Biostatistics 322 Split-Plot Designs 1 Split-plot Designs .
Biostatistics 322ORIGINSplit-Plot DesignsSplit-plot Designs1Split-plot designs involve situations where it is difficult to apply fullrandomization to all crossed factors because some experimental orobservational conditions are harder to apply than others. In agriculturalstudies, such as the classic "Oats" study described here, experimental fieldsare often divided into replicate blocks. Within each block, plots are set uprelated to a hard-to-apply factor (in this case plant variety set in place by anautomated planter) and within each plots, subplots are defined by levels ofmore easily applied treatments (here levels of N fertilizer). Similar datastructures occur in observational studies because data sometimes occurnaturally in a hierarchical arrangement such as teachers within schools withindistricts, etc.Because the factors of the study (Blocks, Plots, Subplots) exist in ahierarchical arrangement, the data collected have a complex covariancestructure between sets of observations that violate the assumptions ofcrossed-factorial design (constant variance and no covariance betweenobservations). Instead, one conducts a mixed-model (Type III) analysis.Example from W. N. Venables & B. D. Ripley 2002. Modern Applied Statisticswith S, p. 282.Example - Balanced Case: OatsData NT 18113828610499119121Although perhaps appearing as a cross-factor study in R's datalong form, the data was collected in the hierarchical format above.OATSY1READPRN( "c:/DATA/Models/Oats.txt" )OATS Only vector Y from the data set is imported here 049799119121
Biostatistics 322Split-Plot Designs2Model:Yijk i(j)where: j k ()jk ijkRestrictions: a constant level (intercept) random variable for Plots within Blocks N(0,i(j)2) 0k 0j fixed constantk fixed constant( )jk fixed constantjijk()jk 0 for all j and for all k random variable N(0, 2)Model Dimensions:s6 number of random blocks observed ina3b4i(j)i1 s number of treatments for fixed factor Vjj1 a number of treatments for fixed factor Nkk1 bGrand Mean:GMmean( Y)GM103.972222 BLOCKSBlock ,][12,]From R:#BLOCKS:B1 Y[B "I"]B2 Y[B "II"]B3 Y[B "III"]B4 Y[B "IV"]B5 Y[B "V"]B6 Y[B "VI"]BLOCKS 105140118156B26191971007010812614996124121144Plot 321337089104117B56290100116808294126637010999Block Means:Bb tapply(oats Y,oats B,mean) #blocking factor means BbIII135.3333 6.2500107.2500B95.916798.166790.9167#PLOT FACTOR V:V1 Y[V "Victory"]V2 Y[V "Golden.rain"]V3 Y[V "Marvellous"]PLOTS 121
Biostatistics 322Split-Plot Designs3Plot Means:Vb tapply(Y,oats V,mean) #PLOT factor level means VbGolden.rain104.500Marvellous109.792 PLOTS97.625Victory97.625V104.5109.791667 SUBPLOTSSubplot Means:N1 N2 N3 N4[1,] 111 130 157 174[2,] 117 114 161 141[3,] 105 140 118 156[4,] 61 91 97 100[5,] 70 108 126 149[6,] 96 124 121 144[7,] 68 64 112 86[8,] 60 102 89 96[9,] 89 129 132 124[10,] 74 89 81 122[11,] 64 103 132 133[12,] 70 89 104 117[13,] 62 90 100 116[14,] 80 82 94 126[15,] 63 70 109 99[16,] 53 74 118 113[17,] 89 82 86 104[18,] 97 99 119 121#SUBPLOT FACTOR N:N1 Y[N "0.0cwt"]N2 Y[N "0.2cwt"]N3 Y[N "0.4cwt"]N4 Y[N "0.6cwt"]SUBPLOTS cbind(N1,N2,N3,N4) tapply(Y,oats N,mean)0.0cwt79.38890.2cwt0.4cwt0.6cwt98.8889 114.2222 123.3889Subplot Means:79.388889NCell 109999799119121114.222222 cbind(C11,C12,C13,C14)123.388889C11 C12 C13 C14[1,] 111 130 157 174[2,] 61 91 97 100[3,] 68 64 112 86[4,] 74 89 81 122[5,] 62 90 100 116[6,] 53 74 118 113#CELLS C:C11 subset(oats,V "Victory"&N "0.0cwt") YC12 subset(oats,V "Victory"&N "0.2cwt") YC13 subset(oats,V "Victory"&N "0.4cwt") YC14 subset(oats,V "Victory"&N "0.6cwt") YC21 subset(oats,V "Golden.rain"&N "0.0cwt") YC22 subset(oats,V "Golden.rain"&N "0.2cwt") YC23 subset(oats,V "Golden.rain"&N "0.4cwt") YC24 subset(oats,V "Golden.rain"&N "0.6cwt") YC31 subset(oats,V "Marvellous"&N "0.0cwt") YC32 subset(oats,V "Marvellous"&N "0.2cwt") YC33 subset(oats,V "Marvellous"&N "0.4cwt") YC34 subset(oats,V "Marvellous"&N "0.6cwt") Y cbind(C21,C22,C23,C24)C21 C22 C23 C24[1,] 117 114 161 141[2,] 70 108 126 149[3,] 60 102 89 96[4,] 64 103 132 133[5,] 80 82 94 126[6,] 89 82 86 104 cbind(C31,C32,C33,C34)C1 cbind(mean(C11),mean(C12),mean(C13),mean(C14))C2 cbind(mean(C21),mean(C22),mean(C23),mean(C24))C3 cbind(mean(C31),mean(C32),mean(C33),mean(C34)) rbind(C1,C2,C3)[,1] [,2] [,3] [,4][1,] 71.5000 89.6667 110.833 118.500[2,] 80.0000 98.5000 114.667 124.833[3,] 86.6667 108.5000 117.167 126.833C31 C32 C33 C34[1,] 105 140 118 156[2,] 96 124 121 144[3,] 89 129 132 124[4,] 70 89 104 117[5,] 63 70 109 99[6,] 97 99 119 121Cell Means:71.5C89.6667 .16667 126.8333
Biostatistics 322Split-Plot Designs4ANOVA Sums of Squares:Degrees of Freedom:BLOCK Factor B:SSBb b13SSVN321.7487(aSSTO51985.944a b siPLOT Factor V:SSVb sVGMj2jSUBPLOT Factor N:SSNa sN2GMkkVN Interactions:SSVNsCjj kNkVjGM21) (b1)6kTotal:m1 length ( 1035a (s1) ( b1)45ANOVA Table in R:LM1 aov(Y V*N Error(B/V),data oats) summary(LM1)Error: BDf Sum Sq Mean Sq F value Pr( F)Residuals 5 158753175Error: B:VDf Sum Sq Mean Sq F value Pr( F)V217868931.490.276013Residuals 10601Error: WithinDf Sum Sq Mean Sq F value Pr( F)N3 20020667437.7 2.5e-12 ***V:N6322540.30.937969Residuals 45177--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Note that Error Sum of Squares ispartitioned in the R output, butsums to SSE above.6013796913982
Biostatistics 322Split-Plot DesignsF-Test:5Three tests shown together here:Hypotheses:H0:H1:all ( ) jk 0 for V with N interactionsall j 0 for fixed factor Vall k 0 for fixed factor N within Vall ( ) jk 0 for V with N interactionsall j 0 for fixed factor Vall k 0 for fixed factor N within VTest Statistics:Mean Squares from ANOVA Table:F for VN Interaction:FNVF for Factor V:FVF for Factor cal Values for the Tests:0.05 Type I error rate must be explicitly set.for VN Interaction:CNVfor V:CVqF 1(a1) a ( s1)for N:CNqF 1(b1) a (s1) ( bqF 1(a1) ( b1) a (s1) ( b1)1)CNV2.308CV3.682CN2.812Decision Rules:If:F NV CNVF V CVF N CNThen Reject H0, otherwise accept H0Probabilities:for VN Interaction:PNVfor V:PV1pF FV ( a1) a ( s1)PN1pF FN ( b1) a ( s1) (bfor N:1pF FNV ( a1) (b1) a ( s1) (b1)1)PNV0.931PV0.258PN2.43610 slight difference for PV noted hereversus R, probably due to rounding.12
Biostatistics 322 Split-Plot Designs 1 Split-plot Designs ORIGIN 1{Split-plot designs involve situations where it is difficult to apply full randomization to all crossed factors because some experimental or observational co
Draft 2 JRM / August 2017 KEF / November 2017 . Ecological Monitoring – Condition Survey . Hedge ref. HMC cycle Additional notes H49 N.East 1 / mid-section 5 / West 1 . PLOT 102 PA33 PLOT 103 PA33 PLOT 104 PA33 PLOT 105 PT36 PLOT 161 PT36 PLOT 162 PA33 PLOT 163 AA33 PLOT 164 AA33 PLOT 165 AA33 PLOT 44
Split-plot designs can also arise in (much) more complicated designs. There can be more than one whole-plot factor. E.g., think of a two-way factorial on the whole-plot level. In addition, there can be more than one factor on th
Master of Science in Biostatistics (MSIBS) Master of Science in Biostatistics and Data Science (MSBDS) 1.1.2. Core Program Coursework for All Programs Statistical Computing with SAS , Introduction to R for Data Science, Biostatistics I, Biostatistics II, Study Design and Clinical Trials, Ethics for
The graph below represents the lap and split times for a workout in which 4 laps were taken. LAP 1 7:11 MIN 7:50 MIN 15:01 MIN SPLIT 2 SPLIT 3 SPLIT 4 7:08 MIN 22:09 MIN 7:30 MIN 29:39 MIN 7:11 MIN SPLIT 1 LAP 2 LAP 3 LAP 4: TAKING A SPLIT While the timer is running, press SPLIT to take a split.
SPLIT 2 SPLIT 3 SPLIT 4 7:08 MIN 22:09 MIN 7:30 MIN 29:39 MIN 7:11 MIN SPLIT 1 LAP 2 LAP 3 LAP 4 taking a Split While the timer is running, press SPLIT to take a split. The lap and split time for the lap you completed is displayed, and the watch begins timing the new lap. Individual lap data is saved when you save and reset the workout.
Split-Plot Designs: What, Why, and How BRADLEY JONES SAS Institute, Cary, NC 27513 CHRISTOPHER J. NACHTSHEIM Carlson School of Management, University of Minnesota, Minneapolis, MN 55455 The past decade has seen rapid advances in the development of new methods for the design and analysis of split-plot experiments.
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