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 3.18

Index]