# Multiway ANOVA and Nested Design with Biomedical Applications

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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

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

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

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!

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.

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?

CASE STUDY: McDEVITT LAB RESEARCH

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

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

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

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

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

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

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

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

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

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

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')

Data Distribution

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

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

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

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]

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

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

Acknowledgements McDevitt Lab