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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 Xbar and Range (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.

  1. 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.
  2. On Options, choose the estimation method and the analysis options.
  3. On Gage Info, optionally record the gage and study details.
  4. 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:

  1. User Input, and the Gage Info block beside it
  2. Stats Advisor
  3. Notes, when there are any
  4. the Gage R&R results table, with the confidence interval table below it
  5. the analysis of variance table, beside them
  6. Probabilities of Misclassification
  7. 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. The header states the multiplier the numbers under it actually used. Six is the default and every study type but Extended runs at it; an Extended study run at a different Study variation multiplier (k): heads the column with that value, so k of 5.15 gives Study Var (5.15 x SD)
% 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
AIAG Statistics a heading of its own, opening the four percentages below it
%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. An upper end the run does not report prints as unbounded; with no lower end there is no interval to show and the bare count stands alone

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, the study variation column under the same header the results table gives it, then % Contribution and % Tolerance. Each cell holds a pair written as lower to upper.

Under restricted maximum likelihood the % Contribution column is not there at all. That method fills no bound on a percent contribution, because the bound needs a confidence interval on a ratio of two sums that the method does not supply, so the column is left out rather than shown empty. The test is the method the run actually used, so an Automatic run that resolved to REML on unbalanced data leaves it out too, and the Estimation method: row near the top of the report is where you confirm which method that was.

Where a bound is not reported for any other reason, 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; Xbar and Range reports none; and a fixed term has 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.
  • The alpha the removal loop used is stated below the block. A line reads Alpha to remove interaction term = 0.25, or whatever threshold the run applied. It appears whenever interaction removal ran, which is by default on every study type, and it appears whether or not anything was actually removed. It is absent when removal did not run at all: the data have no replicates, or an Extended study cleared Remove insignificant interactions.
  • 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, at the very bottom of the report: after every chart above them and directly above the Linearity & Bias charts they describe, so the tables and those charts read together.

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 notes that hover on cells

Several cells on the report carry a note that appears when you hover over them. They are ordinary Excel cell comments, so they print only if you ask Excel to print comments, and the little marker in the cell corner is what tells you one is there. A note appears only when its own cell does, so the two that belong to tolerance are absent from a study with no specification limit.

In the Probabilities of Misclassification table, each probability row's label carries a note describing that probability in words.

In the Gage R&R results table there are up to eight. Four state a rule of thumb, and each sits on the number it judges:

Cell What the note says
% Tolerance on the Total Gage R&R (GRR) row under 10 percent acceptable, 10 to 30 percent marginal and depending on the application, over 30 percent unacceptable, citing AIAG MSA 4th ed., p 78 (3rd ed., p. 77)
the %GRR number the same three bands read against the total standard deviation rather than the tolerance, citing AIAG MSA 4th ed., p 78
the Number of Distinct Categories (ndc) number that ndc counts the non-overlapping 97 percent confidence intervals spanning the expected product variation, and should be 5 or more, citing AIAG MSA 4th ed., p 123. (3rd ed., p. 117)

Four more state a definition and its equation, and each sits on a label rather than on a number, so that the %GRR number is left free for its rule of thumb. One Excel cell carries one comment.

Label cell The equation the note gives
%GRR 100 times the Total Gage R&R standard deviation over the Total Variation standard deviation, with the note adding that the Total Gage R&R standard deviation is the square root of EV squared plus AV squared
%EV 100 times the repeatability standard deviation over the Total Variation standard deviation
%AV 100 times the reproducibility standard deviation over the Total Variation standard deviation
%PV 100 times the part to part standard deviation over the Total Variation standard deviation

The eighth sits on the P/Tol Ratio label and is the one that says what the statistic is not:

This is not an AIAG statistic. It is defined by NIST:

It then gives the ratio as the study variation multiplier times the Total Gage R&R standard deviation over the tolerance width, writing out the multiplier the run actually used, and notes that NIST's own form, three standard deviations of measurement error over the half tolerance, is the same number. It cites the NIST/SEMATECH e-Handbook glossary.

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, and the first three rows are pairs meant to be read across:

Each chart is named below by its own title, as it appears on the chart.

Row Left Right
1 Components of Variation Gage Performance Curve
2 Measurement by Operator Operator Box Plot
3 Measurement by Part Part Box Plot

Each pair puts the same question two ways: the components against the curve those components produce, and each measurement scatter against the box plot that groups the same way.

Then, still in two columns:

  1. Operator x Part Interaction
  2. the misclassification sweep charts, the two joint ones side by side

The bias and linearity content closes the report, in that order: the Bias and Linearity tables come after every chart above, and below them the linearity charts: one titled Linearity & Bias -- Pooled, then one per operator with that operator's name in place of Pooled. With no Reference (Optional) column the tables report nothing and those charts are not drawn, so nothing shifts.

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 deliberate exception is the pairs. The two joint sweeps share a row, and so do each of the three pairs in the table above, which can leave a gap beside the chart above them so that a pair always stays side by side. A pair whose partner did not render sits alone.

The Linearity & Bias charts also start a fresh row rather than filling the slot beside the last chart above them, because the bias and linearity tables sit between the two groups.

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.

Operator x Part 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. Note that its title spells the term the other way round from the analysis of variance table, which joins the two factor names with a multiplication sign, for example Part × Operator.

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 reference value, on an axis titled Reference Value of Measured Part, 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.

Any of these withholds it. It needs at least one specification limit and is absent without one. It is absent when Part Variation is Not Representative of the Population is ticked, because the curve is drawn from the part to part standard deviation and that option withholds it. A run launched from a template draws it only when the template dialog's Display gage performance curve box was ticked, which it is not by default. And, like every chart here, it is absent when the study produced no points for it.

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 those two and the probability-the-part-is-good sweep, three in all. The two conditional probabilities are reported as numbers in the misclassification table at the process mean, and no chart sweeps them.

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 & Bias charts plot bias against reference value, with the fitted line and a confidence band. One chart pools every operator and is titled Linearity & Bias -- Pooled; one more is drawn per operator and titled with that operator's name in place of Pooled.

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.

See Also