Reference Tool Sensitivity Specificity
Bogle Thrbahn Berg balance test 0.56 0.96
Faber et al 2006 Tinetti total 0.64 0.661
Faber et al 2006 Tinetti Balance 0.64 0.625
Raiche et al 2000 Tinetti total 0.70 0.52
Trueblood 2001 Tinetti balance 0.24 0.91
Trueblood 2001 Tinetti gait 0.21 0.95
Trueblood 2001 TUG 0.10 0.95 Morris et al 2007 5m-TUG 0.949 0.106 0.718 0.426 0.513 0.638 0.385 0.766 0.333 0.851 0.205 0.936 0.128 0.979
Sensor-based tools. Main paper of reference [11]. More than one model was extracted from some paper.
Author Year Reference Number of fallers Total number of
subjects
Caby et al 2011 [12] 15 20
Marschollek et al 2011 [14] 19 46 O’Sullivan et al 2009 [15] 12 17 Weiss et al 2011 [16] 23 41 Greene et al 2010 [17] 207 349 Riva et al 2013 [18] 42 131 Greene et al 2014 [19] 11 33 37 91 48 124
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Here we recall the formula for the probability mass function of a negative binomial distribution and how it arises as a mixture of Poisson distributions with a gamma mixing distribution.
The probability mass function of a negative binomial distribution is
ℎ(𝑛𝑛) =𝛤𝛤(𝑛𝑛+𝑘𝑘)𝑛𝑛!𝛤𝛤(𝑘𝑘)�𝜇𝜇+𝑘𝑘𝜇𝜇 �𝑛𝑛�𝜇𝜇+𝑘𝑘𝑘𝑘 �𝑘𝑘, 𝑛𝑛 = 0,1,2, … (A1)
The mean of this distribution is 𝜇𝜇 and the variance is 𝜇𝜇 +𝜇𝜇𝑘𝑘2. As the parameter 𝑘𝑘 increases, the variance shrinks toward the mean and the distribution approaches a Poisson distribution. Accordingly, 1/𝑘𝑘 is often referred to as the dispersion parameter.
Given the conditional distribution for 𝑁𝑁 stated in (1), the assignment of a gamma distribution with shape and scale parameters respectively 𝑘𝑘′ and 𝜃𝜃 for the conditioning variable 𝛬𝛬, and the expression of its probability density function recalled in (5) (with 𝑘𝑘 substituted by 𝑘𝑘′), the marginal distribution of 𝑁𝑁 is
𝑃𝑃(𝑁𝑁 = 𝑛𝑛) = � 𝑔𝑔(𝑛𝑛; 𝜆𝜆𝜆𝜆)𝑓𝑓(𝜆𝜆)𝑑𝑑𝜆𝜆+∞ 0 = � 𝑒𝑒 −𝜆𝜆𝜆𝜆(𝜆𝜆𝜆𝜆)𝑛𝑛 𝑛𝑛! 𝜆𝜆𝑘𝑘′−1𝑒𝑒−𝜆𝜆 𝜃𝜃⁄ 𝛤𝛤(𝑘𝑘′)𝜃𝜃𝑘𝑘′ 𝑑𝑑𝜆𝜆 +∞ 0 =𝑛𝑛! 𝛤𝛤(𝑘𝑘′)𝜃𝜃𝜆𝜆𝑛𝑛 𝑘𝑘′� 𝑒𝑒+∞ −𝜆𝜆(𝜆𝜆+1 𝜃𝜃⁄ )𝜆𝜆𝑛𝑛+𝑘𝑘′−1 0 𝑑𝑑𝜆𝜆 = 𝜆𝜆𝑛𝑛 𝑛𝑛! 𝛤𝛤(𝑘𝑘′)𝜃𝜃𝑘𝑘′� 𝜃𝜃 𝜆𝜆𝜃𝜃 + 1� 𝑛𝑛+𝑘𝑘′ � 𝑒𝑒+∞ −𝑠𝑠𝑠𝑠𝑛𝑛+𝑘𝑘′−1 0 𝑑𝑑𝑠𝑠 =𝛤𝛤(𝑛𝑛+𝑘𝑘′)𝑛𝑛!𝛤𝛤(𝑘𝑘′)�𝜃𝜃𝜆𝜆+1𝜃𝜃𝜆𝜆 �𝑛𝑛(𝜃𝜃𝜆𝜆+1)1 𝑘𝑘′ (A2)
After comparison between (A1) and (A2) we see that marginally 𝑁𝑁 follows a negative binomial distribution, whose parameter 𝑘𝑘 is equal to the shape parameter of the gamma distribution 𝑘𝑘′ and whose parameter 𝜇𝜇 is related to the shape and scale parameters of the gamma distribution 𝑘𝑘′ and 𝜃𝜃, and to the parameter 𝜆𝜆 by the relation 𝜇𝜇 = 𝜃𝜃𝜆𝜆𝑘𝑘′.
The identifiability of Poisson mixture distribution allows to state that the gamma distribution is the only mixing distribution that makes the mixture follow a negative binomial distribution. More properties of Poisson mixture distributions are reviewed in [1].
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This Appendix contains material supplemental to Chapter 3. In particular, a description of FRAT-up risk factors, analyses on the harmonized datasets, a sensitivity analysis on FRAT-up performance. Details about source variables, target variables, and harmonization algorithms for the three datasets are given in Appendix 3-ActiFE, Appendix 3-ELSA, and Appendix 3- InCHIANTI.