INTRODUCTION – Advanced Hypothesis Testing
During this module, students will increase their knowledge of hypothesis testing through the study of further important and common critical statistics such as the Chi-squared test and the analysis of variance (ANOVA). These are part and parcel of a statistician’s tool kit and give him the ability to perform tests and understand various types of data while allowing a more in-depth insight into the data. Real examination of how to apply these tests to different situations allows the participants to increase their statistics and analytical power.
The understanding of two varieties of Chi-squared tests and one and two-way ANOVA has its foundation in earlier theory-based sections. Practice, according to theory, allows participants not only to refine their understanding of statistical methods but also to make the most suited choice for application to various data analysis problems. Such thorough exposure strengthens participants in data-driven decisions and enables them to take the next steps in their evolution as data professionals.
Learning objectives:
- CHAPTER OBJECTIVES
- Define ANOVA, ANCOVA, MANOVA, and MANCOVA
Define ANCOVA, MANOVA, and MANCOVA - Conduct post hoc tests with ANOVA
- Conduct a two-way ANOVA test
- Conduct a one-way ANOVA test
- Mention when to employ an analysis of variance (ANOVA) test
- Conduct a Chi-square test of independence
- Conduct a Chi-Squared (“Chi-Squared”) goodness-of-fit test
PRACTICE QUIZ: TEST YOUR KNOWLEDGE: THE CHI-SQUARED TEST
1. The chi-squared goodness of fit test determines whether an observed categorical variable follows an expected distribution.
- True (CORRECT)
- False
Correct: The chi-squared goodness of fit test determines whether or not the data observed for a categorical variable fits into a certain expected distribution. The null hypothesis states that the variable is in accordance with the expected distribution, while the alternative hypothesis posits that the variable is not in accordance with the expected distribution.
2. Which test determines whether two categorical variables are associated with each other?
- Chi-squared test for independence (CORRECT)
- Chi-squared alternative of fit test
- Chi-squared goodness of fit test
- Chi-squared test for dependence
Correct: The chi-squared test for independence evaluates the association between two categorical variables. The null hypothesis states that the variables are independent, that is, not related. The alternative hypothesis claims that the variables are not independent and, therefore, associated with each other.
3. Fill in the blank: The chi-squared statistic equals the sum of the observed number minus the expected number, squared, divided by the _____ number.
- Observed
- Hypothesis
- Expected (CORRECT)
- predicted
Correct: The chi-squared statistic comprises a summation of squared differences between the observed and expected frequencies, each divided by expected frequency.
PRACTICE QUIZ: TEST YOUR KNOWLEDGE: ANALYSIS OF VARIANCE
1. Which of the following statements accurately describe t-tests and analyses of variance? Select all that apply.
- A t-test can test means between several groups.
- An analysis of variance test can only test the difference of mean between two groups.
- An analysis of variance test can test means between several groups. (CORRECT)
- A t-test can only test the difference of mean between two groups. (CORRECT)
Correct: T-test is for comparing the means of two groups while the ANOVA test is to test for the comparison of means among three or more groups.
2. Which of the following are analysis of variance (ANOVA) tests? Select all that apply.
- Half-way ANOVA
- Five-way ANOVA
- Two-way ANOVA (CORRECT)
- One-way ANOVA (CORRECT)
Correct: One-way ANOVA and two-way ANOVA are two types of tests under the analysis of variance (ANOVA) tests. In General, ANOVA is a statistical method used to compare the means of three or more groups in order to determine whether there are any statistically significant differences between them. One-way ANOVA compares the means of different groups based on one independent variable, while two-way ANOVA examines the effect of two independent variables on the dependent variable and any interaction between those independent variables.
3. Fill in the blank: A post hoc test performs a pairwise comparison between all available groups while controlling for the _____.
- Tukey’s HSD
- variable selection
- error rate (CORRECT)
- confidence interval
Correct: A post hoc test is that test which compares all of the groups with each other once it is shown that the ANOVA has produced a significant result. Hence it tries to control for the extra increase in Type I error (false positive) at which such multiple comparisons can lead. Because by chance alone one may reject the null hypothesis, the post hoc test corrects for this by applying correction procedures to ensure that the overall error rate is controlled and also to discount the chance of false rejections.
PRACTICE QUIZ: TEST YOUR KNOWLEDGE: ANCOVA, MANOVA, and MANCOVA
1. Which statistical technique better isolates the relationship between a single categorical variable of interest and the Y variable?
- One-way ANOVA
- Analysis of covariance (ANCOVA) (CORRECT)
- Multivariate analysis of variance (MANOVA)
- Multivariate analysis of covariance (MANCOVA)
Correct: Analysis of Covariance (ANCOVA) is the blend of analysis of variance and regression. Its main utility is to trap the relationship between a categorical independent variable and the dependent variable (Y) while controlling the influence of one or more other continuous covariates. ANCOVA controls confounding that could occur by adjusting for covariates to clear a better picture about the relationship between the main categorical variable and the dependent variable, adjusted for other influencing factors.
2. Which of the following statements accurately describe ANCOVA and linear regression? Select all that apply.
- Linear regression focuses on a continuous Y variable (CORRECT)
- ANCOVA includes covariates to gain a more clear understanding of the categorical variable. (CORRECT)
- ANCOVA allows for continuous and categorical independent variables (CORRECT)
- Linear regression helps predict the Y variable for unrecognized data. (CORRECT)
Correct: It usually employs covariates within ANCOVA as a method to clarify the relationship between a categorical variable and any outcome. In other words, linear regression is used for predicting the dependent variable (Y) for new and unseen data.
3. What is the key difference between MANCOVA and MANOVA?
- MANCOVA includes a null hypothesis.
- MANOVA has two or more continuous variables.
- MANCOVA controls for covariates. (CORRECT)
- MANOVA includes a categorical variable.
Correct: The difference between MANCOVA and MANOVA is that MANCOVA controls for covariates. Thus, if a data professional is mainly looking at a single categorical variable but wishes to consider the effects of another, then MANCOVA can be employed.
QUIZ: MODULE 4 CHALLENGE
1. Fill in the blank: The _____ determines whether an observed categorical variable follows an expected distribution.
- f-test
- bias-variance test
- chi-squared test for independence
- chi-squared goodness of fit test (CORRECT)
2. What examines the relationship between categorical variables and continuous variables?
- Explanatory variance
- Analysis of variance (CORRECT)
- Adjusted R-squared
- Loss function
3. A data analytics team at a technical support provider works to identify the expected outcome of a customer policy update. They compare the means of one continuous dependent variable based on three groups of two categorical variables. What type of test does this scenario describe?
- One-way analysis of variance
- Two-way analysis of variance (CORRECT)
- Post hoc test
- T-test
4. The post hoc test performs a pairwise comparison between all available groups while controlling for what?
- mean
- bias
- error rate (CORRECT)
- median
5. A data professional needs to answer a question about company financials. They study the relationship between categorical and continuous variables to control for the effect of variables that are unrelated to the financial question. What type of statistical technique do they use?
- Analysis of independence
- Analysis of covariance (CORRECT)
- Analysis of variance
- Analysis of regression
6. Fill in the blank: The acronym MANOVA means _____ analysis of variance.
- Mean
- model
- multiple
- multivariate (CORRECT)
7. A data analyst wants to evaluate the effectiveness of different exercise programs on memory and fitness levels in elderly test subjects, controlling for age. She has two continuous dependent variables: memory score and fitness score. Her independent variable is the exercise program, which can be yoga, tai chi, or swimming. What type of test should she use?
- MANCOVA (CORRECT)
- MANOVA
- ANOVA
- ANCOVA
8. What is the group of statistical techniques that test the difference of means between three or more groups?
- Analysis of variance (CORRECT)
- Interactions of variance
- Linearity of variance
- Variance of selections
9. A data professional at an online retailer wants to understand the expected outcome of an upcoming sale. They perform a test that compares the means of one continuous dependent variable based on five groups of two categorical variables. What type of test does this scenario describe?
- One-way analysis of variance
- Two-way analysis of variance (CORRECT)
- Post hoc test
- T-test
10. What test performs a pairwise comparison between all available groups while controlling for the error rate?
- Bias-variance test
- Post hoc test (CORRECT)
- Analysis of variance test
- Chi-squared test
11. A data professional at an automotive manufacturer is asked to find a solution to a common manufacturing defect. They research the relationship between categorical and continuous variables to ensure all variables are relevant to the specific defect. What type of statistical technique do they use?
- Analysis of covariance (CORRECT)
- Analysis of variance
- Analysis of independence
- Analysis of regression
12. A data professional compares how two or more continuous variables vary according to categorical independent variables. What statistical technique are they using?
- Analysis of variance
- Analysis of variables
- Multivariate analysis of variance (CORRECT)
- Mean analysis of variables
13. Fill in the blank: The chi-squared goodness of fit test determines whether an observed _____ variable follows an expected distribution.
- continuous
- absolute
- dependent
- categorical (CORRECT)
14. A data analytics team wants to solve a problem about employee retention. They study the relationship between categorical and continuous variables to ensure all variables are relevant to the retention issues. What type of statistical technique do they use?
- Analysis of independence
- Analysis of regression
- Analysis of covariance (CORRECT)
- Analysis of variance
15. Fill in the blank: When using _____, the independent variables must be categorical and the outcome variables must be continuous.
- analysis of variance
- multiple analysis of variables
- multivariate analysis of variance (CORRECT)
- analysis of variables
16. A researcher wants to evaluate the effectiveness of different job training programs on various skill outcomes. She has two continuous dependent variables: a technical skills score and a soft skills score. Her independent variable is the training program, which can be either in-person instruction or online instruction. What type of analysis should she use?
- MANOVA (CORRECT)
- ANCOVA
- MANCOVA
- ANOVA
17. A statistician wants to determine if weight loss differs significantly based on certain diets. His dependent variable is amount of weight lost (in kgs), and his independent variable is diet (vegan, low-carb, or omnivore). Which statistical test is most appropriate?
- MANCOVA
- 2-way ANOVA
- 1-way ANOVA (CORRECT)
- MANOVA
18. Fill in the blank: Analysis of variance examines the relationship between _____.
- categorical and continuous variables (CORRECT)
- dependent and independent variables
- null and alternative variables
- initial and second hypothesis variables
19. Fill in the blank: The chi-squared _____ of fit test determines whether an observed categorical variable follows an expected distribution.
- Goodness (CORRECT)
- variance
- bias
- independence
20. A junior data analyst at a fabric supplier works to identify the expected outcome of a new product introduction. They compare the means of one continuous dependent variable based on four groups of two categorical variables. What type of test does this scenario describe?
- One-way analysis of variance
- Post hoc test
- T-test
- Two-way analysis of variance (CORRECT)
21. Fill in the blank: The chi-squared test for independence determines whether _____ categorical variables are associated with each other.
- two or more
- any number of
- three
- two
Correct: A chi-squared test of independence tests whether a relationship or association exists between two categorical variables.
22. Fill in the blank: Analysis of variance is a group of statistical techniques that test the difference of means between _____ groups.
- three
- an infinite number of
- three or more (CORRECT)
- two
Correct: ANOVA, also known as Analysis of Variance, is actually a statistical technique to compare means from three or more populations. t-tests are statistical procedures specifically designed for two populations; therefore, one can easily say that ANOVA is the generalization of t-tests in handling more than two groups.
23. Covariates are the variables that are directly relevant to the question to be answered in an analysis of covariance test.
- True
- False (CORRECT)
Correct: Covariates are variables that, although not the main focus of the analysis, could influence the response. Analysis of covariance (ANCOVA) is a statistical method that compares differences in means for three or more groups while controlling for the effects of their covariates so that they can represent a more accurate evaluation of the different impacts of the primary measured variables.
CONCLUSION – Advanced Hypothesis Testing
This finalizes that this part introduces the participant to an advanced understanding of statistical hypothesis testing with its main areas of focus on Chi-squared testing and analysis of variance (ANOVA) tests. An understanding of such data comes along with the capability of applying complex statistical tests for meaningful outputs.
Conducting such tests on practical situations will further hone a participant’s statistical powers into the capability of making informed, data-driven choices in real-world scenarios. Such breadth of exploration further makes the data person more well-rounded in skills that are important for effective contributions to the data interpretation and decision-making exercises as well.