zscoreT {limma} | R Documentation

## Transform t-statistics to standard normal

### Description

Compute z-score equivalents of t-distributed random deviates.

### Usage

    zscoreT(x, df, approx=FALSE, method = "bailey")
    tZscore(z, df)

### Arguments

`x` | numeric vector or matrix of values from a t-distribution.
`df` | degrees of freedom (>0) of the t-distribution.
`approx` | logical. If `TRUE` then a fast approximation is used otherwise exact z-scores are computed.
`method` | character string specifying transformation to be used when `approx=TRUE`, options being `"bailey"`, `"hill"` or `"wallace"`.
`z` | numeric vector or matrix of values from the standard normal distribution.

### Details

`zscoreT` transforms t-distributed values to standard normal. Each value is converted to the equivalent quantile of the normal distribution so that if `z <- zscoreT(x, df=df)` then `pnorm(z)` equals `pt(x, df=df)`.

`tZscore` is the inverse of `zscoreT` and computes t-distribution equivalents of standard normal deviates.

If `approx=FALSE`, the transformation is done by converting to log tail probabilities using `pt` or `pnorm` and then converting back to quantiles using `qnorm` or `qt`. For numerical accuracy, the smaller of the two tail probabilities is used for each deviate.

If `approx=TRUE`, then an approximate closed-form transformation is used to convert t-statistics to z-scores directly without computing tail probabilities. The `method` argument provides a choice of three transformations. `method="bailey"` is equation (5) of Bailey (1980) or equation (7) of Brophy (1987). `method="hill"` is from Hill (1970) as given by equation (5) of Brophy (1987). `method="wallace"` is from Wallace (1959) as given by equation equation (2) of Brophy (1987). Bailey's transformation is a modification of Wallace's approximation. The Hill approximation is generally the most accurate for df > 2 but is poor for df < 1. Bailey's approximation is faster than Hill's and gives acceptable two-figure accuracy throughout. Bailey's approximation also works for some extreme values, with very large `x` or `df`, for which Hill's approximation fails due to overflow.

### Value

Numeric vector or matrix of z-scores or t-distribution deviates.

### Note

The default approximation used when `approx=TRUE` was changed from Hill to Bailey in limma version 3.41.13.

### Author(s)

Gordon Smyth

### References

Bailey, B. J. R. (1980). Accurate normalizing transformations of a Student's t variate. Journal of the Royal Statistical Society: Series C (Applied Statistics) 29(3), 304–306.

Hill, GW (1970). Algorithm 395: Student's t-distribution. Communications of the ACM 13, 617–620.

Brophy, AL (1987). Efficient estimation of probabilities in the t distribution. Behavior Research Methods 19, 462–466.

Wallace, D. L. (1959). Bounds on normal approximations to Student's and the chi-square distributions. The Annals of Mathematical Statistics, 30(4), 1121–1130.

### See Also

`zscore`.

`zscoreNBinom` in the edgeR package.

### Examples

    zscoreT(4, df=3)
    zscoreT(4, df=3, approx=TRUE)
    zscoreT(4, df=Inf)
    tZscore(2.2, df=3)

[Package limma version 3.42.2 Index]