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Non-normal models
Other discrete and continuous distributions can serve as the basis for ìAOVî, regression, etc. The key statistical concepts are likelihood, maximum likelihood estimation, and likelihood ratio tests.
Sums of squares, ìleast squaresî, etc. are not used (they arise from the normal distribution).
Likelihood (R. A. Fisher): the likelihood (or likelihood function) is the probability model for the data, evaluated at the data. One can think of it as the probability, if the random process could be repeated, that the particular outcome represented by the data would reoccur.
ex. ] μ binomial a 8 , 1 b
T ] œ C œ '
8x Cx 8 ' C x
a b a b a b
1 C^11 8'C
data: 8 œ 10, C œ 6
P œ '
x x x
1 6 a 1 1 b^4
( P depends on the value of the unknown parameter 1 )
ex. Random sample from a Poisson distribution (sample plots in a field; count dandelions in each plot) ] (^) " , ]#, ..., ] 8 , where ] μ 3 Poissona b.
T ] œ C œ C œ
Cx
a 3 b
'. .C
data: 0, 3, 0, 1, 4, 6, 2, 1
P œ T ]a (^) " œ 0 and ] (^) # œ 3 and â and ] (^) ) œ 1 b
œ T ]a (^) " œ 0 b T a ] (^) # œ 3 b âT a ] (^) ) œ 1 b
œ â
x x x
Maximum likelihood parameter estimates (R. A. Fisher): the values of the unknown parameters that maximize the likelihood are the maximum likelihood (ML) estimates.
ML estimates have good statistical properties (small variances, approx. unbiased, etc.)
ex. Binomial P œ (^) 6 4^10 x xx 16 a 1 ' 1 b^4
calculus: 1 s œ 106 ML
ex. Poisson random sample C (^) " , C (^) # , ...,C 8
.s œ C'
ex. Normal random sample C (^) " , C (^) # , ...,C 8
.s minimizes aC (^) " '. b #^1 aC (^) # '. b #^ 1 â 1 aC 8 '.b#
.s œ C'
5 s œ C '. 1 C '. 1 â 1 C '. 8
(^) " s # (^) # s # (^8) s
’ a b a b a b “
(ML uses 8 , not 8 '1, in denominator)
Maximized likelihood: Psis the likelihood value calculated using the ML estimates of the parameters.
ex. Binomial P œs^ 6 4^10 106 1 ' 106 x^6 x x ˆ^ ‰ ˆ^ ‰
ex. Normal
P œs^ Šs 5 #^1 ‹ /
' (^) ' 2
(^82 ) 2
Likelihood ratio hypothesis test (J. Neyman, E. S. Pearson, S. S. Wilks)
) œ a) (^) " , ) (^) # , ..., )in an AOV)
< œ a< (^) " , < (^) # , ..., <=b other parameters in the model (ìnuisance parametersî, ex. 5 # in an AOV)
H : parameters in! )are constrained (ex.. (^) " œ. (^) # œ â œ. (^) > œ.)
H : parameters are not constraineda
Ps^! : likelihood maximized under model H!
Psa : likelihood maximized under model Ha
P ÎPs^! sa : likelihood ratio comparing the two models (if large, H (^)! favored; if small, H (^) afavored)
