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Replications needed to estimate a mean For a single population, you may want to determine the sample size needed to obtain a given level of precision Y 0 t Recall that s2 r Rearrange the formula r t 2 2 α,df 2 s d r is the number of replications t is the critical t with r-1 df d is the half-width of the confidence interval 2 t α,df s r 2 d Example 2 In a preliminary trial, a sample of 5 replicates has a standard deviation of 3.4 units You would like to conduct an experiment to estimate the mean within two units of the true mean with a confidence level of 95% Y2 But we have a problem here... We need to know how many reps in order to calculate degrees of freedom So it is: – pick – plug – adjust t 2α,df s2 r d2 Calculations Given s = 3.4, d = 2, = 0.05 Try starting with r = 5 Know your t table! In Excel, =T.INV.2T(0.05,4) 2.78 (2-tailed distribution) In Kuehl, use =0.025 = 4 2.78 (Prob. of t t,) r (2.782 * 3.42)/22 = 22.3 Additional iterations… r df 23 22 13 12 14 13 t 2.1 2.2 2.2 calculated r 12.4 13.7 13.5 r = 14 Power Analysis How confident are we that we can detect an important difference, if it exists? ES r Power = 1 - , where ES = Effect Size Used to design an experiment (a priori) – The size of an experiment is often limited by factors other than statistics – Nonetheless, it is good experimental technique to try to estimate the degree of precision that will be attained and to present this information as part of the proposal for the experiment Evaluate a completed experiment (a posteriori ) – May provide justification for publishing nonsignificant results – Guide for improving experimental technique in the future Power Analysis Sources of input for power analysis – educated guesses derived from theory – results of previous studies reported in literature – pilot data Effect Size – several options – minimal practical significance – educated guess of the true underlying effect Questions of interest – number of replications – number of subsamples – plot size – detectable difference Detecting differences between means 2(t1 t 2 ) r 2 d 2 Where 2 may be expressed as percentage of mean or on actual scale r = the number of replications t1 = t at the significance level for the test t2 = t at 2(1-P), where P is the selected probability of obtaining a significant result (power) (note that 1-P is , the probability of a type II error) = standard deviation (or CV%) d = the difference to be detected (d = ES = ) (d can also be calculated for a fixed number of reps) What is meaningful? If it can be established that the new is superior to the old by at least some stated amount, say 20%, then we will have discovered a useful result If the experiment shows no significant difference, we will be discouraged from further investigation The experiment should be large enough to ensure a meaningful difference. For example: r > 2(t1 + t2)2CV2 / d2 We have two varieties to compare. A previous experiment with these treatments found a CV of 11% How many replications would be needed to detect a difference of 20% with a probability of .85 using a 5% significance level test? first pick 4 reps (6 df) then test to see if 8 reps (14 df) is correct t1=T.INV.2T(0.05,6)=2.447 t2=T.INV.2T (0.30,6)=1.134 r > 2(2.145+1.076)2(11)2/(20)2 r > 6.28 ~ 7 r > 2(2.447+1.134)2(11)2/(20)2 r > 7.76 ~ 8 then test to see if 7 reps (12 df) is correct r > 6.44 ~ 7 Number of reps for the ANOVA The ANOVA may have more than two treatments, so we must consider differences among multiple means Power curves are commonly used – see Kuehl pg 63 for more information For this class, we will often get approximate estimates of the number of reps needed using the formula for two means – use the error term from ANOVA (MSE) to estimate s2 – use the appropriate degrees of freedom for the MSE • df = #treatments*(r-1) for a CRD • df = (#treatments-1)(r-1) for an RBD Power curve for ANOVA – example (v1=4) Power = 1 - Values of v2 SStreatments # treatments * MSE 2 OR v1= treatment df rd2 2 * # treatments * MSE 2 v2 = error df SAS Power Calculations PROC POWER proc power; Title 'determine #reps'; onewayanova test=overall groupmeans = 50|56.75|58.25 alpha= 0.01 stddev = 7.7 power = 0.80 npergroup = . ; run; PROC GLMPOWER SAS power and sample size application – Stand alone desktop application that utilizes PROC POWER and PROC GLMPOWER through a user friendly interface Determining Plot Size Factors that affect plot size Type of crop Type of experiment Phase of the research program Variability of the experimental site Presence and nature of border effects Type of machinery to be used Number and type of treatments Land area available Cost Factors Affecting Plot Size Increasing Plot Size Factor Small plots Soil variability Uniform Crop Turf--Cereals Large Plots Heterogeneous -- Row crops -- Trees -- Pasture Late Research phase Early Experiment type Breeding -- Fertilizer -- Tillage -- Irrigation Machinery None Research Farm Scale Effect on Variability Variability per plot decreases as plot size increases But large plots may yield higher experimental error because of larger more variable area for the experiment Very small plots are highly variable because: – Losses at harvest and measurement errors have a greater effect – Reduced plant numbers – Competition and border effects are greater How Plot Size Affects Variability Plot Variability Plot Size Plot Size ‘Rule of Thumb’ There are some lower limits: – Should be large enough to permit removal of borders with enough left over to harvest and measure adequately – Should be large enough to handle the machinery needed Once the plots are large enough to be handled conveniently, precision is increased faster by increasing the number of replications Good review of studies on optimum plot size for many crops: LeClerg, Leonard, and Clark, 1964, Field Plot Technique (chapter 9) Smith’s Soil Variability Index V Vx = xb Where: V = variance of a unit plot Vx = variance, on a per unit basis, of plots formed from x adjacent units x = plot size in multiples of adjacent unit plots b = index of soil variability Smith’s Soil Variability Index V Vx = b x The index can vary from 0 to 1 When b=0 then Vx = V When b=1 then Vx = V x This means there is no relation between variance and plot size. Adjacent plots are completely correlated. Nothing can be gained from larger plots. This means that the units that make up the plots are independent of each other. Increasing plot size will reduce variance. This is the case when the soil is highly uniform. This is the case when the soil is highly heterogeneous. Effect of plant sample size However, can only obtain b=0 when plant sample size is above a minimal level For large plants, Variance of the mean sample size may be more critical than soil variability in determining optimal plot size Effect of sample size on variance 8.0 7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0 5 10 15 20 25 30 number of plants 35 40 45 50 Smith’s “Law” – calculation of b b = Smith’s Soil Variability Index = Smith’s coefficient of soil heterogeneity V Vx = b x log Vx = log V - b log x y = a + bx a (intercept) Log of variance per plot b (slope) y x Log of plot size Raw data from Uniformity Trial 488 488 457 440 325 347 424 426 448 476 416 387 470 446 410 388 303 315 406 392 359 403 365 462 387 384 454 430 445 382 418 426 393 420 333 328 491 457 436 388 369 393 408 346 347 490 440 319 353 352 385 389 389 365 479 438 396 450 343 390 354 389 374 428 415 393 402 394 411 419 396 426 438 296 503 420 361 368 386 375 321 435 449 438 470 427 446 414 473 455 367 510 337 384 376 437 402 375 403 452 442 455 410 435 441 348 371 375 976 897 672 850 924 803 916 798 618 798 762 827 771 884 827 844 813 661 948 824 762 754 837 759 705 774 754 917 846 733 743 802 808 796 830 822 734 923 729 761 756 887 897 860 928 877 721 813 777 855 897 845 789 746 Combine yields of plots from adjacent rows (2x1) Combination x 1x1 1x2 1x3 1x6 2x1 2x2 2x3 2x6 3x1 3x2 3x3 3x6 4x1 4x2 4x3 4x6 6x1 6x2 6x3 1 2 3 6 2 4 6 12 3 6 9 18 4 8 12 24 6 12 18 Log(x) 0.0000 0.3010 0.4771 0.7782 0.3010 0.6021 0.7782 1.0792 0.4771 0.7782 0.9542 1.2553 0.6021 0.9031 1.0792 1.3802 0.7782 1.0792 1.2553 Vx 2177.18 1244.46 959.05 631.50 1447.50 920.00 706.10 428.81 1090.53 767.66 592.02 432.91 684.23 472.11 335.64 351.01 339.54 194.84 159.58 Log(Vx) 3.3379 3.0950 2.9818 2.8004 3.1606 2.9638 2.8489 2.6323 3.0376 2.8852 2.7723 2.6364 2.8352 2.6740 2.5259 2.5453 2.5309 2.2897 2.2030 Smith’s Index of Variability Log of variance per plot 4 3.5 log Vx = 3.3261- 0.7026 log x 3 2.5 2 0.0000 0.4771 0.7782 0.9542 Log of plot size log Vx = log V - b log x y = a + bx 1.2553 Is there a better way? Examination of a large number of data sets indicated that a value of b=0.5 may serve as a reasonable approximation. “Finagles” constant b=0.5 Nested designs, Generalized Least Squares may give better estimates (Swallow and Wehner, 1986) Learn from expert knowledge and your own experience – Plot size – Block size – Number of replications Optimum Plot Size Optimum Size will either minimize cost for a fixed variance or minimize variance for a fixed cost You must know certain costs: T = K 1 + K 2x Where: T = total cost per plot ($ or time) K1= cost per plot ($ or time) that is independent of plot size K2= cost per plot ($ or time) that depends on plot size x = number of unit plots Optimum Plot Size Knowing those costs, Smith found the optimum plot size to be: xopt = bK1 (1-b)K2 Where: T = total cost per plot ($ or time) K1= cost per plot ($ or time) that is independent of plot size K2= cost per plot ($ or time) that depends on plot size x = number of unit plots b = Smith’s index of soil variability Optimum Plot Size So in our example, we found b to be 0.70 and our unit plot area was 5m x 2m or 10m2 Assume K1 = $3.00 and K2 = $5.00 for 10 m2 xopt = 2m = 7m bK1 (1-b)K2 (0.70)(3.00) = 1.4 (1-0.70)(5.00) Area = (1.4)(10) = 14.0 m2 Convenient Plot Size Smith’s optimum plot size was based on soil variability and cost. Hathaway developed a formula based on soil variability, the size of difference to be detected, and significance level of the test. xb = 2(t1 + t2)2 2 rd2 Convenient Plot Size xb = 2(t1 + t2)2 2 rd2 Where: x= number of units of plots of size x (plot size) b= Smith’s coefficient of soil variability t1= t at the significance level for the test t2= t in the t table at 2(1-P), where P is the selected probability of obtaining a significant result 2= variance from a previous experiment (may also use the CV2) r= number of replications d= difference to be detected (may be an absolute amount or expressed as percentage of the mean) For Example . . . A variety trial of 50 selections in a randomized block design with three blocks What size plot do we need to detect a difference of 25% of the mean with an 80% probability of obtaining a significant result using a 5% significance level test? Previous experiment had a CV of 11% xb = 2(t1 + t2)2 CV2 rd2 error df = (3-1)(50-1) = 98 t1 = t0.05(98)) = 1.984 t2 = t0.40(98) = 0.845 {t 2(1-P) = t 2(1-0.80) = t0.40} b = 0.70 CV = 11% Then: xb = 2(1.984 + 0.845)2(11)2 = 1936.78 (3)(25)2 1875 xb = 1.0330 or x.70 = 1.0330 x = 1.03301/.70 = 1.0475 basic units Convenient Plot Size Since the basic plot has an area of 10 m2, the required plot size would be: (10.00)(1.0475) = 10.48 m2 2m 5.24 m The Detectable Difference Using Hathaway’s formula for convenient plot size and solving for d2, it is possible to compute the detectable difference possible when you know the number of reps and the plot size. d2 = 2(t1 + t2)2 2 rxb Note that for a standard plot size (X=1), the Xb term drops out and we have the same formula that we used for calculating # reps An Alternative Example We want to find the difference that can be detected 80% of the time at a 5% significance level using a plot 2 m wide and 7 m long in 4 replications d2 = 2(t1 + t2)2 CV2 rxb As before, the CV=11%, and b=0.70 d2 = 2(t1 + t2)2 CV2 rxb d2 = 2(1.976 + 0.844)2(11)2 (4)(1.40.70) = (2)(7.954)(121) (4)(1.2656) because 2m*7m/10m2=1.4 = 1924.90 5.0623 = 380.24 d = 19.50 Therefore, with a plot size of 2m x 7m using 4 replications, a difference of 19.50% of the mean could be detected 35 30 Detectable difference 25 (% of 20 Mean) 15 10 5 2 4 6 8 10 20 Plot Area (m2) 30 Replication Using these tools