Regression analysis is a statistical method that measures the average amount of change in the dependent variable associated with a unit change in one or more independent variables. The method is widely used because it helps managers “get behind the numbers” so they understand why costs behave the way they do and what managers can do to influence them. For example, at Analog Devices, a maker of digital and analog integrated circuits, managers use regression analysis to evaluate how and why defect rates and product quality change over time. Managers who understand these relationships gain greater insights and make better decisions.
Simple regression analysis estimates the relationship between the dependent variable and one independent variable. In the Elegant Rugs example, the dependent variable is total indirect manufacturing labor costs; the single independent variable, or cost driver, is number of machine-hours. Multiple regression analysis estimates the relationship between the dependent variable and two or more independent variables. Multiple regression analysis for Elegant Rugs might use number of machine-hours and number of batches as independent variables. The chapter appendix explores simple regression and multiple regression in more detail.
Exhibit 10-6 shows the regression line that best fits the data in columns B and C of Exhibit 10-3. The cost function is
Exhibit 10-6
Regression Model for Weekly Indirect Manufacturing Labor Costs and Machine-Hours for Elegant Rugs
This graph is also an extension of the shown in exhibit 10-4, depicting the numbered scatter plot of indirect manufacturing labor and machine-hours. Details of the graph are as follows:
The graph depicts the relevant range from 46 to 96 machine-hours.
Regression line: The straight line is depicted by the equation (y equals $300.98 plus the product of $10.31 and X). The line roughly touches the plot for 8th week and 2nd week. It is extended in both directions and intersects the Y-axis at about $300.
Residual term: The line joining the plot for 1st week and the regression line is marked as residual term.
The regression line in Exhibit 10-6 is the line that minimizes the sum of the squared vertical distances from the data points (the various points in the graph) to the line. The vertical distance, called the residual term, measures the difference between actual cost and estimated cost for each observation of the cost driver. Exhibit 10-6 shows the residual term for week 1. The line from the observation to the regression line is drawn perpendicular to the horizontal axis, or x-axis. The smaller the residual terms, the better is the fit between the actual cost observations and estimated costs. Goodness of fit indicates the strength of the relationship between the cost driver and costs. The regression line in Exhibit 10-6 rises from left to right. The positive slope of this line and small residual terms indicate that, on average, indirect manufacturing labor costs increase as the number of machine-hours increases. The vertical dashed lines in Exhibit 10-6 indicate the relevant range, the range within which the cost function applies.
Instructors and students who want to explore the technical details of estimating the least-squares regression line can go to Appendix: Regression Analysis, and return to this point without any loss of continuity.
The estimate of the slope coefficient, b, indicates that, on average, indirect manufacturing labor costs increase by $10.31 for every machine-hour used within the relevant range. Managers can use the regression equation to set budgets. For example, if Elegant Rugs estimates it will use 90 machine-hours in the upcoming week, it will predict indirect manufacturing labor costs to be
The regression method is more accurate than the high-low method because it uses all observations to estimate the cost function while the high-low equation uses only two observations. The inaccuracies of the high-low method can mislead managers. Consider the high-low method equation in the previous section, For 90 machine-hours, the predicted weekly costs using the high-low method equation are Suppose that over the next 12-week period, Elegant Rugs runs its machines for 90 hours each week. Assume the average indirect manufacturing labor costs for those 12 weeks are $1,300. Based on the high-low method prediction of $1,366.48, Elegant Rugs would conclude it has performed well because actual costs are lower than predicted costs. But comparing the $1,300 performance with the more-accurate $1,228.88 prediction of the regression model tells a different story and would prompt Elegant Rugs to search for ways to improve its cost performance.
Managers at Elegant Rugs are interested in evaluating whether changes in the production process (that resulted in the data in Exhibit 10-3) have reduced indirect manufacturing labor costs such as supervision, maintenance, and quality control costs. Using data on number of machine-hours used and indirect manufacturing labor costs of the previous process (not shown here), the manager estimates the prior regression equation to be
The constant ($300.98 versus $545.26) and the slope coefficient ($10.31 versus $15.86) are both smaller for the new process relative to the old process indicating that indirect manufacturing labor costs have decreased.
