# Concepts Related to Hypothesis Tests

### 1 Review of concepts related to hypothesis tests

#### 1.1 Type I and Type II errors

In hypothesis testing, there are two types of errors

**Type I error:**reject null hypothesis when it is true- Type I error rate

P(reject \(H_0\) | \(H_0\) true)

- When testing \(H_0\) at a pre-specified
**level of significance**\(\alpha\), the Type I error rate is controlled to be no larger than \(\alpha\). **Type II error:**accept the null hypothesis when it is wrong.- Type II error rate

P(accept \(H_0\) | \(H_0\) wrong).

**Power :**probability of rejecting \(H_0\) when it is wrong

Power = P(reject \(H_0\) | \(H_0\) wrong)

= 1 - Type II error rate.

#### 1.2 What determines the power?

The power of a testing procedure depends on

- Significance level \(\alpha\) - the maximum allowable Type I error - the larger \(\alpha\) is , the higher is the power.
- Deviation from \(H_0\) - the strength of signal - the larger the deviation is, the higher is the power.
- Sample size: the larger the sample size is, the higher is the power.

### 2 Power of an F-test

#### 2.1 Power calculation for F-test

Test \(H_0\) : \(\mu_1\) = \(\cdots\) = \(\mu_r\) under a single factor ANOVA model: given the significance level \(\alpha\) :

- Decision rule

$$\left\{\begin{array}{ccc}{\rm reject} H_0 & if & F^{\ast}> F(1-\alpha;r-1,n_T-r)\\{\rm accept} H_0 & if & F^{\ast} \leq F(1-\alpha;r-1,n_T-r)\end{array}\right.$$

- The Type I error rate is at most \(\alpha\).
- Power depends on the
**noncentrality parameter**

$$ \phi=\frac{1}{\sigma}\sqrt{\frac{\sum_{i=1}^r n_i(\mu_i-\mu_{\cdot})^2}{r}}.$$

Note \(\phi\) depends on sample size (determined by the \(n_i\)'s) and signal size (determined by the \((\mu_i - \mu.)^2\)'s).

#### 2.2 Distribution of F-ratio under the alternative hypothesis

The distribution of F* under an alternative hypothesis.

- When the noncentrality parameter is \(\phi\), then

$$ F^{\ast} \sim F_{r-1,n_T-r}(\phi), $$

i.e., a noncentral F-distribution with noncentrality parameter \(\phi\).

- Power = P(\(\sim F_{r-1,n_T-r}(\phi)\) > F(1 - \(\alpha\);r - 1, \(n_T - r\))).
- Example: if \(\alpha\) = 0.01, r = 4, \(n_T\) = 20 and \(\phi\) = 2, then Power = 0.61. (Use Table B.11 of the textbook.)

#### 2.3 How to calculate power of the F test using R

- The textbook defines the noncentrality parameter for a single factor ANOVA model as

$$ \phi = \frac{1}{\sigma} \sqrt{\frac{\sum_{i=1}^r n_i (\mu_i - \mu_{\cdot})^2}{r}} $$

where r is number of treatment group (factor levels), \(\mu_i\)'s are the factor level means, \(n_i\) is the sample size (number of replicates) corresponding to the i-th treatment group, and \(\sigma^2\) is the variance of the measurements.

- For a balanced design, i.e., when \(n_1\) = \(\cdots\) = \(n_r\) = n, the formula for \(\phi\) reduces to

$$ \phi = \frac{1}{\sigma} \sqrt{(n/r) \sum_{i=1}^r (\mu_i - \mu_{\cdot})^2}~. $$

Table B.11 gives the power of the F test given the values of the numerator degree of freedom \(v_1\) = r - 1, denominator degree of freedom \(v_2\) = \(n_T - r\), level of significance \(\alpha\) and noncentrality parameter \(\phi\).

- Example: For r = 3, n = 5, (so that \(v_1\) = 2 and \(v_2\) = 12), \(\alpha\) = 0.05 and \(\phi\) = 2, the value of power from Table B.11 is 0.78.

However, if you want to use R to compute the power of the F-test, you need to be aware that the noncentrality parameter for F distribution in R is defined differently. Indeed, compared to the above setting, the noncentrality parameter to used in the function in R will be r x \(\phi^2\) instead of \(\phi\). Here is the R code to be used for computing the power in the example described above: r = 3, n = 5, \(\alpha\) = 0.05 and \(\phi\) = 2:

- Critical value for the F-test when \(\alpha\) = 0.05, \(v_i\) = r - 1 = 2 and \(v_2\) = \(n_T\) - r = 12 is

F.crit = qf(0.95,2,12)

- Then the power of the test, when will be computed as

F.power = 1 - pf(F.crit, 2, 12, 3\(*\)2^2)

- Note that the function qf is used to compute the quantile of the central F distribution. Its second and third arguments are the numerator and denominator degrees of freedom of the F distribution.
- The function pf is used to calculate the probability under the noncentral F- density curve to the left of a given value (in this case F.crit). Its second and third arguments are the numerator and denominator degrees of freedom of the F distribution, while the fourth argument is the noncentrality parameter r x \(\phi^2\) (we specify this explicitly in the above example).
- The values of F.crit and F.power are 3.885294 and 0.7827158, respectively.

### 3 Calculating sample size

God: find the smallest sample size needed to achieve

- a pre-specified
**power**\(\gamma\); - with a pre-specified
**Type I error rate**\(\alpha\); - for at least a pre-specifiec
**signal leval**s.

The idea behind the sample size calculation is as follows:

- On one hand, we want the sample size to be large enough to detect practically important deviations ( with a signal size to be at least s) from \(H_0\) with high probability (with a power at least \(\gamma\)), and we only allow for a pre-specified low level of Type I error rate (at most \(\alpha\)) when there is no signal.
- On the other hand, the sample size should not be unnecessarily large such that the cost of the study is too high.

#### 3.1 An example of sample size calculation

- For a single factor study with 4 levels and assuming a balanced design, i.e., the \(n_1 = n_2 = n_3 = n_4\) (=n, say), the goal is to test \(H_0\): all the factor level means \(\mu_i\) are the same.
- Question: What should be the sample size for each treatment group under a balanced design, such that the F-test can achieve \(\gamma\) = 0.85 power with at most \(\alpha\) = 0.05 Type I error rate when the deviation from \(H_0\) has at least \(s=\sum_{i=1}^{r}(\mu_i-\mu_{\cdot})^2=40\) ?
- One additional piece of information needed in order to answer this question is the residual variance \(\sigma^2\).
- Suppose from a pilot study, we know the residual variance is about \(\sigma^2\) = 10.
- Use a
**trial-and-error strategy**to search Table B.11. This means, for a given n (starting with n = 1),

(i) calculate \(\phi = (1/\sigma) \sqrt{(n/r)\sum_{i=1}^r(\mu_i - \mu_{\cdot})^2} = (1/\sigma) \sqrt{(n/r) s}\);

(ii) fix the numerator degree of freedom \(v_1\) = r - 1 = 3;

(iii) check the power of the test when the denominator degree of freedom \(v_2 = n_T - r\) (where \(n_T\) = nr), with the given \(\phi\) and \(\alpha\) ;

(iv) keep increasing n until the power of the test is closest to (equal or just above) the given value of \(\gamma\).

#### 3.2 An alternative approach to sample size calculation

Suppose that we want to determine the minimum sample size required to attain a certain power of the test subject to a specified value of the maximum discrepancy among the factor level means. In other words, we want the test to attain power \(\gamma\) (= 1 - \(\beta\), where \(\beta\) is the probability of Type II error) when the minimum range of the treatment group means

\[ \Delta = \max_{1\leq i \leq r} \mu_i - \min_{1\leq i \leq r}\mu_i ~. \]

- Suppose we have a balanced design, i.e., \(n_1 = \cdots = n_r\) = n, say. We want to determine the minimum value of n such that the power of the F test for testing \(H_0\) : \(\mu_1 = \cdots = \mu_r\) is at least a prespecified value \(\gamma = 1 - \beta\).
- We need to also specify the level of significance \(\alpha\) and the standard deviation of the measurements \(\sigma\).
- Table B.12 gives the minimum value of n needed to attain a given power 1 - \(\beta\) for a given value of \(\alpha\), for a given number of treatments r and a given "effect size" \(\Delta/\sigma\).
- Example : For r = 4, \(\alpha\) = 0.05, in order that the F-test achieves the power 1 - \(\beta\) = 0.9 when the effect size is \(\Delta/\sigma\) = 1.5, we need n to be at least 14. That is, we need a balanced design with at least 14 experimental units in each treatment group.

### Contributors

- Yingwen Li (UCD)
- Debashis Paul (UCD)