ARTICLES

# Bias Analysis (Multiple Equipment)

Bias is an important characteristic of a measuring system. It can be defined as the difference between the reference (â€˜actualâ€™) value of a part and the measured value. Bias is related to accuracy.

Bias is usually studied one equipment (instrument, device) at a time. However, sometimes there are multiple devices available to the appraiser all the same type and model. It is then important to establish if there are differences in bias amongst the devices.

One at a time studies and comparing is an inefficient way of doing so. A better solution is performing an Analysis of Variance (ANOVA) applied to all instruments of the same model. Different models should not be combined unless it is has been proven that the repeatability (precision) is the same amongst the different models.

The ANOVA will determine if there are statistical differences in the average bias as measured on one part, by one appraiser, and the contribution to total measurement variation.

Input consists of one column of measurements for each device. Bias for each measurement is obtained by subtracting the reference value from the measurements. This is done by the BIS.Net APP.

 27.64 28.4 28.01 27.48 28.24 27.86 26.82 28.54 28.44 27.42 28.61 28.69 27.2 27.82 28.32 27.91 27.93 27.75 28.2 28.45 28.21

The BISNet App provides the following output, each of which are now explained.

### Dart Board

The dart board is a visual tool which for this application enables the analyst to see at a glance how consistent the bias is over the different devices and how good or bad the repeatability is.

The circles are placed at 1 standard deviation (green), 2 standard deviations (yellow), 3 standard deviations (orange) and beyond (red) around zero, which is the target value for bias. The zero-reference value is the large white circle.

The circles have been randomly placed around the centre just as if they were thrown darts. This is an effective way for visualizing repeatability and reproducibility. Each coloured point corresponds to a different appraiser.

The horizontal line is a more conventional, but less effective way of showing the scatter.

The large black circles are the average estimate of bias for each device, highlighting bias inconsistencies, if they exist.

If the black circles cluster around the white circles, as shown in the above image, then the average bias of all instruments is close to zero. The tighter the cluster the better. If all black circles fall to one side of the white circle there is a serious overall bias. An example is shown in the image below.

Bias Evident

### Bias Consistency Chart

The consistency chart is used to visually assess inconsistencies with bias across the instruments. The average bias for each instrument should fall within the two red lines. The large circles are the average biases for each instrument. The small circles are the individual estimates of the bias. Variation between the green circles are probably due to chance. Red circles are due to factors other than chance, usually a faulty instrument.

### Probability Plot

The probability plot is used to establish normality of the measurement error (residuals). BIS.Net MSA uses the Anderson Darling Statistic and will advise if the if there is evidence of non-normality. The ANOVA method does assume normality. Fortunately, measurement error tends to follow a normal distribution.

### Tabular output

The Analysis of Variation table is included for completeness. If the result is significant than there is statistical evidence that the differences in bias between the devices is not due to chance alone. In this instance the conclusion is confirmed by consistency chart where there was one red circle.

Bias Inconsistency refers to differences in bias between the devices. Repeatability is measurement error.

The approximate confidence intervals are the intervals within which the consistency and repeatability, measured by standard deviation, are likely to fall, at the chosen level of significance. If the default of .05 has been used, then the confidence coefficient is equal to 100-.05 *100=95

The Bias table displays the bias for each device if there is a statistically significant difference between the devices. A significant difference from zero, relative to the chosen level of significance is highlighted in the Significance column. The p-value shows the actual significance level. The confidence intervals show where the true values may fall.

The measurement system performance table shows the percentage of variation due to Bias inconsistency relative to either the total study variation, or process variation if entered. The last column uses variance instead of standard deviation. Also shown is the 6 standard deviation range due to variation in bias between the different instruments

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