RES 720 Week 6 Chi-Square and Nonparametric Tests Example

Counting Incidents and Ranking Ratings: Chi-Square and Nonparametric Tests in a Warehouse Study

[Student Name]

University of Phoenix

RES/720: Statistical Research Methods and Design II

Week 6 Assignment

[Instructor Name]

[Date]

The learner, the company, its associates and all data are composites written for a model paper.

The composite Louisville logistics company's safety director has raised a concern that fits the study's focus on rotating shifts: new rotating associates seem to get hurt more often. RQ5 asks whether safety incidents are distributed differently across shifts. Incidents are counts of a yes-or-no event, a categorical outcome, so the parametric tests of earlier weeks do not apply. This paper also analyzes supervisors' ordinal readiness ratings, which call for a rank-based test.

The Contingency Table

Of the 3,600 associates hired over two years, 196 had at least one recordable incident in their first 90 days.

Fixed days: 63 of 1,580 associates, 4.0 percent.

Fixed nights: 52 of 940, 5.5 percent.

Rotating: 81 of 1,080, 7.5 percent.

H0: incident status is independent of shift. H1: incident status is associated with shift.

The Chi-Square Test of Independence

The test sets the observed counts beside the counts one would expect if shift and incidents were unrelated, each found by multiplying a cell's row and column totals and dividing by the overall count. Expected incidents were 86.0 for days, 51.2 for nights and 58.8 for rotating; all expected counts exceed five. The test gave chi-square(2, N = 3,600) = 15.4, p < .001.

McHugh (2013) explains that the test requires independent observations, mutually exclusive categories and adequate expected counts, and that a significant result should be followed by an effect size such as Cramer's V and an examination of which cells contribute to the association. Here, Cramer's V = .07, a small association. With 3,600 cases, even a modest difference reaches significance.

Where the Association Lies

Standardized residuals locate the result; each takes a cell's observed count, subtracts its expected count and scales the gap by the expected count's square root. The rotating incident cell has a residual of 2.9, meaning far more incidents than expected; the day incident cell has -2.5, fewer than expected; the night cell is near zero. The association comes from rotating associates having more incidents and day associates fewer, with nights in between as expected.

Rotating associates had 22 more incidents than independence would predict, while day associates had 23 fewer.

Adjusting for Exposure

Rotating associates work more hours in a typical 90 days because their schedule includes more overtime. Raw counts could reflect more time on the floor rather than more risk per hour. The learner computed incident rates per 200,000 hours worked, the standard used in workplace safety reporting: 4.3 for days, 5.6 for nights and 6.9 for rotating. The gap narrows but remains, and a Poisson model with hours as an exposure term confirmed a higher rate for rotating associates, rate ratio 1.6, 95% CI [1.2, 2.2].

Fisher's Exact Test for a Small Warehouse

At the smallest warehouse, 120 new associates were hired over two years: 30 rotating and 90 on days. Forklift-related incidents numbered four among rotating associates and one among day associates. Expected counts for the incident cells were 1.25 and 3.75, below five, so the chi-square approximation is unreliable. Agresti (2007) recommends Fisher's exact test for such tables, since it computes the exact probability of tables as unusual as the observed one, or more so, given the row and column totals. Fisher's exact test gave p = .014, two-sided. The result is notable but rests on five incidents, so the learner treats it as a signal for the safety director to review forklift training for rotating associates at that site, not as a firm conclusion.

Readiness Ratings and the Mann-Whitney Test

At 30 days, supervisors rate each new associate's readiness on a five-point scale from 1, not ready, to 5, fully independent. During the rollout, ratings were available for 410 associates under the new onboarding and 300 under the old. The ratings are ordinal, with many ties and a ceiling at 5, so a t test on means is questionable.

Mann and Whitney (1947) developed a test of whether values in one population tend to be larger than in another, based on the ranks of the combined observations. The new-program group's median rating was 4 against 3 for the old, and the test gave U = 50,140, z = 5.1, p < .001. The effect size r, which scales z by the overall sample size, was .19, a small-to-moderate effect. A t test on the same data gave the same conclusion, but the rank-based test is the defensible primary analysis for ordinal ratings.

A Goodness-of-Fit Check on Timing

The safety director also asked whether early incidents cluster in the first weeks. Dividing the 90 days into three 30-day periods, the 196 incidents fell 92, 58 and 46 across them. If incidents were spread evenly over time worked, and allowing for the associates who quit and so had fewer hours in later periods, the expected counts would be 72, 64 and 60. A chi-square goodness-of-fit test gave chi-square(2) = 9.4, p = .009: incidents are concentrated in the first month, when associates are least experienced. That finding supports adding safety coaching to the first two weeks of onboarding.

Why Not Treat Ratings as Interval

Some researchers average five-point ratings and run t tests, and with large samples the results often agree with rank-based tests, as they did here. The learner reports the Mann-Whitney test as primary because readiness categories are not equally spaced in meaning: the step from 3 to 4, working without frequent help, is larger in practice than the step from 4 to 5.

Supervisor Bias

Supervisors knew which onboarding each associate received, so their ratings could reflect expectations about the new program rather than actual readiness. Productivity at 30 days, an objective measure, also favored the program, which gives some reassurance, but the learner notes the risk.

What It Means for the Company

New rotating associates have more incidents than other new associates, even per hour worked, though the overall association is small. Combined with Weeks 2 through 5, which found rotating associates slower and more likely to quit, the evidence suggests that the rotating schedule is hard on new associates in several ways. The onboarding program appears to improve readiness, which may also support safety, a question the dissertation can test.

Reporting in APA Style

Incident status was associated with shift, chi-square(2, N = 3,600) = 15.4, p < .001, Cramer's V = .07; standardized residuals showed more incidents than expected among rotating associates. Associates under the new onboarding received higher readiness ratings (Mdn = 4) than those under the old (Mdn = 3), U = 50,140, z = 5.1, p < .001, r = .19.

Conclusion

Chi-square with residuals and an effect size showed a small but real association between shift and early incidents, concentrated among rotating associates and persisting per hour worked. Fisher's exact test handled a small warehouse table, and the Mann-Whitney test compared ordinal readiness ratings appropriately. Week 7 will interpret the full set of complex results together.

References

Agresti, A. (2007). An introduction to categorical data analysis (2nd ed.). Wiley.

Mann, H. B., & Whitney, D. R. (1947). On a test of whether one of two random variables is stochastically larger than the other. The Annals of Mathematical Statistics, 18(1), 50-60. https://doi.org/10.1214/aoms/1177730491

McHugh, M. L. (2013). The chi-square test of independence. Biochemia Medica, 23(2), 143-149. https://doi.org/10.11613/BM.2013.018