# Power and Design Issues in Crossover-Based N-Of-1 Clinical Trials with Fixed Data Collection Periods

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## Abstract

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## 1. Introduction

## 2. Basic Designs For N-Of-1 Trials

## 3. Practical Considerations

## 4. An Analytical Model For N-Of-1 Designs

#### 4.1. The Basic Linear Model

#### 4.2. Evaluating Power

_{t}= −2 times the difference in log-likelihoods, follows a ${\chi}^{2}$ distribution with 1 degree of freedom under the null hypothesis. Under the alternative hypothesis, LR

_{t}follows a non-central ${\chi}_{nc}^{2}$ distribution with the non-centrality parameter, $\lambda $, calculated as a function of the difference in the values of parameters assumed under the null, ${\beta}_{0}$, and alterative, ${\beta}_{1}$, hypotheses, and the inverse of the information matrix, $\mathrm{I}$, for the parameters evaluated under the null hypothesis:

## 5. Simulation Studies

## 6. Design Scenarios Considered

## 7. Results

#### 7.1. The Influence of Serial Correlation and Intervention Period Stacking

#### 7.2. The Influence of Heteroskedasticity (i.e., Unequal Response Variances)

#### 7.3. Detecting Carryover Effects

## 8. Discussion

## 9. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Reduction in serial correlation when washout periods are used based on simulation studies. The settings considered involved a 2 × 200 observation (2 total periods) setting, a 4 × 50 observation (8 total periods) setting and a 10 × 2 observation (20 total periods) setting.

**Figure 2.**Influence of the effect size (i.e., difference between intervention responses in standard deviation units) on power assuming four intervention periods and 400 total observations (i.e., 8 intervention administration periods overall with 50 observations; a ‘4p × 2i’ design as described in the text) with no washout periods based on the proposed analytical model. Each line corresponds to a different assumed serial correlation strength. The upper most line represents a 0.0 serial correlation, with successive lines below it reflecting serial correlations of 0.1, 0.25, 0.5, 0.75, and 0.9, respectively. A type I error rate of 0.05 was assumed.

**Figure 3.**The power to detect the difference between two interventions as a function of the effect size (computed as the difference in means divided by the pooled standard deviation) for serially correlated and potentially heteroskedastic observations based on the proposed analytical model. The gray lines assume equal variances and unequal means and the black lines assume unequal variances. The dashed lines correspond to situations in which a residual serial correlation of 0.5 was assumed and the solid lines correspond to situations in which no residual correlation was assumed. A type I error rate of 0.05 was assumed.

**Figure 4.**Influence of the effect size (i.e., difference between response observations obtained during a fist intervention period and subsequent intervention periods) on the power to detect a carryover effect assuming four intervention periods (i.e., 8 intervention administration periods overall) with no washout periods based on the proposed analytical model. 50 data collections were made during each intervention administration. Each line corresponds to a different assumed serial correlation strength. The upper most line representing a 0.0 correlation, with successive lines below it reflecting serial correlations of 0.1, 0.25, 0.5, 0.75, and 0.9, respectively. A type I error rate of 0.05 was assumed.

**Table 1.**The effect of serial correlation and intervention period stacking with and without washout periods on the power to detect a mean difference of 0.30 standard deviation units between two interventions with 400 total observations based on the proposed analytical model. A type I error rate of 0.05 was assumed.

Design | Serial Correlation Strength | ||||||||
---|---|---|---|---|---|---|---|---|---|

Periods | Obs/Per | 0.000 | 0.000 w | 0.250 | 0.250 w | 0.500 | 0.500 w | 0.750 | 0.750 w |

1p × 2i | 200 | 0.851 | 0.851 | 0.617 | 0.616 | 0.330 | 0.327 | 0.126 | 0.122 |

2p × 2i | 100 | 0.851 | 0.851 | 0.621 | 0.619 | 0.341 | 0.335 | 0.142 | 0.134 |

4p × 2i | 50 | 0.851 | 0.851 | 0.628 | 0.624 | 0.362 | 0.351 | 0.176 | 0.159 |

8p × 2i | 25 | 0.851 | 0.851 | 0.643 | 0.636 | 0.403 | 0.382 | 0.242 | 0.209 |

10p × 2i | 20 | 0.851 | 0.851 | 0.650 | 0.641 | 0.423 | 0.398 | 0.275 | 0.234 |

20p × 2i | 10 | 0.851 | 0.851 | 0.684 | 0.667 | 0.518 | 0.472 | 0.433 | 0.356 |

40p × 2i | 5 | 0.851 | 0.851 | 0.745 | 0.716 | 0.674 | 0.602 | 0.681 | 0.569 |

**Key**: Periods = the number of periods in which two interventions are administered; Obs/Per = the number of observations per intervention period; w=washout period is assumed.

**Table 2.**Simulation study results investigating the power of N-of-1 designs in comparison to the analytical results reflected in Table 1.

Setting | Serial Correlation Strength | ||||||||
---|---|---|---|---|---|---|---|---|---|

Periods | Effect Size | 0.00 s | 0.00 a | 0.25 s | 0.25 a | 0.5 s | 0.5 a | 0.75 s | 0.75 a |

1p × 2i | 0.0 | 0.059 | 0.050 | 0.049 | 0.050 | 0.061 | 0.050 | 0.051 | 0.050 |

4p × 2i | 0.0 | 0.060 | 0.050 | 0.044 | 0.050 | 0.048 | 0.050 | 0.066 | 0.050 |

10p × 2i | 0.0 | 0.061 | 0.050 | 0.063 | 0.050 | 0.051 | 0.050 | 0.060 | 0.050 |

1p × 2i | 0.3 | 0.832 | 0.851 | 0.619 | 0.617 | 0.317 | 0.330 | 0.136 | 0.126 |

4p × 2i | 0.3 | 0.856 | 0.851 | 0.614 | 0.628 | 0.336 | 0.362 | 0.201 | 0.176 |

10p × 2i | 0.3 | 0.862 | 0.851 | 0.656 | 0.650 | 0.425 | 0.423 | 0.288 | 0.275 |

**Key**: Periods = the number of periods in which two interventions are administered; An ‘s’ in the column headings indicates power based on simulation studies and an ‘a’ indicates power based on the analytical model.

**Table 3.**Simulation study results investigating the power of N-of-1 designs as a function of effect size (the difference between the experimental intervention and comparator intervention in standard deviation units) and serial correlation strength.

Setting | Serial Correlation Strength | ||||
---|---|---|---|---|---|

Periods | Effect Size | 0.00 | 0.25 | 0.50 | 0.75 |

1p × 2i | 0.0 | 0.059 | 0.049 | 0.061 | 0.051 |

4p × 2i | 0.0 | 0.060 | 0.044 | 0.048 | 0.066 |

10p × 2i | 0.0 | 0.061 | 0.063 | 0.051 | 0.060 |

1p × 2i | 0.3 | 0.832 | 0.619 | 0.317 | 0.136 |

4p × 2i | 0.3 | 0.856 | 0.614 | 0.336 | 0.201 |

10p × 2i | 0.3 | 0.862 | 0.656 | 0.425 | 0.288 |

1p × 2i | 0.6 | 1.000 | 0.993 | 0.851 | 0.359 |

4p × 2i | 0.6 | 1.000 | 0.992 | 0.905 | 0.528 |

10p × 2i | 0.6 | 1.000 | 0.997 | 0.945 | 0.786 |

1p × 2i | 0.9 | 1.000 | 1.000 | 0.997 | 0.661 |

4p × 2i | 0.9 | 1.000 | 1.000 | 0.994 | 0.842 |

10p × 2i | 0.9 | 1.000 | 1.000 | 0.999 | 0.990 |

**Key**: Periods = the number of periods in which two interventions are administered.

**Table 4.**The effect of serial correlation and intervention period stacking on the power to detect a carryover effect of 0.30 standard deviation units in an N-of-1 crossover trail with 400 total observations based on the proposed analytical model. A type I error rate of 0.05 was assumed.

Design | Serial Correlation Strength | ||||||
---|---|---|---|---|---|---|---|

Periods | Obs/Per | Baseline n | Repeat n | 0.000 | 0.250 | 0.500 | 0.750 |

1p × 2i | 200 | 200 | 0 | 0.050 | 0.050 | 0.050 | 0.050 |

2p × 2i | 100 | 100 | 100 | 0.564 | 0.359 | 0.190 | 0.089 |

4p × 2i | 50 | 50 | 150 | 0.451 | 0.285 | 0.158 | 0.083 |

8p × 2i | 25 | 25 | 175 | 0.289 | 0.188 | 0.116 | 0.074 |

10p × 2i | 20 | 20 | 180 | 0.247 | 0.164 | 0.105 | 0.071 |

20p × 2i | 10 | 10 | 190 | 0.152 | 0.111 | 0.083 | 0.066 |

40p × 2i | 5 | 5 | 195 | 0.102 | 0.083 | 0.070 | 0.063 |

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**MDPI and ACS Style**

Wang, Y.; Schork, N.J. Power and Design Issues in Crossover-Based N-Of-1 Clinical Trials with Fixed Data Collection Periods. *Healthcare* **2019**, *7*, 84.
https://doi.org/10.3390/healthcare7030084

**AMA Style**

Wang Y, Schork NJ. Power and Design Issues in Crossover-Based N-Of-1 Clinical Trials with Fixed Data Collection Periods. *Healthcare*. 2019; 7(3):84.
https://doi.org/10.3390/healthcare7030084

**Chicago/Turabian Style**

Wang, Yanpin, and Nicholas J. Schork. 2019. "Power and Design Issues in Crossover-Based N-Of-1 Clinical Trials with Fixed Data Collection Periods" *Healthcare* 7, no. 3: 84.
https://doi.org/10.3390/healthcare7030084