Model-Averaged Mean Maximized Likelihood Ratio Asymptotic Distribution {harmonicmeanp}

Model-Averaged Mean Maximized Likelihood Ratio Asymptotic Distribution

Description

Density, distribution function, quantile function and random number generation for the mean of L maximized likelihood ratios under their null hypotheses, i.e. the mean of L Pareto(1,1) variables, assuming L is large.

Usage

dmamml(x, L, df, log=FALSE)
pmamml(x, L, df, log=FALSE, lower.tail=TRUE)
qmamml(p, L, df, log=FALSE, lower.tail=TRUE, xmin=1+1e-12, xmax=1e12)
rmamml(n, L, df)

Arguments

x

The value or vector of values of the mean maximized likelihood ratios, for example calculated from data using function mamml.stat.

L

The number of constituent likelihood ratios used in calculating each value of x.

df

Degrees of freedom of the constituent likelihood ratios, assumed all equal for each value of x.

log

If true the log probability is output.

lower.tail

If true (the default) the lower tail probability is returned. Otherwise the upper tail probability.

p

The value or vector of values, between 0 and 1, of the probability specifying the quantile for which to return the mean maximized likelihood ratio.

xmin, xmax

The range of values of the mean maximized likelihood ratio over which to search for the quantile specified by p, used by the numerical root finding algorithm. Must bracket the solution.

n

The number of values to simulate.

Value

dmamml produces the density, pmamml the tail probability, qmamml the quantile and rmamml random variates for the model-averaged mean maximized likelihood ratios when their null hypotheses are true.

Author(s)

Daniel J. Wilson

References

Daniel J. Wilson (2019) The harmonic mean p-value for combining dependent tests. Proceedings of the National Academy of Sciences USA 116: 1195-1200.

See Also

p.hmp

Examples

# For a detailed tutorial type vignette("harmonicmeanp")
# Example: simulate from a gamma distribution mildly enriched for large likelihood ratios.
# Compare the significance of the combined p-value for Bonferroni, Benjamini-Hochberg (i.e. Simes), 
# MAMML with 4 degrees of freedom.
L = 100
df = 4
enrich = 1.5
y = exp(0.5*rgamma(L,shape=df/2,rate=1/2/enrich))
p = pchisq(2*log(y),df,lower.tail=FALSE)
min(p.adjust(p,"bonferroni"))
min(p.adjust(p,"BH"))
(x = mamml.stat(y))
pmamml(x,L,df,lower.tail=FALSE)
p.mamml(y,df,L=L)
# Compare to the HMP - MAMML may act more conservatively because of adjustments for its
# poorer asymptotic approximation
x = hmp.stat(p)
p.hmp(p,L=L)

# Compute critical values for MAMML from asymptotic theory and compare to direct simulations
L = 100
df = 1
alpha = 0.05
(mamml.crit = qmamml(1-alpha,L,df))
nsim = 10000
y.direct = matrix(exp(0.5*rchisq(L*nsim,df)),nsim,L)
mamml.direct = apply(y.direct,1,mamml.stat)
# mamml.crit may be more conservative than mamml.crit.sim because of adjustments for its
# poorer asymptotic approximation
(mamml.crit.sim = quantile(mamml.direct,1-alpha))

# Compare MAMML simulated directly, and via the asymptotic distribution, to the asymptotic density
# Works best for df = 2
L = 30
df = 3
nsim = 1000
y.direct = matrix(exp(0.5*rchisq(L*nsim,df)),nsim,L)
mamml.direct = apply(y.direct,1,mamml.stat)
xmax = quantile(mamml.direct,.95)
h = hist(mamml.direct,c(-Inf,seq(0,xmax,len=60),Inf),col="green3",prob=TRUE,
  main="Distributions of MAMML",xlim=c(1,xmax))
# Slow because rmamml calls qmamml which uses numerical root finding
# mamml.asympt = rmamml(nsim,L,df)
# hist(mamml.asympt,c(-Inf,seq(0,xmax,len=60),Inf),col="yellow2",prob=TRUE,add=TRUE)
hist(mamml.direct,c(-Inf,seq(0,xmax,len=60),Inf),col=NULL,prob=TRUE,add=TRUE)
curve(dmamml(x,L,df),lwd=2,col="red3",add=TRUE)
legend("topright",c("Direct simulation","Asymptotic density"),
  fill=c("green3",NA),col=c(NA,"red3"),lwd=c(NA,2),bty="n",border=c(1,NA))