Analysis of Variance Objective Compute and interpret the results of a one-way ANOVA. Dr. Ahmed M. Sultan

Introduction Most experiments involves a study of the effect of one or more variables on a response. A response can be affected by two types of independent variables: Quantitative Qualitative These independent variables that can be controlled in an experiment are called factors Dr. Ahmed M. Sultan

Factors - Level - Blocks - Treatment To show that: In study of the wear for three types of tires A, B, and C, on each of four automobiles, “tire types” is a factor representing a single quantitative variable at three levels. Automobiles are blocks representing a single quantitative variable at four levels Response depends on the factors that represent treatments Dr. Ahmed M. Sultan

Analysis of Variance: Assumptions Observations are drawn from normally distributed populations. Observations represent random samples from the populations. Variances of the populations are equal. Dr. Ahmed M. Sultan 8

Partitioning Total Sum of Squares of Variation SST (Total Sum of Squares) SSTr (Treatment Sum of Squares) SSE (Error Sum of Squares) Dr. Ahmed M. Sultan

One-Way ANOVA: Sums of Squares Definitions Sum of squares of treatments SSTr: Sum of squares of error SSE: Total sum of squares SST: Dr. Ahmed M. Sultan 10

Cont … Total sum of squares = Treatments sum of squares + Error sum of squares SST = SSTr + SSE Dr. Ahmed M. Sultan

Dr. Ahmed M. Sultan

Dr. Ahmed M. Sultan

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One-Way ANOVA: Computational Formulas Dr. Ahmed M. Sultan 12

One-Way ANOVA: Procedural Overview Dr. Ahmed M. Sultan 9

EX: Valve Openings by Operator 1 2 3 4 6.33 6.26 6.44 6.29 6.36 6.38 6.23 6.31 6.58 6.19 6.27 6.54 6.21 6.4 6.56 6.5 6.34 6.22 Dr. Ahmed M. Sultan

One-Way ANOVA: Preliminary Calculations 1 2 3 4 6.33 6.26 6.44 6.29 6.36 6.38 6.23 6.31 6.58 6.19 6.27 6.54 6.21 6.4 6.56 6.5 6.34 6.22 Tj T1 = 31.59 T2 = 50.22 T3 = 45.42 T4 = 24.92 T = 152.15 nj n1 = 5 n2 = 8 n3 = 7 n4 = 4 N = 24 Mean 6.318000 6.277500 6.488571 6.230000 6.339583 Dr. Ahmed M. Sultan 13

One-Way ANOVA: Sum of Squares Calculations Dr. Ahmed M. Sultan 15

One-Way ANOVA: Sum of Squares Calculations Dr. Ahmed M. Sultan 15

One-Way ANOVA: Mean Square and F Calculations Dr. Ahmed M. Sultan 16

Analysis of Variance for Valve Openings Source of Variations df SS MS F Between Tr 3 0.23658 0.078860 10.18 Error 20 0.15492 0.007746 Total 23 0.39150 Dr. Ahmed M. Sultan 17

A Portion of the F Table for  = 0.05 df1 df 2 1 2 3 4 5 6 7 8 9 161.45 199.50 215.71 224.58 230.16 233.99 236.77 238.88 240.54 … 18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 19 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42 20 4.35 3.49 3.10 2.87 2.71 2.60 2.45 2.39 21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.37 Dr. Ahmed M. Sultan

One-Way ANOVA: Procedural Summary Rejection Region  Critical Value Non rejection Region Dr. Ahmed M. Sultan 19

Output for the Valve Opening Example from Excel Anova: Single Factor SUMMARY Groups Count Sum Average Variance Operator 1 5 31.59 6.318 0.00277 Operator 2 8 50.22 6.2775 0.0110786 Operator 3 7 45.42 6.488571429 0.0101143 Operator 4 4 24.92 6.23 0.0018667 ANOVA Source of Variation SS df MS F P-value F crit Between Groups 0.236580119 3 0.07886004 10.181025 0.00028 3.09839 Within Groups 0.154915714 20 0.007745786 Total 0.391495833 23   Dr. Ahmed M. Sultan

One-Way Analysis of Variance using Minitab Source DF SS MS F P Factor 3 0.23658 0.07886 10.18 0.000 Error 20 0.15492 0.00775 Total 23 0.39150 Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev ---------+---------+---------+------- O1 5 6.3180 0.0526 (-----*------) O2 8 6.2775 0.1053 (----*-----) O3 7 6.4886 0.1006 (-----*----) O4 4 6.2300 0.0432 (------*-------) ---------+---------+---------+------- Pooled StDev = 0.0880 6.24 6.36 6.48 Dr. Ahmed M. Sultan

HW The reaction times for two different stimuli in a psychological word association experiment were compared by using each stimulus on independent random samples of size eight. Thus a total of sixteen people were used in the experiment. Do the following data present sufficient evidence to indicate that there is a difference in the mean reaction times for the two stimulus? Stimulus 1: 1 3 2 1 2 1 3 2 Stimulus 2: 4 2 3 3 1 2 3 3 Dr. Ahmed M. Sultan

Dr. Ahmed M. Sultan