Common statistical tests and applications in epidemiological literature
- 1. Common Statistical Tests and Applications in Epidemiological Literature Dr. Mohammed Jawad
- 2. Introduction Any individual in the medical field will, at some point, encounter instances when epidemiological methods and statistics will be valuable tools in addressing research questions of interest. Examples of such questions might include: Will treatment with a new anti-hypertensive drug significantly lower mean systolic blood pressure? Is a visit with a social worker, in addition to regular medical visits, associated with greater satisfaction of care for cancer patients as compared to those who only have regular medical visits?
- 3. Introduction • There are several steps in evaluating data before addressing the above questions. These steps include description of your data as well as determining what the appropriate tests are for your data.
- 4. Description of data • The type of data one has determines the statistical procedures that are utilized. • Data are typically described in several ways: by type, distribution, location and variation. • There are three different types of data: nominal, ordinal, and continuous data.
- 5. Description of data • Nominal data do not have an established order or rank and contain a finite number of values. Gender and race are examples of nominal data.
- 6. Description of data • Ordinal data have a limited number of values between which no other possible values exist. • Number of children and stage of disease are good examples of ordinal data. • It should be noted that ordinal data do not have to have evenly spaced values as occurs with continuous data, however, there is an implied underlying order. Since both ordinal and nominal data have a finite number of possible values, they are also referred to as discrete data.
- 7. Description of data • The last type of data is continuous data which are characterized by having an infinite number of evenly spaced values. • Blood pressure and age fall into this category. • It should be noted for data collection and analysis that continuous, ordinal, or nominal values can be grouped. • Grouped data are often referred to as categorical. • Possible categories might include low, medium, high, or those representing a numerical range.
- 8. Description of data • A second characteristic of data description, distribution, refers to the frequencies or probabilities with which values occur within our population. • Discrete data are often represented graphically with bar graphs like the one in (Figure 1).
- 9. Description of data • Continuous data are commonly assumed to have a symmetric, bell-shaped curve as shown in (Figure 2). This is known as a Gaussian distribution, the most assumed distribution in statistical analysis.
- 10. Hypothesis testing • Hypothesis testing, also known as statistical inference or significance testing, involves testing a specified hypothesized condition for a population’s parameter. • This condition is best described as the null hypothesis. For example, in a clinical trial of a new anti-hypertensive drug, the null hypothesis would state that there is no difference in effect when comparing the new drug to the current standard treatment.
- 11. Hypothesis testing • Contrary to the null is the alternative hypothesis, which generally defines the possible values for a parameter of interest. • For the previous example, the alternative hypothesis is that there is a difference in the mean blood pressure of the standard treatment and new drug group following therapy. • The alternative hypothesis might also be described as your "best guess" as to what the values are.
- 12. Hypothesis testing • In statistical analysis, the null hypothesis is the main interest, and is the one being tested. • In statistical testing, we assume that the null hypothesis is correct and determine how likely we are to have obtained the sample (or values) we actually obtained in our study under the condition of the null. • If we determine that the probability of obtaining the sample, we observed is sufficiently small, then we can reject the null hypothesis. • Since we can reject the null hypothesis, we have evidence that the alternative hypothesis may be true.
- 13. Hypothesis testing • If the probability of obtaining our study results is not small, we fail to reject the assumption that the null hypothesis is true. • It should be noted that we are not concluding that the null is true. • This is a small, but important distinction. A test that fails to reject the null hypothesis should be considered inconclusive. An example will help to illustrate this point.
- 14. Hypothesis testing • In a sealed bag, we have 100 blue marbles and 20 red marbles. (This bag is essentially representing the entire population). • One individual formulates the null hypothesis that “all the marbles are blue”, and the alternative which is “all the marbles are not blue”. • To test this hypothesis, 10 marbles are sampled from the bag. • All ten marbles selected are indeed blue. Thus the individual has failed to reject the null that all the marbles in the bag are blue. However, because all the marbles were not sampled, you cannot conclude that all the marbles in the bag are blue. (We happen to know this is not true, but it is impossible to know in the real world with populations too large to fully evaluate). • If another individual selects 10 marbles from the bag and finds that 8 are blue and 2 are red, we can reject the null hypothesis that all the marbles are blue since we have selected at least one red marble.
- 15. Error in statistical testing • We can reject the null hypothesis if the probability of obtaining a sample like the one observed in our study is sufficiently small. • You may ask “What is sufficiently small?” “How small” is determined by how willing we are to reject the null hypothesis when it accurately reflects the population from which it is sampled. • This type of error is called a Type I error. This error is also commonly called alpha (α).
- 16. Error in statistical testing • Alpha is the probability of rejecting the null hypothesis when the null is true. • This probability is selected by the researcher and is typically set at 0.05. • It is important to remember that this is an arbitrary cut-point and should be taken into consideration when making conclusions about the results of the study.
- 17. Error in statistical testing • There is a second type of error that can be made during statistical testing. It is known as Type II error, which is the probability of not rejecting the null when the alternative hypothesis is indeed true, or in other words, failing to reject the null when the null hypothesis is false. • Type II error is commonly known as β.
- 18. Error in statistical testing • Beta relates to another important parameter in statistical testing which is power. • Power is equal to (1-β) and is essentially the ability to avoid making a type II error. • Like α, power is also defined by the researcher, and is typically set at 0.80. • Below is a schematic of the relationships between α, β and power.
- 19. Students’ T test • This test is most commonly used to test the difference between the means of the dependent variables of two groups. For example, this test would be appropriate if one wanted to evaluate whether or not a new anti-hypertensive drug reduces mean systolic blood pressure.
- 29. Chi-square analysis • What happens if we don't have continuous data, and are faced with categorical data instead? We could turn to chi-square analysis to evaluate if there are significant associations between a given exposure and outcome (the row and column variables in a contingency table). 2 X 2 contingency tables are one of the most common ways to present categorical data, and we can see this in analyzing data that was collected to address the question presented in this notebook.
- 30. Chi-square analysis • Is a visit with a social worker, in addition to regular medical visits, associated with greater satisfaction of care for cancer patients as compared to those who only have regular medical visits? • Below is a generic 2 X 2 table representing the data. It is important to note the set-up of the table, as cell “a” generally represents the group of interest (diseased and exposed) and cell d represents the referent group (no disease and unexposed).
- 32. Here we have the contingency table with data from our trial:
- 33. Chi-square analysis • In chi-square analysis we are testing the null hypothesis that there is no association between a social worker visit and a greater satisfaction with care. • Generally, in evaluating this type of data, it is important for each of the individual cells to have large values, (i.e. greater than 5 or 10 each). • If these conditions are not met, a special type of chi-square analysis is conducted called the Fisher’s exact test.
- 34. Chi-square analysis • with i representing the frequency in a particular cell of the • 2 X 2 table. Below is the calculation for the frequencies that are expected in each cell.
- 36. Thus, we now have a table that has both the actual and expected (in parentheses) values:
- 38. Chi-square analysis • The chi-square statistic for these data has approximately 1 degree of freedom, an α of 0.05, and it is compared to the critical values on standard Chi-square table. • Note that the degrees of freedom would increase as the number of rows and columns of our tables increases (for instance a 3 X 4 table). • Since our calculated value (χ2 = 6.545) is greater than the critical value (3.841), we can once again reject the null hypothesis that there is no association between the exposure and the outcome of interest, and conclude that in this case seeing a social worker is significantly associated with a greater satisfaction with care.
- 39. Practice Questions Researchers are conducting a study of the association between working in a noisy job environment and hearing loss. The researchers’ null hypothesis is that there is no difference in hearing loss between people who work in a noisy job environment compared with people who work in a quiet job environment. The researchers’ alternative hypothesis is that there is a difference in hearing loss between people who work in a noisy job environment compared with people who work in a quiet job environment. The researchers decided to set their alpha level at 0.05. The researchers’ analysis results show a p-value of 0.0003 (please note that for the purposes of this question you are being provided with just the p-value from the study when in reality a study analysis is much more complex).
- 40. Practice Questions 1. True or false: The alpha level of 0.05 is an arbitrary value. 2. True or false: Based on the results, the researchers can conclude their null hypothesis is true. 3. True or false: Based on these results, the researchers should reject the assumption that their null hypothesis is true. 4. True or false: An alpha level of 0.05 means there is a 0.05 percent chance that the researchers will incorrectly reject the null hypothesis.
- 41. Reference Alexander LK, Lopes B, Ricchetti-Masterson K, Yeatts KB. Common Statistical Tests and Applications in Epidemiological Literature. Epidemiologic Research and Information Center (ERIC) Notebook. Second Edition. 2015