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Questions and Answers
What is the primary purpose of hypothesis testing?
What is the primary purpose of hypothesis testing?
Which of the following statements describes the null hypothesis?
Which of the following statements describes the null hypothesis?
In hypothesis testing, what does rejecting the null hypothesis indicate?
In hypothesis testing, what does rejecting the null hypothesis indicate?
What role do critical values play in hypothesis testing?
What role do critical values play in hypothesis testing?
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Which error occurs when the null hypothesis is incorrectly rejected?
Which error occurs when the null hypothesis is incorrectly rejected?
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What does setting a lower alpha level accomplish in hypothesis testing?
What does setting a lower alpha level accomplish in hypothesis testing?
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Which statement is true regarding Type I and Type II errors?
Which statement is true regarding Type I and Type II errors?
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What is the most common alpha level used in hypothesis testing?
What is the most common alpha level used in hypothesis testing?
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How can one describe a Type II error?
How can one describe a Type II error?
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What type of variables is the Chi Square test most appropriate for?
What type of variables is the Chi Square test most appropriate for?
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What is the purpose of using bivariate tables in Chi Square tests?
What is the purpose of using bivariate tables in Chi Square tests?
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What dimension is represented by the rows in a bivariate table?
What dimension is represented by the rows in a bivariate table?
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What is a characteristic of Chi Square tests?
What is a characteristic of Chi Square tests?
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When discussing the critical region and sampling distribution, what does a higher alpha level imply?
When discussing the critical region and sampling distribution, what does a higher alpha level imply?
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What is the relationship between Type I and Type II errors when alpha is manipulated?
What is the relationship between Type I and Type II errors when alpha is manipulated?
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What is a key characteristic of a research hypothesis?
What is a key characteristic of a research hypothesis?
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Which statement best describes a null hypothesis?
Which statement best describes a null hypothesis?
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What might a hypothesis related to transparent laws and corruption predict?
What might a hypothesis related to transparent laws and corruption predict?
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What is the likely impact of systematic enforcement of laws on unethical behavior?
What is the likely impact of systematic enforcement of laws on unethical behavior?
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What does the notation for a research hypothesis generally indicate?
What does the notation for a research hypothesis generally indicate?
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What is an example of a potential research hypothesis regarding study groups?
What is an example of a potential research hypothesis regarding study groups?
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In hypothesis testing, what role does the alpha level play?
In hypothesis testing, what role does the alpha level play?
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Which of the following is a common mistake related to hypothesis testing?
Which of the following is a common mistake related to hypothesis testing?
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Study Notes
Quantitative Research Methods in Political Science
- Course Instructor: Michael E. Campbell
- Course Number: PSCI 2702 (A)
- Date: 10/31/2024
- Lecture 7: Hypothesis Testing for Nominal and Ordinal Variables (Chi-Square)
Hypotheses
- Hypotheses must be testable; only the most important elements should be identified.
- A testable hypothesis is a clear, measurable statement that can be supported or refuted through experimentation or observation.
- A hypothesis is a statement regarding the relationship between variables, derived from a theory. Hypotheses are more specific than theories, with all terms and concepts fully defined.
Hypotheses Cont'd
- A hypothesis is a statement about the relationship between variables.
- The statement of this relationship is derived from a theory.
- Hypotheses are more specific than theories; all terms and concepts are defined.
- A hypothesis must be straightforward (requires conceptualization/operationalization).
Developing Testable Hypothesis Example
- Research Question (RQ): Does legal enforcement affect the level of unethical behavior in a country?
- Conceptualization includes:
- Legal Enforcement: systematic laws regularly enforced.
- Unethical Behavior: abuse of private office for personal or partisan gain.
- Operationalization and variable selection needed.
Developing Testable Hypothesis Example Cont'd
- Examples of variables: systematic laws that are regularly enforced (X) and abuse of private office for personal or partisan gain (Y).
- Various versions for each variable may exist.
- Questions concerning transparency and predictability of laws need to be addressed.
- Other examples: laws of the land, executive bribery, and corruption exchanges.
Developing Testable Hypothesis Example Cont'd
- A bad hypothesis example suggests that unethical behavior is more problematic where governments enforce laws less.
- This example would be hard to test empirically.
Developing Testable Hypothesis Example Cont'd
- A testable hypothesis: "Transparent Laws with Predictable Enforcement cause lower levels of executive bribery and corruption."
- This means the higher the level of X, the lower the level of Y.
- Variables need to be clear for testing and replication of results.
Null vs. Research Hypotheses
- Researchers typically believe there will be a difference between groups or a relationship between variables.
- Research Hypothesis: Students who study in groups score higher on exams than students who study alone
- Null Hypothesis: There is no difference in exam scores between students who study in groups and students who study alone
Null vs. Research Hypotheses Cont'd
- A null hypothesis is the opposite of a research hypothesis.
- It's a statement of "no difference" or "no relationship."
- Notation for the null hypothesis is ().
Hypothesis Testing
- Hypothesis Testing is an inferential statistical procedure to test for relationships between variables or differences between groups of cases; done at the population level.
- Also called significance testing.
- The goal is to find statistically significant results, which allows rejection of the null hypothesis.
- Empirically compares samples to expectations if no relationship/difference existed.
Five-Step Model for Hypothesis Testing
- Research: any process for gathering information methodically to answer questions, examine ideas, or test theories.
- Steps:
- Make assumptions, meet test requirements.
- State the null hypothesis
- Select sampling distribution and critical region.
- Compute test statistic (obtained score).
- Make a decision and interpret results.
Step 1 - Make Assumptions and Meet Test Requirements
- Assumptions are made when testing hypotheses, depending on the techniques used.
- A constant assumption is that samples are randomly selected.
- Other assumptions include the shape of the population distribution, levels of measurement, etc.
Step 2 - State the Null Hypothesis
- You state the null hypothesis, e.g. "no difference/no relationship".
- It's the opposite of the research hypothesis.
- The goal is to reject the null hypothesis and gather empirical evidence to do so.
Step 3 - Select Sampling Distribution and Establish Critical Region
- Choose a sampling distribution based on the test (Z, t, Chi-Square, F)
- Determine the critical region (areas with unlikely sample outcomes).
- Set alpha at a specific level (e.g., 0.10, 0.05, 0.01).
- Fail to reject the null hypothesis if the sample values are not in the critical region.
Step 4 – Compute the Test Statistic
- Convert sample values into a test score (e.g., Z-score, t-score).
- The resulting score is called the 'obtained score'.
- It's compared to the critical value to determine if it falls within the critical region.
- P-value must also be considered.
The p Value
- The p-value is related but not the same as alpha ( ).
- When alpha is set, it denotes the proportion of the area in the critical sampling distribution
- The p-value is the probability of the obtained score or more extreme outcomes occurring in the sampling distribution if the null hypothesis were true.
- Use the p-value to compare with alpha to decide whether the null hypothesis is rejected.
Step 5 – Make a Decision and Interpret the Results of the Test
- The decision is to either reject or fail to reject the null hypothesis.
- Comparison is made between test statistic (obtained score) and critical value.
- Alternatively, the p-value is compared to alpha.
Type I and II Errors
- Type I error (Alpha error): incorrectly rejecting a true null hypothesis.
- Type II error (Beta error): failing to reject a false null hypothesis.
- These errors are inversely related and cannot be simultaneously minimized. Common alpha values are 0.10, 0.05, and 0.01.
Introduction to Chi Square
- Chi-square is a commonly used hypothesis test
- It can be applied to variables measured at nominal, ordinal, interval, and ratio levels
- It's Non-Parametric-- it doesn't assume anything about the distribution of the population
- Used for nominal and ordinal variables
Bivariate Tables
- Used to compute Chi-Square
- Scores of cases on two different variables are displayed simultaneously.
- Rows and columns denote variables studied.
- Cells represent combined scores from variables within rows and columns
Bivariate Table Example Cont'd
- Independent variable is positioned in columns, while the dependent in rows.
- Bivariate tables show a variety of combinations for the two variables.
The Logic of Chi Square
- Chi-square is useful for various analyses (independence, goodness of fit, homogeneity, model fit testing).
- For tests of independence, variables are considered independent when the classification of a case into a category does not impact the probability of the case falling into any category of another variable.
- Use of expected frequencies to determine if observations are likely due to random chance.
Computing Chi Square
- Produces a test statistic, (obtained).
- (obtained) is compared to a critical value from the chi-square sampling distribution
- Degrees of freedom (df) is a uniquely determined number for each sampling distribution that is based on the number of rows and columns.
- Formula for df: (rows – 1)(columns – 1).
Computing Chi Square Cont'd
- To compute test statistic, use the following formula:
- Use cell frequencies from bivariate tables.
- Expected frequencies describe what frequencies would be expected given each variable was independent.
Computing Test Statistic Example
- Demonstrates a table with employment status and accreditation status in columns.
Computing Test Statistic Example Cont'd
- Independent variable = Accreditation status
- Dependent variable = Employment status.
- Computing expected frequencies using a formula
Computing Test Statistic Example Cont'd
- Computing expected frequencies using a table
- Row and column marginals provide additional information.
Computing Test Statistic Example Cont'd
- Use the formula to determine the obtained chi-square value.
The Chi-Square Test for Independence
- Apply the Chi-Square test to data from the example of employment status and accreditation status.
Step 1 - Make Assumptions and Meet Test Requirements
- Select the random sample using EPSEM.
- Measures variables at the nominal or ordinal level.
Step 2 - State the Null Hypothesis
- The two variables are independent
- The two variables are dependent (alternative hypothesis)
- Example: Accreditation status has no impact on employment status, vs. Accreditation status impacts employment status
Step 3 - Select Sampling Distribution and Establish Critical Region
- Establish the critical region based on the chi-square distribution (positively skewed).
- Degree of freedom is determined by the sampling distribution.
- Setting alpha levels.
Using Distribution of Chi Square
- Appendix C can be used to determine a critical value given a degree of freedom and an alpha level. (e.g. degree of freedom = 1, alpha = 0.05)
Step 4 – Compute the Test Statistic
- Use cell frequencies to determine expected frequencies. (e.g. Table 7.7)
- Apply computational formula.
Step 5 - Make a Decision and Interpret the Results of the Test
- The test statistic is compared to the critical value
- Reject the null if the test statistic is greater than the critical value.
Column Percentages
- Chi-square testing doesn't provide detailed information about the relationship. Column percentages add information to further explain the relationship between the variables.
Column Percentages Cont'd
- Computing column percentages helps explain the relationship better.
Limitations of Chi Square
- Tables with too many categories are hard to interpret.
- A general rule for effective use is that both variables should have four or fewer scores.
- Sample sizes affect Chi-Square applicability. Small sample sizes decrease accuracy.
Computing Chi-Square in SPSS
- Using software like SPSS can simplify Chi-Square calculation
Chi-Square Tests
- SPSS output provides p-value and other statistics to assess degrees of freedom.
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Description
This quiz focuses on hypothesis testing for nominal and ordinal variables using Chi-Square methods. It aims to reinforce your understanding of how to develop testable hypotheses based on theoretical relationships. Prepare to explore the critical elements of crafting clear and measurable hypotheses in political science research.
