Week 6 Reflection And Case Study: Chapter 9 Inference For Tw

Week6 Reflection and Case Study: Chapter 9 Inference for Two-Way Tables

Cleaned Assignment Instructions:

Discuss the use of two-way tables in general and relate to the results obtained for the HMO study using proportions. The HMO study is repeated here for reference. A study was designed to find reasons why patients leave a health maintenance organization (HMO). Patients were classified as to whether or not they had filed a complaint with the HMO. We want to compare the proportion of complainers who leave the HMO with the proportion of those who do not file complaints but who also leave the HMO.

In the year of the study, 639 patients filed complaints, and 54 of these patients left the HMO voluntarily. For comparison, the HMO chose an SRS of 743 patients who had not filed complaints. Twenty-two of these patients left voluntarily. Previously we looked at this business problem using proportions. Discuss how to use the Chi-squared distribution and present specific results comparing to the results obtained using proportions.

Case study: To design effective marketing strategy you need to know your customers. What are the characteristics of people who use the World Wide Web to collect information on travel, and how do they differ from those who do not and use other sources? A survey that collected data to address this question examined the responses of 1401 Web users (WWW) and 1080 people who used other sources for this information (no, other category).The following tables (Week06CaseStudy.xls) give counts of WWW (WWW Yes) and Other (WWW no) for various demographic characteristics. Note that the marginal sums are sometimes less than 1401 and 1080 because of missing data. Use the methods of this chapter to compare the two groups (WWW Yes and Other, WWW No).

Include graphical and numerical summaries along with the results of your significance tests. In some cases you may want to combine some categories for the demographic variables. Be sure to include a discussion of missing values. Write a report summarizing your work. Available data by demographic category in file Week06CaseStudy.xls covers:

  • Age in years
  • Gender
  • Education
  • Occupational category
  • Household income (U.S. $)
  • Race

Use the data to perform analyses comparing Web users and non-Web users across these categories, interpret the results of chi-square tests of independence, discuss the implications, and consider the effect of missing data.

Paper For Above instruction

The utilization of two-way tables in statistical analysis constitutes a fundamental approach to understanding the relationship between categorical variables. These tables enable scholars and practitioners to examine how two categorical variables interact, identify potential associations, and infer whether observed relationships are statistically significant. In the context of the Health Maintenance Organization (HMO) study, two-way tables served as a vital tool for comparing the proportions of patients leaving the HMO, based on whether they had filed complaints. Specifically, the study aimed to analyze whether patients who filed complaints were more likely to leave voluntarily compared to those who did not file complaints. The key advantage of using two-way tables in this setting lies in their ability to organize raw data into an accessible format that highlights joint distributions and marginal totals, thereby facilitating subsequent inferential tests such as the Chi-squared test of independence.

Initially, the raw data indicated that 639 patients filed complaints, among whom 54 left the HMO voluntarily. Conversely, among the 743 patients who did not file complaints, 22 left voluntarily. These data points allowed for the calculation of proportions, which provide a straightforward measure of the likelihood of leaving the HMO within each complaint category. The proportion of complainers who left was approximately 8.45% (54/639), whereas the proportion of non-complainers who left was approximately 2.96% (22/743). A comparison of these proportions suggested that complaint filing might be associated with a higher likelihood of leaving the HMO. While proportions aid in easy interpretation, they do not directly quantify the strength of the association or allow for testing the significance of this relationship. For such purposes, statistical inference via the Chi-squared test is appropriate.

The Chi-squared test assesses whether there is a significant association between complaint filing and voluntary departure by examining the discrepancy between observed counts and expected counts under the assumption of independence. To perform this test, the counts are arranged in a contingency table, and the Chi-squared statistic is computed based on the differences between observed and expected frequencies across all cells. A significant Chi-squared result (typically p-value

Applying the Chi-squared test to the HMO data involves calculating expected counts for each cell, assuming no association exists, and then summing the squared deviations divided by expected counts. This process results in a Chi-squared statistic that can be compared against the Chi-squared distribution with the appropriate degrees of freedom (in this case, 1). Given the data, the test likely reveals a significant association, confirming that filing a complaint is related to a higher rate of voluntary departure. The advantage of using the Chi-squared approach over simply comparing proportions is that it accounts for the entire contingency table simultaneously and provides a formal measure of statistical significance.

Turning to the case study analyzing web-based versus other sources of travel information, the approach involves examining the demographic characteristics of users (Web users) and non-users through two-way tables. This analysis aims to identify whether demographic factors such as age, gender, education, occupation, income, and race are associated with the mode of information source. First, data are summarized graphically and numerically, with tables presenting counts across categories. For example, an analysis of age might compare the number of Web users under 25 versus those over 55, and similarly for non-Web users.

Next, chi-square tests of independence are conducted for each demographic variable to determine whether there are statistically significant differences in distribution between Web users and non-users. The null hypothesis assumes that the demographic attribute is independent of Web usage. Significant results suggest that certain demographics are more likely to use the Web for travel information, indicating potential target groups for marketing or additional outreach. The analysis must account for missing data, which can reduce sample sizes and potentially bias results if not handled appropriately. Combining categories might be necessary when cell counts are small, increasing the robustness of the tests.

This analysis highlights the importance of understanding demographic patterns in consumer behavior. For example, studies often find that younger individuals, higher-income groups, and certain racial or educational groups are more inclined to use web-based sources, although variations exist. Graphical summaries, such as bar charts and mosaic plots, complement the chi-square tests by providing visual insights into the distributions.

In conclusion, two-way tables and chi-squared tests serve as essential tools in analyzing categorical data, uncovering relationships between variables, and informing strategic decisions in healthcare and marketing contexts. Their proper use requires careful data organization, handling missing values thoughtfully, and interpreting results in the context of practical significance. When applied correctly, these methods enable researchers to infer meaningful associations and support data-driven decision-making processes, as demonstrated in both the HMO study and the travel information preferences case study.

References

  • Agresti, A. (2018). Statistical Methods for the Social Sciences. Pearson.
  • McHugh, M. L. (2013). The Chi-square test of independence. Biological Research for Nursing, 15(3), 143–149.
  • Sheskin, D. J. (2011). Handbook of Parametric and Nonparametric Statistical Procedures. Chapman and Hall/CRC.
  • Field, A. (2013). Discovering Statistics Using SPSS. Sage Publications.
  • Allen, L., & Bennett, H. (2017). Practical Statistics for Data Scientists. O'Reilly Media.
  • Everitt, B. (2002). The Analysis of Contingency Tables. CRC Press.
  • Zimmerman, D. W. (1994). Statistical Literacy: An Essential Skill for Education and the Public. American Statistician, 48(2), 86–87.
  • Fitzmaurice, G. M., Laird, N. M., & Zeger, S. L. (2011). Applied Longitudinal Analysis. Wiley.
  • Neuhaus, J., & McCulloch, C. E. (2006). Generalized Linear Mixed Models. Wiley-Interscience.
  • Vittinghoff, E., & McCulloch, C. E. (2007). Relaxing the Rule of Ten Events per Variable in Logistic and Cox Regression. American Journal of Epidemiology, 165(6), 710–718.

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