Logistic regression, survival analysis, model II regression

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

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)

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...

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

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)

Survival curve Kaplan-Meier method:

Míra rizika Hazard rate, l: pravděpodobnost, že jedinec přežije časový úsek t, pokud se jej již dožil Kumulativní funkce míry rizika L(t): ve vztahu ke křivce přežívání platí: L(t) = - log S(t) Využití l u složitějších modelů analýzy přežívání (Coxův model relativního rizika, Cox proportional hazard rate): l(t) = l0(t)*eb0+b1x1+b2x2+…

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

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

Regression model typ II In ordinary Least Squares, in dependence of Y on X, vertical differences are minimized (i.e. (Y-Ypredicted)2 Similarly, if we study X ~ Y, (X-Xpredicted)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

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*L3 a po logaritmování: log(M) = b0 + 3*log(L), kde b0 = log(c) Při užití „normální“ regrese bude ale odhadnutý koeficient b1 < 3

Alometric biomass partitioning Allometric biomass partitioning theory (APT): : Mleaves = b1*Mroots3/4 B.J. Enquist & K.J. Nikolas (2002): Global allocation rules for patterns of biomass partitioning in seed plants. Science 295, 1517-1520.

MA regression: example Vztah biomasy listů a stonků: Mleaves = b1*Mroots3/4 After log transformation, slope should be 0.75 Various herb species RMA program : http://www.bio.sdsu.edu/pub/andy/rma.html