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**Modern Experimental Designs:**Construction and Analysis Replace this box with a picture? Just click : Insert Picture – from file Locate your image Click – insert Position picture over box Crop/scale etc. Select picture, hold down shift key and click on white background then Click Draw –rder – send to back The top of your picture should be hidden by the top shape. Emlyn Williams CSIRO Australia**Topics covered**• Standard Design Types • Completely Randomized • Randomized Block • Factorial • Split-plot • Incomplete Block Designs • Alpha • Row-column • Crossover • Spatial**Analysis of variance**• Completely Randomized Design • Randomized Complete Block Design**Dbh (cm) means for a seed orchard (SO) and a routine**plantation (P) seedlot Seedlot Replication SO P ______________________ 1 30.38 28.16 2 27.91 25.62 3 28.06 26.61 4 31.42 32.59 5 30.11 28.80 6 31.52 28.19 7 31.72 31.23 8 33.53 28.34**GenStat output from analysis of variance ofdbh means for a**Completely Randomized Design Source of variation d.f. s.s. m.s. v.r. F pr. plot stratum seedlot 1 14.269 14.269 3.25 0.093 Residual 14 61.410 4.386 Total 15 75.679 ***** Tables of means ***** Grand mean 29.64 seedlot SO P 30.58 28.69**Estimation - Completely Randomized Design**Overall mean Seedlot effects**Residuals -Completely Randomized Design**Plot residuals vs fitted values**How good is the model?**• If the assumed linear model is good, then the residuals will be small • We need to be able to measure how good the model is**Calculation of sums of squares**Seedlot sum of squares Residual sum of squares**Check assumptions!!!!**• Use plot of residuals vs fitted values • Look for possible outliers • Look for any relationship between the fitted values and the spread of residuals**Example of a plot of residuals against fitted values**I I * * 2 3 5 I 3 2 2 3 3 2 2 2 0.0 I 3 5 3 3 5 * 3 2 4 I * * 3 2 * * 2 I * * 2 2 I I * I -5.0 I I * I I I * I -10.0 I -+---------+---------+---------+---------+---------+---------+-- 0.0 2.0 4.0 6.0 8.0 10.0 12.0 resid v. fitted using symbol ***Layout of plots with seedlot labels (SO or P) and dbh means**Replicate 1 2 3 4 5 6 7 8 ______________________________________________________ SO SO P SO P P SO P 30.38 27.91 26.61 31.42 28.80 28.19 31.72 28.34 P P SO P SO SO P SO 28.16 25.62 28.06 32.59 30.11 31.52 31.23 33.53**GenStat output from analysis of variance of dbh means for a**Randomized Complete Block Design Source of variation d.f. s.s. m.s. v.r. F pr. repl stratum 7 48.867 6.981 3.90 repl.plot stratum seedlot 1 14.269 14.269 7.96 0.026 Residual 7 12.543 1.792 Total 15 75.679 ***** Tables of means ***** Grand mean 29.64 seedlot SO P 30.58 28.69**Estimation**Replicate effects Fitted values**Residuals**Plot residuals vs fitted values**Residual sums of squares**• Completely randomized design=61.410 • Randomized complete block design =12.543 • Residuals are smaller for RCB • It is a better model**Testing**• We test the null hypothesis that the seedlots do not differ from each other • Use the F test • For the RCB design the F value is 7.96 on 1 and 7 d.f. (5% value for F1,7=5.59) • Reject the null hypothesis**Standard errors of difference (SED)**Between two seedlot means is where t- value = 1.89/0.67 = 2.82 (5% value for t7=2.365)**Least significant difference (LSD)**• LSD =t7 * SED=2.365*0.67 =1.585 • this is smaller than the difference between seedlot means (1.89) • Hence seedlot means significantly different**Important issues**• Use of blocking structures (RCB reduces residual mean square from 4.386 to 1.792) • Data needs to be presented in field order so that blocking structures are clear • Check assumptions**Strata**• Each time we place a restriction on the allocation of treatments to plots, we create a stratum • Each replicate contains one plot of each seedlot: replicate stratum • The plots within each replicate create the plot-within-replicate stratum**Layout of seedlings in four replicates of five plots, each**with four trees (each x refers to one tree) Replicate Plot 1 2 3 4 __________________________________________ 1 x x x x x x x x x x x x x x x x 2 x x x x x x x x x x x x x x x x 3 x x x x x x x x x x x x x x x x 4 x x x x x x x x x x x x x x x x 5 x x x x x x x x x x x x x x x x direction of trend**Replicate (repl) stratum**Replicate Plot 1 2 3 4 __________________________________________ 1 x x x xx x x xx x x xx x x x 2 x x x xx x x xx x x xx x x x 3 x x x xx x x xx x x xx x x x 4 x x x xx x x xx x x xx x x x 5 x x x xx x x xx x x xx x x x direction of trend**Plot-within-replicate (repl.plot) stratum**Replicate Plot 1 2 3 4 __________________________________________ 1 x x x x 2 x x x x 3 x x x x 4 x x x x 5 x x x x direction of trend**Tree-within-plot-within-replicate (repl.plot.tree) stratum**Replicate Plot 1 2 3 4 __________________________________________ 1 x xxx 2 3 4 5 direction of trend**Two-dimensional layout with four rows and four columns (each**X refers to a whole plot) Column Row 1 2 3 4 ______________________ 1 x x x x direction 2 x x x x of trend 3 x x x x 4 x x x x direction of trend**What are the strata?**• Row (row) stratum • Column (column) stratum • Plot (row.column) stratum • Could also then have a row.column.tree stratum**Factorial Designs**• More than one treatment factor • e.g. seedlot, fertilizer • or treatments A and B model**Example of interaction**Interaction caused by combined effect of A and B**Germination test**• Treatment A • 6 Acacia mangium seedlots • Treatment B • 4 seed pre-treatments • control • nick • boiling water and soak • boiling water 1 min • 25 seeds per dish • 3 replicates (trays) • 24 dishes per replicate (4 x 6 array) • variate is percentage germination**Control comparison**• In GenStat we can set up an option to compare the control treatment against the other treatments • This gives a test of whether the treatments overall are doing better than the control**GenStat output from analysis of variance of percentage**germination ***** Analysis of variance ***** Variate: v[1]; percent - percent=(count/25)*100 Source of variation d.f. s.s. m.s. v.r. F pr. repl stratum 2 35.11 17.56 0.18 repl.row.column stratum contcomp 1 58542.30 58542.30 601.52 <.001 seedlot 5 2894.44 578.89 5.95 <.001 contcomp.treat 2 5300.15 2650.07 27.23 <.001 contcomp.seedlot 5 1347.04 269.41 2.77 0.029 contcomp.treat.seedlot 10 961.19 96.12 0.99 0.467 Residual 46 4476.89 97.32 Total 71 73557.11**(continued)******* Tables of means ***** Variate: v[1]; percent - percent=(count/25)*100 Grand mean 51.4 contcomp 1 2 2.0 67.9 rep. 18 54 seedlot 18265 18249 18248 18211 18212 18217 59.7 48.7 40.7 58.0 52.3 49.0 contcomp treat control nick bw&s bw1min 1 2.0 2 56.9 65.8 80.9**Plot of residuals against fitted values from analysis of**variance of germination percentage I I I 25.0 I I * I * * I * * * * * I * * * I ** * *** * * * 2 * ** 0.0 I 73 ** * ** * * * 2 I 2** * *** 2 * *** * I 2* ** I * ** * I * * I -25.0 I * -+---------+---------+---------+---------+---------+---------+-- -20.0 0.0 20.0 40.0 60.0 80.0 100.0 resid v. fitted using symbol ***Options to correct for problem with spread of residuals**• Transformation? • Here small fitted values and small range of residuals related just to the control treatment, so better to remove the control from the analysis • Use the restrict option of GenStat**GenStat output from analysis of variance with the control**pre-treatment deleted ***** Analysis of variance ***** Variate: v[1]; percent - percent=(count/25)*100 Source of variation d.f. s.s. m.s. v.r. F pr. repl stratum 2 64.6 32.3 0.25 repl.row.column stratum treat 2 5300.1 2650.1 20.61 <.001 seedlot 5 4148.1 829.6 6.45 <.001 treat.seedlot 10 961.2 96.1 0.75 0.676 Residual 34 4372.7 128.6 Total 53 14846.8**Interpretation of analysis**• Seedlots significantly different (***). Best germination is seedlot 18265 (79.1%)LSD=10.6 • Control treatment removed from the analysis • Other pre-treatments significantly different (***). Best pre-treatment is boiling water for one minute (80.9%)LSD=7.67 • Interaction not significant**Split-plot designs**• More than one treatment factor • Strata normally called replicates, main-plots-within-replicates and sub-plots-within-main-plots-within-replicates • Treatment factors on different strata • e.g. basal nitrogen at the main-plots level, seedlots at the sub-plots level**Irrigation / Fertilizer trial**• Main-plot treatment factors • irrigation (yes or no) • fertilizer (yes or no) • Sub-plot treatment factor • 4 Eucalyptus grandis seedlots • 2 replicates • 42 trees per plot (7 x 6) • Variate is height at 34 months**Layout of plots with seedlot numbers and height means**Replicate 1 2 irrigation: none none plus plus plus plus none none fertilizer: none plus plus none none plus none plus ______________________________________________________ 4 2 1 3 2 1 4 3 4.71 16.36 14.38 4.66 5.41 14.60 4.32 14.98 3 1 2 4 3 4 1 2 6.23 15.29 16.89 4.95 5.73 12.21 4.16 15.98 2 3 4 1 4 2 3 1 7.46 13.99 11.25 5.81 5.80 14.84 5.02 14.40 1 4 3 2 1 3 2 4 6.39 11.08 15.58 7.50 6.39 15.00 6.79 11.98 Seedlots 1 Bulahdelah 2 Coffs Harbour seed orchard 3 Pomona plantation 4 Atherton**Plot of residuals against fitted values from analysis of**variance of height means 1.2 I I * I * I * * * I * ** * * I ** * * * 0.0 I * * I * * ** * I *2 * * I * * * I * I * -1.2 I -+---------+---------+---------+---------+---------+---------+-- 2.5 5.0 7.5 10.0 12.5 15.0 17.5 resid v. fitted using symbol ***GenStat output from analysis of variance of height means**Source of variation d.f. s.s. m.s. v.r. F pr. repl stratum 1 0.7564 0.7564 1.08 repl.mainpl stratum irrig 1 0.1081 0.1081 0.15 0.721 fert 1 590.6485 590.6485 841.11 <.001 irrig.fert 1 0.0072 0.0072 0.01 0.926 Residual 3 2.1067 0.7022 1.05 repl.mainpl.subpl stratum seedlot 3 39.6538 13.2179 19.68 <.001 irrig.seedlot 3 1.1098 0.3699 0.55 0.657 fert.seedlot 3 9.9503 3.3168 4.94 0.018 irrig.fert.seedlot 3 1.7360 0.5787 0.86 0.487 Residual 12 8.0596 0.6716 Total 31 654.1364**(continued)******* Tables of means ***** Grand mean 10.00 irrig none plus 9.95 10.06 fert none plus 5.71 14.30 seedlot Bulahdelah Coffs SO Pomona pltn Atherton 10.18 11.40 10.15 8.29 fert seedlot Bulahdelah Coffs SO Pomona pltn Atherton none 5.69 6.79 5.41 4.95 plus 14.67 16.02 14.89 11.63**Interpretation of analysis**• Fertilizer significant (***) • Irrigation not significant • Irrigation by fertilizer interaction not significant • Seedlots significant (***) • Coff’s harbour seed orchard best (11.40m) • LSD =0.89 • Seedlot by fertilizer interaction significant (*) • due to different behaviour of Atherton seedlot**2 x2 drug trial(Jones and Kenward Ex 2.1)**• 56 patients • Drug (A) vs Placebo (B) • 27 in AB group, 29 in BA group • Variate is exploratory flow rate (response)

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