Detailed STAT 503 Lesson 13 - Experiments with Random Factors Practice Flashcards

A set of 150 vocabulary flashcards covering Lesson 13 on experiment designs with random factors, including model equations, EMS, and variance estimation.

Fixed factor

A factor whose particular levels are deliberately chosen because those specific levels are of interest; 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; inference is intended to apply to that population.

Main difference target (Fixed)

The individual factor-level effects are the targets of inference.

Main difference target (Random)

The variability among a population of possible factor levels is the target.

Population of levels

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

Industrial use of random-factor models

Widely used in measurement-system and gauge capability studies.

Random effect

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

Random factor categorical status

Random factors are usually categorical.

Continuous covariates vs. random effects

Continuous covariates are modeled systematically (e.g., linearly), whereas random effects are not treated as systematic trends.

One-factor random-effects model equation

$y_{ij} = \beta + \tau_{i} + \text{epsilon}_{ij}$ for $i = 1, \text{...}, a$ and $j = 1, \text{...}, n$

$\mu$ in the one-factor random model

The overall population mean.

$\tau_{i}$ in the one-factor random model

The random effect of the $i$-th sampled factor level.

$\epsilon_{ij}$ in the one-factor random model

Random experimental error for observation $j$ at sampled level $i$.

Distribution of experimental error

$\text{epsilon}_{ij}$ is $NID(0, \text{sigma}^{2})$

Distribution of treatment effects

$\tau_{i}$ is $NID(0, \text{sigma}_{\tau}^{2})$

NID

Normally and independently distributed.

Independence assumption in random models

$\tau_{i}$ and $\text{epsilon}_{ij}$ are independent.

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

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

Noise factor approach in robust design

Often treated as random because they are not controlled during normal operation.

Example: Store as a fixed factor

If a company has exactly five stores and studies all five because those specific five are the ONLY levels of interest.

Example: Store as a random factor

If a company randomly selects five stores out of $100$ to generalize inference to future or all stores.

Random factors in $2^{k}$ or $3^{k}$ designs

Two or three levels generally provide too little information for useful estimation of a population variance.

Fixed-effect null hypothesis concept

Concerns equality of treatment means or equivalently zero fixed treatment effects.

Random-effect null hypothesis ($H_{0}$)

$$H_{0}: \text{sigma}_{\tau}^{2} = 0$$

Random-effect alternative hypothesis ($H_{1}$)

$$H_{1}: \text{sigma}_{\tau}^{2} > 0$$

Interpretation of $H_{0}: \sigma_{\tau}^{2} = 0$

There is no population variability attributable to the random factor.

Interpretation of $H_{1}: \sigma_{\tau}^{2} > 0$

The population of factor levels has genuine variability beyond residual error.

ANOVA sums of squares in random models

The usual one-way ANOVA partition of total variation is still used.

Expected Mean Square (EMS)

The theoretical expected value of a mean square under the assumed model.

Role of EMS in ANOVA

Determines the correct denominator for an $F$ test and is used to estimate variance components.

$E(MS_{E})$ (one-factor random model)

$$\text{sigma}^{2}$$

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

$$\text{sigma}^{2} + n \times \text{sigma}_{\tau}^{2}$$

One-factor random-effect $F$ statistic ($F_{0}$)

$$F_{0} = \frac{MS_{Treatments}}{MS_{E}}$$

Numerator degrees of freedom (one-factor random)

$$a - 1$$

Denominator degrees of freedom (one-factor random)

$$N - a$$

ANOVA method of variance-component estimation

Set theoretical expected mean squares equal to observed mean squares and solve for unknown variance components.

Estimate of residual variance (ANOVA method)

$$\text{sigma}\_hat^{2} = MS_{E}$$

Variance-component equation for treatments

$$\text{sigma}\_hat^{2} + n \times \text{sigma}\_hat_{\tau}^{2} = MS_{Treatments}$$

Estimate of random treatment variance formula

$$\text{sigma}\_hat_{\tau}^{2} = \frac{MS_{Treatments} - MS_{E}}{n}$$

Negative variance-component estimate cause

Occurs when a treatment mean square is smaller than the mean square used to remove background variation.

Negative estimate treatment: Zero replacement

Replace the negative estimate by zero.

Negative estimate treatment: Alternative methods

Use another estimation method that constrains variance estimates to be nonnegative.

Confidence interval for $\sigma^{2}$

$\frac{(N-a)MS_{E}}{\text{chi-square}_{\alpha/2, N-a}}$ to $\frac{(N-a)MS_{E}}{\text{chi-square}_{1-\alpha/2, N-a}}$

Closed-form variance-component intervals

Not available for every variance-component parameter.

Example 13.1 subjects

Four looms randomly chosen from a weaving shed; four fabric-strength observations each.

Loom 1 data row sum

$$390$$

Loom 2 data row sum

$$366$$

Loom 3 data row sum

$$383$$

Loom 4 data row sum

$$388$$

Loom ANOVA: Loom source DF

$$3$$

Loom ANOVA: Loom source SS

$$89.188$$

Loom ANOVA: Loom source MS

$$29.729$$

Loom ANOVA: Loom source $F$

$$15.68$$

Loom ANOVA: Loom source $p-value$

$$0.000$$

Loom ANOVA: Error source DF

$$12$$

Loom ANOVA: Error source MS

$$1.896$$

Estimated loom variance component (Example 13.1)

$$6.958$$

Estimated error variance (Example 13.1)

$$1.896$$

$E(MS_{Loom})$ formula (Example 13.1)

$$\text{sigma}^{2} + 4 \times \text{sigma}_{Loom}^{2}$$

Two-factor random factorial setting

Both factors $A$ and $B$ have large populations of possible levels sampled for the experiment.

Two-random-factor model equation

$$y_{ijk} = \beta + \tau_{i} + \text{beta}_{j} + (\tau \text{beta})_{ij} + \text{epsilon}_{ijk}$$

Indices for two-random-factor model

$i = 1, \text{...}, a$; $j = 1, \text{...}, b$; $k = 1, \text{...}, n$

Random $A$ effect (two-factor)

$\tau_{i}$ is $NID(0, \text{sigma}_{\tau}^{2})$

Random $B$ effect (two-factor)

$\text{beta}_{j}$ is $NID(0, \text{sigma}_{\text{beta}}^{2})$

Random interaction effect (two-factor)

$(\tau \text{beta})_{ij}$ is $NID(0, \text{sigma}_{\tau \text{beta}}^{2})$

Total observation variance ($V(y_{ijk})$)

$$\text{sigma}_{\tau}^{2} + \text{sigma}_{\text{beta}}^{2} + \text{sigma}_{\tau \text{beta}}^{2} + \text{sigma}^{2}$$

Two-random factor A hypothesis

$H_{0}: \text{sigma}_{\tau}^{2} = 0$ versus $H_{1}: \text{sigma}_{\tau}^{2} > 0$

$E(MS_{A})$ (two-factor random)

$$\text{sigma}^{2} + n \times \text{sigma}_{\tau \text{beta}}^{2} + b \times n \times \text{sigma}_{\tau}^{2}$$

$E(MS_{B})$ (two-factor random)

$$\text{sigma}^{2} + n \times \text{sigma}_{\tau \text{beta}}^{2} + a \times n \times \text{sigma}_{\text{beta}}^{2}$$

$E(MS_{AB})$ (two-factor random)

$$\text{sigma}^{2} + n \times \text{sigma}_{\tau \text{beta}}^{2}$$

$E(MS_{E})$ (two-factor random)

$$\text{sigma}^{2}$$

$F$ test for random $A$ (two-factor)

$$F_{0} = \frac{MS_{A}}{MS_{AB}}$$

$F$ test for random $B$ (two-factor)

$$F_{0} = \frac{MS_{B}}{MS_{AB}}$$

$F$ test for random interaction $A \times B$

$$F_{0} = \frac{MS_{AB}}{MS_{E}}$$

Logic: A tested against AB

Under $H_{0}: \text{sigma}_{\tau}^{2} = 0$, $E(MS_{A})$ matches $E(MS_{AB})$ ($\text{sigma}^{2} + n \times \text{sigma}_{\tau \text{beta}}^{2}$).

Estimate of $\sigma_{\tau}^{2}$ (two-factor random)

$$\text{sigma}\_hat_{\tau}^{2} = \frac{MS_{A} - MS_{AB}}{bn}$$

Estimate of $\sigma_{\text{beta}}^{2}$ (two-factor random)

$$\text{sigma}\_hat_{\text{beta}}^{2} = \frac{MS_{B} - MS_{AB}}{an}$$

Estimate of $\sigma_{\tau \text{beta}}^{2}$ (two-factor random)

$$\text{sigma}\_hat_{\tau \text{beta}}^{2} = \frac{MS_{AB} - MS_{E}}{n}$$

Main-effect test warning (random factorial)

The main-effect $F$ tests generally use the interaction mean square as the denominator rather than residual error.

Measurement-system capability study

Experiment used to quantify sources of variability in a measurement system; also called gauge capability.

Gauge repeatability

$\text{sigma}^{2}$; within-part/within-operator variation from repeated measurement of the same part by the same operator.

Gauge reproducibility

Variation associated with operators ($\text{sigma}_{\text{beta}}^{2} + \text{sigma}_{\tau \text{beta}}^{2}$).

Product/part variability

Variation among the sampled parts, represented by the part variance component.

Example 13.2 Operator results (Full Model)

$DF = 2$, $MS = 1.308$, $F = 1.84$, $p = 0.173$

Example 13.2 Interaction results (Full Model)

$DF = 38$, $MS = 0.712$, $F = 0.72$, $p = 0.861$

Ex 13.2 Full-model Interaction variance estimate

$$-0.1399$$

Ex 13.2 Reduced-model Pooled error MS

$$0.883$$

Ex 13.2 Reduced-model Operator variance estimate

$$0.0106$$

Ex 13.2 Reduced-model Part variance estimate

$$10.2513$$

Estimated gauge variance after interaction removal

$$\text{sigma}\_hat_{gauge}^{2} = \text{sigma}\_hat^{2} + \text{sigma}\_hat_{\text{beta}}^{2} = 0.88 + 0.01 = 0.89$$

Mixed model

A model containing both fixed and random effects.

Lesson 13 Mixed Setting

Factor $A$ is fixed and factor $B$ is random.

Restricted mixed-model equation

$$y_{ijk} = \beta + \tau_{i} + \text{beta}_{j} + (\tau \text{beta})_{ij} + \text{epsilon}_{ijk}$$

Fixed-effect constraint (Restricted Model)

$$\text{sum}_{i} \tau_{i} = 0$$

Restricted interaction variance convention

$$V((\tau \text{beta})_{ij}) = \frac{a-1}{a} \times \text{sigma}_{\tau \text{beta}}^{2}$$

Restricted interaction constraint

For every $j$, $\text{sum}_{i} (\tau \text{beta})_{ij} = 0$

Why model is called Restricted

The random interaction effects are constrained to sum to zero across the levels of the fixed factor.

Restricted-model Interaction Distribution

Normal but not independent due to the imposed sum-to-zero restriction.