RES 720 Week 2 One-Way ANOVA Example

Do Shift Patterns Differ in Productivity? A One-Way ANOVA With Unequal Variances

[Student Name]

University of Phoenix

RES/720: Statistical Research Methods and Design II

Week 2 Assignment

[Instructor Name]

[Date]

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

Week 1 set out five research questions for a composite HR director at a Louisville, Kentucky, logistics company with nine warehouses. RQ1 asks whether productivity at 90 days differs among associates on fixed days, on fixed nights and on a rotating schedule of four days on and three off that alternates between day and night. Site managers who blame the rotating pattern for early quits also say rotating associates are slower. This paper tests that claim.

Why One ANOVA Instead of Three t Tests

Comparing three groups two at a time requires three t tests. Each carries a 5 percent risk of a false positive, so across all three the risk that one or more is a false alarm climbs to about 14 percent. One-way ANOVA tests the null hypothesis that all group means are equal with a single test at the chosen alpha, and post hoc procedures then compare pairs while controlling the overall error rate.

H0: mean units per hour are equal for fixed days, fixed nights and rotating shifts.

H1: at least one shift's mean differs.

The Data

Historical records cover 3,600 associates hired over two years. Productivity at 90 days exists only for the 2,380 who stayed that long. Units picked per hour are averaged over each associate's third month, measured by handheld scanners.

Fixed days: n = 1,050; M = 118 units per hour; SD = 22.

Fixed nights: n = 620; M = 112; SD = 25.

Rotating: n = 710; M = 104; SD = 28.

The rotating group is slowest by an average of 14 units per hour compared with fixed days, roughly a 12 percent gap.

Checking Assumptions

Observations are independent across associates, though associates are clustered in nine warehouses; Week 4's models will address clustering. Histograms show roughly symmetric distributions within each shift, and with hundreds per group the central limit theorem covers mild departures. Homogeneity of variance is doubtful: the rotating group's standard deviation is 27 percent larger than the day group's, and Levene's test is significant, F(2, 2377) = 21.4, p < .001. The largest group also has the smallest variance, the pattern in which the classic F test becomes too liberal.

Choosing Welch's F

Delacre et al. (2019) compared the classic F test with Welch's F and other alternatives across many simulated conditions and concluded that Welch's test controls error rates well when variances differ, loses very little power when they are equal and should be used by default. The learner reports both, treating Welch's as primary.

Results

The classic ANOVA gave F(2, 2377) = 68.1, p < .001. Welch's ANOVA gave F(2, 1352) = 66.4, p < .001. Both reject the null: shift patterns differ in mean productivity.

The effect size, eta squared, is the between-groups sum of squares divided by the total sum of squares: 83,080 / 1,533,527 = .054. Shift pattern accounts for about 5 percent of the variance in 90-day productivity. Omega squared, which corrects eta squared's slight upward bias, is about .053, almost the same with a sample this large.

Five percent of the variance sounds small until it is translated into 14 units an hour across hundreds of associates.

Post Hoc Comparisons

Games and Howell (1976) developed a pairwise comparison procedure for unequal group sizes and variances and found in Monte Carlo work that it held error rates close to nominal levels under those conditions, unlike methods that assume equal variances. The learner uses it:

Days versus nights: difference 6 units per hour, p < .001, d = 0.26.

Days versus rotating: difference 14 units per hour, p < .001, d = 0.56.

Nights versus rotating: difference 8 units per hour, p < .001, d = 0.30.

All three pairs differ, with the largest and most practically meaningful gap between fixed days and the rotating schedule.

Interpreting the Result in Business Terms

At the company's average labor cost per hour, a 14-unit gap means that a rotating associate costs about 13 percent more per unit picked than a day associate. Across the 710 rotating associates in the data, that gap amounts to roughly 25 full-time equivalents of lost output. Even the smaller nights-versus-rotating gap suggests that working nights alone does not explain the rotating group's lower productivity; the switching itself may matter.

A Survivor Problem

The analysis includes only associates who stayed 90 days. Week 1's records show that rotating shifts lose more new associates early, 41 percent against 29 percent for fixed days. If those who leave rotating shifts are disproportionately the faster pickers, who have better options elsewhere, the remaining rotating associates would look slower than rotating associates in general. If those who leave are the slower ones, the gap would be understated. The learner can partly check this by comparing 30-day productivity, before most quits, across shifts: the gap at 30 days was 11 units per hour, slightly smaller, which suggests that survivor bias may exaggerate the 90-day gap a little but does not create it.

What Leaders Should and Should Not Conclude

The result supports a narrow claim: among associates who stayed 90 days, those on rotating shifts picked fewer units per hour. It does not show that moving an associate from rotating to fixed days would raise output by 14 units, since the groups differ in more than schedule. Before redesigning schedules, leaders should see whether the gap holds after accounting for experience, pay and warehouse, which the next two weeks address.

Selection Into Shifts

Associates choose shifts at hiring, subject to openings. People who choose rotating schedules may differ in experience, age or other commitments. One-way ANOVA cannot separate the shift from the people in it. Week 3's factorial analysis with experience as a covariate and Week 4's regression will control for several of these differences.

A Nonparametric Check

Because variances differed, the learner also ran the rank-based test that Kruskal and Wallis (1952) introduced as an analysis of variance on ranks, which compares groups without assuming normal distributions. It gave H(2) = 131.7, p < .001, consistent with the ANOVA, and pairwise rank comparisons matched the Games-Howell results.

Reporting in APA Style

Ninety-day productivity differed by shift pattern, Welch's F(2, 1352) = 66.4, p < .001, eta squared = .05. Games-Howell comparisons showed that associates on fixed days (M = 118, SD = 22) picked more units per hour than those on fixed nights (M = 112, SD = 25) and rotating shifts (M = 104, SD = 28), and fixed-night associates picked more than rotating associates, all p < .001.

Conclusion

A one-way ANOVA, run in its Welch form because variances differed, showed that shift pattern is associated with productivity, explaining about 5 percent of its variance, with rotating associates 14 units per hour slower than fixed-day associates. Post hoc tests located every difference, and a 30-day comparison suggested survivor bias does not explain the gap. Whether shift causes the difference remains open; Week 3 adds the onboarding program and controls for experience in a factorial design.

References

Delacre, M., Leys, C., Mora, Y. L., & Lakens, D. (2019). Taking parametric assumptions seriously: Arguments for the use of Welch's F-test instead of the classical F-test in one-way ANOVA. International Review of Social Psychology, 32(1), 13. https://doi.org/10.5334/irsp.198

Games, P. A., & Howell, J. F. (1976). Pairwise multiple comparison procedures with unequal n's and/or variances: A Monte Carlo study. Journal of Educational Statistics, 1(2), 113-125. https://doi.org/10.3102/10769986001002113

Kruskal, W. H., & Wallis, W. A. (1952). Use of ranks in one-criterion variance analysis. Journal of the American Statistical Association, 47(260), 583-621. https://doi.org/10.1080/01621459.1952.10483441