Everything else held constant, what is the relationship between alpha and beta?

Question

I want the mathematical formula the statistical programs are using to compute alpha and beta. I typically use G-Power to estimate the necessary sample size for a given power (1-beta) for an a-priori analysis. The input it requires for this calculation is: Type of analysis (t-test, difference between two independent samples) the number of tails (2) alpha (0.05) power (0.8) effect size (0.2) = estimated necessary sample size (788) I have looked everywhere I know to find out what the function/formula is that is being used to compute the sample size value from these variables.

I know that:

• alpha=type I error=FP/(FP+TN)
• beta=type II error= FN/(TP+FN)
• where FP=false positive
• TP=true negative
• FN=false negative
• TN=true negative

But I don't know how to compute false positives or true negatives a-priori.

​Unless Stevens (2009) is wrong, power (the inverse of beta) is a function of alpha therefore β ∝ α (e.g. if, {α = .1, β = .37, power = .63}, {α = .05, β = .52, power = .48}, {α = .01, β = .78, power = .22}, see p.4). I am simply interested in finding the function that describes the relationship between alpha and beta/power. My question: Do you know where I can find the function to describe this relationship?

The reason? To actually understand what I am doing every time I run a statistical analysis.

when I perused the source code for an R package pwr I found:

What looks like the n is obtained numerically using a root finding algorithm computed with the highlighted part of the function below (uniroot looks like a general function to find where a supplied function equals 0).

> pwr.t.test
function (n = NULL, d = NULL, sig.level = 0.05, power = NULL, 
    type = c("two.sample", "one.sample", "paired"), alternative = c("two.sided", 
        "less", "greater")) 
{
    if (sum(sapply(list(n, d, power, sig.level), is.null)) != 
        1) 
        stop("exactly one of n, d, power, and sig.level must be NULL")
    if (!is.null(sig.level) && !is.numeric(sig.level) || any(0 > 
        sig.level | sig.level > 1)) 
        stop(sQuote("sig.level"), " must be numeric in [0, 1]")
    if (!is.null(power) && !is.numeric(power) || any(0 > power | 
        power > 1)) 
        stop(sQuote("power"), " must be numeric in [0, 1]")
    type <- match.arg(type)
    alternative <- match.arg(alternative)
    tsample <- switch(type, one.sample = 1, two.sample = 2, paired = 1)
    ttside <- switch(alternative, less = 1, two.sided = 2, greater = 3)
    tside <- switch(alternative, less = 1, two.sided = 2, greater = 1)
    if (tside == 2 && !is.null(d)) 
        d <- abs(d)
    if (ttside == 1) {
        p.body <- quote({
            nu <- (n - 1) * tsample
            pt(qt(sig.level/tside, nu, lower = TRUE), nu, ncp = sqrt(n/tsample) * 
                d, lower = TRUE)
        })
    }
    if (ttside == 2) {
        p.body <- quote({
            nu <- (n - 1) * tsample
            qu <- qt(sig.level/tside, nu, lower = FALSE)
            pt(qu, nu, ncp = sqrt(n/tsample) * d, lower = FALSE) + 
                pt(-qu, nu, ncp = sqrt(n/tsample) * d, lower = TRUE)
        })
    }
    if (ttside == 3) {
        p.body <- quote({
            nu <- (n - 1) * tsample
            pt(qt(sig.level/tside, nu, lower = FALSE), nu, ncp = sqrt(n/tsample) * 
                d, lower = FALSE)
        })
    }
    if (is.null(power)) 
        power <- eval(p.body)
    else if (is.null(n)) 
        n <- uniroot(function(n) eval(p.body) - power, c(2 + 
            1e-10, 1e+07))$root
        else if (is.null(d)) {
            if (ttside == 2) {
                d <- uniroot(function(d) eval(p.body) - power, c(1e-07, 
                    10))$root
            }
            if (ttside == 1) {
                d <- uniroot(function(d) eval(p.body) - power, c(-10, 
                    5))$root
            }
            if (ttside == 3) {
                d <- uniroot(function(d) eval(p.body) - power, c(-5, 
                    10))$root
            }
        }
        else if (is.null(sig.level)) 
            sig.level <- uniroot(function(sig.level) eval(p.body) - 
                power, c(1e-10, 1 - 1e-10))$root
    else stop("internal error")
    NOTE <- switch(type, paired = "n is number of *pairs*", two.sample = "n is number in *each* group", 
        NULL)
    METHOD <- paste(switch(type, one.sample = "One-sample", two.sample = "Two-sample", 
        paired = "Paired"), "t test power calculation")
    structure(list(n = n, d = d, sig.level = sig.level, power = power, 
        alternative = alternative, note = NOTE, method = METHOD), 
        class = "power.htest")
}
<environment: namespace:pwr>

I'm ​just ​not savvy enough with the notation to transform this into something prettier e.g. f(x)=...

Answer 1

Check out Introduction to sample size determination and power analysis for clinical trials by John M. Lachin. Equations (6) and (7) look like they will be useful to you. $$N = \frac{4\sigma^2(Z_\alpha+Z_\beta)^2}{\mu_1^2}$$ $$Z_\beta=\frac{\left|\mu_1\right|\sqrt{N}-2Z_\alpha\sigma}{2\sigma} $$

This is for equal sample sizes, but a form allowing for unequal sample sizes is found in the linked paper.