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Analyzing Vibration Levels of Motor Bearings using Oneway ANOVA in Minitab, Study notes of Data Acquisition

New England Association of Schools and CollegesData Acquisition

How to use minitab's one-way anova routine to compare the means of different populations or groups (brands of motor bearings in this case) based on the vibration levels. It also covers the assumptions of oneway anova and how to check them using minitab's test for equal variances and normality test. An example of analyzing the motor bearing vibration data and interpreting the results.

Typology: Study notes

2021/2022

Uploaded on 09/12/2022

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Oneway ANOVA: Analysis and Assumption Checking using Minitab
Review: Recall that Oneway ANOVA is a method for comparing the means of Idifferent
populations or groups by testing the ANOVA hypotheses
H0:µ1=µ2=··· =µIvs.
H1:at least two means differ
These hypotheses are tested using the F-ratio test statistic F = MSTr/MSE. The larger the
F, the stronger the evidence against H0in favor of H1. Therefore the p-value for testing H0
vs H1is given by P(F≥Factual), computed assuming H0true, where Factual is the F-ratio
for the actual sample data.
Overview: In the following we detail how to analyze ANOVA data using Minitab’s One-Way
routine to compute the F-ratio test statistic and the ANOVA residuals. Then we state
the ANOVA assumptions and describe how to use Minitab’s Stat ->ANOVA ->Test for
Equal Variances routine and the residuals to check them.
Using One-Way: We will use Minitab’s ANOVA routine One-Way, not One-Way (Unstacked),
because it computes the residuals we need to check the ANOVA assumptions:
Stacking Data: In order to use One-Way we need the data to be in stacked form, that is, one
column (“response”) contains all the values and a second column (“factor”) identifies
the sample to which each value belongs. If the data is not stacked, you can stack it
using
Data -> Stack -> Columns
(For details on stacking, use Help ->Search Help and enter “stack.”)
Calling One-Way:To invoke One-Way do the following:
1. Click on Stat ->ANOVA ->One-Way
2. Select the Response column containing the values
3. Select the Factor column indentifying the samples
4. Check the Store residuals box then click OK.
Interpreting One-Way Output: Below is the output after stacking and analyzing the data
in the Minitab data set www.rose-hulman.edu/∼inlow/bearing.MTW:
One-way ANOVA: Vibration versus Brand
Source DF SS MS F P
Brand 2 20.70 10.35 8.99 0.003
Error 15 17.28 1.15
Total 17 37.98
This data set - which consists of the vibration levels of three brands of motor bearings - was
stacked into a response column called“Vibration” and a factor column called “Brand.” The
F test statistic (F-ratio) is 8.99 and the p-value is 0.003.
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Oneway ANOVA: Analysis and Assumption Checking using Minitab

Review: Recall that Oneway ANOVA is a method for comparing the means of I different populations or groups by testing the ANOVA hypotheses

H 0 : μ 1 = μ 2 = · · · = μI vs. H 1 : at least two means differ

These hypotheses are tested using the F-ratio test statistic F = MSTr/MSE. The larger the F, the stronger the evidence against H 0 in favor of H 1. Therefore the p-value for testing H 0 vs H 1 is given by P (F ≥ Factual), computed assuming H 0 true, where Factual is the F-ratio for the actual sample data.

Overview: In the following we detail how to analyze ANOVA data using Minitab’s One-Way routine to compute the F-ratio test statistic and the ANOVA residuals. Then we state the ANOVA assumptions and describe how to use Minitab’s Stat -> ANOVA -> Test for Equal Variances routine and the residuals to check them.

Using One-Way: We will use Minitab’s ANOVA routine One-Way, not One-Way (Unstacked), because it computes the residuals we need to check the ANOVA assumptions:

Stacking Data: In order to use One-Way we need the data to be in stacked form, that is, one column (“response”) contains all the values and a second column (“factor”) identifies the sample to which each value belongs. If the data is not stacked, you can stack it using Data -> Stack -> Columns (For details on stacking, use Help - > Search Help and enter “stack.”)

Calling One-Way: To invoke One-Way do the following:

  1. Click on Stat -> ANOVA -> One-Way
  2. Select the Response column containing the values
  3. Select the Factor column indentifying the samples
  4. Check the Store residuals box then click OK.

Interpreting One-Way Output: Below is the output after stacking and analyzing the data in the Minitab data set www.rose-hulman.edu/∼inlow/bearing.MTW: One-way ANOVA: Vibration versus Brand

Source DF SS MS F P Brand 2 20.70 10.35 8.99 0. Error 15 17.28 1. Total 17 37.

This data set - which consists of the vibration levels of three brands of motor bearings - was stacked into a response column called“Vibration” and a factor column called “Brand.” The F test statistic (F-ratio) is 8.99 and the p-value is 0.003.

Anova Assumptions

For the p-value associated with the F-ratio to be accurate the following four assumptions must be met

  • The I samples are IID
  • The I samples are independent
  • The I populations/processes have equal variance (homoscedasticity)
  • The I populations/processes are normally distributed

The first two assumptions describe the data acquisition process; the last two describe the populations/processes under study.

Checking the Anova Assumptions

Sampling Assumptions (1 and 2): Determining if the data satisfies assumptions 1 and 2 requires knowledge of how the data was acquired/generated. A common violation of this assumption 2 is the case where the same subjects comprise all I samples, that is, the same subjects are measured multiple times. This results in what is called “Repeated Measures ANOVA,” the I sample analog of paired data.

Population/Process Assumptions:

Equal Variances (3): To check this assumption, i.e., to verify that the populations or processes are homoscedastic (have “same scatter”) you need the data to be in stacked form, that is, one column contains all the values and a second column identifies the sample to which each value belongs. Once the data is in stacked form, you can test for equal variances using Stat -> ANOVA -> Test for Equal Variances Select the Response and Factor columns which are same as those for One-Way. Using the Levene test p-value, reject H 0 : variances equal if the p-value is less than or equal to 0.05. For the motor bearing vibration data above, the Levene p-value is .585, so we don’t reject the equal variance assumption.

Normality (4): To check normality you need to do the following:

  1. Check the Store residuals box on the One-Way dialog box. This tells One-Way to create a column called RESI1 or similar.
  2. Test the normality assumption by testing the residuals in RESI1 using Stat -> Basic Statistics -> Normality Test... Reject H 0 : populations/processes normal if the p-value is less than or equal to 0.05. For the motor bearing vibration data, the p-value is 0.406 so we don’t reject the nor- mality assumption.