Presentation on theme: "Multiway ANOVA and Nested Design with Biomedical Applications"— Presentation transcript:
1Multiway ANOVA and Nested Design with Biomedical Applications Mohamad Ali NajiaUndergraduate Researcher, Engineering Stem Cell Technologies LabCenter for Bioengineering StatisticsThe Wallace H. Coulter Department of Biomedical EngineeringGeorgia Institute of Technology & Emory University
2Statistical methods for comparing multiple groups Binary data: comparing multiple proportionsChi-square tests for r × 2 tablesIndependenceGoodness of FitHomogeneityCategorical data: comparing multiple sets of categorical responsesSimilar chi-square tests for r × c tablesContinuous data: comparing multiple meansAnalysis of variance
3ANOVA: DefinitionStatistical technique for comparing means for multiple (usually ≥ 3) independent populationsTo compare the means in 2 groups, just use t-test to conduct a hypothesis test for the equality of two sample meansPartition the total variation in a response variable intoVariability within groupsVariability between groupsIf the null hypothesis is true the standardized variances are equal to one another
4Assumptions 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 varianceIf unequal variances need to make a correction! This is usually α/2.The errors are independent from each otherChecking the assumptionsUse 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 designFor mild violations of the assumptions, there are options for correctionWhen the assumptions are not met – the p-value is simply wrong!
5Types of ANOVA One-way ANOVA One factor — e.g. smoking status (never, former, current)Two-way ANOVATwo factors — e.g. gender and smoking statusThree-way ANOVAThree factors — e.g. gender, smoking and beer consumptionMulti-way ANOVAThe 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.
6EmphasisOne-way ANOVA is an extension of the t-test to 3 or more samplesfocus analysis on group differencesTwo-way ANOVA (and higher) focuses on the interaction of factorsDoes the effect due to one factor change as the level of another factor changes?
8Regenerative Medicine and Bioprocessing Stem CellEfficient, scalable, and robust bioprocessing technologiesLimitations:Dynamic regulation of cell fateScalable cell productionTherapeutic Application
9Pluripotent Stem Cells IsolationSelf-RenewalESCs 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
10Fluid Shear Stress and Stem Cells Endothelial Cells0.98 to 15 dynes/cm2Ando 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. 2010Stem Cells Exposed to Fluid Shear StressDo embryonic stem cells continue to differentiate towards these phenotypes after exposure to fluid shear stress?Ledran et al. Stem Cell, 2008Hematopoietic cells5 dynes/cm2Daley et al. Nature, 2009
11Embryoid 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 contacts3D culture platform recapitulates morphogenic eventsImproves viability through increased cell-cell contactsAllows higher density suspension culture configurations
12Controlling Differentiation MicroenvironmentGrowth factorsCytokinesExtracellular MatrixCell-cell interactions400umControlling DifferentiationGrowth factorsOxygenHydrodynamicsBratt-Leal et al, Biotechnology Progress, 2009.pre-treatment of embryonic stem cells as a method to control embryoid body differentiationPre-treating ESCsEB DifferentiationESCs
13ObjectiveTo study the effects of vasculogenic cues on fluid shear stress preconditioned embryonic stem cells.HypothesisExposing 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 ShearStress Pre-conditioningEndothelial Differentiation and MorphogenesisEmbryoid Body DifferentiationVasculogenesisVEGFOxygen
14Embryoid Body (EB) Culture Experimental DesignPreconditioning (PC)Embryoid Body (EB) CultureEnd preconditioning and start EB cultureESCs seeded on coll IVPrecondition ESCs w/ fluid shear stress-4-224710Time (Day after Preconditioning)Preconditioning-Parallel Plate Flow Chamber System-Fluid Shear Stress 5 dynes/cm2Assessments-Gene expression-Protein expression-Protein localization-Morphology
15Embryoid Body (EB) Culture Experimental DesignPreconditioning (PC)Embryoid Body (EB) CultureEnd preconditioning and start EB cultureESCs seeded on coll IVPrecondition ESCs w/ fluid shear stress-4-224710Time (Day after Preconditioning)21% OxygenExperimental Conditions at:5 dynes shear PC0 dynes (static) shear PC3% OxygenSoluble VEGF AdditionQuantitative Response Variable: Gene Expression of Endogenous VE-cadherin at Day 7 and 10 of culturen=5
16Nested DesignStatic(i = 1)Shear(i = 2)Preconditioning21% O2(j = 1)3% O2(j = 2)VEGF(j = 3)Treatment.Sample(k = 1)(k = 5)(k = 1)(k = 5)Time-pointD7D10Treatments are crossed and nested between/within PreconditioningSamples are nested within TreatmentSampling at each Time-point is repeated measures design
17Linear Model yijkl = μ + αi + βj(i) + γk(j) + δl(k) + εijkl where, μ overall grand meanαi 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
18Data 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')
20Hypothesis Testing H0: μ1 = μ2 = . . . = μk H1: the μ’s are not all equalFor 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 ANOVAIf the overall null is rejected, must do more specific hypothesis testing to determine which means are different, often referred to as contrasts
24Further AnalysisIf H0 is rejected, we conclude that not all the μ’s are equalCan use planned or unplanned comparisons (or contrasts)Planned comparisons are interesting comparisons decided on before analysisUnplanned comparisons occur after seeing the results (Tukey’s Multiple Comparisons)Interaction or profile plotsAn interaction plot is a way to look at outcome means for two factors simultaneouslyA plot with parallel lines suggests an additive modelA plot with non-parallel lines suggests an interaction model
25Contrast or Post-Hoc ANOVA? With specific a priori predictions about the data, use contrastsWithout specific a priori predictions, use post-hoc comparisonsPost-Hoc comparisons are pairwise comparisons designed to compare all different combinations of treatment groups
26Acknowledgements McDevitt Lab Petit Undergraduate Research Scholars ProgramPresident’s Undergraduate Research Award