01.12.2013 | Research article | Ausgabe 1/2013 Open Access

# Impact of misspecifying the distribution of a prognostic factor on power and sample size for testing treatment interactions in clinical trials

- Zeitschrift:
- BMC Medical Research Methodology > Ausgabe 1/2013

## Electronic supplementary material

## Competing interests

## Authors’ contributions

## Background

## Methods

### Overview

### Specification of key parameters used in the paper

#### Treatment variable

#### Prognostic factor

_{ j }representing the j

^{ th }level of the prognostic factor. When referring to the distribution of the prognostic factor we indicated the percentage in the k

_{ 1 }level of the prognostic factor, defined as p

_{ 1 }. We varied p

_{ 1 }from 10% to 50% in 10% increments.

#### Misspecification of the prognostic factor

#### Outcome variable

_{ 1 }treatment/prognostic factor combination (i.e. APM/mild knee OA severity). The mean improvement in the active/k

_{ 2 }, placebo/k

_{ 1 }, and placebo/k

_{ 2 }groups were held constant at 5, 5, and 0 respectively. We assumed a common standard deviation (σ) of 10 for all four combinations.

#### Magnitude of the interaction

_{ ij }be mean improvement in the i

^{ th }treatment and j

^{ th }level of the prognostic factor. We then defined the treatment efficacy in the j

^{ th }level of the prognostic factor as:

_{11}varied. The magnitudes of the interaction effect that we considered were 15 and 5. The estimate of the interaction effect was defined as follows:

_{ 1 }can be derived as follows (note that N equals the total sample size for the trial):

_{ 1 }) increases, the variance decreases, which would imply that the power increases for a fixed sample size.

### Initial sample size for interaction effects

^{ th }treatment and j

^{ th }prognostic factor level to detect the interaction effect described under a balanced design (i.e. p

_{ 1 }= 0.5) with a two-sided significance level of α and power equal to 1– β has been previously published by Lachenbruch [9].

_{ 1–β }represents the z-value at the 1–β (theoretical power) quantile of the standard normal distribution and z

_{ 1–α/2 }represents the z-value at the 1–

^{ α }/

_{ 2 }(probability of a type I error) quantile of the standard normal distribution. Under a balanced design with p

_{ 1 }= 0.5 we can just multiply n

_{ ij }by four to obtain the total sample size since there are four combinations of treatment and prognostic factor. A limitation of this formula is that it uses critical values from the standard normal distribution rather than the Student’s t-distribution as most statistical tests of interaction are performed using a t-distribution. To account for this we calculated the total sample size required to detect an interaction effect with a two-sided significance level of α and power equal to 1– β:

_{ ij }degrees of freedom.

_{ ij }equal to ${n}_{\mathit{ij}}^{*}$ and repeat step 2.

### Effect of misspecifying the distribution of the prognostic factor

_{ 1 }+ q) and magnitude of the interaction effect was calculated using equation 7 below.

### Strategies for accounting for the misspecification of the distribution of the prognostic factor

#### Quota sampling

_{ 1 }group and 70% in the k

_{ 2 }group then exactly 60 subjects would be recruited in the k

_{ 1 }group and 140 in the k

_{ 2 }group. This method removes the variability in the sampling distribution and ensures that the sampled prognostic factor distribution always matches what was planned for. Because of this approach, the observed distribution of the prognostic factor in the trial will always match the planned distribution and there will be no misspecification. However, this method may require turning away potential subjects because one level of the prognostic factor is already filled, delaying trial completion. Also, it may reduce the external validity of the overall treatment results as the trial subjects can become less representative of the unselected population of interest. Because of these limitations we also considered a modified quota sampling approach.

#### Modified quota sampling

^{ N }/

_{ 2 }subjects. After the first

^{ N }/

_{ 2 }subjects were enrolled we tested to see if the sampling distribution of the prognostic factor was different from what was planned for using a one-sample test of the proportion. If this result was statistically significant at the 0.05 level then a quota sampling approach was undertaken for the second

^{ N }/

_{ 2 }subjects to be enrolled to ensure that the sampling distribution of the prognostic factor matched the planned distribution exactly. If the result was not statistically significant then the study continued to enroll normally, allowing for variability in the distribution of the prognostic factor.

#### Sample size re-estimation using conditional power

^{ N }/

_{ 2 }subjects were enrolle

_{ 2 }) were determined by the O’Brien-Fleming alpha-spending function [11] using the SEQDESIGN procedure in the SAS statistical software package. We also used the SEQDESIGN procedure to calculate a futility boundary at the interim analysis (b

_{ 1 }). Since these critical values are based on a standard normal distribution and not the student’s t-distribution we converted the critical values to those based on the student’s t-distribution. First, we converted the original critical values to the corresponding percentile of the standard normal distribution. We then converted these percentiles to the corresponding critical value of the Student’s t-distribution with N-4 degrees of freedom.

_{ 1 }< b

_{ 1 }) then we stopped the trial for futility and considered the result of the trial to be not statistically significant. If the absolute value of the test statistic was greater than the interim critical value (c

_{ 1 }) then we stopped the trial for efficacy and considered the result of the trial to be statistically significant. If absolute value of the test statistic was greater than b

_{ 1 }but less than c

_{ 1 }then we evaluated the conditional power and determined if sample size re-estimation was necessary. The following paragraphs outline this procedure.

_{ 2 }is the final critical value, n

_{ 2 }is the sample size at the final analysis, n

_{ t }is the originally planned total sample size, z

_{ 1 }is the test statistic for the interaction at the interim analysis, n

_{ 1 }is sample size at the interim analysis, δ is the difference in means, and σ is the common standard deviation for the two groups. We updated the formula by replacing z

_{ 1 }with t

_{ 1 }(because the interaction test uses the Student’s t-distribution), δ (difference in means between groups) with θ (magnitude of the interaction effect), and Φ (cumulative distribution function of a standard normal distribution) with Ψ (cumulative distribution function of a student’s t-distribution). Recall that p

_{ 1 }is the proportion in the k

_{ 1 }group and σ is the common standard deviation:

_{ 2 }= n

_{ t }as conditional power is calculated as if you were to not re-estimate the sample size. The values of θ, σ, and p

_{ 1 }for the conditional power formula were estimated at the interim analysis. If the conditional power was less than 80% then a new n

_{ 2 }was estimated such that conditional power was 80% and a new final critical value, c

_{ 2 }, was calculated as a function of the original final critical value, ${\tilde{c}}_{2}$, and the interim test statistic t

_{ 1 }using the following formula:

_{ 1 }, n

_{ 2 }(the new total sample size), and n

_{ t }(the original total sample size). Since all values except n

_{ 2 }are fixed, we can calculate the new critical value c

_{ 2 }for new final sample sizes n

_{ 2 }. According to Denne, this method for re-estimating the sample size maintains the overall Type I error rate at α (equal to 0.05 in our case) [10]. The final sample size n

_{ 2 }and final critical value c

_{ 2 }were chosen so that the conditional power formula shown in equation 9 was equal to 80%. If the conditional power was greater than 80% at the interim analysis then we used the originally calculated n

_{ t }as the final sample size (n

_{ 2 }= n

_{ t }) so that the final sample size was only altered to increase the conditional power to 80%.

### Validating the conditional power formula

_{ 1 }. For each trial, the second half of the trial was simulated 5,000 times to obtain the empirical conditional power. Since there were 10 different combinations of prevalence of the prognostic factor and magnitude of the interaction effect and 10 trials for each combination, the plot generated 100 points. We generated a scatter plot of the empirical conditional power based on 5,000 replicates against the calculated conditional power (Figure 1). Values that line up along the y = x line demonstrate that formula provided an accurate estimation of the conditional power.

### Simulation study details

_{ 1 }. We first evaluated the empirical power for detecting the interaction effect without accounting for misspecification of the distribution of the prognostic factor. We varied the misspecification of the prognostic factor at −15%, -5%, 0%, +5%, and +15%. For the quota sampling method we did not vary the misspecification of the distribution of the prognostic factor because the definition of the method does not allow for misspecifications. While we did not expect the quota sampling method to have power or type I error estimates that differ from the traditional one-stage sampling design under no misspecification, we conducted the simulation study for this study design method to confirm there was no impact on power and type I error. For the modified quota sampling method and sample size re-estimation using conditional power we used the same misspecifications as described above.

## Results

### Effect of misspecifying the distribution of the prognostic factor on power for the interaction test

### Performance of the quota sampling procedure

θ
| Planned distribution of the prognostic factor | Planned n
_{
t
}
| Quota sampling | Modified quota sampling | Sample size re-estimation using conditional power |
---|---|---|---|---|---|

5 | 10% | 1,418 | 0.8088 | 0.8054 | 0.8836 |

20% | 798 | 0.8152 | 0.8082 | 0.8916 | |

30% | 608 | 0.8178 | 0.8014 | 0.8870 | |

40% | 532 | 0.8116 | 0.8036 | 0.8866 | |

50% | 512 | 0.8156 | 0.8098 | 0.8974 | |

15 | 10% | 178 | 0.8556 | 0.8204 | 0.8890 |

20% | 100 | 0.8490 | 0.8128 | 0.8930 | |

30% | 78 | 0.8442 | 0.8316 | 0.9012 | |

40% | 68 | 0.8556 | 0.8322 | 0.9038 | |

50% | 64 | 0.8412 | 0.8256 | 0.9082 |

θ
| Planned distribution of the prognostic factor | Planned n
_{
t
}
| Quota sampling | Modified quota sampling | Sample size re-estimation using conditional power |
---|---|---|---|---|---|

0 | 10% | 1,418 | 0.0496 | 0.0534 | 0.0210 |

20% | 798 | 0.0484 | 0.0522 | 0.0260 | |

30% | 608 | 0.0494 | 0.0464 | 0.0286 | |

40% | 532 | 0.0510 | 0.0514 | 0.0302 | |

50% | 512 | 0.0532 | 0.0540 | 0.0248 | |

0 | 10% | 178 | 0.0494 | 0.0512 | 0.0248 |

20% | 100 | 0.0488 | 0.0516 | 0.0274 | |

30% | 78 | 0.0536 | 0.0524 | 0.0224 | |

40% | 68 | 0.0472 | 0.0496 | 0.0302 | |

50% | 64 | 0.0482 | 0.0482 | 0.0328 |

### Performance of the modified quota sampling procedure

θ
| Planned distribution of the prognostic factor | Planned n
_{
t
}
| Empirical power | Percent of trials switching to quota sampling |
---|---|---|---|---|

Misspecification of the prognostic factor: -5%
| ||||

5 | 10% | 1,418 | 0.8012 | 99.98% |

20% | 798 | 0.7736 | 74.08% | |

30% | 608 | 0.7802 | 47.52% | |

40% | 532 | 0.7932 | 36.64% | |

50% | 512 | 0.8060 | 35.50% | |

15 | 10% | 178 | 0.6636 | 33.04% |

20% | 100 | 0.7400 | 11.06% | |

30% | 78 | 0.7940 | 11.84% | |

40% | 68 | 0.8164 | 11.90% | |

50% | 64 | 0.8122 | 8.48% | |

Misspecification of the prognostic factor: -15%
| ||||

5 | 10% | 1,418 | -- | -- |

20% | 798 | 0.8022 | 100.00% | |

30% | 608 | 0.8024 | 100.00% | |

40% | 532 | 0.8058 | 99.94% | |

50% | 512 | 0.8090 | 99.82% | |

15 | 10% | 178 | -- | -- |

20% | 100 | 0.7788 | 89.06% | |

30% | 78 | 0.7834 | 63.64% | |

40% | 68 | 0.7904 | 49.70% | |

50% | 64 | 0.8038 | 40.00% | |

Misspecification of the prognostic factor: +5%
| ||||

5 | 10% | 1,418 | 0.8000 | 98.28% |

20% | 798 | 0.8090 | 67.72% | |

30% | 608 | 0.8278 | 49.32% | |

40% | 532 | 0.8092 | 37.08% | |

50% | 512 | 0.7984 | 36.30% | |

15 | 10% | 178 | 0.8880 | 35.14% |

20% | 100 | 0.8656 | 16.62% | |

30% | 78 | 0.8542 | 10.94% | |

40% | 68 | 0.8350 | 7.86% | |

50% | 64 | 0.8198 | 8.60% | |

Misspecification of the prognostic factor: +15%
| ||||

5 | 10% | 1,418 | 0.8738 | 100.00% |

20% | 798 | 0.8010 | 100.00% | |

30% | 608 | 0.8092 | 99.98% | |

40% | 532 | 0.8088 | 99.80% | |

50% | 512 | 0.8064 | 99.86% | |

15 | 10% | 178 | 0.8942 | 97.82% |

20% | 100 | 0.8504 | 72.44% | |

30% | 78 | 0.8572 | 50.72% | |

40% | 68 | 0.8466 | 38.48% | |

50% | 64 | 0.8174 | 40.88% |

θ
| Planned distribution of the prognostic factor | Planned n
_{
t
}
| Empirical type I error | Percent of trials switching to quota sampling |
---|---|---|---|---|

Misspecification of the prognostic factor: -5%
| ||||

0 | 10% | 1,418 | 0.0528 | 99.90% |

20% | 798 | 0.0506 | 74.26% | |

30% | 608 | 0.0498 | 47.34% | |

40% | 532 | 0.0480 | 36.66% | |

50% | 512 | 0.0492 | 36.06% | |

0 | 10% | 178 | 0.0566 | 34.74% |

20% | 100 | 0.0496 | 12.12% | |

30% | 78 | 0.0528 | 11.24% | |

40% | 68 | 0.0452 | 11.30% | |

50% | 64 | 0.0522 | 8.80% | |

Misspecification of the prognostic factor: -15%
| ||||

0 | 10% | 1,418 | -- | -- |

20% | 798 | 0.0500 | 100.00% | |

30% | 608 | 0.0496 | 100.00% | |

40% | 532 | 0.0518 | 99.92% | |

50% | 512 | 0.0568 | 99.76% | |

0 | 10% | 178 | -- | -- |

20% | 100 | 0.0578 | 89.60% | |

30% | 78 | 0.0522 | 62.82% | |

40% | 68 | 0.0542 | 50.56% | |

50% | 64 | 0.0478 | 41.20% | |

Misspecification of the prognostic factor: +5%
| ||||

0 | 10% | 1,418 | 0.0568 | 98.68% |

20% | 798 | 0.0490 | 68.74% | |

30% | 608 | 0.0490 | 48.10% | |

40% | 532 | 0.0508 | 36.48% | |

50% | 512 | 0.0508 | 36.64% | |

0 | 10% | 178 | 0.0504 | 34.46% |

20% | 100 | 0.0534 | 16.32% | |

30% | 78 | 0.0496 | 10.90% | |

40% | 68 | 0.0450 | 8.50% | |

50% | 64 | 0.0502 | 8.94% | |

Misspecification of the prognostic factor: +15%
| ||||

0 | 10% | 1,418 | 0.0462 | 100.00% |

20% | 798 | 0.0490 | 100.00% | |

30% | 608 | 0.0484 | 100.00% | |

40% | 532 | 0.0516 | 99.76% | |

50% | 512 | 0.0522 | 99.86% | |

0 | 10% | 178 | 0.0452 | 97.48% |

20% | 100 | 0.0444 | 70.72% | |

30% | 78 | 0.0526 | 51.00% | |

40% | 68 | 0.0524 | 38.60% | |

50% | 64 | 0.0528 | 40.72% |

### Validating the conditional power formula

### Performance of the sample size re-estimation using conditional power procedure

θ
| Planned distribution of the prognostic factor | Planned n
_{
t
}
| Empirical power | Percent stopping for futility | Percent stopping for efficacy |
---|---|---|---|---|---|

Misspecification of the prognostic factor: -5%
| |||||

5 | 10% | 1,418 | 0.7324 | 16.58% | 5.76% |

20% | 798 | 0.8410 | 10.34% | 11.76% | |

30% | 608 | 0.8616 | 9.02% | 13.72% | |

40% | 532 | 0.8754 | 8.24% | 15.42% | |

50% | 512 | 0.8868 | 7.64% | 16.22% | |

15 | 10% | 178 | 0.7246 | 17.14% | 7.90% |

20% | 100 | 0.8232 | 12.26% | 11.82% | |

30% | 78 | 0.8736 | 8.94% | 14.76% | |

40% | 68 | 0.8936 | 7.72% | 15.80% | |

50% | 64 | 0.9010 | 7.10% | 16.38% | |

Misspecification of the prognostic factor: -15%
| |||||

5 | 10% | 1,418 | -- | -- | -- |

20% | 798 | 0.5614 | 26.06% | 3.5% | |

30% | 608 | 0.7536 | 15.46% | 7.3% | |

40% | 532 | 0.8304 | 10.72% | 10.80% | |

50% | 512 | 0.8750 | 8.34% | 14.32% | |

15 | 10% | 178 | -- | -- | -- |

20% | 100 | 0.6026 | 22.68% | 4.96% | |

30% | 78 | 0.7634 | 15.76% | 8.50% | |

40% | 68 | 0.8412 | 10.78% | 11.86% | |

50% | 64 | 0.8734 | 9.32% | 14.76% | |

Misspecification of the prognostic factor: +5%
| |||||

5 | 10% | 1,418 | 0.9528 | 3.10% | 27.44% |

20% | 798 | 0.9194 | 5.58% | 20.94% | |

30% | 608 | 0.9064 | 6.26% | 18.36% | |

40% | 532 | 0.8898 | 7.18% | 17.16% | |

50% | 512 | 0.8868 | 7.52% | 16.02% | |

15 | 10% | 178 | 0.9476 | 3.94% | 29.36% |

20% | 100 | 0.9194 | 5.70% | 21.78% | |

30% | 78 | 0.9204 | 5.94% | 20.38% | |

40% | 68 | 0.9146 | 6.36% | 18.28% | |

50% | 64 | 0.9056 | 6.96% | 15.88% | |

Misspecification of the prognostic factor: +15%
| |||||

5 | 10% | 1,418 | 0.9868 | 0.94% | 45.24% |

20% | 798 | 0.9494 | 3.38% | 27.26% | |

30% | 608 | 0.9186 | 5.34% | 20.84% | |

40% | 532 | 0.8990 | 6.98% | 17.12% | |

50% | 512 | 0.8822 | 8.00% | 14.30% | |

15 | 10% | 178 | 0.9902 | 0.78% | 49.94% |

20% | 100 | 0.9584 | 3.18% | 29.86% | |

30% | 78 | 0.9412 | 4.32% | 22.32% | |

40% | 68 | 0.9108 | 6.56% | 17.12% | |

50% | 64 | 0.8816 | 8.76% | 13.84% |

_{ 1 }and c

_{ 2 }for all combinations of θ, planned distribution of the prognostic factor, and misspecification of the distribution of the prognostic factor. Under the null hypothesis, the percentage of trials stopping for futility ranged between 42% and 45%, while the percentage of trials stopping for efficacy was at most 0.6% (Table 6).

θ
| Planned distribution of the prognostic factor | Planned n
_{
t
}
| Empirical type I error | Percent stopping for futility | Percent stopping for efficacy |
---|---|---|---|---|---|

Misspecification of the prognostic factor: -5%
| |||||

0 | 10% | 1,418 | 0.0278 | 43.48% | 0.24% |

20% | 798 | 0.0264 | 44.54% | 0.26% | |

30% | 608 | 0.0262 | 42.66% | 0.20% | |

40% | 532 | 0.0280 | 44.16% | 0.24% | |

50% | 512 | 0.0246 | 42.88% | 0.30% | |

0 | 10% | 178 | 0.0304 | 42.94% | 0.40% |

20% | 100 | 0.0276 | 42.44% | 0.28% | |

30% | 78 | 0.0252 | 43.16% | 0.54% | |

40% | 68 | 0.0230 | 43.08% | 0.44% | |

50% | 64 | 0.0298 | 42.00% | 0.56% | |

Misspecification of the prognostic factor: -15%
| |||||

0 | 10% | 1,418 | -- | -- | -- |

20% | 798 | 0.0254 | 43.62% | 0.24% | |

30% | 608 | 0.0278 | 42.94% | 0.30% | |

40% | 532 | 0.0282 | 44.00% | 0.36% | |

50% | 512 | 0.0248 | 42.62% | 0.32% | |

0 | 10% | 178 | -- | -- | -- |

20% | 100 | 0.0326 | 44.24% | 0.50% | |

30% | 78 | 0.0282 | 42.52% | 0.44% | |

40% | 68 | 0.0286 | 42.62% | 0.42% | |

50% | 64 | 0.0296 | 43.26% | 0.50% | |

Misspecification of the prognostic factor: +5%
| |||||

0 | 10% | 1,418 | 0.0314 | 41.46% | 0.36% |

20% | 798 | 0.0284 | 42.94% | 0.22% | |

30% | 608 | 0.0296 | 44.76% | 0.44% | |

40% | 532 | 0.0312 | 43.82% | 0.32% | |

50% | 512 | 0.0290 | 44.16% | 0.36% | |

0 | 10% | 178 | 0.0284 | 43.64% | 0.30% |

20% | 100 | 0.0274 | 43.30% | 0.38% | |

30% | 78 | 0.0296 | 43.66% | 0.44% | |

40% | 68 | 0.0280 | 44.24% | 0.48% | |

50% | 64 | 0.0290 | 43.82% | 0.46% | |

Misspecification of the prognostic factor: +15%
| |||||

0 | 10% | 1,418 | 0.0244 | 43.32% | 0.28% |

20% | 798 | 0.0274 | 44.44% | 0.34% | |

30% | 608 | 0.0232 | 43.98% | 0.20% | |

40% | 532 | 0.0290 | 43.18% | 0.32% | |

50% | 512 | 0.0284 | 44.22% | 0.32% | |

0 | 10% | 178 | 0.0282 | 43.36% | 0.38% |

20% | 100 | 0.0282 | 42.50% | 0.36% | |

30% | 78 | 0.0266 | 43.92% | 0.34% | |

40% | 68 | 0.0266 | 44.30% | 0.46% | |

50% | 64 | 0.0248 | 42.48% | 0.34% |

θ
| Planned distribution of the prognostic factor | Planned n
_{
t
}
| Percent re-estimating sample size | Empirical conditional power | Mean sample size | Median sample size |
---|---|---|---|---|---|---|

Misspecification of the prognostic factor: 0%
| ||||||

5 | 10% | 1,418 | 37.44% | 0.9471 | 2,500 | 1,418 |

20% | 798 | 37.30% | 0.9566 | 1,470 | 798 | |

30% | 608 | 37.74% | 0.9671 | 1,143 | 608 | |

40% | 532 | 38.00% | 0.9689 | 984 | 532 | |

50% | 512 | 39.44% | 0.9660 | 989 | 512 | |

15 | 10% | 178 | 37.44% | 0.9701 | 327 | 178 |

20% | 100 | 40.48% | 0.9674 | 202 | 100 | |

30% | 78 | 39.28% | 0.9695 | 150 | 78 | |

40% | 68 | 42.02% | 0.9719 | 155 | 68 | |

50% | 64 | 43.58% | 0.9748 | 154 | 64 | |

Misspecification of the prognostic factor: -5%
| ||||||

5 | 10% | 1,418 | 49.70% | 0.8881 | 3,568 | 1,418 |

20% | 798 | 43.12% | 0.9429 | 1,696 | 798 | |

30% | 608 | 40.98% | 0.9458 | 1,231 | 608 | |

40% | 532 | 38.56% | 0.9570 | 1,062 | 532 | |

50% | 512 | 38.82% | 0.9629 | 987 | 512 | |

15 | 10% | 178 | 45.00% | 0.8738 | 402 | 178 |

20% | 100 | 42.54% | 0.9342 | 219 | 100 | |

30% | 78 | 43.56% | 0.9660 | 171 | 78 | |

40% | 68 | 43.38% | 0.9779 | 153 | 68 | |

50% | 64 | 44.40% | 0.9770 | 155 | 64 | |

Misspecification of the prognostic factor: -15%
| ||||||

5 | 10% | 1,418 | -- | -- | -- | -- |

20% | 798 | 51.98% | 0.7618 | 2,425 | 865 | |

30% | 608 | 47.34% | 0.8952 | 1,439 | 608 | |

40% | 532 | 44.96% | 0.9346 | 1,155 | 532 | |

50% | 512 | 41.42% | 0.9546 | 1,028 | 512 | |

15 | 10% | 178 | -- | -- | -- | -- |

20% | 100 | 48.70% | 0.7573 | 267 | 100 | |

30% | 78 | 46.80% | 0.9167 | 190 | 78 | |

40% | 68 | 47.38% | 0.9565 | 161 | 68 | |

50% | 64 | 44.50% | 0.9685 | 148 | 64 | |

Misspecification of the prognostic factor: +5%
| ||||||

5 | 10% | 1,418 | 25.94% | 0.9861 | 1,984 | 1,418 |

20% | 798 | 33.26% | 0.9747 | 1,310 | 798 | |

30% | 608 | 35.76% | 0.9676 | 1,096 | 608 | |

40% | 532 | 37.68% | 0.9618 | 975 | 532 | |

50% | 512 | 39.24% | 0.9623 | 969 | 512 | |

15 | 10% | 178 | 28.32% | 0.9845 | 263 | 178 |

20% | 100 | 35.36% | 0.9762 | 177 | 100 | |

30% | 78 | 37.42% | 0.9840 | 153 | 78 | |

40% | 68 | 43.32% | 0.9797 | 150 | 68 | |

50% | 64 | 45.72% | 0.9790 | 157 | 64 | |

Misspecification of the prognostic factor: +15%
| ||||||

5 | 10% | 1,418 | 15.04% | 0.9960 | 1,472 | 1,418 |

20% | 798 | 27.34% | 0.9824 | 1,159 | 798 | |

30% | 608 | 34.22% | 0.9673 | 1,007 | 608 | |

40% | 532 | 38.58% | 0.9725 | 1,026 | 532 | |

50% | 512 | 41.56% | 0.9644 | 1,045 | 512 | |

15 | 10% | 178 | 14.40% | 0.9944 | 190 | 89 |

20% | 100 | 30.38% | 0.9888 | 159 | 100 | |

30% | 78 | 37.62% | 0.9856 | 149 | 78 | |

40% | 68 | 43.78% | 0.9753 | 151 | 68 | |

50% | 64 | 48.00% | 0.9708 | 168 | 64 |

## Discussion

_{ 1 }level of the prognostic factor then enrollment would be capped at 60 subjects in the k

_{ 1 }level and 140 in the k

_{ 2 }level. This method did a good job of maintaining the power at 80% and controlling the type I error at 5%. The modified quota sampling approach did not perform as well in all situations. In summary, this method enrolled subjects randomly for the first half the trial. The sampling method would switch to the quota sampling approach if the distribution of the prognostic factor differed significantly from what was planned. Power was maintained at 80% when the percentage of trials switching to the quota sampling approach was large. However, when the percentage switching was small and there was a negative misspecification of the distribution of the prognostic factor, the power was compromised, but rarely substantially.