STAT 503 Lesson 13 - Experiments with Random Factors Vocabulary

Vocabulary terms and definitions from Lesson 13 on Experiments with Random Factors, covering fixed vs. random factors, expected mean squares, mixed models, and gauge R&R studies.

Fixed factor

A factor whose particular levels are deliberately chosen because those specific levels are of interest, and statistical inference is confined to those levels.

Random factor

A factor whose observed levels are selected at random from a larger population of possible levels, and inference is intended to apply to that population.

Population of levels

The larger collection of possible factor levels from which the levels used in a random-factor experiment are sampled.

Random effect

A factor effect regarded as a random variable because the observed level was sampled from a larger population of possible levels.

One-factor random-effects model

$y_{ij} = \mu + \tau_i + \epsilon_{ij}$, for $i = 1, \dots, a$ and $j = 1, \dots, n$.

$$NID$$

Normally and independently distributed.

Variance component $\sigma_{\tau}^2$

The population variance among random factor-level effects.

Variance component $\sigma^2$

The residual or within-level error variance.

Variance components

Variance parameters that quantify different sources of random variation in a random- or mixed-effects model.

Random-effect null hypothesis

$H_0: \sigma_{\tau}^2 = 0$.

Expected Mean Square (EMS)

The theoretical expected value of a mean square under the assumed model; it determines the correct denominator for an F test and estimates variance components.

One-factor random-effect F statistic

$F_0 = MS_{\text{Treatments}} / MS_E$.

$E(MS_{\text{Treatments}})$ in balanced one-factor random model

$\sigma^2 + n \times \sigma_{\tau}^2$.

ANOVA method of variance-component estimation

A method where theoretical expected mean squares are set equal to observed mean squares to solve for unknown variance components.

$$\hat{\sigma}_{\tau}^2$$

$(MS_{\text{Treatments}} - MS_E) / n$.

Two-random-factor model

$y_{ijk} = \mu + \tau_i + \beta_j + (\tau \beta)_{ij} + \epsilon_{ijk}$.

F test for random A (Two-factor)

$F_0 = MS_A / MS_{AB}$.

Gauge R&R study

A measurement-system capability study used to quantify sources of variability, also known as a gauge repeatability and reproducibility study.

Gauge repeatability

$\sigma^2$, representing the within-part/within-operator variation from repeated measurement of the same part by the same operator.

Gauge reproducibility

Variation associated with operators, represented by $\sigma_{\beta}^2 + \sigma_{\tau \beta}^2$.

Mixed model

A model containing both fixed and random effects.

Restricted mixed-model equation

$y_{ijk} = \mu + \tau_i + \beta_j + (\tau \beta)_{ij} + \epsilon_{ijk}$ with the constraint $\sum_{i} \tau_i = 0$.

Restricted interaction constraint

$\sum_{i} (\tau \beta)_{ij} = 0$ for every $j$.

Unrestricted mixed model

A mixed model that does not impose the restricted-model sum-to-zero constraint on the random interaction.

Satterthwaite procedure

A method proposed in $1946$ that forms an approximate F ratio using linear combinations of original mean squares when no exact F test exists.

Synthetic mean square

A linear combination of original mean squares constructed to approximate the desired expected mean square for an F test.