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# Introduction to Analysis of Variance.docx Introduction to Analysis of Variance.docx

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School
Western University
Department
Statistical Sciences
Course
Statistical Sciences 2244A/B
Professor
Mark Cleveland
Semester
Winter

Description
Introduction to Analysis of Variance 3/20/2012 7:28:00 AM Experimental Design  “blue print” or map of research (directions)  includes: o if the study is experimental or correlational o research hypothesis/question o who will be tested (participants) o how assignment to conditions will take place (if experimental) o how variables will be measured o degree of control (versus naturalness) while collecting data o how will the data be analyzed to answer the question/hypothesis Two and More Group Designs  Simplest experimental design o 2 groups (1 experimental and 1 control) o = 1 IV, 2 levels, and 1 DV  simplest statistic for 2 groups = t-test  if increase number of groups (3 or more), then build onto the design  …move from the t-test statistic to the analysis of variance (ANOVA) o the ANOVA design is defined using the prefix “way” (1 IV 1 “way”, 2 IV “2 ways”)  for example, if o 1 IV, 1 DV and 3 or more groups (eg. 3 levels of the IV), then design is a “1-way” ANOVA o IV “A” A1 A2 A3 o Example is a 1x3 ANOVA design o Results in 3 cells o Cells = IV condition combinations  In a one-way ANOVA, question is whether or not one of the cells is different from the others, termed a “main effect”  Ho: M1 = M2 = … = Mk  Ha: at least 1 mean is different  (controls Type I error rates as not doing multiple t-tests)  The ANOVA title changes according to the number of IV’s o 2 IV’s = 2 way ANOVA o 3 IV’s = 3 way ANOVA etc…  by introducing another IV: o main effects of each IV o examine interactions (how the combination of different IV levels effect performance on the DV) o example: 2 IV’s (A and B) 2 levels of IV A 2 levels of IV B A = chemo, 1 = none 2 = chemo B = radiation, 1 = none 2 = radiation Design is 2x2 A1B1 A1B2 A2B1 A2B2 o design results in tests for: o main effect for A o main effect for B o interaction of AxB  if add another IV, “C”, then now have a 3-way ANOVA  example: 2 levels of A 3 levels of B 2 levels of C = 2x3x2 ANOVA design results in test for: main effect for A, main effect for B, main effect for C AxB interaction, AxC interaction, BxC interaction, AxBxC interaction *for 3 way ANOVA’s only need to understand and know definitions Stats in Conducting an ANOVA  Examines the amount of variance (differences) between the means of the cells (called “Between-Subjects”)  Examines the amount of variance between individuals within a cell (called “Within-Subjects”) o Is then considered to be noise, random error, or error variance o Analysis starts with the toral amount of variance o Then partitions variance into between and within groups o s^2total = ∑(x – Mt)^2 Nt – 1 o ∑(x – Mt)^2 = total Sums of Squares = SST o Mt = Grand Mean o Nt = total number of subjects o Nt-1 = total degree of freedom o When SS is divided by the df, the result is a mean square o S^2 total = SST/DFt = MSt o SST is “broken” up into:  SSW = Within Group Variance (error)  SSB = Between Group Variance  SSW o =sums of squares within groups o =d
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