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PUH 5302, Applied Biostatistics 1

Course Learning Outcomes for Unit VI Upon completion of this unit, students should be able to:

4. Formulate scientific questions into statistical hypotheses. 4.1 Distinguish between null and alternative hypothesis. 4.2 Model the steps involved in statistical hypothesis testing. 4.3 Discriminate the significance of Type I and Type II errors.

Required Unit Resources Chapter 5: Estimation and Hypothesis Testing, Sections 5.4–5.9 Unit Lesson

Introduction Welcome to Unit VI. In this unit, we will discuss the process involved in hypothesis testing. We will also examine how data are prepared for hypothesis testing, define null and research hypothesis, and describe the six-step process involved in hypothesis testing. Furthermore, when performing hypothesis testing, there is potential for reaching a wrong conclusion and we will discuss the probability of committing these types of errors.

Tests of Hypotheses’ A scientific hypothesis is a testable statement about a phenomenon or relationship that requires verification. The process of hypothesis testing starts with the researcher making two specific statements about the population sample based on available information. The two statements oppose each other with one indicating an association between the variables and the other indicating no association between the variables. Some examples of scientific hypotheses are the following:

• Age is positively associated with a person wearing a protective face covering to guard against the COVID-19 virus.

• Rates of COVID-19 are greater in lower income communities. • Drinking sugary drinks leads to being overweight. • Students who eat breakfast will perform better on a math exam than students who don’t eat breakfast.

A hypothesis test makes an assumption about the population and then probability is used to estimate the likelihood of the result obtained from the sample under the assumptions about the population. Hypothesis testing is the process of selecting between the null and alternative hypothesis. In hypothesis testing, we begin by presenting two hypotheses.

1. The NULL Hypothesis is the first and it usually states what is currently believed, expected, claimed, or has been in the past. The NULL hypothesis is assumed to be correct unless enough evidence can be found to show otherwise. Null hypothesis is also represented by H0.

2. The second is the ALTERNATIVE Hypothesis also called the research hypothesis. The researcher always aims to provide support for the alternative hypothesis. Alternative hypothesis is also represented by H1.

UNIT VI STUDY GUIDE Hypothesis Testing

PUH 5302, Applied Biostatistics 2

UNIT x STUDY GUIDE Title

Let’s look an example. Suppose it is commonly believed that 0.8 of the adult population would immediately inform the people around them if they exhibited symptoms of COVID-19. Your goal is to test whether this statement is accurate. To do so, you outline the null and alternative hypothesis based on this statement.

Solution: Null hypothesis H0: p = 0.8 and Alternative hypothesis H1: p ≠ 0.8

The Hypothesis Testing Process

The process of hypothesis testing can be broken down into six steps (Merrill, 2022): Step 1: Formulate the null hypothesis in statistical terms. The null hypothesis denoted by H0, specifies the value of a population parameter. For example:

Null hypothesis, H0: Age is positively associated with a person wearing a protective face covering to guard against the COVID-19 virus

Step 2: Formulate the alternative or research hypothesis in statistical terms. The alternative hypothesis (H1) is the statement that two variables do influence each other. It gives an opposing conjecture to that of the null hypothesis Typically, the researcher wants to prove this hypothesis. The investigator believes, from our example, that there is a statistically significant relationship between age and wearing a protective face covering among adults.

Alternative hypothesis, H1: Age is NOT positively associated with a person wearing a protective face covering to guard against the COVID-19 virus

Step 3: Select the level of significance.

The level of significance α is generally 0.05. It can also be set at 0.01 or 0.1. Step 4: Set the decision rule. Select the appropriate test statistic and identify the degrees of freedom and the critical value for rejecting the null hypothesis. Identify the acceptance or rejection statements by setting up decision rules. The p-value is the strength of evidence in support of the null hypothesis.

If the p-value is less than the significance level (α = 0.05), the null hypothesis can be rejected, and the alternative hypothesis can be accepted. If the p-value is greater than the

significance level (α = 0.05), we fail to reject the null hypothesis. Step 5: Collect the data and calculate the test statistic. A test statistic is a quantity calculated from the sample that is used when making a decision

about the hypothesis of interest. Step 6: Conclusion – reject or fail to reject the null hypothesis. The researchers will make a final conclusive statement about the test based on the results. For example:

• If the p-value is less than the significance level, or (α) is greater than the significant level, reject the null hypothesis in favor of the alternative hypothesis because this would mean that the result is significant.

• If the p-value is greater than the significance level, or (α) is less than the significant level, fail to reject the null hypothesis or accept the null hypothesis, the result is not statistically significant.

PUH 5302, Applied Biostatistics 3

UNIT x STUDY GUIDE Title

Type I and Type II Errors There are two types of errors possible in hypothesis testing that are a result of reaching a wrong conclusion. The following table shows the possible decisions and consequences in hypothesis testing.

Possible Decisions True State of the Population

H0 is true H1 is true Reject H0 Type I error Correct decision Fail to reject H0 Correct decision Type II error

From the table above, we can see that we can make a correct decision if “H0 is true” and we “Fail to reject H0” or if “H1 is true” and we “Reject H0”. However, if we “Reject H0”, when “H0 is true”, we commit a Type I error. Similarly, if we “Fail to reject H0” when in fact “H1 is true”, we commit a Type II error. 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. 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.

𝜶𝜶 = 𝐏𝐏(𝐓𝐓𝐓𝐓𝐓𝐓𝐓𝐓 𝐈𝐈 𝐓𝐓𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞) = 𝐏𝐏(𝐑𝐑𝐓𝐓𝐑𝐑𝐓𝐓𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑 𝐇𝐇𝟎𝟎 | 𝐇𝐇𝟎𝟎 𝐑𝐑𝐢𝐢 𝐑𝐑𝐞𝐞𝐭𝐭𝐓𝐓)

𝜷𝜷 = 𝐏𝐏(𝐓𝐓𝐓𝐓𝐓𝐓𝐓𝐓 𝐈𝐈𝐈𝐈 𝐓𝐓𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞) = 𝐏𝐏(𝐅𝐅𝐅𝐅𝐑𝐑𝐅𝐅𝐑𝐑𝐑𝐑𝐑𝐑 𝐑𝐑𝐞𝐞 𝐞𝐞𝐓𝐓𝐑𝐑𝐓𝐓𝐑𝐑𝐑𝐑 𝐇𝐇𝟎𝟎 | 𝐇𝐇𝟏𝟏 𝐑𝐑𝐢𝐢 𝐑𝐑𝐞𝐞𝐭𝐭𝐓𝐓)

p-value The p-value is the probability that the test statistic used to evaluate a hypothesis has a value as extreme as or more extreme than the sample statistic, given the null hypothesis is true. If the p-value is lower than the level of significance chosen by the investigator, then we say that there is sufficient evidence to reject the null hypothesis. Criteria for evaluating hypothesis based on the p-value is as follows:

𝑰𝑰𝑰𝑰 𝒑𝒑 − 𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗 < 𝜶𝜶, 𝒕𝒕𝒕𝒕𝒗𝒗𝒕𝒕 𝒓𝒓𝒗𝒗𝒓𝒓𝒗𝒗𝒓𝒓𝒕𝒕 𝒕𝒕𝒕𝒕𝒗𝒗 𝒕𝒕𝒗𝒗𝒗𝒗𝒗𝒗 𝒕𝒕𝒉𝒉𝒑𝒑𝒉𝒉𝒕𝒕𝒕𝒕𝒗𝒗𝒉𝒉𝒉𝒉𝒉𝒉,𝑯𝑯𝟎𝟎 𝑰𝑰𝑰𝑰 𝒑𝒑 − 𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗 ≥ 𝜶𝜶, 𝒕𝒕𝒕𝒕𝒗𝒗𝒕𝒕 𝒅𝒅𝒉𝒉 𝒕𝒕𝒉𝒉𝒕𝒕 𝒓𝒓𝒗𝒗𝒓𝒓𝒗𝒗𝒓𝒓𝒕𝒕 𝒕𝒕𝒕𝒕𝒗𝒗 𝒕𝒕𝒗𝒗𝒗𝒗𝒗𝒗 𝒕𝒕𝒉𝒉𝒑𝒑𝒉𝒉𝒕𝒕𝒕𝒕𝒗𝒗𝒉𝒉𝒉𝒉𝒉𝒉,𝑯𝑯𝟎𝟎

Reference

Merrill, R. M. (2022). Principles and applications of biostatistics. Jones & Bartlett Learning. https://online.vitalsource.com/#/books/9781284251159

Suggested Unit Resources In order to access the following resource, click the link below. The following article shows how different the p value is from the corresponding likelihood favoring the null hypothesis under a variety of important conditions; investigates the influence that multiple testing and the overall likelihood that the null hypothesis is true have on these differences; and pays particular attention to p values falling just under .05, the standard threshold for considering a result “statistically significant.”

PUH 5302, Applied Biostatistics 4

UNIT x STUDY GUIDE Title

Anderson, S. F. (2020). Misinterpreting p: The discrepancy between p values and the probability the null hypothesis is true, the influence of multiple testing, and implications for the replication crisis. Psychological Methods, 25(5), 596–609. https://libraryresources.columbiasouthern.edu/login?url=https://search.ebscohost.com/login.aspx?dire ct=true&db=pdh&AN=2019-74997-001&site=ehost-live&scope=site

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. You are encouraged to complete the odd-numbered exercises at the end of Chapter 5 in your eTextbook as they allow you to test your knowledge of the concepts presented in this unit. Answers to these questions may be found in Appendix A.