Sequences of Independent Trials

2018 ◽  
pp. 63-103
Author(s):  
Boris V. Gnedenko
Keyword(s):  
Independent Trials

scholarly journals Recommendations on multiple testing adjustment in multi-arm trials with a shared control group

2016 ◽  
Vol 27 (5) ◽  
pp. 1513-1530 ◽  
Author(s):  
Dena R Howard ◽  
Julia M Brown ◽  
Susan Todd ◽  
Walter M Gregory
Keyword(s):  
False Positive ◽  
Multiple Testing ◽  
Shared Control ◽  
Control Group ◽  
Type I ◽  
False Positive Error ◽  
Control Data ◽  
Experimental Therapies ◽  
Multiple Testing Adjustment ◽  
Independent Trials

Multi-arm clinical trials assessing multiple experimental treatments against a shared control group can offer efficiency advantages over independent trials through assessing an increased number of hypotheses. Published opinion is divided on the requirement for multiple testing adjustment to control the family-wise type-I error rate (FWER). The probability of a false positive error in multi-arm trials compared to equivalent independent trials is affected by the correlation between comparisons due to sharing control data. We demonstrate that this correlation in fact leads to a reduction in the FWER, therefore FWER adjustment is not recommended solely due to sharing control data. In contrast, the correlation increases the probability of multiple false positive outcomes across the hypotheses, although standard FWER adjustment methods do not control for this. A stringent critical value adjustment is proposed to maintain equivalent evidence of superiority in two correlated comparisons to that obtained within independent trials. FWER adjustment is only required if there is an increased chance of making a single claim of effectiveness by testing multiple hypotheses, not due to sharing control data. For competing experimental therapies, the correlation between comparisons can be advantageous as it eliminates bias due to the experimental therapies being compared to different control populations.

Markov chains close to a sequence of independent trials

1979 ◽  
Vol 25 (3) ◽  
pp. 243-248
Author(s):  
A. M. Zubkov
Keyword(s):  
Markov Chains ◽  
Independent Trials

scholarly journals Growth and Development of Purple Nutsedge Based on Days or Thermal Units

2015 ◽  
Vol 33 (2) ◽  
pp. 165-173 ◽  
Author(s):  
R.S.O. Lima ◽  
E.C.R. Machado ◽  
A.P.P. Silva ◽  
B.S. Marques ◽  
M.F. Gonçalves ◽  
...  
Keyword(s):  
Plant Growth ◽  
Mathematical Models ◽  
Growth And Development ◽  
Dry Matter ◽  
Cyperus Rotundus ◽  
Growing Degree Days ◽  
Plant Growth And Development ◽  
Degree Days ◽  
Purple Nutsedge ◽  
Independent Trials

This work was carried out with the objective of elaborating mathematical models to predict growth and development of purple nutsedge (Cyperus rotundus) based on days or accumulated thermal units (growing degree days). Thus, two independent trials were developed, the first with a decreasing photoperiod (March to July) and the second with an increasing photoperiod (August to November). In each trial, ten assessments of plant growth and development were performed, quantifying total dry matter and the species phenology. After that, phenology was fit to first degree equations, considering individual trials or their grouping. In the same way, the total dry matter was fit to logistic-type models. In all regressions four temporal scales possibilities were assessed for the x axis: accumulated days or growing degree days (GDD) with base temperatures (Tb) of 10, 12 and 15 oC. For both photoperiod conditions, growth and development of purple nutsedge were adequately fit to prediction mathematical models based on accumulated thermal units, highlighting Tb = 12 oC. Considering GDD calculated with Tb = 12 oC, purple nutsedge phenology may be predicted by y = 0.113x, while species growth may be predicted by y = 37.678/(1+(x/509.353)-7.047).

scholarly journals Selecting a sequence of last successes in independent trials

2000 ◽  
Vol 37 (2) ◽  
pp. 389-399 ◽  
Author(s):  
F. Thomas Bruss ◽  
Davy Paindaveine
Keyword(s):  
Approximate Solution ◽  
Convenient Method ◽  
Probability Space ◽  
Maximum Probability ◽  
Optimal Rule ◽  
Unknown Number ◽  
Multiple Stopping ◽  
Indicator Functions ◽  
Independent Indicator ◽  
Independent Trials

Let I1,I2,…,In be a sequence of independent indicator functions defined on a probability space (Ω, A, P). We say that index k is a success time if Ik = 1. The sequence I1,I2,…,In is observed sequentially. The objective of this article is to predict the lth last success, if any, with maximum probability at the time of its occurrence. We find the optimal rule and discuss briefly an algorithm to compute it in an efficient way. This generalizes the result of Bruss (1998) for l = 1, and is equivalent to the problem of (multiple) stopping with l stops on the last l successes. We then extend the model to a larger class allowing for an unknown number N of indicator functions, and present, in particular, a convenient method for an approximate solution if the success probabilities are small. We also discuss some applications of the results.

A limit theorem for the number of non-overlapping occurrences of a pattern in a sequence of independent trials

1988 ◽  
Vol 25 (2) ◽  
pp. 428-431 ◽  
Author(s):  
O. Chryssaphinou ◽  
S. Papastavridis
Keyword(s):  
Limit Theorem ◽  
Poisson Distribution ◽  
Recurrent Patterns ◽  
Independent Trials ◽  
General Conditions

A sequence of independent experiments is performed, each producing a letter from a given alphabet. Using a result by Barbour and Eagleson (1984) we prove that under general conditions the number of non-overlapping occurrences of long recurrent patterns has approximately a Poisson distribution.

Estimating a treatment effect: Choosing between relative and absolute measures

2016 ◽  
Vol 23 (2) ◽  
pp. 197-200 ◽  
Author(s):  
Maria Pia Sormani ◽  
Paolo Bruzzi
Keyword(s):  
Multiple Sclerosis ◽  
Clinical Trials ◽  
Treatment Effect ◽  
Numerical Examples ◽  
Relative Risks ◽  
Relative Treatment Effect ◽  
Hazard Ratios ◽  
Numbers Needed To Treat ◽  
Indirect Comparisons ◽  
Independent Trials

The size of a treatment effect in clinical trials can be expressed in relative or absolute terms. Commonly used relative treatment effect measures are relative risks, odds ratios, and hazard ratios, while absolute estimate of treatment effect are absolute differences and numbers needed to treat. When making indirect comparisons of treatment effects, which is common in multiple sclerosis (MS), having now many drugs tested in independent trials, we can have different figures if we use relative or absolute measures, and a frequently asked question by clinicians is which approach should be used. In this report, we will try to define these measures, to give numerical examples of their calculation and specify their meaning and their context of use.

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