# Question 23

In the context of a randomised control trial comparing a trial drug with placebo:

a)  briefly explain the following terms:

• Type 1 error
• Type 2 error
• Study power
• Effect size

b)  List the factors that influence sample size.

Type 1 error
The null hypothesis is incorrectly rejected. Type 1 errors may result in the implementation of therapy that is in fact ineffective or a false positive test result.

Type 2 error
The null hypothesis is incorrectly accepted. Type 2 errors may result in rejection of effective treatment strategies or a false negative test result.

Study power
Power is equal to 1-β. Thus if β = 0.2, the power is 0.8 and the study has 80% probability of detecting a difference if one exists

Effect size
Effect size (∆) is the clinically significant difference the investigator wants to detect between the study groups. This is arbitrary but needs to be reasonable and accepted by peers. It is harder to detect a small difference than a large difference. The effect size helps us to know whether the difference observed is a difference that matters.

Factors influencing  sample size
•    Selected values for significance level, α, power β and effect size ∆ (smaller values mean larger sample size)
•    Variance /SD in the underlying population (larger variance means larger sample size)

## Discussion

The college presents a concise and effective answer to this question, which should serve as a model. Below is a non-model answer overgrown with the unnecessary fat of references and digressions.

a)

Type 1 error: The incorrect rejection of a null hypothesis.

• A false positive study.
• Finding a treatment effect where there actually is none.
• Results in the implementation of an ineffective treatment.

Type 2 error: the incorrect rejection of the alternative hypothesis.

• A false negative study.
• Finding no treatment effect, when there actually is one.
• Results in an effective treatment being wrongly discarded.

Study power: The probability that the study correctly rejects the null hypothesis, when the null hypothesis is false.

• Expressed as (1-β), where β is the probability of Type 2 error (i.e. the probability of incorrectly accepting the null hypothesis).
• Generally, the power of a study is agreed to be 80% (i.e. = 0.2), because anything less would incur too great a risk of Type 2 error, and anything more would be prohibitively expensive in terms of sample size.

Effect size: a quantitative reflection of the magnitude of a phenomenon; in this case, the magnitude of the positive effects of a drug on the study population.

• In this case, it is the difference in the incidence of an arbitrarily defined outcome between the treatment group and the placebo group.
• Effect size suggests the clinical relevance of an outcome
• The effect size is agreed upon a priori so that a sample size can be calculated (as the study needs to be powered appropriately to detect a given effect size)

Factors which influence sample size:

There is a good article on this in Radiology (2003)

• Alpha value: the level of significance (normally 0.05)
• Beta-value: the probability of incorrectly accepting the null hypothesis (normally 0.2)
• The statistical test one plans to use
• The variance of the population (the greater the variance, the larger the sample size)
• Estimated measurement variability (similar to population variance)
• The effect size (the smaller the effect size, the larger the required sample)

## References

There is an online Handbook of Biological Statistics which has an excellent overview of power analysis.

Kelley, Ken, and Kristopher J. Preacher. "On effect size." Psychological methods 17.2 (2012): 137.

Moher, David, Corinne S. Dulberg, and George A. Wells. "Statistical power, sample size, and their reporting in randomized controlled trials." Jama 272.2 (1994): 122-124.

Cohen, Jacob. "A power primer." Psychological bulletin 112.1 (1992): 155.

Dupont, William D., and Walton D. Plummer Jr. "Power and sample size calculations: a review and computer program." Controlled clinical trials 11.2 (1990): 116-128.

Eng, John. "Sample Size Estimation: How Many Individuals Should Be Studied? 1." Radiology 227.2 (2003): 309-313.