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One-way Analysis of Variance (ANOVA)

One-way Analysis of Variance (ANOVA). Data: Response variable: y One factor Factor level i: i=1,…,a Response at level i: y ij Sample size for each level: n i Total sample size n = n 1 + n 2 + . . . + n a Balanced design if n 1 = n 2 = . . . = n a Sample mean for each level

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One-way Analysis of Variance (ANOVA)

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  1. One-way Analysis of Variance (ANOVA) • Data: Response variable: y • One factor • Factor level i: i=1,…,a • Response at level i: yij • Sample size for each level: ni • Total sample size n = n1 + n2 + . . . + na • Balanced design if n1 = n2 = . . . = na • Sample mean for each level • Overall mean 1 Dr. E. S. Soofi

  2. ANOVA Model • Assumptions about Yijat factor levels (groups) i: i=1,…,a • Distribution of Yij at each factor level (group) has mean E(Yij)= mi • Distributions of Yij for all factor levels (groups) have the same variance Var(Yij)= s2 • Distribution of Yij at each factor level (group) is normal f(Yij)=N(mi,s) • All are independent 2 Dr. E. S. Soofi

  3. One Way ANOVA: Factor level mean model yij = mi + eij , j = 1,…,ni , i = 1,…,a Response = Model + Error term • yij, Response at the ith level of the factor (independent variable) • miis the unknown mean (parameter) of the response yij at the factor level i • eij, unknown error 3 Dr. E. S. Soofi

  4. One Way ANOVA: Factor level effect model yij = m +ti + eij , Response = Model + Error term • m,Unknown overall mean response • ti, the unknown effect (parameter) of the factor level j on the mean response • ti = mi - m • t1+ . . . + ta = 0 4 Dr. E. S. Soofi

  5. Regression Model for ANOVA yij = yh = ma+d1 D1h+ . . . + da-1 Da-1h + eh Response = Model + Error term • Di= 1 for the factor level i • = 0 otherwise • di = mi- ma , i = 1,…, a-1 5 Dr. E. S. Soofi

  6. Least Square (LS) Estimation • LS estimate of the over all mean parameterm • LS estimate of the factor level mean parametermi • LS estimate of the factor level effect parameterti 6 Dr. E. S. Soofi

  7. Sum of Square Decomposition SSF, SS for the factor Degrees of freedom: SST = SSE + SSF n - 1 = n - a + a - 1 7 Dr. E. S. Soofi

  8. ANOVA Table and F-ratio • Mean square factor • Mean square error • s2 is the LS estimate of the error variance s 2 8 Dr. E. S. Soofi

  9. Hypothesis of Interest • Are the factor level effects significantly different? H0: t1 = t2 = . . . = ta = 0 H1: At least one tj is not zero • Equivalently H0: m1 = m2 = . . . = ma H1: At least one mj is different • Reject H0 for large values of F-ratio 9 Dr. E. S. Soofi

  10. Pair-wise comparison • Which pairs of means are significantly different, which pairs are not. • Interval estimates for the differences between every pair of means • Multiple comparison problem • Tukey’s method • When the interval includes zero, the two means are not significantly different • When the interval does not includezero, the two means are significantly different 10 Dr. E. S. Soofi

  11. Testing Model Assumptions • Fitted value • Residual • Use the residuals to test assumptions • Normality of the error term e • Constancy of the error variance s2 • Uncorrelated errors Corr(ei ,ej ) = 0 • Outliers 11 Dr. E. S. Soofi

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