1. Multiway ANOVA and Nested Design with Biomedical Applications
Mohamad Ali Najia
Undergraduate Researcher, Engineering Stem Cell Technologies Lab
Center for Bioengineering Statistics
The Wallace H. Coulter Department of Biomedical Engineering
Georgia Institute of Technology & Emory University

2. Statistical methods for comparing multiple groups
Binary data: comparing multiple proportions
Chi-square tests for r × 2 tables
Independence
Goodness of Fit
Homogeneity
Categorical data: comparing multiple sets of categorical responses
Similar chi-square tests for r × c tables
Continuous data: comparing multiple means
Analysis of variance

3. ANOVA: Definition
Statistical technique for comparing means for multiple (usually ≥ 3) independent populations
To compare the means in 2 groups, just use t-test to conduct a hypothesis test for the equality of two sample means
Partition the total variation in a response variable into
Variability within groups
Variability between groups
If the null hypothesis is true the standardized variances are equal to one another

4. Assumptions The errors εij for each factor are normally distributed
Across the conditions, the errors have equal spread. Often referred to as equal variances.
Rule of thumb: the assumption is met if the largest variance is less than twice the smallest variance
If unequal variances need to make a correction! This is usually α/2.
The errors are independent from each other
Checking the assumptions
Use the residuals, which are the estimates of εij
-Look at normal probability plot
-Look at residual versus fitted plot
-Hard to check, often assumed from study design
For mild violations of the assumptions, there are options for correction
When the assumptions are not met – the p-value is simply wrong!

5. Types of ANOVA One-way ANOVA
One factor — e.g. smoking status (never, former, current)
Two-way ANOVA
Two factors — e.g. gender and smoking status
Three-way ANOVA
Three factors — e.g. gender, smoking and beer consumption
Multi-way ANOVA
The two possible means models for two-way ANOVA are the additive model and the interaction model. The additive model assumes that the effects on the outcome of a particular level change for one explanatory variable does not depend on the level of the other explanatory variable. If an interaction model is needed, then the effects of a particular level change for one explanatory variable does depend on the level of the other explanatory variable.

6. Emphasis
One-way ANOVA is an extension of the t-test to 3 or more samples
focus analysis on group differences
Two-way ANOVA (and higher) focuses on the interaction of factors
Does the effect due to one factor change as the level of another factor changes?

7. CASE STUDY: McDEVITT LAB RESEARCH

8. Regenerative Medicine and Bioprocessing
Stem Cell
Efficient, scalable, and robust bioprocessing technologies
Limitations:
Dynamic regulation of cell fate
Scalable cell production
Therapeutic Application

9. Pluripotent Stem Cells
Isolation
Self-Renewal
ESCs are isolated from the inner cell mass of a blastocyst stage embryo. During embryogenesis, stem cells secrete potent combinations of trophic factors that modulate the molecular composition of the environment to evoke responses from resident cells. Through their secretion of these trophic factors, including growth factors, cytokines, chemokines, and mitogens, ESCs create a microenvironment that supports the morphogenic events leading to organ and tissue formation. Thus, a recent paradigm shift has emerged suggesting that the beneficial effects of stem cells may not be restricted to cell restoration alone, but also due to their transient paracrine actions. The delivery of stem cell paracrine factors in vivo has been implicated in neural, myocardial and osteogenic regeneration, as well as in wound healing.
Differentiation
(Germ Linages)
Blastocyst

10. Fluid Shear Stress and Stem Cells
Endothelial Cells
0.98 to 15 dynes/cm2
Ando et al. AM J Physiol Heart Circ Physiol, 2005; Xu et al. J Cell Biol 2006, Nemoto et al. J Artif Organs 2005, Gaetano et al. Circ Res, 2005, Ahsan et al. Tissue Eng. 2010
Stem Cells Exposed to Fluid Shear Stress
Do embryonic stem cells continue to differentiate towards these phenotypes after exposure to fluid shear stress?
Ledran et al. Stem Cell, 2008
Hematopoietic cells
5 dynes/cm2
Daley et al. Nature, 2009

11. Embryoid Bodies 3D culture platform recapitulates morphogenic events
3D culture platform which mimics developmental processes/tissue structure and improves viability in suspension culture through maintenance of cell-cell contacts
3D culture platform recapitulates morphogenic events
Improves viability through increased cell-cell contacts
Allows higher density suspension culture configurations

12. Controlling Differentiation
Microenvironment
Growth factors
Cytokines
Extracellular Matrix
Cell-cell interactions
400um
Controlling Differentiation
Growth factors
Oxygen
Hydrodynamics
Bratt-Leal et al, Biotechnology Progress, 2009.
pre-treatment of embryonic stem cells as a method to control embryoid body differentiation
Pre-treating ESCs
EB Differentiation
ESCs

13. Objective
To study the effects of vasculogenic cues on fluid shear stress preconditioned embryonic stem cells.
Hypothesis
Exposing embryonic stem cells to fluid shear stress prior to EB differentiation will promote embryoid body endothelial differentiation and vasculogenesis in the presence of vasculogenic cues.
Fluid Shear
Stress Pre-conditioning
Endothelial Differentiation and Morphogenesis
Embryoid Body Differentiation
Vasculogenesis
VEGF
Oxygen

14. Embryoid Body (EB) Culture
Experimental Design
Preconditioning (PC)
Embryoid Body (EB) Culture
End preconditioning and start EB culture
ESCs seeded on coll IV
Precondition ESCs w/ fluid shear stress -4 -2 2 4 7 10
Time (Day after Preconditioning)
Preconditioning
-Parallel Plate Flow Chamber System
-Fluid Shear Stress 5 dynes/cm2
Assessments
-Gene expression
-Protein expression
-Protein localization
-Morphology

15. Embryoid Body (EB) Culture
Experimental Design
Preconditioning (PC)
Embryoid Body (EB) Culture
End preconditioning and start EB culture
ESCs seeded on coll IV
Precondition ESCs w/ fluid shear stress -4 -2 2 4 7 10
Time (Day after Preconditioning)
21% Oxygen
Experimental Conditions at:
5 dynes shear PC
0 dynes (static) shear PC
3% Oxygen
Soluble VEGF Addition
Quantitative Response Variable: Gene Expression of Endogenous VE-cadherin at Day 7 and 10 of culture
n=5

16. Nested Design
Static
(i = 1)
Shear
(i = 2)
Preconditioning
21% O2
(j = 1)
3% O2
(j = 2)
VEGF
(j = 3)
Treatment .
Sample
(k = 1)
(k = 5)
(k = 1)
(k = 5)
Time-point D7
D10
Treatments are crossed and nested between/within Preconditioning
Samples are nested within Treatment
Sampling at each Time-point is repeated measures design

17. Linear Model yijkl = μ + αi + βj(i) + γk(j) + δl(k) + εijkl where,
μ overall grand mean
αi effect of preconditioning (at levels i = 1,2)
βj effect of the treatment (at levels j = 1,2,3)
γk effect of the samples (at levels k = 1,2,3,4,5)
δl effect of time (at levels l = 1,2)
εijkl error term

18. Data Distribution clear clc close load('preconditioning.mat');
% Box Plots
%index experimental conditions into individual variables...
figure(1)
subplot(2,1,1)
boxplot([group1 group2 group3 group4 group5 group6]);
title('Day 7')
subplot(2,1,2)
boxplot([group7 group8 group9 group10 group11 group12]);
title('Day 10')

19. Data Distribution

20. Hypothesis Testing H0: μ1 = μ2 = . . . = μk
H1: the μ’s are not all equal
For an N-way ANOVA there are 2n-1 hypotheses (including interactions)
The null hypothesis is called the overall null and is the hypothesis tested by ANOVA
If the overall null is rejected, must do more specific hypothesis testing to determine which means are different, often referred to as contrasts

21. MATLAB: Measurement Matrix
Day 7
Day 10
measurement
Factor A: Preconditioning
Factor B: Treatment
Factor C: Sample
Factor D: Timepoint 1
0.78 2 3 4 5 6
measurement
Factor A: Preconditioning
Factor B: Treatment
Factor C: Sample
Factor D: Timepoint 1 2 3 4 5 6

22. MATLAB: ANOVAN() α β γ δ 1 clear clc close load('preconditioning.mat')
measurements = data(:,1);
time = data(:,2);
pc = data(:,3);
treatmentStart = data(:,4);
treatment = data(:,5);
subjects = data(:,6);
% From the linear model defined, the nesting matrix is...
nested = [ ; ; ; ];
[p,table,stats] = anovan(measurements,{pc, treatment, sample, ...
time},'varnames', {'Preconditioning', 'Treatment', 'Sample', 'Time'},...
'model', 'interaction', 'nested', nested); α β γ δ 1

23. MATLAB: Outputs p = 0.0192 0.0000 NaN 0.0002 stats = source: 'anovan'
resid: [60x1 double]
coeffs: [27x1 double]
Rtr: [12x12 double]
rowbasis: [12x27 double]
dfe: 48
mse:
nullproject: [27x12 double]
terms: [4x4 double]
nlevels: [4x1 double]
continuous: [ ]
vmeans: [4x1 double]
termcols: [5x1 double]
coeffnames: {27x1 cell}
vars: [27x4 double]
varnames: {4x1 cell}
grpnames: {4x1 cell}
vnested: [4x4 double]

24. Further Analysis
If H0 is rejected, we conclude that not all the μ’s are equal
Can use planned or unplanned comparisons (or contrasts)
Planned comparisons are interesting comparisons decided on before analysis
Unplanned comparisons occur after seeing the results (Tukey’s Multiple Comparisons)
Interaction or profile plots
An interaction plot is a way to look at outcome means for two factors simultaneously
A plot with parallel lines suggests an additive model
A plot with non-parallel lines suggests an interaction model

25. Contrast or Post-Hoc ANOVA?
With specific a priori predictions about the data, use contrasts
Without specific a priori predictions, use post-hoc comparisons
Post-Hoc comparisons are pairwise comparisons designed to compare all different combinations of treatment groups

26. Acknowledgements McDevitt Lab
Petit Undergraduate Research Scholars Program
President’s Undergraduate Research Award