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Variance Components¶

This page defines what a measurement system analysis estimates, and the rule that turns those estimates into the rows of the Gage R&R table. Every study uses it, whichever estimation method produced the numbers, and whether the study is Crossed, Nested or Extended.

Notation¶

Term	Description
\(y\)	one measured value
\(\mu\)	the grand mean of every measured value that was kept
\(\tau_t\)	the contribution of model term \(t\)
\(\varepsilon\)	the residual
\(\sigma^2_t\)	the variance component of term \(t\)
\(\hat{\sigma}^2_{t,\text{raw}}\)	the estimate of \(\sigma^2_t\) before the floor at zero
\(\sigma^2_{\text{repeat}}\)	repeatability, the residual component
\(\sigma^2_{\text{reprod}}\)	reproducibility, the sum of the terms that are not part variation
\(\sigma^2_{\text{part}}\)	part to part variation
\(\sigma^2_{\text{gage}}\)	total gage R&R, repeatability plus reproducibility
\(\sigma^2_{\text{total}}\)	total variation, total gage R&R plus part to part
\(\alpha_{tj}\)	an estimated coefficient of a fixed term \(t\)
\(J_t\)	the number of levels of term \(t\)

The model¶

A measurement system analysis fits a variance components model. Every measured value is written as the sum of the study's overall mean, one contribution per model term, and a residual:

\[ y = \mu + \sum_{t} \tau_t + \varepsilon \]

\(\mu\) is the grand mean, \(\tau_t\) is the contribution of model term \(t\) (a part effect, an operator effect, an interaction, or an additional factor in an Extended study), and \(\varepsilon\) is the residual. Each random term is taken to vary with its own variance component, and the residual with \(\sigma^2_{\text{repeat}}\):

\[ \operatorname{Var}(\tau_t) = \sigma^2_t, \qquad \operatorname{Var}(\varepsilon) = \sigma^2_{\text{repeat}} \]

Estimating those \(\sigma^2_t\) is the whole job of the analysis. Which estimation method is used decides how they are computed, but not what they mean, and not how they are then grouped and reported.

The grouping rule¶

Every estimated component is placed into exactly one of three buckets. The rule has three clauses and no exceptions:

Bucket	Which components go in it
Repeatability	the residual, and nothing else
Part to part	every term selected under Part-to-part variation
Reproducibility	every other term in the model

So reproducibility is defined by what is left over rather than by a list of its own. An interaction, an additional factor in an Extended study, and a term whose estimate came from a fixed effect all roll into reproducibility unless the user moved them to part to part.

That last point is what the Part-to-part variation checkbox list on the Model tab does, and the dialog says so in the same words: by default the variation from additional factors and interactions is attributed to reproducibility, and selecting a term attributes its variation to part to part instead.

The five aggregates¶

The three buckets are summed, and two further aggregates are formed from those sums:

\[ \sigma^2_{\text{repeat}} = \text{the residual component} \]

\[ \sigma^2_{\text{reprod}} = \sum_{t \,\notin\, \text{part}} \sigma^2_t \]

\[ \sigma^2_{\text{part}} = \sum_{t \,\in\, \text{part}} \sigma^2_t \]

\[ \sigma^2_{\text{gage}} = \sigma^2_{\text{repeat}} + \sigma^2_{\text{reprod}} \]

\[ \sigma^2_{\text{total}} = \sigma^2_{\text{gage}} + \sigma^2_{\text{part}} \]

\(\sigma^2_{\text{gage}}\) is reported as Total Gage R&R and \(\sigma^2_{\text{total}}\) as Total Variation. Every percentage on the report is a ratio against one of these two, so they are computed once and never recomputed further down the sheet.

The report prints one row per bucket total and one row per member component, in this fixed order: Total Gage R&R, Repeatability, Reproducibility total, each reproducibility member, Part to part total, each part to part member, Total Variation. Members appear in ascending model term order.

Why a component can be reported as zero¶

A variance cannot be negative, but an estimate of one can be. The report therefore carries two numbers for each component: the estimate as computed, and the estimate after a floor at zero. The floored value is what is reported and what every sum, percentage and grouping uses; the raw value is kept beside it so a negative estimate is visible rather than hidden.

\[ \sigma^2_t = \max\left(0,\; \hat{\sigma}^2_{t,\text{raw}}\right) \]

A row whose raw estimate was negative, or whose constrained estimate stopped at the zero boundary, is flagged and carries a footnote. The two methods reach a reported zero for different reasons, and the footnote differs accordingly:

Under the expected mean squares method the moment estimate itself comes out negative, and zero is reported in its place.
Under restricted maximum likelihood nothing negative is ever computed. That method estimates the components under the constraint that they cannot be negative, so a reported zero means the estimate reached that boundary.

A fixed term reports a substitute¶

A term whose factors are all declared Fixed has no variance component, because a fixed effect has no distribution to have a variance. So that it can still take a place on the report, a substitute quantity is reported in the component column: the sum of the term's squared coefficients divided by its number of levels.

\[ \text{substitute}_t = \frac{1}{J_t}\sum_{j=1}^{J_t} \alpha_{tj}^2 \]

\(\alpha_{tj}\) are the term's estimated coefficients and \(J_t\) its number of levels. The row is marked as a fixed term substitute, and two consequences follow that a reader will notice:

It is not a variance component, even though it sits in the variance component column and is summed into an aggregate along with the real ones.
It carries no confidence interval, because a fixed effect has no sampling distribution for an interval to be built from.