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Logistic regression, survival analysis, model II regression

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Logistic regression Response (dependent) variable is either –yes/ no (alive/ dead, flowering/sterile) –Number of positive cases out of total (seed germination, number of flowering individuals out of total no of individuals) – assuming binomial distribution Regression model predicts probability, i.e. value between 0 and 1

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Logistic regression 2 Logit transformation: log( p / (1-p) ) = log (odds ratio) Can not be applied directly to 0/1, applied on predicted probabilities: p in (0, 1) Special case of Generalized linear models (GLM)

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Logistická regrese a Statistica Example – survival of winter depending on flowering and rhizome size Advanced Linear / Nonlinear Models Generalized... Models Logit model Or non-linear estimation...

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Possible application Example – how is the probability of survival over the winter affected by flowering in previous summer, storage of sugars, and length of the winter? Surmalog.xls, list ReprEff

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Survival analysis Survival analysis, mainly in medicine Useful for data (usually about time) with censoring Most often right censoring: I have finished the experiment, but some individuals are still alive (or did not germinate yet etc.] Left censoring For data without censoring are probably simpler methods available - mostly generalized linear models)

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Survival curve Kaplan-Meier method:

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Míra rizika Hazard rate, : pravděpodobnost, že jedinec přežije časový úsek t, pokud se jej již dožil Kumulativní funkce míry rizika (t): ve vztahu ke křivce přežívání platí: (t) = - log S(t) Využití u složitějších modelů analýzy přežívání (Coxův model relativního rizika, Cox proportional hazard rate): (t) = 0 (t)*e b 0 +b 1 x 1 +b 2 x 2 +…

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Use of survival analysis? Comparison of survival curves among groups Estimate “halftime” (of life, survival time, time to germination) with confidence interval Testing effects of both quantitative and qualitative predictors

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Survival analysis - exercises Germination dynamics affected by chilling, file Surmalog.xls, sheet Germination, method Comparing two samples Effect of radio-collars on survival of antilops -obojků na úmrtnost antilop, file Surmalog.xls, sheet RadioCollars, method Regression / Proportional hazard (Cox) regression

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Regression model typ II In ordinary Least Squares, in dependence of Y on X, vertical differences are minimized (i.e. (Y-Y predicted ) 2 Similarly, if we study X ~ Y, (X- X predicted ) 2 is minimized. The angel among the two lines decreases with increasing (r) Major axis (MA) regression – symmetric – what is perpendicular depends on units – various standardizations

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MA regression: motivation Zkoumáme vztah mezi délkou (L) a hmotností (M) jedinců určitého druhu Pokud se tvar těla s růstem nemění (isometrický růst), lze vztah popsat takto: M = c*L 3 a po logaritmování: log(M) = b 0 + 3*log(L), kde b 0 = log(c) Při užití „normální“ regrese bude ale odhadnutý koeficient b 1 < 3

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Alometric biomass partitioning Allometric biomass partitioning theory (APT): : M leaves = 1 *M roots 3/4 B.J. Enquist & K.J. Nikolas (2002): Global allocation rules for patterns of biomass partitioning in seed plants. Science 295,

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MA regression: example Vztah biomasy listů a stonků: M leaves = 1 *M roots 3/4 After log transformation, slope should be 0.75 Various herb species RMA program :

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