Anova-Type Statistics, - NCS2014 Non-Clinical Statistics Conference

29 Pages·2010·0.67 MB·English

Anova-Type Statistics,
a good alternative to parametric methods for analyzing
repeated data from preclinical experiments
Odile Coudert Berthion
(Keyrus Biopharma)
Véronique Onado and Guy Mathieu
(Sanofi-Aventis)
INTRODUCTION
In preclinical field: A solution for some
classical design:
Frequent departure from
Anova-type statistics
hypotheses required by
parametric analyses
Normal distribution
Variances homogeneity
Small sample size
Repeated design
Analyzing data = challenge
INTRODUCTION
Anova-Type Method
Simulation results
Conclusion
ANOVA-TYPE METHOD (1)
Non parametric method
P(Y<y)
1
NON PARAMETRIC ANALYSIS:
F1 F2
Comparison of distributions: F1 and F2
0
y
m1 m2
PARAMETRIC ANALYSIS:
Comparison of positional parameters: m1 and m2
ANOVA-TYPE METHOD (2)
Distribution functions of the response variable: F … F … F
11 it aT
Group: i=1…a
Time: t=1…T
Null hypothesis: H0: CF=0
C: Contrast matrix
F=(F ): Vector of distribution
i t
Weighted average of all the F : H=1/N ∑∑ n F
i t i t i i t
Relative marginal effect: p =∫ H dF
i t i t
1 ⎛ 1 ⎞
Ranks are used for estimating the p pˆ = ⎜R − ⎟
it :
it it.
N ⎝ 2 ⎠
Themselves used for testing the null hypotheses
ANOVA-TYPE METHOD (3)
Illustration for a response measured on 3 timepoints following
known distribution:
PARAMETRIC:
Difference
LEGEND:
of means
m : mean at time 1
1
m: mean over time Time 1: F1~N(-2,1) m -m=-2
1
m m
1
m : mean at time 2
2
Time 2: F2~N(0,1) m -m=0
m: mean over time 2
m = m
2
m : mean at time 3
3 Time 3: F3~N(2,1) m -m=2
m: mean over time 3
m m
3
ANOVA-TYPE METHOD (3)
Illustration for a response measured on 3 timepoints following
known distribution:
ANOVA-TYPE:
Relative marginal
LEGEND:
effect
_: distribution function at T1
_: average distribution F H
Time 1: F1~N(-2,1) 1 p =0.194
1
m m
1
_: distribution function at T2
F H
2
_: average distribution Time 2: F2~N(0,1) p =0.5
2
m = m
2
_: distribution function at T3
Time 3: F3~N(2,1) H F p =0.806
_: average distribution 3 3
m m
3
ANOVA-TYPE METHOD (4)
Test of H0: CF=0
n
( ) [ ]−
F C = pˆ'C' CC' Cpˆ
Anova Type Statistic: ( )
n [ ]− ˆ
tr C' CC' CV
C n
: contrast matrix
pˆ
: vector of estimated relative marginal effects
ˆ
V
: estimated covariance matrix
n
F (C) ~>Fisher(DFnum,DFden)
n
» DFden= ∞
» When C does not depend on the repeated factor, Box
approximation for DFden
The covariance matrix is allowed to be singular
since only its trace is used.
ANOVA-TYPE METHOD (5)
SAS implementation:
Proc rank
Estimation of the
Followed by proc mixed:
covariance matrix
Anova-Type Statistics and pvalues
proc mixed data=dataset ANOVAF METHOD=MIVQUE0 ;
class Time Treatment id ;
model RANK = Time Treatment Time* Treatment / ddfm=kr;
repeated Time / subject=id(Treatment );
Contrast “Comparison” / …..;
run;
For obtaining non-parametric tests for factors levels comparisons
ANOVA-TYPE METHOD (6)
Anova-Type main advantages:
Easy implementation under SAS software
No underlying hypothesis on the response
distribution function (shape, variability…)
It is allowed to have no variability in some
groups
The covariance matrix being allowed to be
singular