspecified rate-based models for Markov processes
\(\phi \)
|
\(\zeta = 0\)
|
\(\zeta = \log 1.5\)
|
\(\zeta = \log 3.0\)
| |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
BIAS | ESE | ASE\(^{1}\) | ASE\(^2\) | ECP\(^1\) | ECP\(^2\) | BIAS | ESE | ASE\(^{1}\) | ASE\(^2\) | ECP\(^1\) | ECP\(^2\) | BIAS | ESE | ASE\(^{1}\) | ASE\(^2\) | ECP\(^1\) | ECP\(^2\) | |
Marginal model, \(P(Z=1) = 0.25\) | ||||||||||||||||||
0.5 | \(-\) 1.40 | 5.24 | 5.13 | 5.34 | 93.4 | 94.7 | \(-\) 7.30 | 5.15 | 4.83 | 5.24 | 67.3 | 72.8 | \(-\) 20.70 | 6.26 | 4.15 | 6.01 | 2.5 | 8.7 |
1.0 | \(-\) 1.40 | 5.24 | 5.13 | 5.34 | 93.4 | 94.7 | \(-\) 1.43 | 5.17 | 4.82 | 5.22 | 92.3 | 94.9 | \(-\) 2.19 | 6.02 | 4.10 | 6.03 | 78.5 | 93.3 |
2.0 | \(-\) 1.40 | 5.24 | 5.13 | 5.34 | 93.4 | 94.7 | 4.58 | 5.11 | 4.81 | 5.21 | 82.4 | 85.6 | 16.14 | 6.03 | 4.09 | 5.98 | 10.0 | 23.8 |
4.0 | \(-\) 1.40 | 5.24 | 5.13 | 5.34 | 93.4 | 94.7 | 9.98 | 5.24 | 4.81 | 5.20 | 45.9 | 51.6 | 33.26 | 5.83 | 4.12 | 5.88 | 0.0 | 0.0 |
Marginal model, \(P(Z=1) = 0.50\) | ||||||||||||||||||
0.5 | \(-\) 1.40 | 5.24 | 5.13 | 5.34 | 93.4 | 94.7 | \(-\) 9.28 | 5.03 | 4.58 | 5.01 | 46.9 | 54.2 | \(-\) 20.59 | 5.28 | 3.56 | 5.20 | 0.8 | 2.1 |
1.0 | \(-\) 1.40 | 5.24 | 5.13 | 5.34 | 93.4 | 94.7 | \(-\) 1.56 | 4.86 | 4.56 | 5.00 | 92.0 | 95.0 | \(-\) 2.47 | 5.19 | 3.52 | 5.19 | 77.5 | 91.9 |
2.0 | \(-\) 1.40 | 5.24 | 5.13 | 5.34 | 93.4 | 94.7 | 5.68 | 5.00 | 4.55 | 4.99 | 72.6 | 77.9 | 15.56 | 5.21 | 3.51 | 5.19 | 5.0 | 14.9 |
4.0 | \(-\) 1.40 | 5.24 | 5.13 | 5.34 | 93.4 | 94.7 | 12.57 | 4.94 | 4.55 | 4.98 | 22.7 | 28.0 | 32.87 | 5.41 | 3.53 | 5.17 | 0.0 | 0.0 |
Partially conditional model, \(P(Z=1) = 0.25\) | ||||||||||||||||||
0.5 | \(-\) 0.21 | 5.06 | 5.19 | 5.17 | 95.4 | 95.6 | \(-\) 4.46 | 4.84 | 4.91 | 4.90 | 86.2 | 86.0 | \(-\) 5.06 | 4.31 | 4.28 | 4.32 | 79.2 | 79.9 |
1.0 | \(-\) 0.21 | 5.06 | 5.19 | 5.17 | 95.4 | 95.6 | 0.94 | 4.82 | 4.88 | 4.87 | 94.4 | 94.3 | 7.63 | 4.23 | 4.17 | 4.15 | 55.5 | 54.7 |
2.0 | \(-\) 0.21 | 5.06 | 5.19 | 5.17 | 95.4 | 95.6 | 6.41 | 4.78 | 4.85 | 4.85 | 73.6 | 73.5 | 19.58 | 4.16 | 4.12 | 4.10 | 0.2 | 0.2 |
4.0 | \(-\) 0.21 | 5.06 | 5.19 | 5.17 | 95.4 | 95.6 | 11.33 | 4.87 | 4.85 | 4.84 | 35.9 | 35.9 | 31.17 | 4.21 | 4.14 | 4.17 | 0.0 | 0.0 |
Partially conditional model, \(P(Z=1) = 0.50\) | ||||||||||||||||||
0.5 | \(-\) 0.21 | 5.06 | 5.19 | 5.17 | 95.4 | 95.6 | \(-\) 5.94 | 4.69 | 4.66 | 4.66 | 75.4 | 74.7 | \(-\) 4.83 | 3.72 | 3.68 | 3.72 | 74.3 | 75.6 |
1.0 | \(-\) 0.21 | 5.06 | 5.19 | 5.17 | 95.4 | 95.6 | 1.13 | 4.51 | 4.62 | 4.61 | 94.5 | 94.5 | 7.45 | 3.49 | 3.58 | 3.53 | 44.9 | 44.2 |
2.0 | \(-\) 0.21 | 5.06 | 5.19 | 5.17 | 95.4 | 95.6 | 7.73 | 4.61 | 4.59 | 4.58 | 59.8 | 59.3 | 19.01 | 3.56 | 3.54 | 3.49 | 0.0 | 0.0 |
4.0 | \(-\) 0.21 | 5.06 | 5.19 | 5.17 | 95.4 | 95.6 | 14.00 | 4.55 | 4.58 | 4.56 | 14.1 | 14.3 | 30.43 | 3.75 | 3.55 | 3.58 | 0.0 | 0.0 |
5 Application to a trial in cystic fibrosis
\(\beta \)
|
\(\exp (\beta )\)
| Robust S.E. | P value | |
---|---|---|---|---|
Marginal model | ||||
Without FEVC | \(-\) 0.271 | 0.763 | 0.124 | 0.029 |
With FEVC | \(-\) 0.267 | 0.766 | 0.120 | 0.027 |
Partially conditional model | ||||
Without FEVC | \(-\) 0.234 | 0.791 | 0.108 | 0.030 |
With FEVC | \(-\) 0.246 | 0.782 | 0.109 | 0.024 |
\(\zeta \)
| Marginal model | Partially conditional model | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
BIAS | ESE | ASE\(^{1}\) | ASE\(^2\) | ECP\(^1\) | ECP\(^2\) | BIAS | ESE | ASE\(^{1}\) | ASE\(^2\) | ECP\(^1\) | ECP\(^2\) | |
\(\sigma _z = 20\)
| ||||||||||||
0.20 | 14.47 | 13.19 | 3.22 | 12.89 | 19.8 | 79.6 | 21.61 | 5.01 | 3.27 | 4.92 | 0.2 | 0.8 |
0.00 | \(-\) 3.29 | 12.14 | 11.00 | 12.22 | 92.2 | 94.7 | \(-\) 0.68 | 10.94 | 11.06 | 11.03 | 96.1 | 96.1 |
\(-\) 0.01 | \(-\) 3.00 | 12.11 | 10.84 | 12.27 | 91.4 | 94.5 | 0.02 | 10.69 | 10.90 | 10.88 | 95.0 | 94.8 |
\(-\) 0.10 | 6.34 | 15.41 | 4.41 | 15.46 | 37.0 | 93.2 | 18.47 | 6.96 | 4.46 | 6.93 | 9.0 | 25.0 |
\(-\) 0.50 | 20.40 | 12.15 | 2.74 | 11.70 | 8.3 | 59.9 | 22.52 | 4.36 | 2.83 | 4.12 | 0.0 | 0.0 |
\(\sigma _z = 26\)
| ||||||||||||
0.20 | 16.21 | 12.30 | 3.01 | 12.36 | 17.6 | 74.1 | 21.79 | 4.83 | 3.07 | 4.57 | 0.0 | 0.7 |
0.00 | \(-\) 3.29 | 12.14 | 11.00 | 12.22 | 92.2 | 94.7 | \(-\) 0.68 | 10.94 | 11.06 | 11.03 | 96.1 | 96.1 |
\(-\) 0.01 | \(-\) 3.46 | 12.54 | 10.73 | 12.32 | 89.8 | 94.0 | \(-\) 0.03 | 10.96 | 10.80 | 10.77 | 94.7 | 94.6 |
\(-\) 0.10 | 10.22 | 14.12 | 3.80 | 14.26 | 30.4 | 89.9 | 20.00 | 6.14 | 3.85 | 5.91 | 2.3 | 9.5 |
\(-\) 0.50 | 21.00 | 11.71 | 2.68 | 11.57 | 8.8 | 54.6 | 22.74 | 4.21 | 2.80 | 4.04 | 0.0 | 0.0 |
\(\sigma _z = 30\)
| ||||||||||||
0.20 | 18.56 | 12.25 | 2.93 | 12.17 | 13.7 | 66.3 | 22.19 | 4.62 | 3.00 | 4.44 | 0.0 | 0.3 |
0.00 | \(-\) 3.29 | 12.14 | 11.00 | 12.22 | 92.2 | 94.7 | \(-\) 0.68 | 10.94 | 11.06 | 11.03 | 96.1 | 96.1 |
\(-\) 0.01 | \(-\) 2.69 | 12.50 | 10.63 | 12.38 | 89.5 | 93.6 | 0.88 | 10.82 | 10.70 | 10.67 | 94.8 | 94.9 |
\(-\) 0.10 | 12.40 | 13.62 | 3.56 | 13.72 | 28.4 | 85.6 | 20.95 | 5.58 | 3.61 | 5.50 | 0.8 | 5.1 |
\(-\) 0.50 | 21.40 | 11.25 | 2.65 | 11.48 | 7.1 | 53.9 | 22.42 | 4.29 | 2.78 | 4.01 | 0.0 | 0.0 |