第六因設(shè)計分析

1、第六章 析因設(shè)計分析生物統(tǒng)計與實(shí)驗設(shè)計1浙江大學(xué) 朱軍 徐海明6.1 雙因素析因設(shè)計實(shí)例分析 (1)某面包公司為城市里的許多超市提供包裝好的意大利薄片面包設(shè)計一個試驗用來研究貨架的高度（底部、中部、上部）和貨架的寬度（常規(guī)、寬型）對該 種面包銷售量的影響2第六章 析因設(shè)計分析6.1 雙因素析因設(shè)計6.2 多因素析因設(shè)計1第六章 析因設(shè)計分析生物統(tǒng)計與實(shí)驗設(shè)計2浙江大學(xué) 朱軍 徐海明雙因素析因設(shè)計方差分析模型：Yijk = m + ai + b j + abij + eijki = 1, 2,K, a; j = 1, 2,K, b; k = 1, 2,K, r,4因素B 因素AB1B2¼

2、;Bj¼BbA1AB11AB12¼AB1j¼AB1bA2AB21AB22¼AB2j¼AB2bAiABi1ABi2¼ABij¼ABib#AaABa1ABa2¼ABaj¼ABab試驗單元: 12個具有相似銷售量和顧客群的超市因素:貨架的高度，三個水平 (底部、中部、上部)貨架的寬度，兩個水平 (常規(guī)、寬型)6個處理 (3 ´ 2)每個處理被隨機(jī)地安排到兩個超市，面包在每個超市的擺放按照處理的要求放置3第六章 析因設(shè)計分析生物統(tǒng)計與實(shí)驗設(shè)計3浙江大學(xué) 朱軍 徐海明總平方和及自由度的分解å&#

3、229;å(Y- Y.)2 =brå(Y . - Y.)2 +arå(Y. . - Y.)2ijkijijkij+råå(Y . - Y . - Y. . +Y.)2 +ååå(Y- Y .)2ijijijkijijijkSSTO = SSA + SSB + SSAB + SSEdfTO= df A+dfB+df AB+dfeabr - 1 = (a - 1) + (b - 1) + (a - 1)(b - 1) + ab(r - 1)65第六章 析因設(shè)計分析生物統(tǒng)計與實(shí)驗設(shè)計4浙江大學(xué) 朱軍 徐海明固定模型Yij

4、k =m+ai +b j +abij +eijk N (m+ai +b j +abij , s 2 )ai = 1, 2,K, a; j = 1, 2,K, b; k = 1, 2,K, r;eijk N (0, s 2 )eåai = åb j =åabij = åabij = 0iiij8兩因素析因設(shè)計的方差分析表異 自由度E(MS) 固定E(MS)E(MS) 混合源隨機(jī)Aa-1s 2+ br åa 2s 2+rs 2 + brs 2 s 2+rs 2 + br å a 2eabaea - 1 iieaba - 1 iiBb-1s

5、 2+ ar å b 2s 2+rs 2 + ars 2 s 2+ars 2eabbebeb - 1 jjB(a-1) s 2+råå ab 2 s 2+rs 2s 2+rs 2eabeab(b-1)e(a - 1)(b - 1) ijijror ab(r-1) s 2s 2s 2eee7第六章 析因設(shè)計分析生物統(tǒng)計與實(shí)驗設(shè)計5浙江大學(xué) 朱軍 徐海明Data Bread;Input Height $ Width $ Sales ; Datalines;A1 B1 47 A1 B2 46A1 B1 43 A1 B2 40A2 B1 62 A2 B2 67A2 B1

6、68 A2 B2 71A3 B1 41 A3 B2 42A3 B1 39 A3 B2 46;PROC GLM;Class Height Width;MSales = Height Width Height*Width; Means Height /LSD;Means Height /Tukey; Means Width/LSD; Means Width /Tukey;Means Height*Width/LSD; Means Height*Width/Tukey; RUN;10變異來源 自由度均方期望均方 固定A因素a-1MSA s 2 + br åa 2ea - 1i iB因素b-

7、1MSB s 2 + ar å b 2eb - 1j j互作(a-1)(b-1) MSAB s 2 +rååab 2e(a - 1)(b - 1)ij ij誤差ab(r1)MSE s 2eF 測驗H0: 所有 ai = 0; H1: 不是所有 ai = 0F* = MSA/MSE F(a-1, ab(r-1) 如果H0 是真H0: 所有 bj = 0; H1: 不是所有 bj = 0F* = MSB/MSE F(b-1, ab(r-1) 如果H0 是真H0: 所有 abij = 0; H1: 不是所有 abij = 0F* = MSAB/MSE F(a-1) (b

8、-1), ab(r-1) 如果H0 是真9第六章 析因設(shè)計分析生物統(tǒng)計與實(shí)驗設(shè)計6浙江大學(xué) 朱軍 徐海明t GroupingMeanNHeight A67.0004A2B44.0004A1B42.0004A3WidthA52.0006B2A50.0006B1Height WidthNMeanStd DevA1B1245.00000002.82842712A1B2243.00000004.24264069A2B1265.00000004.24264069A2B2269.00000002.82842712A3B1244.00000002.82842712A3B2244.00000002.82842

9、71212Dependent Variable: SalesSum ofSourceDFSquaresMean Square F ValuePr > F M51580.00316.0030.580.0003Height21544.00772.0074.71<.0001Width112.0012.001.160.3226Height*Width 224.0012.001.160.3747Error662.0010.3311第六章 析因設(shè)計分析生物統(tǒng)計與實(shí)驗設(shè)計7浙江大學(xué) 朱軍 徐海明14實(shí)例分析 (2)o 水稻品種和栽培措施的試驗:o 因素:n 區(qū)組 (3 次重復(fù))n A: 品種 (

10、A1 A2 A3)n B:密度(108cm2, 163cm2, 218cm2)n 9個處理 (3 ´ 3)o 試驗單元: 27 個小區(qū) (20m2)o 依變量: 產(chǎn)量 (kg)13第六章 析因設(shè)計分析生物統(tǒng)計與實(shí)驗設(shè)計8浙江大學(xué) 朱軍 徐海明SourceDFSum SquaresMean Square F ValuePr > F M1054.592592595.459259267.610.0002Variety29.851851854.925925936.860.0070Stand225.4074074112.7037037017.70<.0001Variety*Stan

11、d418.814814814.703703706.550.0025Block20.518518520.259259260.360.7023Error1611.481481480.71759259ANOVA Without Block EffectsM854.074074076.7592592610.14<.0001Variety29.851851854.925925937.390.0045Stand225.4074074112.7037037019.06<.0001Variety*Stand418.814814814.703703707.060.0013Error1812.0000

12、00000.6666666716Data Rice;Input Variety $ Stand $ Block Yield ; Datalines;A1 B1 1 8 A1 B1 2 8 A1 B1 3 8A1 B2 1 7 A1 B2 2 7 A1 B2 3 8A1 B3 1 4 A1 B3 2 5 A1 B3 3 4A2 B1 1 9 A2 B1 2 9 A2 B1 3 8A2 B2 1 7 A2 B2 2 9 A2 B2 3 6A2 B3 1 6 A2 B3 2 5 A2 B3 3 4A3 B1 1 9 A3 B1 2 7 A3 B1 3 8A3 B2 1 8 A3 B2 2 7 A3

13、B2 3 8A3 B3 1 8 A3 B3 2 8 A3 B3 3 9;PROC GLM;Class Variety Stand Block;MYield = Variety Stand Variety*Stand Block;Means Variety /LSD;基于最小二乘的均值估計；STDERR:Means Variety /Tukey; 打印出t檢驗(H0: LSM=0); PDIFF: Means Stand /LSD;打印出各平均數(shù)對比較后的統(tǒng)計顯著程Means Stand /Tukey; 度; TDIFF: 打印出t值及顯著程度LSMeans Variety*Stand / ST

14、DERR PDIFF; RUN;15第六章 析因設(shè)計分析生物統(tǒng)計與實(shí)驗設(shè)計9浙江大學(xué) 朱軍 徐海明如果AB互作是顯著的，就沒有必要檢驗 A or B主效是否顯著.我們應(yīng)該檢驗 A因素在特定的 B水平中的表現(xiàn), 或B因素在特定的 A水平中的表現(xiàn)18Tukey Grouping MeanNVariety A8.00009A3B7.00009A2B6.55569A1A8.22229B1A7.44449B2B5.88899B3VarietyStandNMeanStd DevA1B138.000000000.00000000A1B237.333333330.57735027A1B334.33333333

15、0.57735027A2B138.666666670.57735027A2B237.333333331.52752523A2B335.000000001.00000000A3B138.000000001.00000000A3B237.666666670.57735027A3B338.333333330.5773502717第六章 析因設(shè)計分析生物統(tǒng)計與實(shí)驗設(shè)計10浙江大學(xué) 朱軍 徐海明Least Squares Means for effect Variety*Stand Pr > |t| for H0: LSMean(i)=LSMean(j)20i/j12345678910.3495&

16、lt;.00010.34950.34950.00051.00000.63640.636420.34950.00050.07181.00000.00390.34950.63640.16753<.00010.0005<.00010.00050.3495<.00010.0002<.000140.34950.0718<.00010.0718<.00010.34950.16750.636450.34951.00000.00050.07180.00390.34950.63640.167560.00050.00390.3495<.00010.00390.00050.

17、00140.000271.00000.3495<.00010.34950.34950.00050.63640.636480.63640.63640.00020.16750.63640.00140.63640.349590.63640.1675<.00010.63640.16750.00020.63640.3495191A1B18.000000001A1B18.000000002A1B27.333333334A2B18.666666673A1B34.333333337A3B18.000000004A2B18.666666672A1B27.333333335A2B27.33333333

18、5A2B27.333333336A2B35.000000008A3B27.666666677A3B18.000000003A1B34.333333338A3B27.666666676A2B35.000000009A3B38.333333339A3B38.33333333第六章 析因設(shè)計分析生物統(tǒng)計與實(shí)驗設(shè)計11浙江大學(xué) 朱軍 徐海明Data Rice;Input Variety $ Stand $ Block Yield ; Datalines;A1 B1 1 8 A1 B1 2 8 A1 B1 3 8A1 B2 1 7 A1 B2 2 7 A1 B2 3 8A1 B3 1 4 A1 B3 2

19、 5 A1 B3 3 4A2 B1 1 9 A2 B1 2 9 A2 B1 3 8A2 B2 1 7 A2 B2 2 9 A2 B2 3 6A2 B3 1 6 A2 B3 2 5 A2 B3 3 4A3 B1 1 9 A3 B1 2 7 A3 B1 3 8A3 B2 1 8 A3 B2 2 7 A3 B2 3 8A3 B3 1 8 A3 B3 2 8 A3 B3 3 9;PROC GLM;Class Variety Stand Block;MYield = Variety|Stand Block; Contrast 'Linear' Stand -1 0 1;Contrast

20、'Quadratic' Stand 1 -2 1; Means Variety /LSD;RUN;22數(shù)量型因素的分析o 因素:n A 是質(zhì)量型:品種 (A1 A2 A3)n B 是數(shù)量型:密度 (108cm2, 163cm2, 218cm2)SAS 命令:Class Variety Stand Block;MYield = Variety|Stand Block; Contrast 'Linear' Stand -1 0 1;Contrast 'Quadratic' Stand 1 -2 1;21第六章 析因設(shè)計分析生物統(tǒng)計與實(shí)驗設(shè)計12浙江大

21、學(xué) 朱軍 徐海明24SourceDFSum SquaresMean SquareF ValuePr>F M1054.592592595.459259267.610.0002Variety29.851851854.925925936.860.0070Var*Sta418.814814814.703703706.550.0025Block20.518518520.259259260.360.7023Error1611.481481480.71759259因Linear和Quadratic的線性組合的系數(shù)正交， 所以Stand的平方和分解成了歸因于Linear和Quadratic的平方和23S

22、tand225.4074074112.7037037017.70<.0001Linear124.5000000024.5000000034.14<.0001Quadratic10.907407410.907407411.260.2774第六章 析因設(shè)計分析生物統(tǒng)計與實(shí)驗設(shè)計13浙江大學(xué) 朱軍 徐海明比較 X 是線性的還是二次型的Xi = ci /å c2iContrast Test (df = 1)X1X2X3 Linear Contrast XL-1/201/2 Quadratic Contrast XQ1/6-2/61/6Contrast Test (df = 1)X

23、1X2X3 Linear Contrast XL-0.7071070.0000000.707107Quadratic Contrast XQ0.408248-0.8164970.408248效應(yīng)的正交分解需根據(jù)效應(yīng)的水平數(shù)查相應(yīng)的正交分解系數(shù)表26第六章 析因設(shè)計分析生物統(tǒng)計與實(shí)驗設(shè)計14浙江大學(xué) 朱軍 徐海明27第六章 析因設(shè)計分析生物統(tǒng)計與實(shí)驗設(shè)計15浙江大學(xué) 朱軍 徐海明第六章 析因設(shè)計分析生物統(tǒng)計與實(shí)驗設(shè)計16浙江大學(xué) 朱軍 徐海明第六章 析因設(shè)計分析生物統(tǒng)計與實(shí)驗設(shè)計17浙江大學(xué) 朱軍 徐海明SourceDF Sum Squares Mean Square F Value Pr &g

24、t; F M1054.592592595.459259267.610.0002Variety29.851851854.925925936.860.0070XL124.5000000024.5000000034.14<.0001XQ10.907407410.907407411.260.2774XL*Var216.000000008.0000000011.150.0009XQ*Var 22.814814811.407407411.960.1731Block20.518518520.259259260.360.7023Error1611.481481480.71759259SourceDF S

25、um Squares Mean Square F Value Pr>F M1054.592592595.459259267.610.0002Variety29.851851854.925925936.860.0070Stand225.4074074112.7037037017.70<.0001Linear124.5000000024.5000000034.14<.0001Quadratic 10.907407410.907407411.260.2774Var*Sta418.814814814.703703706.550.0025Block20.518518520.259259

26、260.360.7023Error1611.481481480.7175925934Data Rice;Input Variety $ Block XL XQ Yield ; Datalines;A1 1 -0.707107 0.408248 8 A1 2 -0.707107 0.408248 8 A1 3 -0.707107 0.408248 8A1 1 0.000000 -0.816497 7 A1 2 0.000000 -0.816497 7 A1 3 0.000000 -0.816497 8A1 1 0.707107 0.408248 4 A1 2 0.707107 0.408248

27、5 A1 3 0.707107 0.408248 4A2 1 -0.707107 0.408248 9 A2 2 -0.707107 0.408248 9 A2 3 -0.707107 0.408248 8A2 1 0.000000 -0.816497 7 A2 2 0.000000 -0.816497 9 A2 3 0.000000 -0.816497 6A2 1 0.707107 0.408248 6 A2 2 0.707107 0.408248 5 A2 3 0.707107 0.408248 4A3 1 -0.707107 0.408248 9 A3 2 -0.707107 0.408

28、248 7 A3 3 -0.707107 0.408248 8A3 1 0.000000 -0.816497 8 A3 2 0.000000 -0.816497 7 A3 3 0.000000 -0.816497 8A3 1 0.707107 0.408248 8 A3 2 0.707107 0.408248 8 A3 3 0.707107 0.408248 9;PROC GLM;Class Variety Block;MYield = Variety XL XQ Variety*XL Variety*XQ Block;Means Variety /LSD; RUN;33第六章 析因設(shè)計分析生

29、物統(tǒng)計與實(shí)驗設(shè)計18浙江大學(xué) 朱軍 徐海明o 確定B因素的取值Contrast (df = 2)B1B2B3å c2iX3(Linear)-1/201/22X4(Quadratic)1/6-2/61/66o 確定AB互作因素的取值o X1×X3, X2×X3 for A × Linearo X1×X4, X2×X4 for A × Quadratic36二因素析因設(shè)計的回歸分析方法Y = b0 + b1 X1 + b2 X 2 +K+ bp X p + eo 確定A因素的取值ContrastA1A2A3 (df = 2)X1

30、10-1X201-135第六章 析因設(shè)計分析生物統(tǒng)計與實(shí)驗設(shè)計19浙江大學(xué) 朱軍 徐海明Data Rice;Input X1 X2 X3 X4 Block Yield ; A1L = X1 * X3;A2L = X2 * X3;Datalines;1 0 -0.707107 0.408248 1 8 1 0 -0.707107 0.408248 2 8 1 0 -0.707107 0.408248 3 81 0 0.000000 -0.816497 1 7 1 0 0.000000 -0.816497 2 7 1 0 0.000000 -0.816497 3 81 0 0.707107 0.4

31、08248 1 4 1 0 0.707107 0.408248 2 5 1 0 0.707107 0.408248 3 40 1 -0.707107 0.408248 1 9 0 1 -0.707107 0.408248 2 9 0 1 -0.707107 0.408248 3 80 1 0.000000 -0.816497 1 7 0 1 0.000000 -0.816497 2 9 0 1 0.000000 -0.816497 3 60 1 0.707107 0.408248 1 6 0 1 0.707107 0.408248 2 5 0 1 0.707107 0.408248 3 4-1

32、 -1 -0.707107 0.408248 1 9 -1 -1 -0.707107 0.408248 2 7 -1 -1 -0.707107 0.408248 3 8-1 -1 0.000000 -0.816497 1 8 -1 -1 0.000000 -0.816497 2 7 -1 -1 0.000000 -0.816497 3 8-1 -1 0.707107 0.408248 1 8 -1 -1 0.707107 0.408248 2 8 -1 -1 0.707107 0.408248 3 9;PROC REG;MYield = X1 X2 X3 X4 A1L A2L; RUN;38X

33、1X2X3X4YX1X2X3X4YX1X2X3X4Y10-0.7071070.408248810-0.7071070.408248810-0.7071070.4082488100.000000-0.8164977100.000000-0.8164977100.000000-0.8164978100.7071070.4082484100.7071070.4082485100.7071070.408248401-0.7071070.408248901-0.7071070.408248901-0.7071070.4082488010.000000-0.8164977010.000000-0.8164

34、979010.000000-0.8164976010.7071070.4082486010.7071070.4082485010.7071070.4082484-1-1-0.7071070.4082489-1-1-0.7071070.4082487-1-1-0.7071070.4082488-1-10.000000-0.8164978-1-10.000000-0.8164977-1-10.000000-0.8164978-1-10.7071070.4082488-1-10.7071070.4082488-1-10.7071070.408248937第六章 析因設(shè)計分析生物統(tǒng)計與實(shí)驗設(shè)計20浙江

35、大學(xué) 朱軍 徐海明SourceDFSum SquaresMean SquareF ValuePr > F M449.8888888912.4722222216.95<.0001X119.388888899.3888888912.760.0017X3124.5000000024.5000000033.30<.0001A1L14.000000004.000000005.440.0293A2L14.000000004.000000005.440.0293Error2216.185185190.7356902440ParameterEstimateSEt ValuePr > |

36、t|Intercept7.1851851850.1650690443.53<.0001X1-0.7222222220.20216745-3.570.0017X3-1.6499153120.28590787-5.77<.0001A1L-0.9428087500.40433478-2.330.0293A2L-0.9428087500.40433478-2.330.029339ParameterEstimateSEt ValuePr > |t|Intercept7.1851850790.1656346743.38<.0001X1-0.6296296300.23424279-2

37、.690.0141X2-0.1851851850.23424279-0.790.4385X3-1.6499153120.28688757-5.75<.0001X4-0.3175264150.28688763-1.110.2815A1L-0.9428087500.40572029-2.320.0308A2L-0.9428087500.40572029-2.320.0308SourceDFSum SquaresMean SquareF ValuePr > FM651.259259268.5432098811.53<.0001X119.388888899.3888888912.68

38、0.0020X210.462962960.462962960.620.4385X3124.5000000024.5000000033.08<.0001X410.907407410.907407411.230.2815A1L112.0000000012.0000000016.200.0007A2L14.000000004.000000005.400.0308Error2014.814814810.74074074第六章 析因設(shè)計分析生物統(tǒng)計與實(shí)驗設(shè)計21浙江大學(xué) 朱軍 徐海明42ABX1X2X3X1X3X2X3YA111010-0.681-0.6810.0008.230A112010-0.

39、553-0.5530.0007.896A113010-0.424-0.4240.0007.563A114010-0.296-0.2960.0007.230A115010-0.167-0.1670.0006.896A116010-0.039-0.0390.0006.563A1170100.0900.0900.0006.230A1180100.2190.2190.0005.896A1190100.3470.3470.0005.563A1200100.4760.4760.0005.230A1210100.6040.6040.0004.896A1220100.7330.7330.0004.563o B

40、因素有三個水平： 108cm2, 163cm2, 218cm2.m = 163, D = 55, l1 = 1, C1 = 2o 播種密度為 200cm2,X= ( S - 163) /2;355o 產(chǎn)量Y =7.185 - 0.722X1 - 1.650X3 - 0.943X1X3 - 0.943X2X341第六章 析因設(shè)計分析生物統(tǒng)計與實(shí)驗設(shè)計22浙江大學(xué) 朱軍 徐海明Yield隨機(jī)模型Yijk =m+ai +b j +abij +eijk N (m, s 2 + s 2+ s 2 + s 2 )eabbai = 1, 2,", a; j = 1, 2,", b; k

41、= 1, 2,", r;eijk N (0, s 2 ), a N (0, s 2 )eiab j N (0, s 2 ), ab N (0, s 2 )bijab44密度品種1品種2品種3 1108.2308.9527.7461207.8968.6187.7771307.5638.2857.8071407.2307.9527.8371506.8967.6187.8681606.5637.2857.8981706.2306.9527.928101805.8966.6187.9591905.5636.2857.98992005.2305.9528.0192104.8965.6188.0

42、5082204.5635.2858.0807654100120140160180200220240Stand43品種 1品種 2品種 3第六章 析因設(shè)計分析生物統(tǒng)計與實(shí)驗設(shè)計23浙江大學(xué) 朱軍 徐海明實(shí)例分析 (2)水稻品種和栽培措施的試驗:因素:區(qū)組 (3 次重復(fù))A: 品種 (A1 A2 A3)B:密 度 (108cm2,163cm2, 218cm2)9個處理 (3 ´ 3)試驗單元: 27 個小區(qū) (20m2)依變量: 產(chǎn)量 (kg)46SourcedfMSE(MS) Random Factor Aa-1MSAs 2 +rs 2 + brs 2eabaFactor Bb-1MS

43、Bs 2 +rs 2 + ars 2eabbInteraction (a-1)(b-1) MSAB s 2 +rs 2eabErrorab(r1)MSEs 2eF 檢驗H0: s 2 = 0 ; H1: s 2 > 0aaF* = MSA/MSAB F(a-1, (a-1)(b -1) 如果 H0 是真H0: s 2 = 0 ; H1: s 2 > 0bbF* = MSB/MSAB F(b-1, (a-1)(b-1) 如果 H0 是真H0: s 2= 0 ; H1: s 2 > 0ababF* = MSAB/MSE F(a-1)(b-1), ab(r-1) 如果 H0 是真4

44、5第六章 析因設(shè)計分析生物統(tǒng)計與實(shí)驗設(shè)計24浙江大學(xué) 朱軍 徐海明Not Selecting for Error MSSource DF Type III SS Mean Square F Value Pr > F48SourceDFType I SSMean SquareF ValuePr > FVariety29.8518524.9259261.050.4308Stand225.40740712.7037042.700.1810Error(V×S)418.8148154.703704SourceDFType I SSMean SquareF ValuePr >

45、FVariety29.851851854.925925936.860.0070Stand225.4074074112.7037037017.70<.0001Variety*Stand418.814814814.703703706.550.0025Block20.518518520.259259260.360.7023Data Rice;Input Variety $ Stand $ Block Yield ; Datalines;A1 B1 1 8 A1 B1 2 8 A1 B1 3 8Random: 要求GLM程序?qū)1 B2 1 7 A1 B2 2 7 A1 B2 3 8Random

46、中指定的隨機(jī)效應(yīng)做A1 B3 1 4 A1 B3 2 5 A1 B3 3 4適當(dāng)?shù)腇檢驗，分母完全根據(jù)效A2 B1 1 9 A2 B1 2 9 A2 B1 3 8應(yīng)均方的期望而定A2 B2 1 7 A2 B2 2 9 A2 B2 3 6A2 B3 1 6 A2 B3 2 5 A2 B3 3 4確定作F檢驗的效應(yīng)變量A3 B1 1 9 A3 B1 2 7 A3 B1 3 8A3 B2 1 8 A3 B2 2 7 A3 B2 3 8A3 B3 1 8 A3 B3 2 8 A3 B3 3 9確定作F檢驗的分母的效應(yīng)變量;PROC GLM data=Rice; Class Variety Stand

47、Block;MYield = Variety Stand Variety*Stand Block; Random Variety Stand Variety*Stand Block/test; Test H = Variety Stand E = Variety*Stand;RUN;PROC VARCOMP method=REML;Class Variety Stand Block;MYield = Variety Stand Variety*Stand Block; RUN;47第六章 析因設(shè)計分析生物統(tǒng)計與實(shí)驗設(shè)計25浙江大學(xué) 朱軍 徐海明混合模型Yijk =m+ai +b j +abij

48、 +eijk N (m+ai , s 2 + s 2+ s 2 )eabbi = 1, 2,", a; j = 1, 2,", b; k = 1, 2,", r;åai = 0ieijk N (0, s 2 ), b N (0, s 2 ), ab N (0, s 2 )ejbijab50REML EstimatesAsymptotic Covariance Matrix of EstimatesVar(Variety)Var(Stand)Var(Var*Sta)Var(Block)Var(Error)Var(Variety)0.436140.13657-0.409720-1.256E-9Var(Stand)0.136572.12898-0.409720-4.5215E-8Var(Variety*Stand)-0.40972-0.409721.234650-0.0