Question

By considering the asymptotic power lemma, Theorem \(10.4 .2\), show that the equal sample size situation \(n_{1}=n_{2}\) is the most powerful design among designs with \(n_{1}+n_{2}=n, n\) fixed, when level and alternatives are also fixed. Hint: Show that this problem is equivalent to maximizing the function $$ g\left(n_{1}\right)=\frac{n_{1}\left(n-n_{1}\right)}{n^{2}} $$ and then obtain the result.

Short answer

In order to maximize \(g\left(n_{1}\right)=\frac{n_{1}\left(n-n_{1}\right)}{n^{2}}\), the first derivative was computed, set to zero and solved for \(n_{1}\), resulting in \(n_{1} = \frac{n}{2}\). The second order derivative was negative, indicating a maximum point. It was thus concluded that the equal sample size scenario \(n1 = n2 = \frac{n}{2}\) is indeed the most powerful design.

Step-by-step solution

Compute the first order derivative

Compute the first derivative of the function \(g(n1)\). This can be done using the quotient rule, applied to the function \(g(n1) = \frac{n_1(n-n_1)}{n^2}\). We obtain \(g'(n1) = \frac{(n-2n1)n}{n^2}\).

Set the derivative to zero

We want to maximize \(g(n1)\), so we set its derivative equal to zero and solve for \(n1\). This gives us \(g'(n1) = \frac{(n-2n1)n}{n^2} = 0\). The result is \(n1 = \frac{n}{2}\).

Compute the second order derivative

To ensure the result from Step 2 indeed maximizes the function, compute the second derivative of \(g(n1)\), which results in \(g''(n1) = \frac{-2n}{n^2} < 0\).

Conclude the analysis

Since the second order derivative \(g''(n1) < 0\), this indicates that the function \(g(n1)\) is maximized when \(n1 = \frac{n}{2}\). We thus conclude that the situation where \(n1 = n2 = \frac{n}{2}\) gives the most powerful design, given \(n1 + n2 = n\).

Key concepts

Statistical Hypothesis Testing

In the realm of statistics, hypothesis testing is a critical procedure used to decide whether a specific statement about a parameter in a population is likely to be true based on sample data. This process involves two hypotheses: the null hypothesis (ullh) and the alternative hypothesis (\(H_a\)). The null hypothesis typically represents a default position, such as no effect or no difference. On the other hand, the alternative hypothesis is what researchers wish to convincingly argue for.

Hypothesis testing includes setting a significance level (\(\alpha\)), which defines the probability of rejecting the null hypothesis when it is actually true (a type I error). Researchers perform a test statistic calculated from the sample data and compare it with a critical value determined by the significance level. If the test statistic exceeds the critical value, the null hypothesis is rejected, indicating that the observed effect is statistically significant.

In the given exercise, maximizing the function \(g(n_1)\) while maintaining a fixed total sample size \(n\) can be linked to the power of a test. The power of a test is the probability of correctly rejecting the null hypothesis when the alternative hypothesis is true. Higher power leads to more reliable testing outcomes, hence the importance of identifying the most powerful design.

First Order Derivative

The first order derivative of a function corresponds to the slope of the function at any given point. It is a fundamental concept in calculus that represents how the function value changes with respect to changes in the independent variable. Symbolically, if we have a function \(y = f(x)\), then its first derivative is denoted as \(f'(x)\) or \(\frac{dy}{dx}\).

Calculating the first derivative provides insights into the behavior of the function: whether the function is increasing, decreasing, or at a stationary point where the slope is zero. For instance, when a function's first order derivative is set to zero, it can help identify critical points, which potentially are maxima, minima, or inflection points.

In the context of the exercise, computing the first order derivative of the function \(g(n1)\) and setting it to zero is utilized to find the value of \(n1\) that maximizes the function under the given constraints. The step-by-step solution properly applies this method to identify that when \(n_1 = \frac{n}{2}\), \(g(n1)\) reaches its maximum.

Second Order Derivative

Moving a step further, the second order derivative tells us about the curvature of the function's graph. In mathematical terms, if \(y = f(x)\) and we know the first derivative \(f'(x)\), then the second derivative is denoted as \(f''(x)\). It describes the rate at which the first derivative is changing, providing information about the concavity of the function.

A second order derivative that is positive indicates that the function is concave upwards, suggesting a local minimum. Conversely, if the second order derivative is negative, as in the exercise, the function is concave downwards, which typically signifies a local maximum.

In the presented problem, the negative value of the second order derivative confirms that the value of \(n1\) found by setting the first derivative to zero does, indeed, correspond to a maximum. This sort of verification is crucial in optimization problems to ascertain that the critical points identified indeed lead to the expected outcome, in this case, the most powerful test design.

Quotient Rule

The quotient rule is a technique for differentiating functions that are divided by each other. In essence, if we have a function that is expressed as a quotient \(u(x) / v(x)\), the quotient rule provides us with a systematic way to find the derivative of the whole function. The rule states that the derivative of \(u(x) / v(x)\) is given by:\[\frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{u'(x) v(x) - u(x) v'(x)}{\left(v(x)\right)^2}\]
This rule is a consequence of the product rule and the chain rule of differentiation and it's important to apply it correctly to avoid mistakes.

In the exercise solution, the quotient rule is used to differentiate the function \(g(n1)\) which involves both a product and a quotient of \(n1\) and \(n\). By correctly applying the quotient rule, the first derivative \(g'(n1)\) can be found, which is a necessary step for finding the critical points and determining the design that maximizes the power of the statistical test.