2.29.2 Algorithm for Gage R&R Study

Contents

Type 1 Gage Study

Basic Statistics

  • Mean
    \[\bar{X} = \frac{1}{N}\sum_{i=1}^{N}x_i\]
  • StdDev
    \[S = \sqrt{\frac{1}{N-1}\sum_{i=1}^{N}(x_i-\bar{x})^2}\]
  • Study Variation(SV)
    \(SV = k_1 * S\), where \(k_1\) is the number of SD specified in the dialog. Default is 6.
  • Tolerance
    Tolerance = USL – LSL specified in the dialog.
  •  % of Tol: Calculate whether the gage resolution (specified in the dialog) is less than(good), greater(bad), or equal to 5% of the tolerance.

Bias

  • Bias
    \(\bar{X} - X_m\), where \(X_m\) is the reference mean value specified in the dialog.
  • T
    t-statistics to test the null hypothesis \(bias = 0\) vs alternative hypothesis \(bias \neq 0\):
    \[t = \frac{\sqrt{N}|\bar{X} - X_m|}{S}\]

Capability

  • Cg
    The capability of the gage: \(C_g = \frac{K/100*Tolerance}{SV}\), where \(K\) is the percent of the tolerance for calculating \(C_g\) which is specified in the dialog.
  • Cgk
    The capability of the gage, considering both the gage variation and the bias: \(C_{gk} = \frac{(K/200*Tolerance) - |\bar{X} - X_m|}{SV/2}\)
  •  %Var (Repeatability)
    Compare the gage repeatability with the tolerance: \(\frac{K*SV}{Tolerance}*100\)
  •  %Var (Repeatability and Bias)
    Compare the gage repeatability and bias with the tolerance: \(\frac{(K*SV)/2}{(K/200*Tolerance) - |\bar{X} - X_m|}\)

Gage Linear Bias Analysis

Utilize the Bias versus Reference Value plot to observe the variation in bias values \(x_{ij}-ref_i\) for each part. Subsequently, apply linear regression to the Bias versus Reference Value plot to estimate the slope and intercept.

Gage Linearity

  • S
    \(S\) estimates the standard deviation around the regression line. \(S = sqrt(RSS/df)\), where \(RSS\) is the residual sum of squares and \(df\) is the degree freedom of the error terms of the linear regression.
  • Linearity
    Linearity assesses whether the gage maintains consistent accuracy across all sizes of objects being measured.
    \( Linearity = |slope| * PV\), where \(PV\) is \(Process\;Variation\) which represents 6 * the process standard deviation and is specified in the dialog if user has it.
  •  %Linearity
     %Linearity represents linearity as a percentage of the process variation.
    \[ \% Linearity = \frac{Linearity}{PV}*100\]

Gage Bias

  • Bias
    Bias refers to the disparity between the part's reference value and the measurements taken by the operator.
    \[ Average\;Bias = \frac{\sum_{i=1}^{r}\sum_{j=1}^{N_i}(x_{ij}-ref_i)}{\sum_{i=1}^{r}N_i}\]
    where \(x_{ij}\) is the \(j^{th}\) measurement of the \(i^{th}\) part, \(ref_i\) is the reference value of the \(i^{th}\) part, \(N_i\) is the number of replicates of the \(i^{th}\) part, \(r\) is the number of parts.
  •  %Bias
     %Bias represents bias as a percentage of the process variation.
    \[ \% Bias= \frac{|Average\;Bias|}{PV}*100\]

Methods to estimate repeatability standard deviation

  • Use the p-values to test whether the bias is 0 at each reference value, and whether the average bias is 0.
  • sample range method
    If each reference value corresponds to a unique part, \(s = \frac{max(X_i) - min(X_i)}{d_2}\).
    If more than one part has the same reference value, \(s = \frac{\bar{R}}{d_2}\), where \(\bar{R}\) is the average range of the bias of each part and \(d_2 = d_2^*[m_i, i]\).
    The t-statistic for testing bias is \(\frac{(average\;bias\;at\;a\;reference\;value)}{\frac{s}{\sqrt{r}}}\), where \(r\) is number of parts.
  • sample standard deviation method
    If each reference value corresponds to a unique part, \(s = \sqrt{\frac{1}{N-1}\sum_{i=1}^{N}(x_i-\bar{x})^2}\). The t-statistics for testing bias is \(\frac{(average\;bias\;at\;a\;reference\;value)}{\frac{s}{\sqrt{r}}}\).
    If more than one part has the same reference value, \(s = \sqrt{\frac{(N_1-1)*S_1^2 + ...+ (N_r-1)*S_r^2}{(N_1-1)+ ...+ (N_r-1)}}\). The t-statistics for testing bias is \(\frac{(average\;bias\;at\;a\;reference\;value)}{{s}*{\sqrt{1/N_1 + ...+ 1/N_r}}}\).

Crossed Gage R&R Study

ANOVA Table

  • When you enter Operators and Parts, the data is analyzed with a balanced two-factor factorial design. Both factors are considered to be random. The Operator by Part interaction is included in the model first:
    \[SS_{Total}=SS_{Rep eatability}+SS_{Part}+SS_{Operator}+SS_{Part*Operator} \,\!\]
  • If the p-value for the interaction is greater than the significance level, the interaction term will be ignored and the data is then fitted with a reduced model with only main terms.
    \(SS_{Total}=SS_{Rep eatability}+SS_{Part}+SS_{Operator} \,\!\),
    where
    \[SS_{Total}=\sum_{i=1}^r\sum_{j=1}^s\sum_{k=1}^t(y_{ijk}-\bar y)^2\]
    \[SS_{Part}=st\sum_{i=1}^r(\bar y_{i\cdot\cdot}-\bar y)^2\]
    \[SS_{Operator}=rt\sum_{j=1}^s(\bar y_{\cdot j\cdot}-\bar y)^2\]
    \[SS_{Part*Operator}=t\sum_{i=1}^r\sum_{j=1}^s(\bar y_{ij\cdot}-\bar y_{i\cdot\cdot}-\bar y_{\cdot j\cdot}+\bar y)^2\]
  • When the interaction term is in the ANOVA model:
    \[SS_{Rep eatability} = SS_{Total} - SS_{Part} - SS_{Operator} - SS_{Part*Operator}\]
  • When the interaction term is not in the ANOVA model:
    \[SS_{Rep eatability} = SS_{Total} - SS_{Part} - SS_{Operator}\]
    \(SS_{Total}\) is the total sum of square, \(SS_{Part}\) represents the variability of the average differences from factor Part, \(SS_{Operator}\) represents the variability of the average differences from factorOperator, \(SS_{Part*Operator}\) represents the variability of interaction, and \(SS_{Rep eatability}\) represents the variability of all individual samples. \(r\) represents the number of parts. \(s\) represents the number of operators. \(t\) represents the number of replicates.
  • Two-way ANOVA table with interaction:
Source of Variation Degrees of Freedom (DF) Sum of Squares (SS) Mean Square (MS) F Value Prob > F
Part r - 1 \[SS_{Part}\] \[MS_{Part}\] \[\frac{MS_{Part}}{MS_{Part*Operator}}\] \[P\{F\geq F_{(r-1,(r-1)(s-1),\alpha )}\}\]
Operator s - 1 \[SS_{Operator}\] \[MS_{Operator}\] \[\frac{MS_{Operator}}{MS_{Part*Operator}}\] \[P\{F\geq F_{(s-1,(r-1)(s-1),\alpha )}\}\]
Part*Operator (r- 1) (s - 1) \[SS_{Part*Operator}\] \[MS_{Part*Operator}\] \[\frac{MS_{Part*Operator}}{MS_{Rep eatability}}\] \[P\{F\geq F_{((r-1)(s-1),rs(t-1),\alpha )}\}\]
Repeatability rs (t - 1) \[SS_{Rep eatability}\] \[MS_{Rep eatability}\]
Total rst - 1 \[SS_{Total}\]
  • Two-way ANOVA table without interaction:
Source of Variation Degrees of Freedom (DF) Sum of Squares (SS) Mean Square (MS) F Value Prob > F
Part r - 1 \[SS_{Part}\] \[MS_{Part}\] \[\frac{MS_{Part}}{MS_{Rep eatability}}\] \[P\{F\geq F_{(r-1,rst - r - s + 1,\alpha )}\}\]
Operator s - 1 \[SS_{Operator}\] \[MS_{Operator}\] \[\frac{MS_{Operator}}{MS_{Rep eatability}}\] \[P\{F\geq F_{(s-1,rst - r - s + 1,\alpha )}\}\]
Repeatability rst - r - s + 1 \[SS_{Rep eatability}\] \[MS_{Rep eatability}\]
Total rst - 1 \[SS_{Total}\]

Number of Distinct Categories

  • The number of distinct categories represents the number of groups that the measurement system can differentiate.
    \[Number\;of\;distinct\;categories\;=\;\frac{Stddev\;for\;Parts}{Stddev\;for\;Gage}*1.41\]

Gge R&R Table

Variance for ANOVA method

The variance components are calculated based on the ANOVA table. The value will be reported as zero if is negative.

  • With interaction:
    \[VARCOMP_{Rep eatability} = MS_{Rep eatability}\]
    \[VARCOMP_{Operator} = \frac{MS_{Operator} - MS_{Operator*Part}}{rt}\]
    \[VARCOMP_{Operator*Part} = \frac{MS_{Operator*Part} - MS_{Rep eatability}}{t}\]
    \[VARCOMP_{Part-to-Part} = \frac{MS_{Part} - MS_{Operator*Part}}{st}\]
    \[VARCOMP_{Reproducibility} = VARCOMP_{Operator} + VARCOMP_{Operator*Part}\]
    \[VARCOMP_{Total-Gage} = VARCOMP_{Rep eatability} + VARCOMP_{Reproducibility}\]
    \[VARCOMP_{Total-Variation} = VARCOMP_{Total-Gage} + VARCOMP_{Part-to-Part}\]
  • Without interaction:
    \[VARCOMP_{Rep eatability} = MS_{Rep eatability}\]
    \[VARCOMP_{Operator} = \frac{MS_{Operator} - MS_{Rep eatability}}{rt}\]
    \[VARCOMP_{Part-to-Part} = \frac{MS_{Part} - MS_{Rep eatability}}{st}\]
    \[VARCOMP_{Reproducibility} = VARCOMP_{Operator}\]
    \[VARCOMP_{Total-Gage} = VARCOMP_{Rep eatability} + VARCOMP_{Reproducibility}\]
    \[VARCOMP_{Total-Variation} = VARCOMP_{Total-Gage} + VARCOMP_{Part-to-Part}\]

Variance for Xbar and R method

  • For variance contributed by each source, the standard deviation is calculated as:
    \[STDDEV_{Rep eatability} = (\sum_{i=1}^r\sum_{j=1}^s\frac{R_{ij}}{rs})\times \frac{1}{d_2}\]
    where \(R_{ij}\) is the range of measurements by operator j for part i. \(d_2 = d_2^*[rs, t]\).
    \[STDDEV_{Reproducibility} = \sqrt{\biggr[\bar{X}_{diff}*\frac{1}{d_2}\biggr]^2 - \biggr[\frac{(STDDEV_{repe atability})^2}{rt}\biggr]}\]
    where \(\bar{X}_{diff} = max(\bar{X}_1, ..., \bar{X}_k) - min(\bar{X}_1, ..., \bar{X}_k)\), \(d_2 = d_2^*[1, s]\)
    \[STDDEV_{Part-to-Part} = R_p\frac{1}{d_2}\]
    where \(R_p\) is the range of part average values, \(d_2 = d_2^*[1, r]\)
    \[STDDEV_{Total-Gage} = \sqrt{(STDDEV_{Rep eatability})^2 + (STDDEV_{Reproducibility})^2}\]
    \[STDDEV_{Total-Variation} = \sqrt{(STDDEV_{Total-Gage})^2 + (STDDEV_{Part-to-Part})^2}\]

%Contribution

\[ \% Contribution = \frac{VARCOMP}{VARCOMP_{Total-Variation}} \]

StdDev

If historical standard deviation \(\sigma\) is specified and is larger than the gage standard deviation \(\hat{\sigma}_{Gage}\), then the total standard deviation is \(\sigma\) and \(\hat{\sigma}_{Part} = \sqrt{\sigma^2 - \hat{\sigma}^2_{Gage}}\). Otherwise, total standard deviation calculated from the data is used: \(StdDev = \sqrt{VarComp}\)

Study Var

The study variation is calculated as the standard deviation for each source of variation multiplied by 6 or the multiplier specified in Study variation. \(Study\;Var\;=\;k\;*\;StdDev\)

%Study Var

\[ \%StudyVar = \frac{Study\;Var}{Total\;Variation}*100\]

%Tolerance

\( \%Tolerance = \frac{Study\;Var}{Tolerance}*100\), where \(Tolerance\) is user entered.

%Process

\( \%Process = \frac{Study\;Var}{6 * (Historical\;Stddev)}*100\), where \(Historical\;Stddev\) is user entered.

Nested Gage R&R Study

ANOVA Table

  • Partition of the variation into components for the ANOVA table:
    \[SS_{Total}=SS_{Rep eatability} + SS_{Operator}+SS_{Part(Operator)} \,\!\]
    where
    \[SS_{Total}=\sum_{i=1}^r\sum_{j=1}^s\sum_{k=1}^t(y_{ijk}-\bar y)^2\]
    \[SS_{Operator}=rt\sum_{j=1}^s(\bar y_{\cdot j\cdot}-\bar y)^2\]
    \[SS_{Part(Operator)}=t\sum_{i=1}^r\sum_{j=1}^s(\bar y_{ij\cdot}-\bar y_{\cdot j\cdot})^2\]
    \[SS_{Rep eatability} = \sum_{i=1}^r\sum_{j=1}^s\sum_{k=1}^t(y_{ijk}-\bar y_{ij\cdot})^2\]

\(SS_{Total}\) is the total sum of square, \(SS_{Operator}\) represents the variability of the average differences from factorOperator, \(SS_{Part(Operator)}\) represents the variability of nested factors, and \(SS_{Rep eatability}\) represents the variability of all individual samples. \(r\) represents the number of parts. \(s\) represents the number of operators. \(t\) represents the number of replicates.

ANOVA table with the nested term:

Source of Variation Degrees of Freedom (DF) Sum of Squares (SS) Mean Square (MS) F Value Prob > F
Operator s - 1 \[SS_{Operator}\] \[MS_{Operator}\] \[\frac{MS_{Operator}}{MS_{Part*Operator}}\] \[P\{F\geq F_{(s-1,(r-1)(s-1),\alpha )}\}\]
Part(Operator) (r- 1) s \[SS_{Part(Operator)}\] \[MS_{Part(Operator)}\] \[\frac{MS_{Part(Operator)}}{MS_{Rep eatability}}\] \[P\{F\geq F_{((r-1)s,rs(t-1),\alpha )}\}\]
Repeatability rs (t - 1) \[SS_{Rep eatability}\] \[MS_{Rep eatability}\]
Total rst - 1 \[SS_{Total}\]

Gge R&R Table

  • The variance components are calculated based on the ANOVA table. The value will be reported as zero if is negative.
    \[VARCOMP_{Rep eatability} = MS_{Rep eatability}\]
    \[VARCOMP_{Reproducibility} = \frac{MS_{Operator} - MS_{Part(Operator)}}{rt}\]
    \[VARCOMP_{Part-to-Part} = \frac{MS_{Part(Operator)} - MS_{Rep eatability}}{t}\]
    \[VARCOMP_{Total-Gage} = VARCOMP_{Rep eatability} + VARCOMP_{Reproducibility}\]
    \[VARCOMP_{Total-Variation} = VARCOMP_{Total-Gage} + VARCOMP_{Part-to-Part}\]

Expanded Gage R&R Study

The app uses the general linear regression model to perform Gage R&R studies with three types of ANOVA models: the random-effects model, the mixed-effects model, and the nested designs model. By default the random-effects model is used. The mixed-effects model is used if any fixed factor is specified. The nested term will be involved if nested term is specified.

The model used for the gage study includes the main effects and the significant highest order interactions and the relevant interactions between. The app uses Fit General Linear Model to generate the ANOVA table and estimate the variance components of the factors and their interactions.

Variance Components for random effects

  • Partition of the variation into components for the ANOVA table:
    \[VARCOMP_{Rep eatability} = MS_{Rep eatability}\]
    \[VARCOMP_{Reproducibility} = VARCOMP_{Operator}\;+\;VarComp\;of\;other\;factors\;and\;interactions\]
    \[VARCOMP_{Part-to-Part} = VARCOMP_{Part}\]
    \[VARCOMP_{Total-Gage} = VARCOMP_{Rep eatability} + VARCOMP_{Reproducibility}\]
    \[VARCOMP_{Total-Variation} = VARCOMP_{Total-Gage} + VARCOMP_{Part-to-Part}\]

Variance Components for fixed effects

  • For fixed terms, the variability across the levels of the term is estimated to represent the variance components. After fitting with general linear model, the fitted coefficients for the first \(J-1\) levels of the factor are calculated. The coefficient for Jth level is \(Coef_J = –\sum_{j=1}^{J-1}coef_j\). Then: \( VARCOMP = \sum{coef_j}^2\).

Attribute Gage Study

Bias

  • Bias
    \( limit + b/a \), where \(limit\) is tolerance limit provided by user. \(b\) and \(a\) are the intercept and slope from the fitted line on the probability plot.
  • Pre-adjusted repeatability
    \(X_t(P_a0.995)-X_t(P_a0.005)\), where \(X_t\) is the estimated reference values at acceptance probabilities of 0.995 and 0.005 on the fitted line.
  • Repeatability
    \[\frac{X_t(P_a0.995)-X_t(P_a0.005)}{1.08}\]

Test of Bias = 0

  • AIAG method
    T: \(\frac{31.3 \times |bias|}{(X_t(P_a0.995)-X_t(P_a0.005))/1.08}\)
    DF: \(N - 1\), where \(N\) is the number of trials.
  • Regression Method
    T: \(\frac{a + b \times LL}{s\sqrt{\frac{1}{K}+\frac{(LL - \bar{x})^2}{\sum(x_i-\bar{x})^2}}}\), where \(LL\) is the lower tolerance limit, \(s\) is the error standard deviation from the fitted line, \(K\) is the number of parts, \(x_i\) is the reference value of each part, \(\bar{x}\) is the mean of the reference values.
    DF: \(N - 2\), where \(N\) is number of points used for the fitting.

Reference