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, (XXpredicted)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