pchisqsum {survey} | R Documentation

## Distribution of quadratic forms

### Description

The distribution of a quadratic form in p standard Normal variables is a linear combination of p chi-squared distributions with 1df.

### Usage

    pchisqsum(x, df, a, lower.tail = TRUE, method = c("satterthwaite", "integration","saddlepoint"))

### Arguments

`x` | Observed values
`df` | Vector of degrees of freedom
`a` | Vector of coefficients
`lower.tail` | lower or upper tail?
`method` | See Details below

### Details

The `"satterthwaite"` method uses Satterthwaite's approximation, and this is also used as a fallback for the other methods.

`"integration"` inverts the characteristic function numerically. This is relatively slow, and not reliable for p-values below about 1e-5 in the upper tail, but is highly accurate for moderate p-values.

`"saddlepoint"` uses a saddlepoint approximation when `x>1.05*sum(a)` and the Satterthwaite approximation for smaller `x`. This is fast and is accurate in the upper tail, where accuracy is important.

### Value

Vector of cumulative probabilities

### References

Davies RB (1973). "Numerical inversion of a characteristic function" Biometrika 60:415-7

Kuonen D (1999) Saddlepoint Approximations for Distributions of Quadratic Forms in Normal Variables. Biometrika, Vol. 86, No. 4 (Dec., 1999), pp. 929-935

### See Also

`pchisq`

### Examples

    x <- 5*rnorm(1001)^2+rnorm(1001)^2
    x.thin<-sort(x)[1+(0:100)*10]
    p.invert<-pchisqsum(x.thin,df=c(1,1),a=c(5,1),method="int" ,lower=FALSE)
    p.satt<-pchisqsum(x.thin,df=c(1,1),a=c(5,1),method="satt",lower=FALSE)
    p.sadd<-pchisqsum(x.thin,df=c(1,1),a=c(5,1),method="sad",lower=FALSE)

    plot(p.invert, p.satt,type="l",log="xy")
    abline(0,1,lty=2,col="purple")
    plot(p.invert, p.sadd,type="l",log="xy")
    abline(0,1,lty=2,col="purple")

[Package survey version 3.19 Index]