Search in book...
Toggle Font Controls
Create new playlist

Name your new playlist

Playlist description (optional)
Sign In

Email address

Password

Forgot Password?

or

Continue with Facebook

Continue with Google
Sign Up

Full Name

Email address

Confirm Email Address

Password

or

Continue with Facebook

Continue with Google

4.4 Assumptions of the Regression Model

If we can make certain assumptions about the errors in a regression model, we can perform statistical tests to determine if the model is useful. The following assumptions are made about the errors:

The errors are independent.
The errors are normally distributed.
The errors have a mean of zero.
The errors have a constant variance (regardless of the value of X).

In Action Using Regression as Part of Improvement Initiative at Trane/Ingersoll Rand

Trane U.S. Inc. is a brand of Ingersoll Rand and is a leader in both home and commercial air conditioning. In an effort to improve quality and efficiency, the company consolidated its manufacturing of evaporator and condenser coils into one location in South Carolina. A key part of the coil unit is a header, which controls the flow of water or refrigerant through the coil. With about 120 unique headers produced on each shift, with increased demand for their air conditioners due to federal refrigerant standards, and with growing pressure to reduce costs, Trane began an improvement initiative in 2010.

Value stream mapping, a lean manufacturing technique, was used to visualize the entire process and identify waste in the system. A new layout was suggested to improve processing and reduce the overall time of production. To predict how well the new layout would perform in the manufacture of the headers, data were collected about the time to move materials into position, time to set up a job, time to set up a machine, and time to process a job. Several regression models were developed to determine which job characteristics were most significant in the various times involved in processing and to measure the variability in times. These were used in simulation models that were developed to determine the overall impact of the new layout and to determine where bottlenecks might occur. One of the bottlenecks, at a robotic machine for welding, was eliminated by retooling the robot to allow the machine operator to operate both sides of the robot. This enables the robot to process two headers simultaneously.

The improvement initiatives from this project are estimated to save over $700,000 per year. This project highlighted the importance of data-based analytics and lean manufacturing tools in identifying the improvement tactics for complex value streams. The company plans similar improvement initiatives for other value streams in the future.

Source: Based on John B. Jensen, Sanjay L. Ahire, and Manoj K. Malhotra, “Trane/Ingersoll Rand Combines Lean and Operations Research Tools to Redesign Feeder Manufacturing Operations,” Interfaces 43, 4 (July–August 2013): 325–340, © Trevor S. Hale.

It is possible to check the data to see if these assumptions are met. Often a plot of the residuals will highlight any glaring violations of the assumptions. When the errors (residuals) are plotted against the independent variable, the pattern should appear random.

Figure 4.4 presents some typical error patterns, with Figure 4.4A displaying a pattern that is expected when the assumptions are met and the model is appropriate. The errors are random and no discernible pattern is present. Figure 4.4B demonstrates an error pattern in which the errors increase as X increases, violating the constant variance assumption. Figure 4.4C shows errors consistently increasing at first and then consistently decreasing. A pattern such as this would indicate that the model is not linear and some other form (perhaps quadratic) should be used. In general, patterns in the plot of the errors indicate problems with the assumptions or the model specification.

Scatter diagram with X on the horizontal axis and Error on the y axis.

Figure 4.4A Full Alternative Text

Figure 4.4B Full Alternative Text

Figure 4.4C Full Alternative Text

Estimating the Variance

While the errors are assumed to have constant variance $(σ^{2}),$ $(σ^{2}),$ this is usually not known. It can be estimated from the sample results. The estimate of $σ^{2}$ $σ^{2}$ is the mean squared error (MSE) and is denoted by $s^{2} .$ $s^{2} .$ The MSE is the sum of squares due to error divided by the degrees of freedom:¹

s^{2} = MSE = \frac{SSE}{n - k - 1}

$s^{2} = MSE = \frac{SSE}{n - k - 1}$ (4-12)

where

\begin{array}{l} n & = & number of observations in the sample \\ k & = & number of independent variables \end{array}

$\begin{array}{l} n & = & number of observations in the sample \\ k & = & number of independent variables \end{array}$

In the Triple A Construction example, $n = 6$ $n = 6$ and $k = 1$ $k = 1$ , so

s^{2} = MSE = \frac{SSE}{n - k - 1} = \frac{6.8750}{6 - 1 - 1} = \frac{6.8750}{4} = 1.7188

$s^{2} = MSE = \frac{SSE}{n - k - 1} = \frac{6.8750}{6 - 1 - 1} = \frac{6.8750}{4} = 1.7188$

From this, we can estimate the standard deviation as

s = \sqrt{MSE}

$s = \sqrt{MSE}$ (4-13)

This is called the standard error of the estimate or the standard deviation of the regression. In this example,

s = \sqrt{MSE} = \sqrt{1.7188} = 1.31

$s = \sqrt{MSE} = \sqrt{1.7188} = 1.31$

This is used in many of the statistical tests about the model. It is also used to find interval estimates for both Y and regression coefficients.²

..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.

Table of Contents for
4.4 Assumptions of the Regression Model

4.4 Assumptions of the Regression Model

Figure 4.4A Pattern of Errors Indicating Randomness

Figure 4.4B Nonconstant Error Variance

Figure 4.4C Pattern of Errors Indicating Relationship Is Not Linear

Estimating the Variance

Table of Contents for 4.4 Assumptions of the Regression Model

Create new playlist

Sign In

Sign Up

Table of Contents for
4.4 Assumptions of the Regression Model