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# 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)