350LectureM_AnovaRCBD_Student

350LectureM_AnovaRCB - Lecture M Randomized Complete Block Designs Text Section 9.4 Completely Randomized Design(CRD Ex Want to study effect of 4

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Unformatted text preview: Lecture M: Randomized Complete Block Designs Text Section 9.4 Completely Randomized Design (CRD) Ex. Want to study effect of 4 formulations of fertilizer (call them A, B, C, and D) on yield in corn. Potential experimental designs: 1. Put seeds in individual pots. Apply the fertilizer to the pots (the experimental unit). 2. Plant corn in farm plots. Fertilizer applied with sprayers to 1 acre-plots (the experimental units). Will measure "yield" as a single value per 1-acre plot. Therefore will need multiple 1-acre plots for each fertilizer for replicates. 2a. Randomly select 20 farms, each farm will get one fertilizer 2b. Randomly select 5 farms, each farm will have 4 1-acre plots, one for each fertilizer. Knapp Stat 350 Spring 2009 Lecture M: RCBD Page 1 of 1 Randomized Complete Block Design (RCBD) There is one observation per Treatment-Block combination If there are a Treatments and b blocks, then the total number of observation is n = ab. x22 ... x2 b ... a xa 1 xa 2 ... xab xa . Sum Mean x.1 x .1 x.2 x .2 ... ... x.b x .b ... x21 ... 2 Sum Mean x1 . x1. x2 . x2 . ... b x1 b ... Blocks ... ... ... 2 x12 ... Treatment 1 1 x11 x.. xa . x .. Note: xij represents the response of the ith treatment in the jth block Assumption of RCBD: There is no interaction between the blocks and the treatments Sums of Squares SST = SSTr + SSB + SSE SST: sum of squared deviations of all individual response values xij from the overall average x SST = ∑∑ ( xij − x ..) a b 2 i =1 j =1 SSTr: Represents the variation due to the differences in the treatment levels a SSTr = b∑ ( xi i − x ..) , 2 i =1 where xi . is the mean of the data for the ith treatment SSB: Represents the variation due to the differences in the blocks SSB = a ∑ ( xi j − x ..) , b 2 j =1 where x . j is the mean of the data for the jth block SSE: Represents the variation in the data unexplained by the treatments and blocks SSE = SST – SSTr – SSB Knapp Stat 350 Spring 2009 Lecture M: RCBD Page 2 of 2 Mean Squares ANOVA decomposition: Degrees of Freedom: MSTr = SST ab - 1 = = SSTr (a - 1) + + SSB (b – 1) + + SSE (a - 1)(b - 1) SSTr a −1 MSB = SSB b −1 MSE = SSE ( a − 1)( b − 1) Hypothesis Testing Null Hypothesis H0: There is no treatment effect H0: There is no block effect Test Statistic MSTr F= MSE MSB F= MSE df1 a–1 df2 (a - 1)(b – 1) Decision Reject H0 if p-value < α b-1 (a - 1)(b – 1) Reject H0 if p-value < α Example: Testing the effectiveness of fertilizer on corn yields. The following table gives the yield in bushels per 1-acre plot (experimental unit). Fertilizer A B C D Mean: 1 2 Farm 3 4 5 Mean 157.11 136.85 145.08 172.63 152.92 136.54 145.75 135.82 157.24 143.84 155.50 145.01 144.87 157.27 150.66 158.51 160.60 167.63 164.02 162.69 141.94 162.94 154.78 183.37 160.76 149.92 150.23 149.64 166.91 154.17 Individual Value Plot of A, B, C, D 190 180 Data 170 160 150 140 130 A Knapp Stat 350 Spring 2009 B C D Lecture M: RCBD Page 3 of 3 Analysis in SAS Special notes: PROC ANOVA is only for "balanced" data – data with an equal number of observations for each combination of factors. (An exception is one-way analysis of variance, you can use PROC ANOVA if you have unequal numbers of observations for the different treatments.) option nodate pageno=1; title 'Lecture 14 - Block Design Corn Example'; data corn; input fertilizer $ farm yield; cards; A 1 157.11 B 1 136.85 C 1 145.08 D 1 172.63 A 2 136.54 B 2 145.75 C 2 135.82 D 2 157.24 A 3 155.50 B 3 145.01 C 3 144.87 D 3 157.27 A 4 158.51 B 4 160.60 C 4 167.63 D 4 164.02 A 5 141.94 B 5 162.94 C 5 154.78 D 5 183.37 ; run; proc anova data = corn; class fertilizer farm; model yield = fertilizer farm; means fertilizer/tukey; means fertilizer/dunnett ('A'); means farm; run; Lecture 14 - Block Design Corn Example 1 The ANOVA Procedure Class Level Information Class Levels Values fertilizer 4 ABCD farm 5 12345 Number of Observations Read Number of Observations Used Knapp Stat 350 Spring 2009 20 20 Lecture M: RCBD Page 4 of 4 Lecture 14 - Block Design Corn Example 2 The ANOVA Procedure Dependent Variable: yield DF Sum of Squares Mean Square F Value Pr > F Model 7 2028.214100 289.744871 3.37 0.0314 Error 12 1031.319120 85.943260 Corrected Total 19 3059.533220 Source R-Square Coeff Var Root MSE yield Mean 0.662916 6.013088 9.270559 154.1730 Source DF Mean Square F Value Pr > F 3 4 fertilizer farm Anova SS 1081.744580 946.469520 360.581527 236.617380 4.20 2.75 0.0302 0.0778 Lecture 14 - Block Design Corn Example 3 The ANOVA Procedure Tukey's Studentized Range (HSD) Test for yield NOTE: This test controls the Type I experimentwise error rate, but it generally has a higher Type II error rate than REGWQ. Alpha 0.05 Error Degrees of Freedom 12 Error Mean Square 85.94326 Critical Value of Studentized Range 4.19852 Minimum Significant Difference 17.407 Means with the same letter are not significantly different. Tukey Grouping N fertilizer A A A A A A A Knapp Stat 350 Spring 2009 Mean 166.906 5 D 150.230 5 B 149.920 5 A 149.636 5 C Lecture M: RCBD Page 5 of 5 Lecture 14 - Block Design Corn Example 4 The ANOVA Procedure Dunnett's t Tests for yield NOTE: This test controls the Type I experimentwise error for comparisons of all treatments against a control. Alpha 0.05 Error Degrees of Freedom 12 Error Mean Square 85.94326 Critical Value of Dunnett's t 2.68292 Minimum Significant Difference 15.731 Comparisons significant at the 0.05 level are indicated by ***. fertilizer Comparison Difference Between Means D-A B-A C-A 16.986 0.310 -0.284 Simultaneous 95% Confidence Limits 1.255 -15.421 -16.015 32.717 16.041 15.447 *** Lecture 14 - Block Design Corn Example 5 The ANOVA Procedure Level of farm N 1 2 3 4 5 4 4 4 4 4 ------------yield-----------Mean Std Dev 152.917500 143.837500 150.662500 162.690000 160.757500 15.5536778 10.0136320 6.6474124 4.0005416 17.3773538 ANOVA Table – Corn Example Source Fertilizer Farm Error Total Knapp Stat 350 Spring 2009 df 3 4 12 19 SS 1081.74 946.47 1031.32 3059.55 MS 360.58 236.62 85.94 F 4.20 2.75 p-value 0.0302 0.0778 Lecture M: RCBD Page 6 of 6 Main Effects Plot 170 165 Yield 160 155 150 145 140 A B C D Fertilizer 190 Interaction Plot 180 A B C D 170 Yield 160 150 140 130 120 110 100 1 2 3 4 5 Farm 200 Interaction Plot 190 180 Yield 170 160 150 Farm 1 140 Farm 2 Farm 3 130 120 Farm 4 Farm 5 110 100 A B C D Fertilizer Knapp Stat 350 Spring 2009 Lecture M: RCBD Page 7 of 7 ...
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This note was uploaded on 02/16/2010 for the course MA 350 taught by Professor Sellke during the Spring '10 term at Purdue University-West Lafayette.

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