Treatment Structure: Considerations, Factors, Examples, and Factorial ANOVA, Study notes of Animal Biology

Download Treatment Structure: Considerations, Factors, Examples, and Factorial ANOVA and more Study notes Animal Biology in PDF only on Docsity! 3/2/04 1 Treatment Structure TREATMENT STRUCTURE 1 Considerations about treatment structure 1.1 Primarily objectives and/or hypotheses driven 1.2 Statistician may be able to help with consideration of the following as they relate to treatment structure 1.2.1 number of treatment levels 1.2.2 spacing of treatments 1.2.3 alternative designs 1.2.4 single vs multiple factor approach 1.2.5 unstructured vs structured 1.2.6 complete vs incomplete factorials 1.2.7 equal vs unequal replication 1.2.8 qualitative vs quantitative factors 1.2.9 response surface designs 1.2.10 mixture designs 3/2/04 2 Treatment Structure 2 Qualitative factors 2.1 Classification of qualitative factors 2.1.1 specific - unstructured, unordered management practices, different stressors, diets, etc. 2.1.2 observed - race, gender, previous education, varieties, locations, etc. 2.1.3 ordered - high, medium, low or disease level (free, slight, ..., dead) 2.2 Replication of qualitative factors 2.2.1 Equally replicate if all pairwise mean comparisons are to be made 2.2.2 Suppose that not all comparison are of interest. The following two examples will demonstrate how to determine appropriate replication 3/2/04 5 Treatment Structure Let's look at for the two solutions plus the ideal distribution of n Ideal distribution of n Solution 1 Solution 2 3/2/04 6 Treatment Structure 2.4 Example 2: Suppose we have 5 treatments and 20 EUs. We are interested in the following 5 comparisons Contrast: A vs B, A vs C, A vs D, A vs E, B vs C Treatments A B C D E # of times used 4 2 2 1 1 _ Optimum = T# 2 : 1.4: 1.4: 1 : 1 For 20 EUs 6.0: 4.1: 4.1: 2.9: 2.9 Solution 6 : 4 : 4 : 3 : 3 3/2/04 7 Treatment Structure For the ideal distribution Nw ——. ale + wile M—_— + Nw ———_. ale + wle M—__— + ——_. wie + wl M——_— " Nw w w 3/2/04 10 Treatment Structure Numbers of replication needed to achieve precision equal to the best standard error of the slope Linear Approximate numbers of replications Number of Equal levels of X 3:6:3 reps 5:2:5 12 31 6 26 4 21 3 24 18 14 2 12 Quadratic Approximate number of replications Number of Equal Levels of X 3:6:3 reps 5:2:5 12 25 6 19 4 15 3 12 13 22 3/2/04 11 Treatment Structure 4 Factorial treatment structure 4.1 Types of factorials 4.1.1 Complete factorials P two or more treatment factors P every factor has two or more levels (zero is a valid level) P every combination of the levels of each factor with the levels of every other factor are included in the study. 4.1.2 Incomplete factorials 4.1.3 Response surface designs P finding the optimum P describing the entire surface 3/2/04 12 Treatment Structure 4.2 Factorials, some simple examples 4.2.1 Symbolic - 2x2 Estimation of interaction and main effects from treatment means 3/2/04 15 Treatment Structure 4.3 Advantages 4.3.1 Factorial treatment designs provide a test of hypotheses concerning interaction between factors. If interactions are present, then the single factor (non-factorial) approach is likely to provide a number of disconnected pieces of information which may lead to mis-interpretation of the inter-relationship and incorrect conclusions about factors. 4.3.2 The factorial approach is a more efficient use of the experiment material, since each factorial source of variation is estimated with all experimental units. That is in a two factor experiment every experimental unit is used to test three hypotheses: 1) to estimate the main effect of the first factor, 2) to estimate the main effect of the second factor, and 3) to estimate the interaction effect of the two factors. If factors are independent (interaction not significant) then the sensitivity of the test of main effects is increased since it is appropriate to average across the levels of the other factor. This has been referred to as hidden replication since it is not known when the experiment is designed if the interaction will be significant or non-significant. 3/2/04 16 Treatment Structure For example, suppose we have a 2x3 factorial with 4 reps for a total of 2x3x4 EUs. effective Source #levels #reps/mean A 3 2x4 = 8 if non-sig interaction B 2 3x4 = 12 if non-sig interaction AxB 6 4 Reps 4 3/2/04 17 Treatment Structure 4.4 Disadvantages 4.4.1 As the number of factors and factor levels increase, the number of treatments, and therefore the size of the experiment, may become prohibitively large. Consider the number of treatments required for the following factors and levels: complete response factors levels factorials surface design 3 2 23 = 8 3 3 33 = 27 15 4 2 24 = 16 4 3 34 = 81 25 6 3 36 = 729 77 4.4.2 Large factorials are not easy to interpret when interactions are present, since even for a two factor experiment 3-D graphics may be required. Interactions involving more then 2 factors are therefore very difficult to present since even 3 dimensional graphics are often inadequate. 3/2/04 20 Treatment Structure 6 The Factorial Analysis of Variance 6.1 Linear additive model Yijk = : + "i + $j + "$ij + ,ijk where: Yijk is the observed value for the k th replicate of the ith level of factor A and the jth level of factor B (where I=1 to a, j=1 to b and k=1 to r). : is the grand mean. "i is the effect of the i th level of factor A; the effect may be either fixed or random. $j is the effect for the j th level of factor B; the effect may be either fixed or random. "$ij is the interaction effect of the i th level of factor A with the jth level of factor B; the interaction effect may be either fixed or random ,ijk is the random error associated with the Yijk experimental unit. 3/2/04 21 Treatment Structure 6.2 Estimate of effects and degrees of freedom Sources of variation df Estimate of effects _ _ A a-1 Yi.. - Y... _ _ B b-1 Y.j. - Y... _ _ _ _ A*B (a-1)(b-1) Yij. - Yi.. - Y.j. + Y... _ Error ab(r-1) Yijk - Yij. _ Total rab-1 Yijk - Y... 3/2/04 22 Treatment Structure 6.3 Expected Mean Squares (complete the F ratios for the fixed effects test of hypotheses) 6.3.1 Random Model Sources of Components variation of variance F ---------- --------------------- -------------- A Fe 2 + rFab 2 + rbFa 2 B Fe 2 + rFab 2 + raFb 2 A*B Fe 2 + rFab 2 Error Fe 2 3/2/04 25 Treatment Structure P two way mean for factor AB (interaction mean of A*B) 6.5 Standard Errors for the Fixed Effects in Mixed Models. Suppose factor B is fixed 3/2/04 26 Treatment Structure 7 Interpretation of factorials (3 factor) 7.1 Traditional approach P Start with highest order source (ABC) IF significant interpret 3 factor by examination of 3 way combination means and/or use graphic techniques Don’t interpret lower order sources. P IF ABC is non-significant examine the two factor sources (AB, AC, BC) IF any 2 factor sources are significant then interpret corresponding 2 way combination means and/or use graphics techniques Don’t interpret lower order sources included in the significant interaction. P Interpret any main effects (A, B, and/or C) that are not involved in any significant interaction and examine the main effect means and/or use graphic techniques. 3/2/04 27 Treatment Structure 7.2 A possible alternative: Use the above approach except for the following modification P For each factor determine if the interaction is 1) Magnitude or 2) Rank order IF the interaction is magnitude then you may cautiously make general statements about lower order effects if significant. IF the interaction is rank order do not attempt to interpret the lower order sources even if significant. 3/2/04 30 Treatment Structure 8.3 Example 2: P Structure a0 a1 c0 c1 c0 c1 b0 X X X b1 X X X X P Complete (r=4) S of V df Reps A 1 16 B 1 16 C 1 16 A*B 1 8 A*C 1 8 B*C 1 8 A*B*C 1 4 Trt total 7 3/2/04 31 Treatment Structure P Incomplete (r=4) Sources of Variation df Reps A @ b0c1, b1c0, b1c1 1 12 B @ a0c1, a1c0, a1c1 1 12 C @ a0b1, a1b0, a1b1 1 12 AB @ c1 1 4 AC @ b1 1 4 BC @ a1 1 4 ABC No information 3/2/04 32 Treatment Structure 8.4 Quantitative dosages 8.4.1 Structures I. Drugs a b c 0 X X X 9 groups, but only Quant. 1 X X X 7 diff. trts. 2 X X X ------ OR ------ II. Drugs a b c 0 ------X------ Just 7 diff. trt. groups Quant. 1 X X X 2 X X X 8.4.2 Anovas P Complete (3x3 factorial) (r=4) S of V df Reps D 2 12 Q 2 12 D*Q 4 4 3/2/04 35 Treatment Structure 8.6 Response Surface Designs 8.6.1 Example 1: P Structure A 1 2 3 1 X* X X* B 2 X X* X 3 X* X X* P Complete (all X) S of V df Contrast A 2 L,Q B 2 L,Q A*B 4 LL,LQ,QL,QQ 3/2/04 36 Treatment Structure P Response surface (* marked Xs only) S of V df Contrast A 1 L B 1 L A*B 1 LL Other 1 All quad & quad interactions 3/2/04 37 Treatment Structure 8.6.2 Example 2: P Structure 3x3 A a1 a2 a3 b1 X X X B b2 X X X b3 X X X P Complete S of V df Contrast A 2 L,Q B 2 L,Q AB 4 LL, LQ, QL, QQ