RES 720 Week 4 Multiple Regression Example

Weighing Several Predictors at Once: Multiple Regression of Warehouse Productivity

[Student Name]

University of Phoenix

RES/720: Statistical Research Methods and Design II

Week 4 Assignment

[Instructor Name]

[Date]

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

Weeks 2 and 3 compared shifts and onboarding conditions at the composite Louisville logistics company. Shift, though, travels with other differences: rotating associates have less prior experience and slightly lower pay. RQ3 asks which factors predict 90-day productivity when considered together. Multiple regression answers that question by estimating each predictor's association while holding the others constant.

The Model

The outcome is units picked per hour in the third month for the 2,380 associates who stayed 90 days. Predictors were chosen from Week 1's framework and from site managers' explanations:

Pay rate, in dollars per hour.

Commute distance, in miles.

Prior warehouse experience, in months.

Age, in years.

Shift, coded as two dummy variables, nights and rotating, with fixed days as the reference.

Warehouse, coded as eight dummy variables, with the largest warehouse as the reference.

Cohen et al. (2003) explain that dummy coding represents a categorical predictor with one fewer variable than categories, so each coefficient compares a category with the reference group, and that coefficients in multiple regression are partial: each estimates a predictor's association with the outcome when the other predictors are held constant.

Blocks of Entry

The learner entered predictors in two blocks. Block 1 contains warehouse fixed effects, which absorb differences between sites such as layout and equipment. Block 2 adds the individual predictors. Warehouse alone explained 6 percent of the variance in productivity; adding the individual predictors raised R squared to .21, a change of .15, F change(6, 2365) = 74.9, p < .001. The full model explains 21 percent of variance, adjusted R squared = .205, F(14, 2365) = 45.0, p < .001.

Coefficients

Unstandardized coefficients, with robust standard errors:

Experience: b = 0.32 units per month, p < .001, beta = .29.

Pay rate: b = 2.1 units per dollar, p < .001, beta = .14.

Commute: b = -0.15 units per mile, p = .002, beta = -.08.

Age: b = -0.05 units per year, p = .31.

Nights versus days: b = -4.2 units, p = .001.

Rotating versus days: b = -9.8 units, p < .001.

Each additional year of prior experience, 12 months, is associated with about 4 more units per hour, holding pay, commute, age, shift and warehouse constant. Each dollar of hourly pay is associated with about 2 more units. Age adds nothing once other variables are in the model.

Holding experience and pay constant cuts the rotating-shift gap from 14 units to about 10.

What Happened to the Shift Gap

Week 2 found that rotating associates picked 14 fewer units per hour than day associates. In the regression, the gap is 9.8 units. About 30 percent of the raw gap is associated with differences in experience, pay and warehouse among associates on each shift. The remaining gap is still substantial, and it persists when other observed differences are held constant, though unmeasured differences such as sleep or family demands could still contribute.

Checking Assumptions

Plotting residuals against predicted values revealed no curve, but the spread of residuals widened at higher fitted values. Long and Ervin (2000) showed through simulation that ordinary standard errors can be badly biased when residual variance is unequal and recommended heteroscedasticity-consistent standard errors, especially the version known as HC3, for samples of any size. The learner reports HC3 standard errors; they were about 10 percent larger than ordinary ones and changed no conclusions.

Residuals were close to normal in a Q-Q plot. Experience showed a slight curve, with gains leveling off after about four years; adding a squared term improved fit only marginally, so the linear term is kept for clarity, with the curve noted.

Multicollinearity

Variance inflation factors ranged from 1.1 to 3.1, the highest for pay and warehouse, since pay scales differ by site. Belsley et al. (1980) developed diagnostics for collinearity and for observations with outsized influence, and their work underlies the point that collinearity mainly inflates standard errors and makes individual coefficients unstable rather than spoiling the model's overall predictions; fixed VIF cutoffs are conventions, not laws. At these levels, collinearity is not a concern, but the learner notes that pay and warehouse effects are partly entangled.

Why Warehouse Fixed Effects

With only nine warehouses, cluster-robust standard errors would be unreliable, since those methods need many clusters to work well. Warehouse dummy variables instead remove every stable difference between sites, so the remaining coefficients compare associates within the same warehouse. The cost is that the model cannot say anything about site-level factors such as building age.

Influential Cases

Influence diagnostics of the kind Belsley and colleagues described, here Cook's distance and DFBETAS, identified 14 associates with unusual influence, mostly very experienced pickers with extreme productivity. Rerunning the model without them changed coefficients by less than 5 percent, so they were retained.

Practical Interpretation

For leaders, the model suggests three things. Experience matters most, so recruiting from other warehouses and retaining associates through their first months, when they gain experience, pays off. Pay is associated with productivity, though, as the model cannot separate the two, this may reflect better-paying sites attracting stronger associates as much as pay motivating effort. And the rotating shift carries a productivity cost of roughly 10 units per hour that does not disappear when experience and pay are considered.

Testing the Shift Difference Within the Model

The nights and rotating coefficients each compare a shift with fixed days. To compare nights with rotating directly, the learner tested whether the two coefficients differ: the 5.6-unit gap was significant, F(1, 2365) = 14.2, p < .001. Rotating associates therefore trail night associates as well as day associates once other predictors are held constant, which supports the Week 2 suggestion that switching between days and nights carries its own cost.

Limits of the Model

Seventy-nine percent of the variance remains unexplained. The model omits factors such as training quality, team composition and associates' health. All associations are observational; the pay coefficient, in particular, should not be read as the gain from a raise. And the model includes only stayers, so it describes productivity among people who did not leave.

Reporting in APA Style

A hierarchical regression showed that individual predictors explained variance in 90-day productivity beyond warehouse, change in R squared = .15, F(6, 2365) = 74.9, p < .001. In the full model, R squared = .21, experience (b = 0.32, beta = .29), pay (b = 2.1, beta = .14) and commute (b = -0.15, beta = -.08) were significant predictors, and rotating-shift associates picked 9.8 fewer units per hour than fixed-day associates, all p < .01.

Conclusion

Multiple regression weighed pay, commute, experience, age and shift together within warehouses. Experience was the strongest predictor, and the rotating-shift gap shrank to about 10 units per hour once other differences were held constant but remained substantial. Robust standard errors, collinearity checks and influence diagnostics support those results. Week 5 turns to the binary outcome that matters most to leaders: whether an associate quits within 90 days.

References

Belsley, D. A., Kuh, E., & Welsch, R. E. (1980). Regression diagnostics: Identifying influential data and sources of collinearity. Wiley.

Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences (3rd ed.). Lawrence Erlbaum Associates.

Long, J. S., & Ervin, L. H. (2000). Using heteroscedasticity consistent standard errors in the linear regression model. The American Statistician, 54(3), 217-224. https://doi.org/10.1080/00031305.2000.10474549