COURSE CONTENT: COURSE REQUIREMENTS: READING LIST: LECTURE NOTES COURSE CODE: STS 352 COURSE TITLE: EXPERIMENTAL DESIGN 1 NUMBER OF UNIT: 2 UNITS

Transcription

1 COURSE CODE: STS 35 COURSE TITLE: EXPERIMENTAL DESIGN NUMBER OF UNIT: UNITS COURSE DURATION: TWO HOURS PER WEEK. COURSE COORDINATOR: MR G.A. DAUDU LECTURER OFFICE LOCATION: AMREC COURSE CONTENT: Basc concepts of expermentaton, Completel randomzed desgn, Randomsed complete block desgn, Latn Square Desgn, Graeco Latn Square Desgn, Smple factoral Desgn COURSE REQUIREMENTS: Ths s a compulsor course for all statstcs students. Students are expected to have a mnmum of 75% attendance to be able to wrte the fnal examnaton. READING LIST:.) Statstcal Desgn and Analss of Experments b P.W.M. John..) Expermental Desgns b Cochran and Cox. 3.) Desgns and Analss of Experments for Bolog and Agrc. Students b Oejola, B.A. 4.) Statstcal Methods b Snedecor and Cochran. 5.) Statstcal Procedures for Agrcultural Research b Gomez and Gomez. LECTURE NOTES Introducton An experment nvolves the plannng, executon and collecton of measurements or observatons. Examples of smple experment. Comparson of two teachng methods. Comparson of two varetes of maze The dfference among expermental unts treated alke s called expermental error, ths error s the prmar bass for decdng whether an observed dfference s real or

2 just due to chance. Clearl ever experment must be desgned to have a measure of the expermental error. Defntons Expermental Unt/plot Ths s the smallest to whch a treatment s appled, and on whch an observaton s made e.g. an anmal brd, an object, a cage, a feld plat and so on. - Defnton of a unt depend on the objectve of the experment. Factors These are dstnct tpes of condton that are manpulated on the expermental unt e.g. age, group, gender, varet, fertlzer and so on. Factor Levels Dfferent mode of the presence of a factors are called factor levels. - Factors can be quanttatve or qualtatve. Treatments Each specfc combnaton of the levels of dfferent factors s called the treatment. Replcaton These are the numbers of expermental unts to whch a gven treatment s appled. MAIN ASPECT OF DESIGNING EXPERIMENT a. Choose the factor to be studed n the experment and the levels of each factor that are relevant to the nvestgaton. b. Consder the scope of nference and choose the tpe of expermental unt on whch treatment are to be appled. c. From the perspectve of cost and desred precson of nference, decde on the number of unts to be used for the experment. d. Fnall, and most mportant, determne the manner n whch the treatments are to be appled to the expermental unts (.e. desgn of the experment). PRINCIPLES OF EXPERIMENTAL DESIGN

3 There are three basc statstcal requrements for a good experment: Randomzaton Replcaton Local Control or Blockng. RANDOMIZATION: Ths s the process b whch t s ensured that each treatment has an equal chance of beng assgned to an expermental unt e.g. Suppose two maze varetes, Yellow (Y) and Whte (W) are to be compared usng four expermental unts for each (I) (II) In laout (II) f the feld has fertlt gradent so that there s a gradual productvt from top to bottom. Then the whte varet wll be at advantage been n a relatvel more fertle area hence, the comparson wthn the varet would be based n favour of varet W. A better laout s obtaned b randomzaton as shown n laout (I).. REPLICATION: Each treatment beng appled to more than one expermental unt. Expermental error can be measured onl f there are replcatons. Also the more the expermental unts used for each treatment, the lower would be the standard error for the estmate for treatment effect and hence, the more precse the experment. Precson s the measurement of how close the observed values are to each other. 3. BLOCKING OR LOCAL CONTROL: Ths s the process of groupng together expermental unts that are smlar and assgnng all treatments nto each group or block separatel and ndependentl. Ths allows the measurement of varaton among blocks whch can be removed from the expermental error. Blockng s therefore one of the measure for reducng or mnmzng expermental error. The

4 ablt of detectng exstng or real dfferences among treatments ncrease as the sze of the expermental error decreases. COMPLETELY RANDOMIZE DESIGN (CRD) Introducton A CRD s a desgn n whch the treatments are assgned completel at random so that each expermental unt has equal chance of recevng an one treatment. An dfference among the expermental unts recevng the same treatment s consdered to be expermental error. Model:

5 j = μ + e j = μ + α + e j =,,, t and j =,,, r Where s the observed value for replcate j of treatment, μ s the populaton mean for treatment, μ s the populaton mean, s the effect of treatment and e j s the expermental error resultng from replcate j of treatment. Assumpton: are assumed normall dstrbuted about the mean, μ, and varance, σ or N (0, σ ).e. ndependentl and dentcall normall dstrbuted wth mean 0 and constant varance σ. Also α = 0, Estmaton of the Parameters ds = dμ j ( j μ α ) ( ˆ μ α ) = 0 j j j j j j ˆ μ α = 0 j j rt ˆ μ r α = 0 Impose the constran α = 0 ds = dα rtμˆ = j ( j ˆ j μ = =.. rt j μ α )

6 ˆ α = ( ˆ ˆ j μ α ) = 0 j ˆ ˆ j μ α = 0 j j j r j j j r ˆ μ r ˆ α = 0 ˆ μ j =... Randomzaton Procedure. Determne the total number of expermental unts or plots (N) where N = rt wth r beng the number of replcatons and t the number of treatments.. Assgn a plot number to each expermental unt n an convenent manner consecutvel to N. 3. Assgn the treatments to the expermental unts b an chosen randomzaton scheme e.g. usng table of random numbers, random number generator, drawng of lots and so on. Data Structure Treatments T L t.

7 L t. r L r r tr. r.. L t.. Analss of Varance The total varaton n CRD s parttoned nto two sources of varaton.e. varaton due to treatment and varaton due to the error. The relatve sze of the two varatons s used to ndcate whether the observed dfference among the treatment means s sgnfcant or due to chance, the treatment dfference s sad to be sgnfcant f the treatment varaton s sgnfcantl larger than the expermental error. Total sum of squares, SST, SST t n = j = j= j (..) = N.. Sum of squares due to treatment SSB t n SSB =. (...) = n (...) = n = j= t = t = N.. Sum of Square due to Error, SSE t n SSE = ( ) = j= j... C. F = = correctng factor N SST = SSB + SSE. e. SSE = SST SSB ASSIGNMENT

8 Show that: ( ) = ( ) + ( ) j..... j. j j j ANOVA TABLE Source of varaton Degree of freedom Sum of Squares Between treatment t- SSB Error N-t SSE Total N- SST Means squares SSB MSB = t SSE MSE = N t F-rato MSB Fc = = Fc MSE If there are no dfferences n the effect of the treatment Fc follows the F- dstrbuton. Hence, f F c > F T where F T s the table value from the F-Table wth t and N t degrees of freedom at a gven sgnfcance level, then the effect are sad to be sgnfcantl dfferent Or Reject H 0 f F C F T COMPARISON OF MEANS If a sgnfcant result s declared then there s need to dentfed the mean that are dfferent and ths can be done usng multple comparson of means such as LSD Least Sgnfcant Dfference DMRT Duncan s Multple Range Test Turke Scheffee etc. LSD = tsed

9 = r If the observe dfference between an two means s greater than the LSD value then those two means are sad to be sgnfcantl dfferent. COEFFICIENT OF VARIATION Ths s a measure of precson of the estmates obtaned from the data. It s also used to assess the qualt of the management of an experment. A low coeffcent of varaton ndcates hgh precson of estmate or effcent management of the experment. Example: In an effort to mprove the qualt of recordng tapes, the effect of four knds of coatng A, B, C, D on the reproducng qualt of sound are to be compared. The measurements of sound dstorton are gven below. A B C D Recommend the best coatng for the sound producton. ADVANTAGE OF CRD. The desgn s ver flexble. The statstcal analss s smple

10 3. It has hgh degrees of freedom relatve to other desgns 4. It s best for small experments DISADVANTAGE OF CRD Desgn s ver neffcent f unts are not homogenous. ASSIGNMENT. Analze the followng data from a feld experment wth four treatments usng % sgnfcance level. Carrout mean comparson f necessar. How good s the management of the experment. A B C D Three fertlzer sources A, B, C, were each appled to seven plots chosen at random n a feld of carrot. Analze the data usng 5% sgnfcance level. Carrout mean comparson f necessar. How effcent was the management of the experment A B C RANDOMISED COMPLETE BLOCK DESIGN (RCBD)

11 Introducton The desgn s used when the expermental unt can be grouped such that the number of unts n a group s equal to number of treatments. The groups are called blocks or replcates and the purpose of groupng s to have unts n a group as homogeneous as possble so that observed dfferences n a group are manl due to treatments. Varablt wthn group s expected to be lower than varablt between groups. Snce the number of unts per block equal the number of treatments, the blocks are of equal sze hence, the desgn s a complete block desgn. The prmar purpose of blockng s to reduce the expermental error b elmnatng the known sources of varablt. Model: j = μ + α + β j + e j =,,, t and j =,,, r where s the observed value for block j of treatment, μ s the populaton mean, s the effect of treatment, β j s the effect of block j and e j s the expermental error resultng from block j of treatment. Assumpton: - block and treatment effect are addtve, - N (0, σ ) - α = 0, β j = 0, Estmaton of Parameters A procedure smlar to that used n CRD can be utlzed here to obtan the desred estmates. Randomzaton and Laout The randomzaton process for randomsed complete block desgn s appled separatel and ndependentl to each of the block. - Dvde the expermental area nto r-blocks. - Sub-dvde the block nto t-expermental unts. Where t s the number of treatments.

12 - Number the plot consectvel from I to t and assgn the t-treatment at random to the t-unt wthn each block followng an randomzaton scheme. DATA STRUCTURE Blocks Treatment 3 r Total 3 t Total r r.. Analss of Varance The total varaton s parttoned nto the varaton due to blocks, varaton due treatments, and varaton due to error..e... j.. j SSTotal = ( ) = N SS SS Trt Block. = (...) = r j =. ( j ) =.. t SSE = SSTotal SSTrt SSBlock N.. N..

13 ANOVA TABLE Source Df SS MS F Block r SSB MSB = r Treatment t MST Error (r-)(t-) SSE SST = t Total rt Hpothess Or Comparng the calculated F-ratos to the table F-value at a gven sgnfcance level, we decde to reject or fal to reject the null hpothess..e. Reject f Reject f COMPARISON OF MEANS Use LSD to compare the treatments f the F-rato s found to be sgnfcant. where Coeffcent of varaton CAUSES OF MISSING VALUES AND THEIR ESTIMATIONS A mssng data can occur whenever a vald observaton s not avalable for an one of the expermental unts, occurrence of mssng data result n two major problems.e. loss of nformaton and non applcablt of standard analss of varance.

14 COMMON CAUSES OF MISSING DATA nclude:. When ntended treatment s not appled.e. mproper treatment.. When expermental plants are destroed probabl due to poor germnaton, phscal damage, pest damage etc. Ths causes total or hgh percentage of the plants n a plot to be destroed such that no meanngful observaton can be made on the plot. 3. Loss of harvested sample: Ths ma result from the fact that some plant characters cannot be convenentl recorded ether n the feld or mmedatel after harvest due to some other process requred. Hence some samples wll be lost between the tme of harvestng and actual recordng of data. 4. Ths happens after data have been recorded and transcrbed generall referred to a llogcal data. The value ma be too extreme as a result of msread observaton or ncorrect transcrpton. ESTIMATION OF MISSING VALUE FROM RCBD Let x be the mssng value,,, xˆ ˆ μ ˆ α β = 0, j =

15 , = 0 where G 0 s the grand total excludng the mssng value. T 0 s the total observed value for the treatment that contaned the mssng value. B 0 s the total observed value for the Block that contaned the mssng value. Note that: the degree of freedom must be adjusted b the number of mssng values.e. reduce the number of degrees of freedom b the number of mssng values. ADVANTAGES OF RANDOMIZED COMPLETE BLOCK DESIGNS. A reducton of experment error due to blockng s expected.. Estmaton of mssng value s eas to compute. 3. The ANOVA s also eas to compute. DISADVANTAGES OF RCBD. Not best for large number of treatments.. More taskng n the executon of the desgn than the CRD. 3. Mssng value can create problem especall n estmaton and non formal analss. 4. The precson wll be affected due to mssng values. RELATIVE EFFICIENCY Blockng maxmzes the dfference among blocks. Hence t s necessar to examne how much s ganed b the ntroducton of blockng nto the desgn. The magntude of the

16 reducton n the expermental error due to blockng over the CRD can be obtaned b computng relatve effcenc. Where s the block mean square and the s the error mean square.. Example: In an experment to examne the respond of maze to nutrent fertlzer applcaton. Sx treatments were used n four blocks. Analze the data and recommend the approprate fertlzer rate. F F F 3 F 4 F 5 F 6 TOTAL I II III IV TOTAL The followng data are eld of groundnut n a varet tral nvolvng fve varetes of groundnut usng four replcatons n randomzed complete block desgn. The data has one mssng value. Analze the data and make our recommendaton. V V V 3 V 4 V 5 I II * III IV ASSIGNMENT In an experment to test the effect of fve level of potash (ABCDE) on the eld of cotton the followng strength ndces were obtan as gven below. One of the data pont s mssng. Analze the data and compare the mean of the fve level of potash and make necessar recommendaton. How effectvel was the experment carred out. Was there an gan n precson n usng RCBD over CRD

17 A (7.6) C (*) E (7.46) D (7.7) B (8.4) E (7.68) B (8.5) C (7.73) D (7.57) A (800) C (7.47) E (7.0) A ( (7.93) B (7.87) D (7.80) LATIN SQUARE DESIGN Introducton The major feature of the latn square desgn s ts capact to smultaneousl handle two known sources of varaton among the expermental unts. These are commonl referred to as row blockng and column blockng. It s ensured that ever treatment occurs onl once n each row and once n each column. Hence the varaton due to row and column can be estmated and removed from the expermental error. Note that: the presence of row and column blockng also consttute a restrcton. Ths s due to the requrement that all treatment appear n each row and n each column. Ths s onl satsfed f the number of replcatons equal the number of treatments. Hence, for large number of treatments, the desgn s not practcable. Also when the number of treatments s small the degrees of freedom for the error becomes too small for the error to be relabl estmated, the desgn s therefore not generall, wdel adopted.

18 Model: jk = μ + α + β j + δ k + e jk where, j, k =,,, t where s the observed value from row j and column k recevng treatment. - s the overall mean - s the effect of treatment - s the effect of row j - s the effect of row k - s the random error component for row j and column k recevng treatment. Assumptons: The model s completel addtve.e. there s no nteracton between the rows, columns and treatments. N (0, σ ) and Estmaton of Parameters A procedure smlar to that used n CRD can be utlzed here to obtan the desred estmates. RANDOMIZATION PROCEDURE. Obtan a square feld parttoned nto t rows and t columns.. Arrange the treatment nto the unt n a standard form. 3. Randomze between the columns 4. Randomze between the rows Example: Consder an experment wth four treatments to be compared usng latn square desgn.e. 4 x 4 LS

19 col Standard form * A B C D * B C D A * C D A B * D A B C 3 4 Randomze btw B D C A 43 C A D B 3 D B A C 4 A C B D Randomze between row C A D B D B A C 34 3 A C B D 4 B D C A ANALYSIS OF VARIANCE The total varaton s parttoned nto components for row, column, treatment and error. The sum of squares are obtaned n the usual form. ANOVA TABLE Source Df SS MS F Rows t- SSR MSR Columns t- SSC MSC Treatments t- SS Trt MS Trt MSR / MSE MSC / MSE MS Trt / MSE Error (t-) (t-) SSE MSE Total t - SST BLOCKING EFFICIENCY The effcenc of both row and column blockngs n a latn square desgn ndcate the gan n precson relatve to ether the CRD or RCBD.

20 RELATIVE EFFICIENCY OF LSD TO CRD Relatve effcenc of a latn square desgn as compared to CRD s gven b Where Er, E c, E e are the mean squares row, column, and error respectvel wth t as the number of treatment. For an R.E = 35% t ndcates that the use of LSD s estmated to ncrease the expermental precson b 5% whle f the R.E s less than 00% means that there s no gan. RELATIVE EFFICIENCY LSD TO RCBD Relatve effcenc of latn square desgn as compared to RCBD can be computed n two was.e. when rows are consdered as blocks and when columns are consdered as blocks of the RCBD. I II III IV I II III IV ) = ) = Example: Suppose we have Result ndcate that, the addtonal column blockng b use of latn square desgn s estmated to have ncreased the precson over that of the RCBD wth row as block b 94%. However, the addtonal row blockng n the LS Desgn dd not ncrease the precson over the RCBD wth column as blocks. Hence for ths experment the Randomzed Complete Block Desgn wth column as blocks would have been as effcent as the LS Desgn. MISSING VALUE ESTIMATION Mssng value n the latn square experment can be estmated as follows

21 Where the R 0, C 0, T 0 are the total of the row, column and treatment respectvel that contan the mssng observaton. Agan one degree of freedom s subtracted from both total and error degrees of freedom n the case of one mssng value. ADVANTAGES AND DISADVANTAGES. The elmnaton of two sources of varaton often lead to a smaller error mean square than would be obtaned b use of CRD and RCBD.. ANOVA s smple 3. Mssng values can easl be handled Dsadvantages.. Assumpton of no nteracton between dfferent factors ma not hold.. Unlke the CRD and RCBD, the number of treatments s restrcted to the number of replcatons. Hence t s lmted n applcaton. 3. For large number of treatments such as t >, the square becomes too large and does not reman homogeneous. 4. For small number of treatments such as t < 3, degrees of freedom for the error s usuall too small for an meanngful comparson or concluson. 5. A square feld s often requred for the desgn and ths ma not be practcable. Example: The followng show the feld laout and eld of a 5 x 5 latn square experment on the effect of spacng on eld of mllet, the spacng are: A(cm), B(4cm), C(6cm), D(8cm) and E(0cm) Column Row Total B: 57 E: 30 A: 79 C: 87 D: 0 55 D: 45 A: 83 E: 45 B: 80 C: E: 8 B: 5 C: 80 D: 46 A: A: 03 C: 04 D: 7 E: 93 B: C: 3 D: 7 B: 66 A: 334 E:

22 Total GRAECO LATIN SQUARE DESIGN (GLSD) Introducton In the Graeco latn square desgn, the treatments are grouped nto replcates n three dfferent was: ths trple groupng s to elmnate from the error, three sources of blockng of blockng varaton. Recall the earler desgn: CRD - No blockng varaton RCBD - Sngle blockng varaton Latn Square - Double blockng varaton Graeco latn - Trpple blockng varaton Thus GLSD provde more opportunt than the other desgns n the reducton of error through skllful plannng. The expermental unt should be arranged and the experment conducted so that dfferences n the three drectons represent major sources of varaton. RANDOMIZATION Arrange the rows and the columns ndependentl at random ncludng the treatment. Assgn the subscrpt at random to ther respectve classfcaton. The treatment and subscrpt must appear once n a column and once n a row, and must appear together onl once. The laout can be dffcult to desgn but lke n the latn square desgn, once we obtan a standard form then we can randomze between the columns and between the rows. I II III I II III I A B 3 C C 3 A B II B C A 3 A B 3 C III C 3 A B B C A 3 Randomze between rows I II III A B C 3

23 B 3 C A C A 3 B Randomze between columns T = 4 A, B, C. D I II III IV I A C 4 B 3 D II B D 3 A 4 C III C 3 A D B 4 IV D 4 B C A 3 T = 5 A, B, C, D, E I II III IV V A B 3 C 5 D E 4 B C 4 D E 3 A 5 C 3 D 5 E A 4 B D 4 E A 3 B 5 C E 5 A B 4 C D 3 MODEL jkl = μ + α + β j + δ k + τ l + e jkl where, j, k, l =,,, t where s the observed value from row j and column k recevng treatment. s the grand mean s the effect of treatment

24 s the effect of row j s the effect of column k τ l s the effect of subscrpt factor l. s the random error component Assumptons: The model s completel addtve.e. no nteracton between the row, the column, the subscrpt factor and the treatment N (0, σ ), and τ l = 0 Estmaton of Parameters A procedure smlar to that used n CRD can be utlzed here to obtan the desred estmates. STATISTICAL ANALYSIS The total varaton s petton nto fve components.e. the row, column, subscrpt factor, treatment and error. The sum of square are obtan n the usual form.e. SSR = t. j..... N

25 ANOVA TABLE Source Df SS MS F Rows t- SSR MSR Column t- SSC MSC MSR / MSE MSC / MSE Subscrpt t- SS Subscrpt MS Subscrpt MS Subscrpt / MSE Treatments t- SS Trt MS Trt Error (t-) (t-3) SSE MSE Total t - SST MS Trt / MSE Note the followngs:. The number of replcatons equals the number of treatments hence for large number of treatments, the desgn s not practcable (t > ).. The expermental error s lkel to ncrease wth the sze of the square. 3. Small square provde onl a few degree of freedom for the error. 4. Expermental unts are dffcult to balance convenentl n all the three groupngs. Example Consder the followng data whch was obtan from an experment to stud the samplng error, the data consst of fve samplers (A E) beng controlled for order of samplng, area of samplng and qualfcaton of sampler. Analse the data. AREA Order Total A (3.5) B 3 (4.) C 5 (6.7) D (6.6) E 4 (4.) 5. B (8.9) C 4 (.9) D (5.8) B 3 (4.5) A 5 (.4) C 3 (9.6) D 5 (.7) E (.7) A 4 (3.7) B (6.0) D 4 (0.6) E (0.) A 3 (4.6) B 5 (3.7) C (5.) E 5 (3.) A (7.) B 4 (4.0) C (3.3) D 3 (3.5). Total

26 ASSIGNMENT The data below s obtaned from an experment usng Graeco Latn Square Desgn wth four det, (A, B, C, D). breed I, II, III, IV weght group {,, 3, 4} feed concentraton { v}. Is there an sgnfcant dfference between the dets. If an, compare them and make necessar recommendaton. Also comment on the management of the experment. Breeds I II III IV A (5.9) B 3 (4.) C 4 (0.) D (6.6) B (8.9) A 4 (4.5) D 3 (6.0) C (3.0) 3 C 3 (9.6) D (5.8) A (7.) B 4 (4.6) 4 D 4 (0.5) C (4.) B (3.5) A 3 (6.7) SIMPLE FACTORIAL EXPERIMENT Introducton Factoral experments are used n the stud of the effects of two or more factors. In factoral experments, all the possble combnatons of the level of the factors make up the treatments. For example, f there are two factor A, B each wth a and b levels respectvel, then we have ab treatment combnatons. A Maze varet B Fertlzer Rate There are 0 treatment combnatons. Also factoral experments allow us to nvestgate the nteracton between the factors. That s, how the levels of a factor perform n the presence of the levels of another factor. We are able to answer the queston on how the responses to one factor were affected b another. A B Maze Varet Fertlzer Rate a 0 Whte b 0 0 kg/ha a Yellow b 30 kg/ha

27 In analzng data from a factoral experment, we would be nterested n the man effect and the nteracton effect of a factor. The man effect of a factor s a measure of the change n response n the level of a factor averaged over all levels of the other factors. For example, let two factors A and B be at two level a 0, a and b 0, b respectvel wth treatment combnatons a 0 b 0, a 0 b, a b 0, a b. The man effect of A s a measure of change n A from a 0 to a averaged over the two levels of B..e. At level b 0 of B: the smple effect of A s a b 0 a 0 b 0. Smlarl, at level b of B: the smple effect of A s a b a 0 b.. Man effect of A = The r represent the replcaton, where each treatment total response s from r unts. Also at level a 0 of A: smple effect of B s a 0 b a 0 b 0. Smlarl, at level a of A smple effect of B s Thus Man effect of B = Each effect of a factor at a gven level of the other factor s known as smple effect. The nteracton effect s the dfferental response to one factor n combnaton wth varng levels of a second factor. That s, an addtonal effect due to the combned nfluence of two or more factors. For example, nteracton between A and B (AB) s estmated as the dfference between two smple effects. and.e. the nteracton effect AB = BA = Consder the followng result Mean Responses b 0 b b 0 b b 0 b a a () () ()

28 From above llustraton () shows the case of no nteracton, () shows the case of mld nteracton and () shows the case of strong nteracton. LAYOUT An of the earler desgn dscussed can be used, n partcular, the RCBD. The treatment combnatons are assgned to each block randoml For example, consder the case of two factors A and B, each at two levels a 0, a and b 0, b respectvel. The treatment combnatons are a 0 b 0, a 0 b, a b 0, a b. a 0 b

29 RCBD a b CRD MODEL CRD where s the observed value from the k th unt correspondng to level of factor A and the j th level of B s the grand mean s the effect of level of factor A s the effect of the j th of factor B s the effect of nteracton between and. s the random error component. MODEL RCBD Where T k s the effect of the k th block H 0 : H : atleast one H 0 : H : atleast one H 30 : H 3 : {there s nteracton between A and B} ANALYSIS OF VARIANCE

30 The total varaton s parttoned nto that due to factor A, factor B, the nteracton AB and the error.e. where the SST, SSA, SSB, SSAB and SSE are obtan n the usual form.e. -SSA- SSB ANOVA TABLE Source Df SS MS F A a- SSA MSA B b- SSB MSB AB (a-) (b-) SSAB MSAB ERROR ab(r-) SSE MSE Total abr- SST Example MSA / MSE MSB / MSE MSAB / MSE An engneer desgnng a batter for use n a devce that would be subjected to some extreme varaton n temperature has three tpes of plate materals to use. He decded to test the plate materals under three temperature settngs (5 0 F, 70 0 F, 5 0 F) to see ther effect on the lfe of a batter. Four test runs are to be made at each treatment combnaton. Test the effect of temperature and plate materal and ther possble nteracton on the batter lfe.

31 Temperature Tpe 5 0 F 70 0 F 5 0 F A B C ASSIGNMENT Three tpes of tres are to be compared usng four dfferent brand of cars. The threadng on the tres are measured after a perod of use. Below s the rescaled data. Are the tres sgnfcantl dfferent? Also does the performance of the tres depend on the brand of the car? Tres Car brand A B C 3

32 4