# Two-sample normal sample size

## Introduction

Limited support is provided for 2-sample design with a normally distributed random variable as the outcome. Users are encouraged to look at guidance such as in Jennison and Turnbull (2000). We provide a tool where for a large sample case where a reasonable estimate of standard deviation is available, a reasonable sample size can be computed based straightforward distribution theory outlined below.

## The problem considered

The overall sample size notation used for gsDesign is to consider a standardized effect size parameter which is referred to as \(\theta\) in Jennison and Turnbull (2000). We begin with the 2-sample normal problem where we assume a possibly different standard deviation in each treatment group. For \(j = 1, 2\), we let \(X_{j, i}\), \(i = 1, 2, \ldots n_j\) represent independent and identically distributed observations following a normal distribution with mean \(\mu_j\) and standard deviation \(\sigma_j\). The natural parameter for comparing the two distributions is

\[\delta = \mu_2 - \mu_1\]

and we wish to test if \(\delta > 0\) in a one-sided testing scenario to test for superiority of treatment 2 over treatment 1. We could also consider testing, say, \(\delta > \delta_0\) for a non-inferiority scenario with \(\delta_0<0\) or super superiority if \(\delta_0>0\). While normally a t-test would be used for this, for large sample sizes this would be nearly equivalent to a Z-test defined by:

\[Z=\frac{\bar X_2 - \bar X_1-\delta_0}{\sqrt{\sigma^2_2/n_2 + \sigma_1^2/n_1}}\approx \frac{\bar X_2 - \bar X_1}{\sqrt{s^2_2/n_2 + s_1^2/n_1}}=t\] where \(\bar X_j\) is the sample mean and \(s_j^2\) is the sample variance for group \(j=1,2\). The far right hand side of this is Welch’s t-test. For our examples we use this \(t\)-test and show that the sample size computation based on the \(Z\)-test above works well for the chosen problems.

## Sample size

Thus, \(n_2=rn/(1+r)\), \(n_1=n/(1+r)\) and when \(r=1\) we have \(n_1=n_2=n/2\). Now that we have completed needed notation, those not interested in the theory behind the sample size and power calculation used may skip the rest of this section.

We let \[\sigma^2=(1+r)(\sigma_1^2+\sigma_2^2/r)\] and define \[ \theta= (\delta -\delta_0)/\sigma.\] Under the given assumptions, \[Z \sim \text{Normal}\left(\sqrt n\theta,1\right).\] Under the null hypothesis that \(\delta=\delta_0\), we have \(Z\sim \text{Normal}(0,1)\). Thus, regardless of \(n\) we have \[P_0[Z\ge \Phi^{-1}(1-\alpha)]=\alpha.\] Under the alternate hypothesis that \(\delta=\delta_1\) and we denote a corresponding \(\theta_1\). We define the type II error \(\beta\) and power \(1-\beta\) by

\[ \begin{align} 1-\beta =& P_1[Z\ge \Phi^{-1}(1-\alpha)]\\ =& P[Z-\sqrt n\theta_1\ge \Phi^{-1}(1-\alpha)-\sqrt n\theta_1]\\ =&\Phi(\Phi^{-1}(1-\alpha)-\sqrt n\theta_1)). \end{align}\]

If the power \(1-\beta\) is fixed, we can invert this formula to compute sample size with:

\[n= \left(\frac{\Phi^{-1}(1-\beta)+\Phi^{-1}(1-\alpha)}{\theta_1}\right)^2.\]

For 2-sided testing, we simply substitute \(\alpha/2\) for \(\alpha\) in the above two formulas.

## Examples

We consider two examples to check the above formulas vs. `nNormal()`. We then confirm that the approximation is working well by simulating and confirming that the power and Type I error approximations are useful. Finally, we provide a simple group sequential design example.

### Sample size

We consider an example with \(\sigma_2=1.25\), \(\sigma_1=1.6\), \(\delta=0.8\) and \(\delta_0=0\). We let the sample size ratio be 2 experimental group observations per control observation. We compute sample size with `nNormal()` assuming one-sided Type I error \(\alpha=0.025\) and 90% power (\(1-\beta=0.9\)).

Checking using the sample size formula above, we have:

    r <- 2
    sigma <- sqrt((1 + r) * (1.6^2 + 1.25^2 / r))
    theta <- 0.8 / sigma
    ((qnorm(.9) + qnorm(.975)) / theta)^2
    #> [1] 164.5684

### Power

Now, assume we let the sample size be 200 and compute power under the same scenario.

    nNormal(delta1 = 0.8, sd = 1.6, sd2 = 1.25, alpha = 0.025, n = 200, ratio = 2)
    #> [1] 0.9466825

From the power formula above, we duplicate this with:

    pwr <- pnorm(qnorm(.975) - sqrt(200) * theta, lower.tail = FALSE)
    pwr
    #> [1] 0.9466825

If we want to plot power for a variety of sample sizes, we can input `n` as a vector:

    n <- 100:200
    pwrn <- nNormal(delta1 = 0.8, sd = 1.6, sd2 = 1.25, alpha = 0.025, n = n, ratio = 2)
    plot(n, pwrn, type = "l")

Alternatively, you could fix sample size at 200 and plot power under different treatment effect assumptions:

    delta1 <- seq(.5, 1, .025)
    pwrdelta1 <- nNormal(delta1 = delta1, sd = 1.6, sd2 = 1.25, alpha = 0.025, n = 200, ratio = 2)
    plot(delta1, pwrdelta1, type = "l")

### Verification with simulation

Rather than simulate individual observations, we will take advantage of the fact that for \(j=1,2\)

\[\bar X_j\sim \text{Normal}(\mu_j,\sigma_j^2/n_j)\]

and

\[(n_j-1)s_j^2/\sigma_j^2=\sum_{i=1}^{n_j} (X_{ij}-\bar X_j)/\sigma^2 \sim \chi ^2_{n_j-1}\]

are independent. Thus, we can simulate trial power with \(n=200\) 1 million times with a t-statistic with unequal variances quickly as follows under the alternate hypothesis:

    nsim <- 1000000
    delta <- 0.8
    sd1 <- 1.6
    sd2 <- 1.25
    n1 <- 67
    n2 <- 133
    deltahat <- rnorm(n = nsim, mean = delta, sd = sd1 / sqrt(n1)) -
      rnorm(n = nsim, mean = 0, sd = sd2 / sqrt(n2))
    s <- sqrt(
      sd1^2 * rchisq(n = nsim, df = n1 - 1) / (n1 - 1) / n1 +
        sd2^2 * rchisq(n = nsim, df = n2 - 1) / (n2 - 1) / n2
    )
    z <- deltahat / s
    mean(z >= qnorm(.975))
    #> [1] 0.946816

The standard error for this simulation power calculation is approximately

    sqrt(pwr * (1 - pwr) / nsim)
    #> [1] 0.0002246659

suggesting we should be within less than about 0.001 if the actual power, which suggests the normal power approximation is reasonable for this scenario.

### Group sequential design

Now we derive a group sequential design under the above scenario. We will largely use default parameters and show two methods. For the first, we plug in the fixed sample size above as follows:

    d <- gsDesign(
      k = 2,
      n.fix = nNormal(delta1 = 0.8, sd = 1.6, sd2 = 1.25, alpha = 0.025, beta = .1, ratio = 2),
      delta1 = 0.8
    )
    d |>
      gsBoundSummary(deltaname = "Mean difference") |>
      kable(row.names = FALSE)

Analysis | Value | Efficacy | Futility
--- | --- | --- | ---
IA 1: 50% | Z | 2.7500 | 0.4122
N: 86 | p (1-sided) | 0.0030 | 0.3401
| ~Mean difference at bound | 0.9399 | 0.1409
| P(Cross) if Mean difference=0 | 0.0030 | 0.6599
| P(Cross) if Mean difference=0.8 | 0.3412 | 0.0269
Final | Z | 1.9811 | 1.9811
N: 172 | p (1-sided) | 0.0238 | 0.0238
| ~Mean difference at bound | 0.4788 | 0.4788
| P(Cross) if Mean difference=0 | 0.0239 | 0.9761
| P(Cross) if Mean difference=0.8 | 0.9000 | 0.1000

A textual summary of the design is given by:

    cat(summary(d))

Asymmetric two-sided group sequential design with non-binding futility bound, 2 analyses, sample size 172, 90 percent power, 2.5 percent (1-sided) Type I error. Efficacy bounds derived using a Hwang-Shih-DeCani spending function with gamma = -4. Futility bounds derived using a Hwang-Shih-DeCani spending function with gamma = -2.

We can get the same answer by plugging in the standardized effect size we computed above:

    gsDesign(
      k = 2,
      delta = theta,
      delta1 = 0.8
    ) |>
      gsBoundSummary(deltaname = "Mean difference") |>
      kable(row.names = FALSE)

Analysis | Value | Efficacy | Futility
--- | --- | --- | ---
IA 1: 50% | Z | 2.7500 | 0.4122
N: 86 | p (1-sided) | 0.0030 | 0.3401
| ~Mean difference at bound | 0.9399 | 0.1409
| P(Cross) if Mean difference=0 | 0.0030 | 0.6599
| P(Cross) if Mean difference=0.8 | 0.3412 | 0.0269
Final | Z | 1.9811 | 1.9811
N: 172 | p (1-sided) | 0.0238 | 0.0238
| ~Mean difference at bound | 0.4788 | 0.4788
| P(Cross) if Mean difference=0 | 0.0239 | 0.9761
| P(Cross) if Mean difference=0.8 | 0.9000 | 0.1000

We leave it to the reader to verify the properties of the above design using simulation as in the fixed design example.

## References

Jennison, Christopher, and Bruce W. Turnbull. 2000. Group Sequential Methods with Applications to Clinical Trials. Boca Raton, FL: Chapman; Hall/CRC.