Download ANOVA and ANCOVA for Statistical Analysis of Experimental Data - Prof. Mervyn G. Marasingh and more Exams Statistics in PDF only on Docsity! Oneway Classification One Treatment (Factor) in a CRD; may be unequal replications yij = μ + αi︸ ︷︷ ︸ μi + ij i = 1, 2, . . . , t, j = 1, . . . , ni, ij ∼ iid N(0, σ2) equivalent to assuming yij ∼ N(μi, σ2), j = 1, . . . , ni for each treatment i. Also involves the assumption of homogeneity of variance i.e., same variance in each population. Estimation μ̂i = ȳi. = ( ∑ j yij)/ni, i = 1, . . . , t σ̂2 = s2 = ∑ i ∑ j(yij − ȳi.)2 N − t , N = ∑ i nîαp − αq = ȳp. − ȳq., p = q SE(ȳp. − ȳq.) = sd = s √ 1 np + 1 nq A (1 − α)100% C.I. for αp − αq (or μp − μq.) is (ȳp. − ȳq.) ± tα/2(ν) · sd where tα/2(ν) = upper α/2 percentage point of the t-distribution with ν d.f. ν = N − t Testing Hypotheses AoV Table SV d.f. SS MS F p-value Trt t − 1 MSTrt Fc = MSTrt/MSE Pr(F > Fc) Error N − t MSE(= s2) Total N − 1 The F-statistic tests H0 : μ1 = μ2 = · · ·μt vs. Ha : at least one ineq. or equivalently H0 : α1 = α2 = · · · = αt vs. Ha : at least one ineq. Testing H0 : μp = μq vs. Ha : μp = μq or equivalently H0 : αp = αq vs. Ha : αp = αq Use the t-statistic tc = |ȳp. − ȳq.| sd Rej. H0 iff tc > tα/2(ν) ν = N − t Contrasts (or Comparisons)∑ i ciμi is said to be a contrast of means μ1, μ2, . . . , μt if c1, c2, . . . , ct are constants such that ∑ i ci = 0. Examples μ1 − μ2, 2μ1 − μ2 − μ3, μ1 − 1 3 μ2 − 1 3 μ4 − 1 3 μ5 Test for Preplanned Comparisons (Equal Sample Size Case i.e., n1 = n2 = · · · = n) H0 : ∑ i ciμi = 0 vs. Ha : ∑ i ciμi = 0 tc = |∑i ciȳi.| s( ∑ c2i n ) 1 2 Rej. H0 : if tc > tα/2(N − t) or Fc = n( ∑ ciȳi.) 2/( ∑ c2i ) s2 Rej. H0 : if Fc > Fa(1, N − t) Pairwise Comparison of Means Individual Comparisons: * By the t-test of H0 : μp = μq * By the C.I.’s for μp − μq * Equivalently, using the Least Significance Difference (LSD) when sample sizes are equal. t-test for H0 : μp − μq = 0 gives Rej: H0 if |ȳp. − ȳq.| > tα/2(N − t) · s · √ 2/n︸ ︷︷ ︸ LSDα , n = sample size Multiple Comparisons: * Tukey’s procedure for all possible pairwise comparisons simultaneously (HSD). * Scheffe’s procedure for several comparisons (contrasts of the type ∑ ciμi) simultaneously. Oneway Analysis of Covariance One factor experiment in a CRD; a single covariate is also measured. Assume equal replication. yij = μ + τi + β(xij − x̄..) + ij i = 1, . . . , tj = 1, . . . , n ij ∼ iidN(0, σ 2) ⇐⇒ Assuming straight line regressions for each treatment with the same slope β Treatment 1: y1j = α1 + βx1j + 1j , j = 1, . . . , n Treatment 2: y2j = α2 + βx2j + 2j , j = 1, . . . , n ... ... Treatment t: ytj = αt + βxtj + tj , j = 1, . . . , n where αi = μ + τi − x̄.. Estimation μ̂i = ȳi.(Adj.) = ȳi − b(x̄i. − x̄..) ‘Adjusted Treatment Means’ b = Exy Exx Exy = ∑ i ∑ j(xij − x̄i.)(yij − ȳi.) Exx = ∑ i ∑ j(xij − x̄i.)2 ȳi. = ∑ j yij n x̄i. = ∑ j xij n x̄.. = ∑ i ∑ j xij nt σ̂2 = s2 MS Error from the ‘Adjusted AoV’ A (1 − α) 100% C.I. for μp − μq is (ȳp.(Adj.) − ȳq.(Adj.)) ± tα/2(ν)sd where sd = s { 2 n + (x̄p. − x̄q.) Exx 2}1/2 and ν = t(n − 1) − 1 Testing Hypotheses An analysis of covariance table SV df SS MS F Trt t − 1 SSTrt MSTrt MSTrt/MSEUnadj. Error(Unadj.) t(n − 1) SSEUnadj. MSEUnadj. Regression 1 SSReg MSReg MSReg/MSE Error(Adj.) t(n − 1) − 1 SSE MSE(= s2) Total tn − 1 SSTot Trt(Adj.) t − 1 SSTrt MSTrt MSTrt/MSE Error(Adj.) t(n − 1) − 1 SSE MSE(= s2) The F -statistic for Trt tests the hypothesis H0 : μ1 = μ2 = · · · = μt versus Ha : at least one inequality when the covariate is not present in the model. The F -statistic for Regression tests the hypothesis H0 : β = 0 versus Ha : β = 0 The F -statistic for Trt(Adj.) tests the hypothesis H0 : τ1 = τ2 = · · · = τt versus Ha : at least one inequality when β is not zero. This test is equivalent to comparing the intercepts of the regression lines i.e., H0 : α1 = α2 = · · · = αt versus Ha : at least one inequality If this hypothesis is rejected, then at least one pair of treatment effects (equiv- alently, adjusted treatment means) is different.
