Home / Shared Math Details / MSA / Bias and Linearity

Bias and Linearity¶

This page defines bias, its test against zero, and the linearity fit of bias against reference value. Together they are the only part of a measurement system analysis that can say whether the system reads correctly, as opposed to how much it varies.

Requires a reference column

Both are reported only when a Reference (Optional) column was supplied, and both are absent together otherwise.

Notation¶

Term	Description
\(x_{\text{ref}}\)	the reference value, the known true value of a part
\(\bar{y}\)	the mean of the measurements in the group being reported
\(b\)	the bias, \(\bar{y} - x_{\text{ref}}\)
\(SE(b)\)	the standard error of the bias
\(\nu\)	degrees of freedom
\(t\)	the t statistic of a bias row
\(\alpha\)	the significance level, the dialog's Significance level:, default \(0.05\)
\(\beta_0, \beta_1\)	the intercept and slope of the linearity fit
\(\varepsilon\)	the residual of that fit

Why a reference column is needed¶

Bias and linearity are the only part of the analysis that needs to know the true value of a part. Everything else in a measurement system analysis compares measurements against each other, so it can say how much the system varies without knowing whether it is right. These two say whether it is right.

That is why both are reported only when a Reference (Optional) column was supplied. Without one the whole block is absent, and its absence is not a problem to be reported.

Bias¶

Bias is the difference between what the system measures and what the part actually is:

\[ b = \bar{y} - x_{\text{ref}} \]

\(\bar{y}\) is the mean of the measurements in the group being reported and \(x_{\text{ref}}\) the reference value. A positive bias means the system reads high.

The four levels of grouping¶

Bias is reported at four levels of grouping, so a bias that only one operator has, or that only appears at one end of the range, is visible rather than averaged away:

Row kind	One row per
one reference value, one operator	each combination of reference value and operator
overall, one operator	each operator, across all their reference values
one reference value, pooled	each reference value, across all operators
overall, pooled	one row for the whole study

An overall row spans several reference values, so it has no single reference value of its own and reports none. Its bias is still a difference of measurement from reference, taken per measurement and then averaged, not a difference of two averages of unlike things.

Testing bias against zero¶

Each bias row is tested against zero with a t test:

\[ t = \frac{b}{SE(b)}, \qquad \nu = \text{the row's degrees of freedom} \]

and a two-sided p value is read from the t distribution on \(\nu\) degrees of freedom. A confidence interval is reported on the bias itself, at the confidence level set in the options:

\[ b \pm t_{1 - \alpha/2,\; \nu}\; SE(b) \]

The significance level the p value is judged against is the one set by Significance level:, which defaults to 0.05.

Reading the test the way the product does: a p value at or above the significance level means the bias is statistically zero, which is not the same as the bias being zero. It means the study did not have enough evidence to distinguish it from zero.

Linearity¶

Linearity asks whether the bias is the same everywhere in the measuring range, or whether it changes as the parts get bigger. It is a straight-line fit of bias against reference value:

\[ b = \beta_0 + \beta_1 x_{\text{ref}} + \varepsilon \]

\(\beta_1\), the slope, is how much the bias changes per unit of reference value. A slope of zero means the bias is the same across the range, whatever its size.
\(\beta_0\), the intercept, is the bias the line gives at a reference value of zero.

One row is reported per operator plus one pooled row, so an operator whose bias drifts across the range can be told from one whose bias is merely offset.

The fit uses one point per individual measurement, not one point per part average. A study with ten parts measured three times by each of three operators contributes ninety points to the pooled fit, not ten.

What each linearity row reports¶

Each linearity row reports the fit and then two quantities derived from the slope:

Reported	Meaning
intercept and slope	each with its standard error, t statistic and p value
R squared	the fraction of the variation in bias the line accounts for
regression standard error	the spread of the points about the line
residual standard deviation	what sets the half width of the confidence band drawn on the chart
degrees of freedom	the point count minus two
process variation used	the spread of the process the two quantities below are scaled against

\[ \text{Linearity} = \lvert \beta_1 \rvert \times \text{process variation} \]

\[ \%\text{Linearity} = 100 \times \frac{\text{Linearity}}{\text{process variation}} = 100 \times \lvert \beta_1 \rvert \]

Percent linearity reduces exactly to the absolute slope, because the process variation it is divided by is the same one it was multiplied by. That is worth knowing for two reasons: percent linearity does not depend on the process variation at all, and linearity does. The process variation that was used is reported alongside them for exactly that reason.

The acceptance verdict¶

Each linearity row also carries a single acceptable or not acceptable verdict. It is reported as acceptable when both the intercept and the slope t statistics fall below the critical value at the significance level, meaning neither a constant bias nor a changing bias could be distinguished from zero.

Both conditions are required. A row with a flat but clearly non-zero bias fails on the intercept, and a row whose bias passes through zero but tilts across the range fails on the slope.

The Linearity and Bias chart shows the same verdict graphically: the fitted line is drawn with a confidence band, and the verdict is about whether the horizontal line at zero bias lies inside that band.

When the block is refused¶

The block is refused in four situations, and the report prints the reason in these words:

Condition	What the report says
no reference column supplied	nothing; the section is simply absent
every reference value identical	`Bias and linearity were not computed: every reference value is identical.`
fewer than three distinct reference points	`Bias and linearity were not computed: fewer than three distinct reference points were supplied.`
the reference column is present but not usable	`Bias and linearity were not computed: the reference column is present but not usable.`

The first case prints no note on purpose. The other three all describe something the user supplied that could not be used, which is exactly the condition the Notes section exists to report.

Two of them are the same requirement seen from either side: a straight line through reference values needs at least two distinct reference values to have a slope at all, and the fit needs a third to have any degrees of freedom left to judge it by.