Gauge Bias Linearity (multiple equipment)

The BIS.Net Team BIS.Net Team

Bias in the measurement system is the difference between the average of measured values and the actual value of a part. The actual value of the part is called the reference value. Ideally the bias should be zero.

The above image shows the variation in the different measurements on the same part about the average and the deviation of the average from the reference value. In this instance the bias is negative.

It cannot be assumed that bias is constant over all ranges of measurements. It is possible that bias increases, or reduces, as the magnitude of measurements increases or decreases. This can be due to several reasons, such as poor calibration, poor maintenance. For example, weighing scales used to weigh light weights, medium and heavy weights may provide different biases at each level, if the scales have not been maintained.

A linearity study is performed to establish if there is a linear relationship with magnitude of the measured item. For such a study reference parts are chosen varying between the normal operating range. Usually, for a simple analysis several measurements are taken for each reference part by the same appraiser.

A multiple equipment study is an extension of a single linearity study performed by one appraiser using one equipment. It is not uncommon for multiple measuring devices to be available for testing the same parts. Due to faulty equipment, it cannot be assumed that each measuring device has the same linearity. To establish differences, or consistency, or absence of linearity amongst all measuring devices a multiple equipment linearity analysis must be carried out.

BIS.Net Analyst provides this information through an Analysis of Covariance which includes the following output.

Tabular Input

The tabular output includes an analysis of variance (covariance). For the non-statistician, the important column is the Significance Column. A significant overall linearity result means that on a combined basis there is linearity in the bias over the measurement range.

A significant difference in equipment means that there are differences in bias, not due to differences in linearity.

A significant difference in slopes means that the linearity differs amongst the measuring devices.

The Regression Equations table can be used to view the slopes and strength of the relationship from the last column. A value of 100% is a perfect relationship and 0 no relationship. The closer to zero the better.

Scatter Chart

The scatter chart enables one to see the different lines of best fit for each measuring device.

Slope Consistency Chart

This chart visualizes the analysis of covariance. There are two sets of limits. One is red and the other is orange. Each circle corresponds to the slope of the corresponding device. The red limits are those limits within which all slopes should fall, if there is no linearity (zero slope). [Due to measurement error, there will always be some variation in slopes. Circles falling inside these limits (shown in green) imply that these devices have no linearity.

The orange limits are used to better detect differences in slopes amongst the different measuring devices. The limits are places around the overall slope, not zero slope, as for the red lines. If the orange limits are near equivalent to the red limits, such as shown below then the overall slope is zero.

As an example for interpretation, consider the first of the slope consistency charts. The orange limits are clearly below the red limits. This means that the overall slope is not zero, confirming the Analysis of Covariance. All points are red meaning that every device has a non-zero slope, i.e. every device has linearity. All are within the orange limits, hence implying that there is no statistically significant difference in slopes between the devices.

Now, consider the image above. The orange limits are equivalent to the red limits. Thus, there is no overall linearity. No point falls outside the red limits, meaning that no measuring device has linearity. No point falls outside the orange limits (in this case same as red) and hence all devices are the same, as already concluded.

These results are consistent with the Analysis of Covariance were all sources of variation are insignificant.

Please note that the orange limits can also fall above the red limits.

Probability Plot

The probability plot on the residuals is used to establish normality of the residuals.

The Analysis of Covariance in-particularly relies on the normality assumption. However if the points fall closely to the straight line, even if not perfectly normal, this analysis will provide sufficient evidence of linearity, if used in combination with the Slope Consistency Chart and Scatter Diagram.

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