Multigroup Dose-Finding Trial

Dunnett Test

The Dunnett test is used when there are multiple active arms and one common control arm in a clinical trial. It involves multiple tests; each test compares an active treatment arm to the control arm. The null hypotheses for these tests are given by:

\[ H_i : \mu_i = \mu_0, \quad i = 1, \ldots, K. \tag{2.14} \]

Distribution Assumption

Dunnett’s test assumes the test statistics follow a K-variate normal distribution:

\[ T \sim N(\mu, \Sigma). \tag{2.15} \]

Critical Value and Error Rate

Dunnett proposed using a single critical value, \(c_\alpha\), for all \(K\) tests, ensuring the rejection probability under the global null hypothesis \(H_G\) is \(\alpha\). This method involves using the maximum test statistic among all \(T_i (i = 1, \ldots, K)\) to determine \(c_\alpha\), thereby controlling the type-I error rate under the global null hypothesis at \(\alpha\) because: 1. If any \(H_i\) is rejected, the one with the maximum test statistic \(T\) is also likely to be rejected. 2. If the \(H_i\) with the maximum test statistic \(T\) is not rejected, then no other \(H_i\) should be rejected.

This method effectively controls the familywise type-I error in multiple comparisons scenarios.

Mathematical Representation

Chang (2014) provided that the cumulative distribution function (c.d.f.) for the maximum test statistic \(T\) can be expressed as:

\[ F_T(t) = \int_{-\infty}^{\infty} \left[ \Phi(z) \right]^K \phi(\sqrt{2t} - z) \, dz. \tag{2.16} \]

Critical Values Table

Tables often accompany this explanation to show the critical values for the maximum Dunnett test for significance levels such as \(\alpha = 0.025\) and \(\alpha = 0.05\).

Additional Explanation

Dunnett’s test is valuable in situations where multiple treatment groups are compared against a single control, common in pharmaceutical studies testing different dosages of a drug against a placebo.

The formula for \(F_T(t)\) integrates the probability density function \(\phi\) of a standardized normal and raises the cumulative distribution function \(\Phi\) to the power of \(K\). This accounts for the correlation among the test statistics due to the shared control group, adjusting the critical value to maintain the familywise error rate at a desired level.

This method ensures robust control over Type I errors across multiple comparisons, preventing the potential inflation of error rates that could occur if each test were conducted independently without adjustments for multiple testing.

## R Function: Dunnett Test for Multiple Arms with Common Control
powerDunnett <- function(n0, u0, sigma0, nArms, cAlpha, nSims) {
    powers <- rep(0, nArms)
    tValue <- rep(0, nArms)
    yCtl <- rep(0, nArms)
    yTest <- rep(0, nArms)
    power <- 0

    for (i in 1:nSims) {
        yCtl <- rnorm(n0, mean = u0, sd = sigma0)
        OverSig <- 0

        for (j in 1:nArms) {
            yTest <- rnorm(ns[j], mean = us[j], sd = sigmas[j])
            tValue <- t.test(yTest, yCtl)$statistic

            if (tValue >= cAlpha) {
                powers[j] <- powers[j] + 1 / nSims
                OverSig <- 1
            }
        }
        power <- power + OverSig / nSims
    }
    return (c("power=" , power, "Rejection Probs = ", powers))
}

## Determine critical value Zalpha for alpha (power) =0.025 ##
ns <- c(288, 288, 288)
us <- c(0, 0, 0)
sigmas <- c(2, 2, 2)
powerDunnett(n0 = 288, u0 = 0, sigma0 = 2, nArms = 3, cAlpha = 2.361, nSims = 100000)
## [1] "power="              "0.0246499999999992"  "Rejection Probs = "  "0.00936999999999979" "0.00948999999999978" "0.0090499999999998"
## Determine Power ##
ns <- c(288, 288, 288)
us <- c(0.3, 0.4, 0.5)
sigmas <- c(2, 2, 2)
powerDunnett(n0 = 288, u0 = 0, sigma0 = 2, nArms = 3, cAlpha = 2.361, nSims = 10000)
## [1] "power="             "0.819599999999926"  "Rejection Probs = " "0.295499999999984"  "0.517599999999959"  "0.742099999999935"