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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

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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

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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

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Assumptions 1.The errors ε ij for each factor are normally distributed 2.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. 3.The errors are independent from each other

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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.

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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?

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CASE STUDY: McDEVITT LAB RESEARCH

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Regenerative Medicine and Bioprocessing Stem Cell Therapeutic Application Efficient, scalable, and robust bioprocessing technologies

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Pluripotent Stem Cells Isolation Self-Renewal Blastocyst Differentiation (Germ Linages)

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Fluid Shear Stress and Stem Cells Stem Cells Exposed to Fluid Shear Stress Hematopoietic cells 5 dynes/cm 2 Daley et al. Nature, 2009 Endothelial Cells 0.98 to 15 dynes/cm 2 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 Ledran et al. Stem Cell, 2008 Do embryonic stem cells continue to differentiate towards these phenotypes after exposure to fluid shear stress?

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Embryoid Bodies 3D culture platform recapitulates morphogenic events Improves viability through increased cell-cell contacts Allows higher density suspension culture configurations

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

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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 VasculogenesisEmbryoid Body Differentiation VEGF Oxygen

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

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

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Nested Design Static (i = 1) 21% O2 (j = 1) Shear (i = 2) Preconditioning Treatments are crossed and nested between/within Preconditioning Samples are nested within Treatment Sampling at each Time-point is repeated measures design Treatments are crossed and nested between/within Preconditioning Samples are nested within Treatment Sampling at each Time-point is repeated measures design 3% O2 (j = 2) VEGF (j = 3) (k = 1) (k = 5) Time-point Treatment Sample (k = 1) (k = 5) D7D10

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Linear Model y ijkl = μ + α i + β j(i) + γ k(j) + δ l(k) + ε ijkl where, μ overall grand mean α i effect of preconditioning (at levels i = 1,2) β j effect of the treatment (at levels j = 1,2,3) γ k effect of the samples (at levels k = 1,2,3,4,5) δ l effect of time (at levels l = 1,2) ε ijkl error term

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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')

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Data Distribution

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Hypothesis Testing H 0 : μ 1 = μ 2 =... = μ k H 1 : 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

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MATLAB: Measurement Matrix measurement Factor A: Preconditioning Factor B: Treatment Factor C: Sample Factor D: Timepoint measurement Factor A: Preconditioning Factor B: Treatment Factor C: Sample Factor D: Timepoint Day 7Day 10

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MATLAB: ANOVAN() 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); αβγδ α 0000 β 1000 γ 0100 δ 0010

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MATLAB: Outputs p = NaN 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]

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Further Analysis If H 0 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

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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

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

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