This function provides the empirical power/reduction in variance in an updated meta-analysis for a vector of number of clusters to randomize per group and a vector of estimates of the ICC.

1 2 | ```
n4meansMeta(data, k, ICC, ICCDistn="unif", lower=0, upper=0.25, varRed=FALSE, m, sdm,
meanC, sdC, sdT=sdC, iter=1000, alpha=0.05)
``` |

`data` |
A matrix of individual studies (each row). The first column contains the estimate of the relative risks, second column contains the lower 95 % confidence limit and the third contains the upper 95 % confidence limit. |

`k` |
A vector of potential number of clusters to randomize to each of the treatment and control groups. Note that this function assumes an equal allocation to treatment and control group status. |

`ICC` |
A vector of potential values of the ICC, these can be obtained from the cluster trials themselves, or from the literature. |

`ICCDistn` |
The hypothetical distribution of the ICC values. This can be set to "fixed" (note that only one ICC value is accepted for this option), "unif" on the range [lower, upper], "normal", corresponding to the truncated normal distribution (Turner et al, 2004), and "smooth" corresponding to the empirical smoothing option. (Rotondi and Donner, 2009) |

`lower` |
The lower bound for the smoothing or unif options. Default value is zero. |

`upper` |
The upper bound for the smoothing or unif options. Default value is 0.25. |

`varRed` |
Logical; If varRed is set to TRUE, the proportionate reduction of variance is displayed for the fixed effects meta-analysis. |

`m` |
The mean cluster size. |

`sdm` |
The standard deviation of the mean cluster size. This adds additional real-world variation in the simulated study. |

`meanC` |
The anticipated mean response level in the control group. The anticipated treatment mean is calculated from the simulated effect size of the preliminary meta-analysis. |

`sdC` |
The standard deviation of the control rate. This adds real-world variation in the simulated study and can be precise or imprecise depending on the investigators preference. |

`sdT` |
The standard deviation of the treatment rate. By default, this is set to the same sdC. |

`iter` |
The number of iterations for each value of k and the ICC. This has a large impact on computational time. Default is 1000. |

`alpha` |
The desired type I error rate for calculation of confidence limits for the fixed effects level. For simplicity, this function assumes that each of the inputed lower and upper limits that are 95 % confidence limits. |

This function calculates the empirical power of an updated meta-analysis by a generalization of Sutton et al. (2007). The procedure is summarized in the accompanying manuscript. In short, a hypothetical new study of a given size is simulated, then added to the meta-analysis. The results are re-meta-analyzed and it is verified whether the result is statistically significant. Note that the proportion of variance reduction and power may not always strictly decrease with k, as the simulation exhibits individual-level variation.

`power` |
The power of the updated meta-analysis. Presented as a matrix of number of clusters by ICC values. |

`data` |
The data matrix is returned. |

`newMean` |
The preliminary fixed effects mean difference. |

`newVar` |
The variance of the preliminary fixed effects mean difference (MD). |

`lF` |
The 100(1 - |

`uF` |
The 100(1 - |

`Var` |
The variance of the effect measure. |

`Sig` |
Is the result statistically significant (Binary zero or one). |

`k` |
The number of clusters randomized per group (vector). |

`ICC` |
A vector of ICC values. |

`ICCDistn` |
The distributional assumption about the ICC. |

`varRed` |
Variance Reduction: Logical. |

`varianceReduction` |
The proportionate reduction in variance for the number of clusters in the fixed effects meta-analysis. |

`m` |
The mean cluster size. |

`sdm` |
The standard deviation of the mean cluster size. |

`meanC` |
The control mean. |

`sdC` |
The standard deviation of the control mean. |

`alpha` |
The desired type I error rate. |

`iter` |
The total number of iterations. |

Michael Rotondi, mrotondi@yorku.ca

Matthews JNS. Introduction to Randomized Controlled Clinical Trials (2nd Ed.) Chapman & Hall: New York, 2006.

Donner A and Klar N. Design and Analysis of Cluster Randomization Trials in Health Research. Arnold: London, 2000.

Sutton AJ et al. (2007) Evidence-based sample size calculations based upon updated meta-analysis. Statistics in Medicine, 26(12):2479-2500.

Turner R et al. (2004) Allowing for imprecision in the intracluster correlation coefficient in the design of cluster randomized trials. Statistics in Medicine, 23(8):1195-1214.

Rotondi M and Donner A. (2009) Sample Size Estimation in Cluster Randomized Trials: An Empirical Bayes Approach. Journal of Educational and Behavioral Statistics. DOI: 10.3102/1076998609332756.

Rotondi M and Donner A. (2012) Sample Size Estimation in Cluster Randomized Trials: An Evidence-Based Perspective. Computational Statistics and Data Analysis 56:1174-1187.

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