## Confidence Limit Calculations for Variance Components

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The following calculations of confidence limits are used for GR&R Reports based on the GR&R and ANOVA test types.

### ANOVA Confidence Limit Calculations

Datapage+ uses the non-additive model when the interaction between parts and operators is significant; it uses the additive model when interaction is not significant.

Non-Additive Model

This model is used when the interaction between parts and operators IS SIGNIFICANT.

Variance Component | Confidence Limit (CL) | Degrees of Freedom
`tau2` (repeatabliity) | `CL = (MS_e)/(MS)` | `nk(r-1)`
`gamma2` (interaction) | `CL = (MS_e)/r \ \ [(MS_(po))/(F\ MS_e) -1]` | `(n-1)(k-1),nk(r-1)`
`alpha2` (part-to-part) | `CL = (MS_(po))/(kr) \ \ [(MS_(p))/(F\ MS_(po)) -1]` | `n-1, (n-1)(k-1)`
`omega2 + t2 + gamma2` (R&R) | `CL = (1)/(nr) \ \ [(MS_o)/(MS) +\ n\ (r-1)MS_e + (n-1)MS_(po)]` | `k-1`

`tau2`, `gamma2`, `omega2`, and `alpha2` are notations for the master variance components.

Additive Model

This model is used when the interaction between parts and operators is INSIGNIFICANT.

Variance Component | Confidence Limit (CL) | Degrees of Freedom
`tau2` (repeatability) | `CL = (MS_(p\o\o\l))/(MS)` | `nkr-n-k+1)`
`omega2` (reproducibility) | `CL = (MS_(p\o\o\l))/(nr) \ \ [(MS_o)\/(F\ MS_(p\o\o\l))-1 ] | `k-1, nkr-n-k+1`
`alpha2` (part-to-part) | `CL = (MS_(p\o\o\l))/(kr) \ \ [(MS_p)\/(F\ MS_(p\o\o\l))-1 ] | `n-1, nkr-n-k+1`
`tau2 + omega2` (R&R) | `CL = (1)/(nr) \ \ [(MS_o)/(MS) +\ (nr-1)\ MS_(p\o\o\l)]` | `k-1`

Degrees of Freedom: Degrees of freedom for the ANOVA test type are based on:

Operator `k-1`

Parts `n-1`

Oper X Parts `(n-1)(k-1)`

Gage (Error) `nk (r-1)`

where `n` is the number of parts measured; `r` is the number of times the part was measured; and `k` is the number of operators

For the ANOVA test type, degrees of freedom for errors associated with variance components will differ depending upon whether the additive or non-additive model is used (i.e., whether the interaction between parts and operators is significant or insignificant). Refer to the tables above for the calculations used in each model.

### GR&R Confidence Limit Calculations

| Lower Limit | Upper Limit
Repeatability (EV) | `EV sqrt((MS)(v1))` | `EV sqrt((MS)(v1))`
Reproducibility (AV) | `AV sqrt((MS)(v2))` | `AV sqrt((MS)(v2))`
Gage GR&R | `R&R sqrt((MS)(v1+v2))` | `R&R sqrt((MS)(v1+v2))`

Degrees of Freedom: For reports using the GR&R test type, degrees of freedom (n) are based on the number of samples (g) and the size of the samples (m). Refer to the Table of d*2 Values & Degrees of Freedom for specific values.