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Crossed¶

A crossed study is the ordinary gage repeatability and reproducibility study: every operator measures every part. Because the same part is measured by more than one operator, the operators can be compared on identical parts, which is what lets the analysis separate operator differences from part differences.

Find it at QXL Stat Tools > MSA / Gage R&R > Crossed.

When to use it¶

Use Crossed when the same physical parts went to every operator. If measuring destroys or changes the part, so that no part can be handed to a second operator, use Nested instead. If there is a factor besides part and operator, or you need to set a factor as fixed, use Extended.

In a crossed study every operator measures every part. Part 1 that operator A measured is the same physical part 1 that operator B measured:

Part 1 ── measured by ── Operator A, Operator B, Operator C
Part 2 ── measured by ── Operator A, Operator B, Operator C
Part 3 ── measured by ── Operator A, Operator B, Operator C

Because the same part appears under more than one operator, two operators can be compared on the same part. That is what makes two things possible that a nested study cannot offer:

Reproducibility can be separated from part to part variation. A difference between operators measured on identical parts cannot be a difference between the parts.
An operator by part interaction can be estimated, which is the question of whether the operators disagree more about some parts than about others.

What Crossed can do that Nested cannot¶

Two things follow from parts being shared between operators, and both appear on the report:

Reproducibility is separated from part to part variation. A difference between operators measured on identical parts cannot be a difference between the parts.
The operator by part interaction can be estimated, and its chart drawn. That is the question of whether the operators disagree more about some parts than about others.

A crossed study is also the only study that offers the XbarR estimation method.

The page sequence¶

The dialog has four tabs: Data, Options and Gage Info are used here. The Model tab belongs to an Extended study and is not shown.

Every control is described on Options.

On Data, name the measurement column or columns, the Part column, and the Operator column. Add a Reference column if you know the true values, and specification limits if you have them.
On Options, choose the estimation method and the analysis options.
On Gage Info, optionally record the gage and study details.
Choose Finish.

What you get¶

Each analysis writes its own worksheet, named for the study type: MSA Crossed, MSA Nested or MSA Extended. Where more than one sheet would take the same name, the later ones are numbered.

One analysis means one measurement column within one group, so the number of sheets is the number of measurement columns ticked multiplied by the number of groups.

Every sheet is laid out in the same order, top to bottom:

User Input, and the Gage Info block beside it
Stats Advisor
Notes, when there are any
the Gage R&R results table, with the confidence interval table below it
the analysis of variance table, beside them
Probabilities of Misclassification
the charts, with the bias and linearity tables inside that band

Anything the study did not produce is left out, and the sheet closes up rather than leaving a gap.

The tables¶

The User Input section records what the analysis was given: the data source, the study type, each role column that was supplied, the group where there is one, and an Estimation method: row.

The role rows are present only when the role was filled: Measurement column:, Part column:, Operator column:, Reference column:, Additional Factors: and Group:.

The Estimation method row names the method that was actually used, which is not always the one that was selected. It differs when Automatic made the choice and when a model REML could not fit caused a fallback to expected mean squares.

Beside the User Input section is the Gage Information block: Gage Name, Gage No., Gage Type, Part Name, Part No., Date and Performed By, plus the LSL and USL that were supplied.

Nothing here enters a calculation. It is there so a printed sheet identifies its own gage and study.

The Stats Advisor prints up to seven short verdicts on the study, each a heading and a sentence, colour coded. The seven are Total GR&R as a percent of tolerance, the number of distinct categories, Total GR&R as a percent of total standard deviation, the average chart, the range chart, bias, and linearity. A verdict appears only when the study produced the number it is about.

It closes with a Color Code Table giving the meaning of the three colours:

Colour	Meaning, in the table's own words
black	`Complies with rules of thumb`
blue	`Doesn't violate rules of thumb, but could be improved`
red	`Does not meet rules of thumb`

Each verdict sentence cites the AIAG Measurement Systems Analysis manual, with the page number, for the rule of thumb it applied.

The Advisor's verdicts are rules of thumb, and it says so

The Advisor compares the study against published rules of thumb and reports the comparison. Its own colour code table calls them rules of thumb, and each sentence names the reference it came from.

They are not thresholds Quantum XL sets, and a verdict is not a decision. Whether a measurement system is fit for a particular job depends on what the measurement is for, which is outside anything the study measured.

The Notes section reports inputs that were supplied but could not be used. It is absent when there are none.

The distinction is deliberate and worth knowing when reading a sheet: an input never supplied produces no note at all, just an absent section. So no note about the reference column means either that it was fine or that there was not one, and the User Input section is where to look to tell those apart.

What does produce a note: a term dropped because an empty cell made it unestimable; a misclassification block refused for a zero part variance, a historical standard deviation that was too small, or a failed calculation; and bias and linearity refused because the reference values were all identical, too few, or unusable.

Two further notes report a decision the analysis took on your behalf, and both concern a study with no replicates, meaning no part was measured twice by the same operator:

The data has no replicates, so interaction removal was turned off.

REML cannot fit a study without replicates; the analysis was rerun with expected mean squares.

The second is the clearest case of the Estimation method: row naming something other than what was selected, so the two are worth reading together.

The main table is headed Gage R&R Results. Its columns are:

Column	What it reports
`Source`	the component's name
`Variance`	the variance component
`Std Dev`	its square root
`Study Var (6 x SD)`	the study variation
`% Contribution`	percent of the total variance
`% Tolerance`	percent of the tolerance width, present only when a specification limit was supplied

The rows are indented into a hierarchy, widest first, so the report reads as a breakdown rather than a list:

Total Variation (TV)
  Total Gage R&R (GRR)
    Repeatability (EV)
    Reproducibility (AV)
      (one row per reproducibility member term)
  Part to part (PV)
    (one row per part to part member term)

Note that this is not the order the components are computed in. The table leads with Total Variation because it is the whole, and puts Part to part at the same level as Total Gage R&R because those two are what the whole divides into.

Beneath the component rows come four named percentage rows, then the two ratios:

Row	What it reports
`%GRR`, `%EV`, `%AV`, `%PV`	percent study variation of Total Gage R&R, repeatability, reproducibility and part to part. `NA` where the study did not produce it
`P/Tol Ratio`	the precision to tolerance ratio, present only with a specification limit
`Number of Distinct Categories (ndc)`	the count, with its confidence interval appended as `(CI: lower to upper)` when the run reports intervals

A component reported as zero, or reported as a substitute, carries a marker on its row label and a matching footnote below the table:

Marker	Footnote
`*`	the method-dependent zero footnote, quoted below
`(fixed)**`	`Fixed-term rows report a substitute quantity, not a variance component.`
`***`	`A negative reproducibility estimate was set to zero (XbarR method).`

The * footnote itself depends on the method, because a reported zero arises differently under each:

Under expected mean squares: A negative variance component estimate was set to zero. Consider the REML method, which estimates variance components under the constraint that they cannot be negative.
Under REML: This variance component was estimated as zero. REML constrains variance components to be non-negative, and this estimate reached that boundary.

The confidence interval table sits directly below the results table and reports the bounds on the same rows, with the same labels and in the same order, so the two read together. It is a separate table so the results table stays readable.

Its heading names the level, for example 95% Confidence Intervals, and its columns are Source, Variance, Std Dev, Study Var (6 x SD), % Contribution and % Tolerance. Each cell holds a pair written as lower to upper.

Where a bound is not reported, the table prints the reason as a sentence rather than leaving the cell blank. The reasons are all real conditions rather than failures, and the common ones are: an Extended study reports no intervals; the expected mean squares method on unbalanced data reports none; XbarR reports none; a fixed term has none; and under REML the percent contribution and percent study variation columns have none.

The analysis of variance table sits beside the results table, headed ANOVA (All Terms). Its columns are Source, DF, Seq SS, Adj SS, Adj MS, and then the F statistic with its p value. A p value below one thousandth prints as <0.001.

Two properties are worth knowing:

The F test is not always exact. Where a term has no single mean square to test against, a denominator is synthesized from several, and the degrees of freedom that go with it are not in general a whole number. The table prints Not an exact F-test. for such a term and reports the denominator it used and those degrees of freedom, naming the mean squares it was built from.
There can be two tables. When the interaction removal option removed something, a second table headed ANOVA (Terms used for the Gage R&R calculation) appears alongside the first, and a removed term is marked (removed) in the all-terms table. With nothing removed there is one table.

The table is absent entirely under the XbarR method, which computes no mean squares.

Probabilities of Misclassification reports what the measurement system does to accept and reject decisions: the two joint probabilities, the two conditional probabilities, and the probability that a part is good. An Extended study reports all of them; a Crossed or Nested study reports the joint pair.

It is headed Probabilities of Misclassification and groups its rows under Joint Probability and Conditional Probability, with % Parts Truly Good beside them. A Parameters block below records Mean Used, Process Std Dev Used and Measurement Std Dev Used, so every number can be traced and a historical standard deviation that replaced the estimated process variation is visible.

The whole block is absent, with a reason, when no specification limit was supplied, when the estimation method is XbarR, when the part to part variance is zero, or when a supplied historical standard deviation was not larger than the gage standard deviation.

The bias and linearity tables sit inside the chart band rather than with the other tables, below the box plots and above the gage performance curve, so they read alongside the Linearity and Bias charts they describe.

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

The bias table is keyed by Operator and Reference, with a pooled row labelled All, and reports Bias. The linearity table reports Intercept, Intercept P, Slope, Slope P, R-Sq, Linearity, % Linearity and an Acceptable verdict of Yes or No.

The charts¶

The charts appear in a fixed order. The average and range charts run the full width at the top; the rest follow in two columns:

Components of Variation
Part by Operator Interaction
Measurement by Part
Measurement by Operator
Box plot by Operator
Box plot by Part
(the bias and linearity tables)
Gage Performance Curve
the misclassification sweep charts
Linearity and Bias, pooled and then one per operator

A chart the study cannot produce is left out and the ones after it move up, so the grid never shows an empty slot where a chart would have been. The one deliberate exception is that the two joint sweeps share a row and the two conditional sweeps share a row, which can leave a gap beside the chart above them so that each pair stays side by side.

The average and range charts are drawn across the full width above the rest. They are ordinary average and range control charts with one series per operator, and their limits come from repeatability, so they ask whether each operator's measurements are consistent with the repeatability the study measured.

Because the limits come from repeatability rather than from the plotted points, the two charts compare each operator's results against the measurement error the study measured, not against their own spread.

The Stats Advisor reports on both. Its average chart verdict is based on what fraction of the plotted averages fall outside the limits, and it names a concern when fewer than half do. Its range chart verdict distinguishes three cases: all ranges in control, one or more operators with ranges out of control, and all operators with ranges out of control. The verdict sentences themselves are printed on the report.

Both are absent when the study has fewer than five measurements, which is the fewest a control limit can be computed from. The rest of the analysis still runs and reports normally.

With no operator column they are still drawn, grouped by part alone, with a single series each.

Components of Variation is a bar chart of the same percentages the results table reports, grouped so that percent contribution, percent study variation and percent of tolerance can be compared across components at a glance. A percentage whose inputs were not supplied has no bars.

Measurement by Part plots every individual measurement against its part, with a mean line, so a part that was measured inconsistently stands out from one that was not.

Measurement by Operator does the same by operator, so an operator who reads consistently high or low, or who is more variable than the others, stands out.

Part by Operator Interaction plots each operator's average against part, one line per operator. Lines that run parallel mean the operators agree about which parts are larger; lines that cross mean they disagree, which is what an interaction is.

The interaction chart needs parts measured by more than one operator, so it is absent from a nested study, where no part is measured twice. It is absent from any study with no operator column too.

The two box plots show the distribution of measurements by operator and by part, rather than the individual points.

A box needs at least four measurements in its group. A group with fewer gets no box, and where that happens the chart carries a printed note under it saying so, so a missing box is never left to be guessed at. The note distinguishes the two cases: some groups short, or every group short and the chart therefore empty.

This bites hardest on the box plot by part in a nested study. A nested part is measured by one operator only, so its group holds just that operator's trials: a study with two or three trials per part leaves every part below the minimum, and the part box plot draws no boxes at all. The box plot by operator is usually unaffected, because an operator's group gathers all of their parts.

The Gage Performance Curve plots the probability that a part is accepted against the part's true value, with the specification limits marked. A perfect gage would step from certain acceptance to certain rejection exactly at each limit; a real one slopes, and how steeply it slopes is the measurement error.

It needs at least one specification limit and is absent without one.

The sweep charts each plot one misclassification probability against a shifted process mean, so they answer what would happen to the accept and reject decisions if the process drifted.

Each is titled with the matching row label from the Probabilities of Misclassification table, and each carries a printed caption beneath it. The single number that table reports is the point on the curve where the shift is zero.

A Crossed or Nested study gets the two joint-probability sweeps; an Extended study gets all five. They need at least one specification limit, and they are withheld along with the misclassification table when part variation has been marked as not representative, unless a historical standard deviation was used.

The Linearity and Bias charts plot bias against reference value, with the fitted line and a confidence band. One chart pools every operator, and one more is drawn per operator.

The horizontal line at zero bias is what the chart is read against: where the band contains it across the whole range, no bias could be distinguished from zero there. The linearity verdict in the tables is that same comparison as a yes or no.

These charts are present only when a reference column was supplied.