## Abstract

Often in randomized clinical trials and observational cohort studies, a non-negative continuously distributed response variable is measured in treatment and control groups. In the presence of true zeros for the response variable, a two-part zero-inflated log-normal model (which assumes that the data has a probability mass at zero and a continuous response for values greater than zero) is usually recommended. However, in some environmental health and human immunodeficiency virus (HIV) studies, quantitative assays for metabolites of toxicants, or quantitative HIV RNA measurements are subject to left-censoring due to values falling below the limit of detection (LD). Here, a zero-inflated log-normal mixture model is often suggested since true zeros are indistinguishable from left-censored values due to the LD. When the probabilities of true zeros in the two groups are not restricted to be equal, the information contributed by values falling below LD is used only to estimate the probability of true zeros in the context of mixture distributions. We derived the required sample size to assess the effect of a treatment in the context of mixture models with equal and unequal variances based on the left-truncated log-normal distribution. Methods for calculation of statistical power are also presented. We calculate the required sample size and power for a recent study estimating the effect of oltipraz on reducing urinary levels of the hydroxylated metabolite aflatoxin M_{1} (AFM_{1}) in a randomized, placebo-controlled, double-blind phase IIa chemoprevention trial in Qidong, China. A Monte Carlo simulation study is conducted to investigate the performance of the proposed methods.

Original language | English (US) |
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Pages (from-to) | 2647-2657 |

Number of pages | 11 |

Journal | Statistics in Medicine |

Volume | 25 |

Issue number | 15 |

DOIs | |

State | Published - Aug 15 2006 |

### Bibliographical note

Funding Information:Authors acknowledge the financial support of the Czech Science Foundation (Project No. 13-12477S, Transport of pharmaceuticals in soils). Pharmaceutical concentrations were measured using devices financially supported by the Ministry of Education, Youth and Sports of the Czech Republic, projects CENAKVA (No. CZ.1.05/2.1.00/01.0024) and CENAKVA II (No. LO1205 under the NPU I program). Authors also thank Zuzana Schmidtov? for technical assistance and Christopher Ash for English language correction.

## Keywords

- Detection limit
- Left-censoring
- Left-truncation
- Mixture models
- Power
- Sample size