Chapter 5: Random Effects Models

• Chapter 5 discusses experiments to study variances
• Chapter 5 covers Random Effects Models, Statistical Models for Variance Components, Intraclass Correlation Measures, Unbalanced Experiments, and includes Exercises

Fixed Effects vs Random Effects

• Fixed factors in experiments study differences in the average response caused by differences in treatment levels and are chosen explicitly by the experimenter before data collection
• Random factors in experiments study the variance caused by changing levels of a factor
• The levels of random factors are samples of possible levels from a larger population and are used in random effects models

Milk Fat Example

• An experimenter aims to determine how to manage milk fat by measuring each cow over ten days
• The purchased cows represents the levels of the experiment
• The focus is on these cows as a representative sample
• Inferences are made to all cows (population) from which they were drawn

Reasons to study variability

• Experiments study the sources of variability
• Descriptive reasons: Variance components have intrinsic value.
• To reduce response variance: Insight is gained by quantifying sources of variability.
• To test causes: Experiments stimulate ideas about variability.

Use cases for Variance Measures

• In dairy cow breeding, milk production variability is partitioned into sire and daughter contributions
• Heritability, the ratio of sire variance to total variance (4σs² / (σs² + σd²),) is significant to dairy farmers.
• In psychological testing, test score variability can be divided into person-to-person and repeat test score variability
• Intraclass correlation (σp² / (σp² + σr²)) indicates test reliability with high values in testing procedures.
• In industrial quality control, measurement variability can be attributed to equipment (gage) and operator differences (σT² = σg² + σo²)
• Managing measurement variability involves concentrating efforts based on the major cause

Random Effects Model

• A random effects model differs slightly from a fixed effects model
• A fixed effects model has the form: yij = μ + τi + eij
  • i ranges from 1 to t, j ranges from 1 to r, and the sum of τi is 0
• A random effects model has the form: yij = μ + ai + eij
  • i ranges from 1 to t, and j ranges from 1 to r, with μ as the overall mean, ai as the random effects, and eij as random experimental errors.

Assumptions for Random Effects Models

• The ai are a random sample with a mean of 0 and variance σa², represented as ai ~ N(0, σa²) for hypothesis tests
• The eij are a random sample with a mean of 0 and variance σe², represented as eij ~ N(0, σe²) for hypothesis tests
• ai and eij are independent of each other.
• σa² and σe² represent among-group and within-group variance components, respectively
• The variance of an observation is V(yij) = σT² = σa² + σe², representing the total variance

ANOVA

• ANOVA is computed the same as a fixed effects model: with derived mean squares under a random effects model
• The table includes: Source, DF, SS, MS, F, and p-Value
• Includes values such as t-1, SSA, MSA, MSA/MSW, N-t, SSW, MSW, N-1, and SSTotal

Analysis of Variance for Variance Components

• Mean squares from the ANOVA are equated to their expected values to estimate variance components
• Observed mean squares are used as estimates
• Equations include: MSA = σ̂e² + r σ̂a² and MSW = σ̂e²
• σ̂e² = MSW and σ̂a² = (MSA - MSW) / r are called the ANOVA estimators or method of moments estimators.

Hypothesis Testing

• H0 : σa² = 0 is tested against H1 : σa² > 0 using the F test
• The test statistic is F = MSA / MSW
• Reject H0 if F > Fα,(t-1,N-t)

Confidence Intervals

• Exact confidence intervals (CIs) for σa² and σe² require normality and equal replicates per random factor level
• The CI for σe² is (SSW / χ²(α/2,N-t)), (SSW / χ²(1-α/2,N-t))
• The CI for σa²: [SSA(1 - Fα/2,(t-1,N-t) / F) / rχ²(α/2,t-1)], [SSA(1 - F1-α/2,(t-1,N-t) / F) / rχ²(1-α/2,t-1)]

Negative Variance Components

• If MSA < MSW, then σa² < 0, but variance components must be positive by definition
• Actions for negative σa²
  • Accept the negative estimate as evidence
  • Retain the negative estimator, but it may not make sense.
  • Attribute the negative estimator as an incorrect statistical model
  • Use estimation methods like maximum likelihood
  • Gather more data

Intraclass Correlation

• The intraclass correlation coefficient (ICC) measures correlation between observations in the same class
• ICC measures similarity within groups relative to that among groups.
• ICC: ρI = σa² / σT² = σa² / (σa² + σe²)
• ρI measures the proportion of total variance explained by among-group variation

Intraclass Correlation Characteristics

• High similarity within groups results in small σe², so σa² increases in the total variation of σT²
• Point estimate: ρ̂I = (MSA − MSW) / (MSA + (r − 1) MSW)
• Value of ρ̂I can be negative
• A 100 x (1 – α)% CI for ρI: [F − Fα/2,(t-1,N-t) / F + (r − 1) Fα/2,(t-1,N-t)], [F − F1-α/2,(t-1,N-t) / F + (r − 1) F1-α/2,(t-1,N-t)]
• If all Cl do not include zero, reject H0 : ρI = 0

Applications of Intraclass Correlation

• Genetics studies to test heritability of quantitative traits
• Reliability studies measure similarity of products from the same machine
• Medical studies measure repeatability of successive measurements on patients
• Survey sampling measures similarity among people contacted by the same interviewer

Unbalanced Experiments and Replicates

• The random effects model with unequal replicates between groups is: yij = μ + ai + eij
  • i ranges from 1 to t and j ranges from 1 to ri.
• Assumptions from the balanced random effects model also apply here.
• ANOVA computations are done like the fixed effects model with unequal replicates between groups.
• The mean squares for among-groups and within-groups are E(MSA) = σe² + rο σa² and E(MSE) = σe²
  • rο = 1/(t-1) (N - Σri²/N)
• Cls for σa² from previous sections are not applicable, but Cls for σe² are still valid

Experiment Example

• An experiment uses three bars of a high-temperature alloy where each casting is broken down into smaller bars.
• The three castings are generic
• This experiment is a random effects model
• Upper tail area critical values, therefore, are used where: F1-α/2,(t−1,N−t) = 1/Fα/2,(N−t,t−1)
• Data:
  • SSA = 147.88 (analogous to SSTreatment)
  • MSA = 73.94 (analogous to MSTreatment)
  • SSW = 157.10 (analogous to SSE)
  • MSW = 5.82 (analogous to MSE)
• σ̂e² = MSW = SSW / (N − t) = 5.82 is the point estimate of the variance.
• The mean square for "among" groups is a point estimate of variance.
  • σ̂a² = (MSA – MSW)/r = (73.94 – 5.82)/10 = 6.81
• Estimated total variance: V(yij) = σ̂Τ² = σ̂a² + σ̂e² = 12.63
• σ̂a² is larger than σ̂e² indicating more variation between castings.
• To improve reliability, decrease variability by adjusting facility processes.
• The overall variance is 2, is σ̂Τ² = 12.63 with a deviation of σ̂Τ = 3.554 psi
• Among castings amounts for 54% ((6.81/12.63) x 100) of total variance
• The percentage for within each casting totals to 46%
• Given χ²(0.05,27) = 40.10 and χ²(0.95,27) = 16.2, the 90% CI for σe² is calculated and equals (3.92, 9.70)
• Given χ²(0.025,2) = 7.38, χ²(0.975,2) = 0.05, F0.025,(2,27) = 4.24, and F0.975,(2,27) = 0.025, the 90% CI for σa²= (1.34, 295.18)
• The Cl for variance components is wide due to small degrees of freedom.
• The addition of castings would have provided precise estimations for the variance along with narrower Cls
• H0 : σa² = 0 versus H1 : σa² > 0, α = 0.05
  • Test statistic: F = MSA/MSW = 12.71
• F = 12.71 exceeding the value of 3.35 and the variation contributes to tensile strength
• The point estimate of the ICC is approximately 0.54
• Values in the CI do not include zero and the null hypothesis is declined