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Think about the term national census. Why is it important? Does it   have an effect on your life? Do you see any association between the national   census and the contents of this lesson? Explain.

The journal entry must be at least 300 words in length. APA format   required.

See attached study guide for reference materials.

PUH 5302, Applied Biostatistics 1

Course Learning Outcomes for Unit VI

Upon completion of this unit, students should be able to:

6. Summarize the major principles for determining sample size and power. 6.1 Explain how margin of error, effect size, and variability of outcome affect sample size

computations. 6.2 Develop a scholarly paper that compares different types of sample sizes.

Course/Unit Learning Outcomes

Learning Activity

6.1 Unit Lesson Chapter 8 Unit VI Scholarly Activity

6.2 Unit Lesson Chapter 8 Unit VI Scholarly Activity

Chapter 8: Power and Sample Size Determination

Unit Lesson

Welcome to Unit VI. In the previous unit, we discussed the process of data preparation and hypothesis testing. We also identified and defined the null and research hypotheses and their critical role in hypothesis testing.

In this unit, you will demonstrate our understanding of the major principles for determining sample size and power by computing margin of error, effect size, and variability of outcome affect size. Finally, you will be required to develop a scholarly paper comparing different types of sample sizes.

Power and Sample Size Determination

Power and sample size determinations are important in statistical research. Error in sample size calculations or selecting the right sample size will negatively impact the results of a study. A researcher would not want to analyze a biased sample or data as the results may not reflect the entire population. Another important element in this discussion is that of Type I and Type II errors. We briefly discussed Type I and Type II errors in the previous lesson.

Let’s consider this example; suppose we need to decide whether we should apply a given condition such as a medical treatment to a group of people. We will randomly select a group of people, apply the treatment to them, and then note their responses to the treatment. To analytically test our treatment, we come up with the following hypotheses:

H0: The mean treatment of all persons in the trial is not equal (null hypothesis).

H1: The mean treatment of all person in the trial is equal (research hypothesis).

The significant thing we want to know is the number of people (sample) needed for this treatment. In order for us to answer this question, the concept of Type I error and Type II error must be given serious attention.

UNIT VI STUDY GUIDE

Power and Sample Size Determination

PUH 5302, Applied Biostatistics 2

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A Type I error happens when we wrongly reject the null hypothesis when it is true, and Type II error happens when we wrongly accept the null hypothesis when it is false (Sullivan, 2018). The concept of power comes into play. The power of a statistical procedure is its ability to show that the null hypothesis is false when it is really false, or that the null hypothesis is true when it is true. In research, we should be cognizant of the fact that the sample size is related to the desired Type I and Type II error rates. Type I error rate is usually set at 5%, while Type II error rate at β, and the power is set at (1 − β). It is important to note that a small sample size gives us little power to reject the null hypothesis, whereas a large sample size gives us more statistical power to do the same. Often, researchers prefer larger sample sizes because it gives more power to reject or accept the null hypothesis. In doing so, statisticians prefer to use universally accepted standards of measurements such as 5% and at (1 − β) for Type I and Type II error calculations respectively. Power When a test is conducted, and the results rightfully show rejection of the null hypothesis when it is wrong, or accepts the alternative hypothesis when it is right, it indicates a strong power. The range of the statistical power is from zero to one. The power and likelihood of making an error are related; as the power increases, the likelihood of making an error is decreased.

Power analysis can be used to compare different testing procedures, and it can also be used to calculate the following elements:

 the minimum sample size required to detect a size effect;

 the minimum effect size that would be identified in a study with a particular sample size; and

 statistical sensitivity, or the measure of how likely a given test gives the correct result. Power analysis can be done before data are collected. This is often referred to as a priori power analysis. It can also be done after data are collected, which is known as post hoc power analysis. Post-hoc analysis is conducted after the completion of the study, and it is considered flawed most of the time, which makes it of less use. Issues in Estimating Sample Size In calculating sample sizes, there are other factors to consider aside from power analysis and Type 1 and Type II errors. These other factors include the following. Confidence interval (CI): We discussed confidence interval in the previous lessons extensively. CI expresses the range within which a result for the whole population would occur for a particular proportion of times an experiment or test was repeated in a sample of the population. Confidence interval is a standard way of articulating the statistical accuracy of an experiment-based assessment. Confidence interval can be calculated from the formula below: Confidence intervals = point estimate ± margin of error, as illustrated below:

𝜌 ± 1.96 × √ 𝜌(1 − 𝜌)

𝑛

For example, if you use a confidence interval of 5 and 30% percent of your sample say “yes” to an answer, it is likely that if that question is asked among the relevant population between 25% (30-5) and 35% (30+5) would answer “yes.” Confidence level has the same principle as confidence interval. It tells the researchers how confident they are or should be about the population. For example, a 95% or 99% confidence level means there is 95% or 99% chance that the selection is right.

PUH 5302, Applied Biostatistics 3

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Factors that affect the confidence interval include:

 sample sizes,

 percentages, and

 population sizes. Margin of error (ME): The general formula for the margin of error for a sample proportion (if certain conditions are met) is shown below:

𝑧∗ √ �̂� (1 − �̂�)

𝑛

Where �̂� = is the sample proportion, n is the sample size, and z* is the appropriate z*-value for your desired level of confidence . Here are the steps for calculating the margin of error for a sample proportion.

1. Find the sample size, n, and the sample proportion. 2. The sample proportion P = is the number in the sample with the characteristic of

interest, divided by n. 3. Multiply the sample proportion by 1 – P. 4. Divide the result by n. 5. Take the square root of the calculated value. 6. You now have the standard error.

√ �̂� (1 − �̂�)

𝑛

7. Multiply the result by the appropriate z*-value for the confidence level desired.

Calculation of Sample Size The standard score (z-score) is a useful statistic because it helps to calculate the probability of a score occurring within our normal distribution, and it allows the researcher to compare two scores that come from different normal distributions. Z–scores are standardized scores in a normal distribution. The z-scores for most common confidence levels used in sample size calculations are shown below (Smith, 2018).

 90% z-score = 1.645

 95% z-score = 1.96

 99% z-score = 2.576 The formula for sample size = (z-score)2 x StdDev x (1 – StdDev) / (margin of error)2 Size Effect

Size effect is the estimated difference between the groups under investigation in a sample. In order for the researcher to detect a difference with a specified power, a smaller effect size will require a larger sample size. To determine size effect, you must divide the difference between two independent groups by the standard deviation of one group (Sullivan, 2018). It can also be determined by using an effect size calculator software. Results are interpreted using Cohen’s h.

PUH 5302, Applied Biostatistics 4

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Cohen’s h standard sizes are as follows (Cohen, 1988).

 < 0.1 = trivial effect

 0.1 – 0.3 = small effect

 0.3 – 0.5 = moderate effect

 > 0.5 = large difference effect Cohen’s h can be used for the following:

 to determine if the difference between two proportions is "meaningful,”

 in calculating the sample size for a future study, and

 to describe the difference between two proportions as large, medium, or small (Cohen, 1988). Comparing Sample Sizes In biostatistical research studies, it is vital that the researcher makes sure that adequate sample sizes are obtained. This is because you want the results of your study to reflect the population under study. A very small or large sample may affect the results of the study. An adequate sample is always the right choice. In statistics, many tests have been designed to compare two or multiple samples using different parameters or tests. These tests can be done either by using parametric or nonparametric testing procedures. You will be required to compare two articles with two different sample sizes and determine if the sample sizes were adequate and what effect a small or larger sample would have had on the study. Conclusion In most cases, public health research requires the use of research subjects who are selected by the process of sampling. The researcher is under the obligation to select the most appropriate sample and deploy the necessary testing to obtain results. In the lesson, we have highlighted some of the critical factors that affect such testing and the importance for selecting certain standard scores and other parameters to enhance the study.

References Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence

Erlbaum Associates. Smith, S. (2018, January 31). Determining sample size: How to ensure you get the correct sample size [Blog

post]. Retrieved from https://www.qualtrics.com/blog/determining-sample-size/ Sullivan, L. M. (2018). Essentials of biostatistics in public health (3rd ed.). Burlington, MA: Jones & Bartlett

Learning.

Learning Activities (Nongraded) Nongraded Learning Activities are provided to aid students in their course of study. You do not have to submit them. If you have questions, contact your instructor for further guidance and information. Develop a two- to three-page paper to identify issues estimating the sample size for the following items:

 confidence interval estimates,

 sample sizes for two interdependent samples, and

 hypothesis testing.