Practical Data Analysis for Designed Experiments

D. Dealing with Imbalance

10. Unbalanced Experiments

10.1 Unequal Samples

10.2 Additive Model

10.3 Types I, II, III and IV

11. Missing Cells

11.1 What are Missing Cells?

11.2 Connected Cells and Incomplete Designs

11.3 Type IV Comparisons

11.4 Latin Square Designs

11.5 Fractional Factorial Designs

12. Linear Models Inference

12.1 Matrix Preliminaries

12.2 Ordinary Least Squares (OLS)

12.3 Weighted Least Squares (WLS)

12.4 Maximum Likelihood (ML)

12.5 Restricted Maximum Likelihood (REML)

12.6 Inference for Fixed Effects Models

12.7 Anova and Regression Models

10. Unbalanced Experiments
data lost, destroyed, invalid for whatever reason
poor planning, loss of resources

10.1 Unequal Samples
	balanced treatment structure (all factor combinations)
	unbalanced design structure (unequal cell sample sizes)
		no empty cells for now
	cell means inference
	population marginal means
		usual hypotheses
		least squares estimates
		variance and harmonic mean of sample sizes
	sample marginal means
		expectation = weighted average of cell means
		variance
		hypotheses depend on sample sizes
	hint at type I vs. III SS

10.2 Additive Model
	recall effects in balanced design
	Scheffe's reduction notation
	normal equations
	least squares solutions
	hint at type I vs. II SS

10.3 Types I, II, III and IV
	all are equivalent for balanced data (equal sample sizes)
	III & IV agree when no empty cells (sample sizes > 0)
	I: sequential
		order important
		SS add to total
		hypotheses depend on sample sizes
		sequential model building
	II: hierarchical
		main effects before interactions
		OK for additive model
		SS do not add to total with unbalanced data
		used in multiple regression (last in)
		hypotheses depend on imbalance if interaction present
			weighted (tilde) means
		hierarchical model building
	III: orthogonal
		hypotheses concern population marginal means
		hypotheses do NOT depend on sample size
		SS do not add to total with unbalanced data
		general form of hypothesis (full vs. reduced)
		SS adjusted for all other terms (last in)
	IV: balanced
		same as III if no empty (missing) cells
		but see next chapter

11. Missing Cells
11.1 What are Missing Cells?
	cell means inference
	interpreting population marginal means
		some not estimable
		what are appropriate comparisons?
	linear constraints

11.2 Connected Cells and Incomplete Designs
	some marginal means may not be estimable
	balanced / unbalanced contrasts
	confounding from missing levels

	connected subsets / balanced subsets
	Latin square design
	fractional factorial design
	
11.3 Type IV Comparisons
	I,II: sequential, hierarchical
		not appropriate because unbalanced, as before
		marginal means not always defined
	III: orthogonal
		hypotheses depend on pattern of empty cells
		not the usual ones
	IV: balanced
		balanced contrasts for main effects & interactions
		choice is not unique
		packages make automated choice
			which depend on labels for factor levels!
	options
		compare all means with data (one-factor approach)
		consider additive model to achieve estimability
		balanced comparisons (automated Type IV)
		select balanced subsets
		verify that analyses give consistent (similar) results

11.4 Latin Square Designs
	interactions assumed negligible
		but can sometimes test -- replicated LS (later)
	few EUs -- save time or money
	examine many factors at once

11.5 Fractional Factorial Designs
	saturated design (often)
	interactions confounded
		with main effects (resolution III)
		or other interactions
	potential problems of interpretation
		clarify with further experiments
		rethink if expecting several interactions

12. Linear Models Inference
12.1 Matrix Preliminaries
	generalized inverse
	rank, trace, determinant
	quadratic forms

12.2 Ordinary Least Squares (OLS)
	fixed effects models (as encountered so far)
	estimable functions & BLUEs
	projection matrix

12.3 Weighted Least Squares (WLS)
	generalized LS -- unequal variance
	recast as OLS by proper weighting

12.4 Maximum Likelihood (ML)
	WLS <-> ML with normality (almost)
	weighted projection matrix
	ML estimate of variance is biased (divide by sample size)

12.5 Restricted Maximum Likelihood (REML)
	several variance components -- random & mixed models
	project out fixed effects
	REML estimate of variance (components) agree with usual
	recent work of J Jiang (Case Western U)

12.6 Inference for Fixed Effects Models
	quadratic forms & noncentrality parameters
	partition of SS, model
	general linear hypotheses

12.7 Anova and Regression Models
	one-factor & two-factor models
	regression & analysis of covariance (Part F)