Definitions of terms and examples. Experimental Design. Sampling versus experiments. For each experimental unit, measures of the variables of

Transcription

1 Experimetal Desig Samplig versus experimets similar to samplig ad ivetor desig i that iformatio about forest variables is gathered ad aalzed experimets presuppose itervetio through applig a treatmet (a actio or absece of a actio) to a uit, called the experimetal uit. The experimetal uit is a item o which the treatmet is applied. Defiitios of terms ad examples For each experimetal uit, measures of the variables of iterest (i.e., respose or depedet variables) are used to idicate treatmet impacts. Treatmets are radoml assiged to the experimetal uits. Replicatio is the observatio of two or more experimetal uits uder idetical experimetal coditios. A factor is a groupig of related treatmets. The goal is to obtai results that idicate cause ad effect.

2 Examples:.,000 seedligs i a field. Half of the seedligs get a tea bag of utriets, others do ot, radoml assiged. Experimetal uit: the seedlig. Treatmets are: o tea bag, ad tea bag. Factor: ol oe fertilizer (oe, tea bag) Replicatios: 500 seedligs get each treatmet. 300 plat pots i a greehouse: Each pot gets either ) stadard geetic stock; ) geetic stock from aother locatio; 3) improved geetic stock. Treatmets: the three tpes of geetic stock Experimetal Uit: The pot Factor(s): Geetic Stock (oe factor ol) Replicatios: 300 pots /3 treatmets 00 pots /treatmet 3. The umber of tailed frogs i differet forest tpes is of iterest. There are six areas. Three are cut ad the other three are ot cut. Treatmets: cut, ucut Experimetal Uit: each of the six areas Factor(s): ol oe, cuttig with two levels Replicatios: six areas/ two cuttig levels 3 replicates per treatmet. 4. Two forest tpes are idetified, Coastal wester hemlock ad iterior Douglas fir. For each, a umber of samples are located, ad the growth of each tree i each sample is measured. Treatmets: NOT AN EXPERIMENT!! Experimetal Uit: Factor(s): Replicatios: 3 4

3 What does it mea that treatmets are radoml assiged to experimetal uits? Haphazard vs. radom allocatio Practical problems ad implicatios Other terms: The ull hpothesis is that there are o differeces amog the treatmet meas. For more tha oe factor, there is more tha oe hpothesis The sum of squared differeces (termed, sum of squares) betwee the average for the respose variable b treatmet versus the average over all experimetal uits represets the variatio attributed to a factor. The degrees of freedom, associated with a factor, are the umber of treatmet levels withi the factor mius oe. Example of hpotheses: Factor A, fertilizer: oe, medium, heav (3 levels) Factor B, species: spruce, pie ( levels) Number of possible treatmets: 6 e..g, spruce, oe is oe treatmet. Experimetal Uit: 0.00 ha plots Replicates plaed: per treatmet (cost costrait). How ma experimetal uits do we eed? Variable of iterest: Average 5-ear height growth for trees i the plot Null hpotheses: There is o differet betwee the 6 treatmets. This ca be broke ito: ) There is o iteractio betwee species ad fertilizer. ) There is o differece betwee species. 3) There is o differece betwee fertilizers. 5 6

4 Experimetal error is the measure of variace due to chace causes, amog experimetal uits that received the same treatmet. The degrees of freedom for the experimetal error relate to the umber of experimetal uits ad the umber of treatmet levels. The impacts of treatmets o the respose variables will be detectable ol if the impacts are measurabl larger tha the variace due to chace causes. To reduce the variabilit due to causes other tha those Radom allocatio of a treatmet to a experimetal uit helps isure that the measured results are due to the treatmet, ad ot to aother cause. Example: if we have applied the o fertilizer treatmet to experimetal uits o orth facig sites, whereas moderate ad heav fertilizer treatmets are applied ol to south facig sites, we would ot kow if differeces i average height growth were due to the applicatio of fertilizatio, the orietatio of the sites, or both. The results would be cofouded ad ver difficult to iterpret. maipulated b the experimeter, relativel homogeous experimetal uits are carefull selected. 7 8

5 Variatios i experimetal desig Itroductio of More Tha Oe Factor: Iterested i the iteractio amog factors, ad the effect of each factor. A treatmet represets a particular combiatio of levels from each of the factors. Whe all factor levels of oe factor are give for all levels of each of the other factors, this is a crossed experimet. Example: two species ad three fertilizatio levels six treatmets usig a crossed experimet. Fixed, Radom, or Mixed Effects: Fixed factors: the experimeter would like to kow the chage that is due to the particular treatmets applied; ol iterested i the treatmet levels that are i the experimet (e.g., differece i growth betwee two particular geetic stocks) [fixed effects] Radom factors: the variace due to the factor is of iterest, ot particular levels (e.g., variace due to differet geetic stocks radoml select differet stock to use as the treatmet) [radom effects] Mixture of factor tpes: Commol, experimets i forestr iclude a mixture of factors, some radom ad some fixed [mixed effect]. 9 0

6 Restricted Radomizatio Through Blockig: Radomized Block (RCB), Lati Square, ad Icomplete Blocks Desigs: Radomize treatmets with blocks of experimetal uits Reduces the variace b takig awa variace due to the item used i blockig (e.g., high, medium ad low site productivit Results i more homogeeous experimetal uits withi each block. Restricted Radomizatio Through Splittig Experimetal Uits: Called split plot A experimetal uit is split. Aother factor is radoml applied to the split. Example: The factor fertilizer is applied to 0.00 ha plots. Each of the 0.00 ha plot is the split ito two, ad two differet species are plated i each. Fertilizer is applied to the whole plot, ad species is applied to the split plot. Species is therefore radoml assiged to the split plot, ot to the whole experimetal uit.

7 Nestig of Factors Treatmet levels for oe factor ma be particular to the level of aother factor, resultig i estig of treatmets. Example, for the first level of fertilizer, we might use medium ad heav thiig, whereas, for the secod level of fertilizer, we might use o thiig ad light thiig. Hierarchical Desigs ad Sub-Samplig: Commol i forestr experimets, the experimetal uit represets a group of items that we measure. E.g. several pots i a greehouse, each with several plats germiatig from seeds. Treatmets are radoml assiged to the larger uit (e.g, to each plot ot to each seedlig). The experimetal uit is the larger sized uit. Ma wat variace due to the experimetal uit (pots i the example) ad to uits withi (plats i the example). These are ) ested i the treatmet; ) radom effects; ad 3) hierarchical A commo variatio o hierarchical desigs is measurig a sample of items, istead of measurig all items i a experimetal uit. 3 4

8 Itroductio of Covariates The iitial coditios for a experimet ma ot be the same for all experimetal uits, eve if blockig is used to group the uits. Site measures such as soil moisture ad temperature, ad startig coditios for idividuals such as startig height, are the measured (called covariates) alog with the respose variable These covariates are used to reduce the experimetal error. Covariates are usuall iterval or ratio scale (cotiuous). Desigs i use The most simple desig is oe fixed-effects factor, with radom allocatio of treatmets to each experimetal uit, with o ) blockig; ) sub-samplig; 4) splits; or 5) covariates Most desigs use combiatios of the differet variatios. For example, oe fixed-effects factor, oe mixed-effects factor, blocked ito three sites, with trees measured withi plots withi experimetal uits (sub-samplig/hierarchical), ad measures take at the begiig of the experimet are used as covariates (e.g., iitial heights of trees. 5 6

9 Wh? Wat to look at iteractios amog factors ad/or is cheaper to use more tha oe factor i oe experimet tha do two experimets. Experimets ad measuremets are expesive use samplig withi experimetal uits to reduce costs Fidig homogeeous uits is quite difficult: blockig is eeded Mai questios i experimets Do the treatmets affect the variable of iterest? For fixed effects: Is there a differet betwee the treatmet meas of the variable of iterest? Which meas differ? What are the meas b treatmet ad cofidece itervals o these meas? For radom effects: Do the treatmets accout for some of the variace of the variables of iterest? How much? BUT ca ed up with problems: some elemets are ot measured, radom allocatio is ot possible, or measures are correlated i time ad/or space. I this course, start with the simple desigs ad add complexit. 7 8

10 Completel Radomized Desig (CRD) Homogeeous experimetal uits are located Treatmets are radoml assiged to experimetal uits No blockig is used We measure a variable of iterest for each experimetal uit CRD: Oe Factor Experimet, Fixed Effects Mai questios of iterest Are the treatmet meas differet? Which meas are differet? What are the estimated meas ad cofidece itervals for these Notatio: Populatio: i μ + τ + ε i OR i μ + ε i i respose variable measured o experimetal uit i ad treatmet to J treatmets μ the grad or overall mea regardless of treatmet μ the mea of all measures possible for treatmet τ the differece betwee the overall mea of all measures possible from all treatmets ad the mea of all possible measures for treatmet, called the treatmet effect ε i the differece betwee a particular measure for a experimetal uit i, ad the mea for the treatmet that was applied to it ε i i μ estimates? 9 0

11 For the experimet: i + τˆ + ei OR i + ei the grad or overall mea of all measures from the experimet regardless of treatmet; uder the assumptios for the error terms, this will be a ubiased estimate of μ the mea of all measures for treatmet ; uder the assumptios for the error terms, this will be a ubiased estimate of μ τˆ the differece betwee the mea of experimet measures for treatmet ad the overall mea of measures from all treatmets; uder the error term assumptios, will be a ubiased estimate of τ e i the differece betwee a particular measure for a experimetal uit i, ad the mea for the treatmet that was applied to it e i i the umber of experimetal uits measured i treatmet Example: Fertilizatio Trial A forester would like to test whether differet site preparatio methods result i differece i heights. Twet five areas each 0.0 ha i size are laid our over a fairl homogeeous area. Five site preparatio treatmets are radoml applied to 5 plots. Oe hudred trees are plated (same geetic stock ad same age) i each area. At the ed of 5 ears, the heights of seedligs i each plot were measured, ad averaged for the plot. i a particular 0.0 ha area i treatmet, from to 5. Respose variable i : 5-ear height growth (oe average for each experimetal uit) Number of treatmets: J5 site preparatio methods T the umber of experimetal uits measured over all 5 treatmets experimetal uits measured each treatmet T the umber of experimetal uits measured over all treatmets J

12 Schematic of Laout: Data Orgaizatio ad Prelimiar Calculatios For eas calculatios b had, the data could be orgaized i a spreadsheet as: Obs: Treatmet, to J i to 3 J 3 J 3 J J 3 J Sum J.. Averages 3 J J i i i i i T NO Example: J 5 site preparatio treatmets radoml applied to 5 plots. Respose Variable: Plot average seedlig height after 5 ears Plot Average Heights (m) Treatmets Overall Observatio SUMS Meas Example Calculatios: 5 5 i ( ) / 5 i 4. i 5 5 i ( ) / 5 96./ TE: ma ot be the same umber of observatios for each treatmet. 3 4

13 We the calculate: ) Sum of squared differeces betwee the observed values ad the overall mea (SS): SS J J ( i ) df i Also called, sum of squares total (same as i regressio) Alterative formulae for the sums of squares that ma be easier to calculate are: SS SS TR J J i i SSE SS SS TR T T ) Sum of squared differeces betwee the treatmet meas, ad the grad mea, weighted b the umber of experimetal uits i each treatmet (SS TR ) SS TR J J ( ) ( ) df J i 3) Sum of squared differeces betwee the observed values for each experimetal uit ad the treatmet meas (SSE) SSE J ( i ) i df T SS SS + TR SSE J 5 6

14 For the example, differeces from treatmet meas (m): Treatmets Overall Obs SUMS Sum of Squares Error s Differeces from grad mea (m) Treatmets Overall Obs SUMS Sum of Squares Total Example Calculatios: SSE for treatmet s 5 i ( i ) (4.6 4.) + (4.3 4.) + (3.7 4.) + (4.0 4.) + (4.0 4.) SSE for treatmet SS J ( i ) i SS for treatmet + SS for treatmet SS 6.6 for treatmet 5 SSE J ( i ) i SSE for treatmet + SSE for treatmet SSE for treatmet

15 Differece betwee treatmet meas ad grad mea (m) Treatmets Overall Mea Differece Sum of Squares Treatmet Example Calculatios: SS TR + J ( ) ( 5 ( ) ) + ( 5 ( ) ) ( 5 ( ) ) + ( 5 ( ) ) + ( 5 ( ) ) 3.86 Test for differeces amog treatmet meas The first mai questio is: Are the treatmet meas differet? H 0 : μ μ μ J H : ot all the same OR: H 0 : τ τ L τ J 0 H : ot all equal to 0 OR: H 0 : (φ TR+ σ ε) /σ ε H : (φ TR+ σ ε)/σ ε > Where σ ε is the variace of the error terms; φ TR is the effect of the fixed treatmets (see page 34 for more details o what this is). If the treatmet does ot accout for a of the variace i the respose variable, the treatmet effects are likel all 0, ad all the treatmet meas are likel all the same. 9 30

16 Usig a aalsis of variace table: Source df SS MS F p-value Treatmet J- SS TR MS TR F Prob F> SS TR /(J-) MS TR /MSE F (J-),( T -J), Error T -J SSE MSE SSE/( T -J) Total T - SS (- α) For example, if we have 4 treatmets, ad experimetal uits, ad we wat α0.05: F (3,8; 0.95)4.07 Reectio Regio SS F SSE / /( J ) SSTR /( J ) J SSE /( T J ) ( ) TR MSTR MSE Uder H 0, ad the assumptios of aalsis of variace, this follows a F-distributio. If F > F( J, T J, α ) We reect H 0 ad coclude that there is a differece betwee the treatmet meas. Notice that this is a oe-sided test, usig -α If the calculated F is larger tha 4.07, we reect H 0 : The treatmets meas are likel differet, uless a 5% error has occurred. OR: We take our calculated F value from our experimet ad plot it o this F curve. The, fid the area to the right of this value (p-value). We reect a hpothesis if the probabilit value (p-value) for the test is less tha the specified sigificace level. This is because we are testig if the ratio of variaces is >. 3 3

17 For the example: If assumptios of ANOVA are met the iterpret the F-value. H 0 : μ μ μ 3 μ 4 μ 5 H : ot all equal Aalsis of Variace (ANOVA) Table: Source df SS MS F p-value Treatmet Error Total If assumptios of ANOVA are met the iterpret the F-value. NOTE: Fcritical for alpha0.05, df treatmet4 ad df error0 is.87. Sice the p-value is ver smaller (smaller tha alpha0.05), we reect H0 ad coclude that there is a differece i the treatmet Assumptios regardig the error term For the estimated meas for this experimet to be ubiased estimates of the meas i the populatio, ad the MSE to be a ubiased estimate of the variace withi each experimetal uit, the followig assumptios must be met:. Observatios are idepedet ot related i time or i space [idepedet data]. There is ormal distributio of the -values [or the error terms] aroud each treatmet mea [ormall distributed] 3. The variaces of the s aroud each treatmet mea [or the error terms] are the same (homogeeous) for all treatmet meas [equal variace] meas. BUT this is ol a good test if the assumptios of aalsis of variace have bee met. Need to check these first (as with regressio aalsis)

18 Similar to regressio: a ormal probabilit plot for the error terms ca be used to check the assumptio of ormalit, ad a residual plot ca be used to visuall check the assumptio of equal variace. OR, these ca be tested usig () ormalit tests (as with regressio); () Bartlett s test for equal variaces (for more tha oe factor or for other desigs with blockig, etc. this becomes difficult). Trasformatios to meet assumptios Similar to regressio: logarithmic trasformatios ca be used to equalize variaces arcsie trasformatio ca be used to trasform proportios ito ormall distributed variables rak trasformatio ca be used whe data are ot ormall distributed ad other trasformatios do ot work [oparametric aalsis of variace usig raks] Ulike regressio ou must trasform the -variable Process: do our aalsis with the measured respose variable if assumptios of the error term are ot met, trasform the -variable do the aalsis agai ad check the assumptios; if ot me, tr aother trasformatio ma have to switch to aother method: geeralized liear models, etc

19 Expected values: Uder the assumptios of aalsis of variace, MSE is a ubiased estimate of σ ε ad MS TR is a ubiased estimate of For the example, before iterpretig the ANOVA table, we must check assumptios of ANOVA: Is there equal variace across treatmets? (estimated b MSE as 0.49 o our ANOVA table). Usig a residual plot ad EXCEL: Residual Plot φ TR+ σ ε. Therefore, this F-test will give the correct probabilities uder the assumptios. This is the same as saig that the expected value of MSE is Residuals (Error i m) Pred. Values (Treat. Meas i m) σ ε, ad the expected value of MS TR is φ TR+ σ ε. The F-test is the a measure of how much larger the value is whe the treatmet meas are accouted for

20 Are residuals ormall distributed? Agai usig EXCEL: Cumulative probabilit Residuals vs. ormal z(0,) z-values Where stadardized residuals are calculated b: e (stadardized) i ei 0 MSE Stad. Res z(0,) Compare these to z-values for a stadard ormal distributio with a mea of zero ad a variace of (z(0,)) Differeces amog particular treatmet meas If there are differeces amog meas detected, which meas differ? Ca use: Orthogoal cotrasts see textbook Multiple comparisos Multiple comparisos (or cotrasts): Ma differet tpes, e.g. o T-test for ever pair of meas; must adust the alpha level used b dividig b the umber of pairs. o Scheffe s multiple comparisos o Boferoi s adustmets Tr to preserve the alpha level used to test all the meas together (the F-test) For the example, give that there is a differece amog treatmet meas, which pairs of meas differ? t-test for pairs of meas: determie the umber of pairs possible 5 5! 3!! 0 possible pairs of meas 39 40

21 Comparig Treatmets (largest estimated mea) versus 4 (smallest estimated mea): H : μ μ 0 OR H : μ μ 0 H : μ μ Cofidece limits for treatmet meas Uder the assumptios, cofidece itervals for each treatmet mea ca be obtaied b: t ( 4 ) 0 MSE + 4 ± t( T J ), α MSE Sice MSE estimates the variace that is assumed to be equal, t Uder H 0 : This follows: t α /, T J ( ) Usig alpha0.005 (0.05/00.005), for 5 treatmets ad 5 observatios, the t-value is Result? Aother wa to assess this is to obtai the p-value for t3.686, with 0 degrees of freedom (5-5). This is Sice this is less tha 0.005, we reect H 0 ad coclude that these two meas differ. ad the observatios are ormall distributio ad idepedet. For the example: ± t ( T ), α / MSE All For treatmet : MSE are all the same sice (3.74,4.46) 0.73 t 0, are all equal 4.± ±

22 Usig SAS ad R: For etr ito statistical programs like SAS ad R, the data should be orgaized as: Treatmet Obs: Respose to J i to 3 3 () 3 3 Y () J J J J J 3 3J J J (J) 3 For the example, we ca put the data ito a EXCEL file: Treatmet Observatio AveHt

23 Power of the Test: A Tpe I error rate (α, sigificace level), the chace of reectig a ull hpothesis whe it is true (ou reect whe the If the differece betwee populatio meas (real treatmet meas) is ver large, tha a small umber of experimetal uits will result i reectio of the ull hpothesis. meas are actuall the same) must be selected. Give: a particular umber of experimetal uits sizes of the differeces betwee true populatio meas, ad If the umber of experimetal uits is ver large, the eve a small differece betwee populatio meas will be detected. variatio withi the experimetal uits this will set the Tpe II error rate (β), the chace of acceptig a ull hpothesis whe it is false (ou fail to reect whe the meas are actuall differet) The power of the test is - β, the probabilit ou will reect If the variatio withi experimetal uits is ver small, the the differece will be detected, eve with a small differece betwee populatio meas, ad eve with ol a few treatmet uits. the ull hpothesis ad coclude that there is a differece i meas, whe there IS a differece betwee populatio meas

24 Statistical Sigificace is ot the same as differeces of Practical importace! UNLESS ou: have some idea of withi experimetal uit variatio from a previous stud with the same coditios (e.g., MSE from a previous stud) kow the size of the differece that ou wish to detect have selected the α level Methods based o maximum likelihood rather tha least squares ML methods ca be used whe: Treatmets are radom rather tha fixed (more o this later) Trasformatios do ot result i assumptios beig met Your depedet variable is a cout, or it is a biar variable (e.g., es or o; dead or alive; preset or abset) The: You ca calculate the umber of experimetal uits per treatmet that will result i reectio of H 0 : whe the differeces are that large or greater. Alterativel: You ca calculate the power of the test for a experimet ou have alread completed. [see examples i course materials for FRST 430/533] 47 48

25 CRD: Two Factor Factorial Experimet, Fixed Effects Itroductio Treatmets ca be combiatios of more tha oe factor For -factor experimet, have several levels of Factor A ad of Factor B All levels of Factor A occur for Factor B ad vice versa (called a Factorial Experimet, or crossed treatmets) Example: Schematic ad Measured Respose for the Example: AB0 A3B5 A3B435 AB3 AB4 AB34 AB44 AB AB5 AB48 A3B33 A3B5 A3B7 AB43 A3B39 A3B6 AB37 AB A3B435 AB3 AB4 AB AB34 A3B330 AB39 AB8 AB430 A3B33 AB33 AB4 A3B AB49 A3B3 AB8 AB5 A3B3 AB5 A3B437 AB9 A3B4 A3B436 AB48 AB37 AB8 AB0 AB8 AB36 AB38 Factor A, (three levels of fertilizatio: A, A, ad A3) Factor B (four species: B, B, B3 ad B4) Crossed: treatmets Four replicatios per treatmet for a total of 48 experimetal uits AB0 idicates that the respose variable was 0 for this experimetal uit that received Factor A, level ad Factor B, level. Treatmets radoml assiged to the 48 experimetal uits. Measured Resposes: height growth i mm 49 50

26 Orgaizatio of data for aalsis usig a statistics package: A B result Mai questios. Is there a iteractio betwee Factor A ad Factor B (fertilizer ad species i the example)? Or do the meas b Factor A remai the same regardless of Factor B ad vice versa?. If there is o iteractio, is there a differece a. Betwee Factor A meas? b. Betwee Factor B meas? 3. If there are differeces: a. If there is a iteractios, which treatmet meas differ? b. If there is o iteractio, the which levels of Factor A meas differ? Factor B meas? 5 5

27 Notatio, Assumptios, ad Trasformatios Models Populatio: ik μ + τ + τ + τ + ε A Bk AB k ik respose variable measured o experimetal uit i ad factor A level, factor B level k to J levels for Factor A; k to K levels for Factor B μ the grad or overall mea regardless of treatmet τ A the treatmet effect for Factor A, level τ Bk the treatmet effect for Factor B, level k τ ABk ε ik the iteractio for Factor A, level ad Factor B, level k the differece betwee a particular measure for a experimetal uit i, ad the mea for a treatmet: ε ik ik μ + τ A + τ Bk + τ AB ) ik ( i For the experimet: + ˆ τ + ˆ τ + ˆ + e ik A Bk τ AB k the grad or overall mea of all measures from the experimet regardless of treatmet; uder the assumptios for the error terms, this will be a ubiased estimate of μ k the mea of all measures from the experimet for a particular treatmet k the mea of all measures from the experimet for a particular level of Factor A (icludes all data for all levels of Factor B) k the mea of all measures from the experimet for a particular level k of Factor B (icludes all data for all levels of Factor A) ˆ τ A, ˆ τ ˆ Bk, τ ABk uder the error term assumptios, will be ubiased estimates of correspodig treatmet effects for the populatio e ik the differece betwee a particular measure for a experimetal uit i, ad the mea for the treatmet k that was applied to it e ik ik k k the umber of experimetal uits measured i treatmet k T the umber of experimetal uits measured over all K J treatmets k k ik 53 54

28 Meas for the example: Factor A: 6 observatios per level A6.5, A3.38, A38.75 Factor B: observatios per level B7.08, B0.83, B34.7, B49.08 Treatmets (A X B): 4 observatios per treatmet Sums of Squares: SS SSTR + SSE as with CRD: Oe Factor. BUT SSTR is ow divided ito: SS TR SSA + SSB + SSAB SS: The sum of squared differeces betwee the observatios ad the grad mea: SS K J k ( ik ) df T k i SSA: Sum of squared differeces betwee the level meas for factor A ad the grad mea, weighted b the umber of experimetal uits for each treatmet: SSA K J k k ( ) df J 55 56

29 SSB: Sum of squared differeces betwee the level meas for factor B ad the grad mea, weighted b the umber of experimetal uits for each treatmet: SSB K J k k ( ) df k K SSAB: Sum of squared differeces betwee treatmet meas for SSE: Sum of squared differeces betwee the observed values for each experimetal uit ad the treatmet meas: SSE K J k ( ik k ) k i Alterative computatioal formulae: df T JK k ad the grad mea, mius the factor level differeces, all weighted b the umber of experimetal uits for each treatmet: SSAB K J k k (( )) Si k ) ( k ) ( ce some of the estimated grad meas cacel out we obtai: SS SS TR K k i K J J k TR k SSAB SS k ik k T T SSA SSB SSA SSB [See Excel Spreadsheet for the Example] K k K J J k k k SSE SS SS k TR T T SSAB K J k k ( k k + ) df ( J )( K ) 57 58

30 Assumptios ad Trasformatios: Assumptios regardig the error term Must meet assumptios to obtai ubiased estimates of populatio meas, ad a ubiased estimate of the variace of the error term (same as CRD: Oe Factor) o idepedet observatios (ot time or space related) o ormalit of the errors, o equal variace for each treatmet. Use residual plot ad a plot of the stadardized errors agaist the expected errors for a ormal distributio to check these assumptios. Trasformatios: As with CRD: Oe Factor, ou must trasform the -variable Test for Iteractios ad Mai Effects The first mai questio is: Is there a iteractio betwee the two factors? H 0 : No iteractio H : Iteractio OR: H 0 : (φ AB+ σ ε) /σ ε H : (φ AB+ σ ε)/σ ε > Where σ ε is the variace of the error terms; φ AB is the iteractio effect of the fixed treatmets. Process: do our aalsis with the measured respose variable if assumptios of the error term are ot met, trasform the -variable do the aalsis agai ad check the assumptios; if ot me, tr aother trasformatio ma have to switch to aother method: geeralized liear models, etc

31 Usig a aalsis of variace table: Source df SS MS F p-value A J- SSA MSA SSA/(J-) F MSA/MSE Prob F> F (J-),(dfE), - α B K- SSB MSB SSB/(K-) A X B (J-)(K-) SSAB MSAB SSAB/ (J-)(K-) Error T -JK SSE MSE SSE/( T -J) Total T - SS F MSB/MSE F MSAB/MSE Source df MS E[MS] A J- MSA σ ε + φ A B K- MSB A X B (J-)(K-) MSAB Error T -JK MSE σ Total T - σ ε + φ B σ ε + φ AB ε Prob F> F (K-),(dfE),- α Prob F> F dfab,dfe,,- α See Neter et al., page 86, Table 9.8 for details o expected mea squares; φ is used here to represet fixed effects. For the iteractios: SSAB /( J )( K ) F SSE /( JK) T MSAB MSE Uder H 0, this follows F df,df, - α where df is from the umerator (J-)(K-), ad df is from the deomiator ( T - JK) If the F calculated is greater tha the tabular F, or if the p- value for F calculated is less tha α, reect H 0. o The meas of Factor A are iflueced b the levels of Factor B ad the two factors caot be iterpreted separatel. o Graph the meas of all treatmets o Coduct multiple comparisos all treatmets (rather the o meas of each Factor, separatel o Not as much power (reect H 0 whe it is false), if this occurs. 6 6

32 If there are o iteractios betwee the factors, we ca look at each factor separatel fewer meas, less complicated. Factor A: H 0 : μ μ μ J OR: H 0 : (φ A+ σ ε))/σ ε H : (φ A+ σ ε)/σ ε > Where σ ε is the variace of the error terms; φ A is fixed effect for Factor A. From the ANOVA table: SSA/( J ) F SSE /( JK) T MSA MSE Uder H 0, this follows F df,df, - α where df is from the umerator (J-) ad df is from the deomiator ( T -JK) If the F calculated is greater tha the tabular F, or if the p- value for F calculated is less tha α, reect H 0. o The meas of Factor A i the populatio are likel ot all the same o Graph the meas of Factor A levels o Coduct multiple comparisos betwee meas for the J levels of Factor A, separatel The aalsis ad coclusios would follow the same patter for Factor B

33 Aalsis of Variace Table Results for the Example Source Degrees of Freedom Sum of Squares Mea Squares F p A <0.000 B <0.000 A X B Error Total If assumptios met, (residuals are idepedet, are ormall distributed, ad have equal variaces amog treatmets), we ca iterpret the results. Iterpretatio usig α 0.05: No sigificat iteractio (p0.0539); we ca examie species ad fertilizer effects separatel. Are sigificat differeces betwee the three fertilizer levels of Factor A (p<0.000), ad betwee the four species of Factor B (p<0.000). The mea values based o these data are: A6.5, A3.38, A38.75 B7.08, B0.83, B34.7, B49.08 Did ot have to calculate these for each of the treatmets sice there is o iteractio

34 Further aalses, for each Factor separatel: Scheffé s test for multiple comparisos, could the be used to compare ad cotrast Factor level meas. o The umber of observatios i each factor level are: 6 for Factor A, ad for Factor B o Use the MSE for both Factor A ad for Factor B (deomiator of their F-tests) Factor A: t-tests for pairs of meas Determie the umber of pairs possible 3 3!!! 3 possible pairs of meas Use a sigificace level of 0.05/3 pairs0.07 for each t-test Comparig Factor Levels ad : A vs. A H0 : μ μ 0 H : μ μ 0 t-tests for each pair of meas could be used istead. o Agai, use MSE, ad 6 observatios for Factor A versus for Factor B t MSE ( K k k ) 0 + K k 4 k o Must split alpha level used i the F-tests b the umber of pairs t ( )

35 Critical t value from a probabilit table for: df(error) 36 based o ( T JK), ad 0.07 sigificace level (For α 0.05 use 0.05/3 pairs for each t-test), -sided test Usig a EXCEL fuctio: tiv(0.07,36), returs the value of.50 (this assumes a -sided test). Sice the absolute value of the calculated t is greater tha.50 we reect H0. OR eter our t-value, df (error), ad (for -sided) ito the EXCEL fuctio tdist(8.58,36,) Returs a p-value of < (NOTE that ou must eter the positive value, ad the p-value is for the two eds (area greater tha 8.58 plus area less tha -8.58) Sice p<0.07, we reect H0 A Differet Iterpretatio usig α 0.0: There is a sigificat iteractio (p0.0539) usig α 0.0; caot iterpret mai effects (A ad B) separatel. The mea values based o these data are: [Excel] AB0.5 AB4.5 AB AB4.75 AB8.00 AB.50 AB3 4.5 AB48.75 A3B 3.00 A3B5.75 A3B A3B mea values as there is a sigificat iteractio The mea of treatmet A differs from the mea of A. For Factor B Recalculate the umber of possible pairs for 4 factor levels (will be 6 pairs; divide alpha b this for each test ) The observatios per factor level is, rather tha 6 Df(error) ad MSE are the same as for Factor A

36 Further aalses: Cofidece limits for factor level ad treatmet meas Scheffé s test for multiple comparisos (or others) could the be used to compare ad cotrast treatmet meas (pairs or other groupigs of meas). The umber of observatios i Treatmet meas: k ± t( JK ), α MSE k each treatmet are 4 [lower power tha if there was o iteractio], ad use the MSE. Usig t-tests for pairs of meas, the umber of observatios are 4 for each k treatmet, use the MSE, ad recalculate the umber of possible pairs out of treatmets (will be 66 pairs! Retaiig α 0.0, we would use 0.0/ for each t-test ) Factor A meas: Factor B meas: t ± ( JK ), α K t MSE k k ± ( JK ), α J k MSE k [see course materials for FRST 430/533 for other desigs] 7 7