pchisqsum {survey} R Documentation

### 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

`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)