## Describe power of a test

The power of a significance test measures the ability to identify an alternative hypothesis. In statistical language, power of a test is the probability that the test will reject the null hypothesis when the alternative hypothesis is true. To calculate such a probability, we need to have information about the sampling distribution of the test statistic under alternative hypothesis.

## Explain Statistical Power of a Test Analysis

Statistical power of a test is basically the probability of not committing a Type II error. As the power of the test increases, the probability or the chances of committing Type II error decreases. And this probability of committing Type II error is called the false negative rate (β). Therefore, power of a test is then equal to 1- β. This power is often referred to as sensitivity.

 Mathematically,statistical power=P(rejecting null hypothesis|null hypothesis is false)

This power analysis can be used to compute the minimum sample size that is required to detect the effect of a given sample size. It is also used to calculate the minimum effect size in the case of a given sample size. It also helps in comparing the parametric and nonparametric tests of a same hypothesis.

## Describe the Factors Influencing Statistical power.

It has been observed that power comes into picture when there is actually a treatment effect and we are able to detect it also.

Some of the factors pertaining to statistical testing situations are stated below:

Size of the Type I error: The size of the type I error is actually a statistical significance criterion, which predominately says how unlikely the positive result is if the null hypothesis is rejected. The most commonly used significance values are 0.01, 0.05 and 0.10. If we increase the Type 1 error, it means that there are greater chances of accepting the type I error, the probability of rejecting a true null hypothesis. For instance, if the level of significance is 0.05, the probability implies that the effect must be at least as large as the observed effect; when the null hypothesis is actually true, it must be less than 0.05, for the null hypothesis of no effect to be rejected. One way to calculate the power of the test is by applying the larger significance criterion; for instance, 0.15 instead of 0.05. The increased significance value increases the chances of rejecting the null hypothesis when it is actually false, reduces the risk of committing the type II error (false negative regarding whether an effect exists).

Magnitude of the effect: The magnitude of the effect is quantified by an effect size. This means greater power is linked to detect larger effects. An effect size in turn could be a direct estimate of the quantity of interest. For instance, while comparing the values between the treatment and control population, the difference between the outcome means Y-X would then be considered as the direct measure of the effect size, whereas (Y − X)/σ where σ is the standard deviation of the output in the treatment and control group, would then be considered as the standardized effect size. Thus, an appropriate standardized effect size using the sample size helps determine the power of the test but it does not have information about the variability of those measurements.

Sample Size: Sample size helps determine the amount of sampling error present in a test result.  The power of the test depends on sample size. Everything else being equal, the larger the sample size, the higher is the power of a test. Therefore, as the sample sizes increase, within-group variability also known as the error variances decrease.

### Solved Illustration

Is a 6-month regular exercise program increasing the total body bone mineral content among young women?

Given the previous results, we will assume that σ = 2, which accounts for the percentage change in the total body bone mineral content (TBBMC) over the period of six months. However, a 1% change is considered quite significant and the researchers would like to have a reasonable chance of detecting a change this large or larger.

### Are 25 observations sufficient for this study?

STEPS

• State the hypotheses: let µ denote the mean percent change.

H0: µ = 0

Ha: µ > 0

• Determine the rejection region: The z test rejects H0 at the µ = 0:05 level whenever

Thus, we reject the null hypothesis when

• Now, compute the power at a specific alternative: the power of the test at µ =1

Therefore, the power of the test is 0.80.

Our experts have solved many questions on Power of a test. Following are some of the problems that our experts have worked on:

 Q A researcher had originally planned to conduct a study while allowing only 1% Type 1 Error. However, now he has decided to allow 5% Type 1 error.

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