Density, distribution function, quantile function and random generation for mixtures of chi-squared distributions that corresponds to the null distribution of the Likelihood Ratio between 2 nested mixed models.
Usage
rchisqmix(n, s, q)
dchisqmix(x, s, q)
qchisqmix(p, s, q)
pchisqmix(quant, s, q, lower.tail = TRUE)Arguments
- n
number of observations.
- s
number of fixed effects to be tested.
- q
number of random effects to be tested.
- x, quant
a quantile.
- p
a probability.
- lower.tail
logical; if
TRUE(default), probabilities are \(P[X \le x]\); otherwise, \(P[X > x]\).
Value
A vector of random independent observations of the \(\chi^2\) mixture
identified by the values of s and q.
Details
The approximate null distribution of a likelihood ratio for 2 nested mixed models, where both fixed and random effects are tested simultaneously, is a very specific mixture of \(\chi^2\) distributions [Self & Liang (1987), Stram & Lee (1994) and Stram & Lee (1995)]. It depends on both the number of random effects and the number of fixed effects to be tested simultaneously: $$LRT_{H_0}\sim\sum_{k=q}^{q+r}{{r}\choose{k-q}}2^{-r}\chi^2_{(k)}$$
References
Self, S. G. and Liang, K., 1987, Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions, Journal of the American Statistical Association 82: 605--610.
Stram, D. O. and Lee, J. W., 1994, Variance components testing in the longitudinal mixed effects model, Biometrics 50: 1171--1177.
Stram, D. O. and Lee, J. W., 1995, Corrections to "Variance components testing in the longitudinal mixed effects model" by Stram, D. O. and Lee, J. W.; 50: 1171--1177 (1994), Biometrics 51: 1196.
