ANALYSIS OF COVARIANCE

Transcription

1 ANALYSIS OF COVARIANCE YOGITA GHARDE M.Sc. (Agrcultural Statstcs), Roll No I.A.S.R.I., Lbrary Avenue, New Delh Charperson: Dr. V.K. Sharma Abstract: Analyss of covarance (ANCOVA) s a statstcal technque to account for varablty n response varable usng lnear regresson. When heterogenety between epermental plots does not follow a defnte pattern, then blockng may not be a very effectve method for controllng error varaton. In such stuaton ANCOVA s an alternatve. ANCOVA s used to mprove the estmate of error varance and to adust treatment means for the value of the covarate. It s also used to estmate the mssng observaton(s). Here, some specfc applcatons of the ANCOVA technque n controllng epermental error and n adustng treatment means have been dealt wth. Use of ANCOVA technque for RBD to control epermental error and to estmate mssng observaton has been gven. Key words: Analyss of Varance, Analyss of Covarance, Unformty Tral, Analyss wth Mssng Observatons 1. Introducton The test of sgnfcance based on t-dstrbuton s an adequate procedure for testng the sgnfcance of the dfference between two sample means. In a stuaton when there are three or more samples to consder at a tme, an alternatve procedure s needed for testng the hypothess that all the samples are drawn from the same populaton.e. they have the same mean. For eample, 5 fertlzers are appled to four wheat plots and yeld of wheat on each of the plot s recorded. The nterest s to fnd whether effects of these fertlzers on the yelds are sgnfcantly dfferent or n other words, whether the samples have come from the same normal populaton. The answer to ths s provded by Analyss of Varance (ANOVA). The basc purpose of ANOVA s to test the homogenety of several means.e. to test the hypothess that several means are equal. Ths technque s an etenson of the two-sample t-test. ANOVA s the technque of separaton of varance ascrbable to one group of causes from the varance ascrbable to other group. It conssts of the estmaton of the amount of varaton due to each of the ndependent factors (causes) separately and then comparng these estmates due to assgnable factors wth the estmate due to chance factor or epermental error. ANOVA nvolves the F statstc whch s defned as follows: F Varaton among the samplemeans Varaton wthn thesample ANOVA s an effectve tool under the stuaton when response varable s not affected by other varables called concomtant varables (covarates). If covarate s there then to

2 Analyss of Covarance handle ths stuaton, another technque known as Analyss of Covarance (ANCOVA) s an alternatve.. Analyss of Covarance Analyss of covarance s an etenson of the analyss of varance technque. The queston s whether the varaton n the response varable over the classes s due to class effects or due to ts dependence on the other varables s, called the ndependent or concomtant varables, whch also vary from class to class. ANCOVA controls the epermental error by takng nto consderaton the dependence of y on the s. Forester (1937) llustrated the applcaton of covarance to error control on data from felds where the varablty has been ncreased by prevous eperments. Some of the eamples where the technque of ANCOVA may be used are as follows: The yeld of a crop may depend on the number of plants per plot. Consderng the number of plants as the covarate, ANCOVA can be performed on yeld. In a study of effect of drugs or dets on the growth of anmals, the growth may depend on the ntal condton (say, ntal weght) of the anmals. The man purpose of the ANCOVA s statstcal control of varablty besdes local control and makes use of lnear regresson. It s a statstcal method for reducng epermental error or for removng the effect of covarate(s). The analyss of covarance s lke an analyss of varance on the resduals of the values of the dependent varable, after removng the nfluence of the covarate, rather than on the orgnal values themselves. In so far as the measures of the covarate are taken n advance of the eperment and they correlate wth the measures of the dependent varable they can be used to reduce epermental error (the sze of the error term) or control for an etraneous varable by removng the effects of the covarate from the dependent varable. Uses of ANCOVA To control epermental error and to adust treatment means for the value of the covarate To estmate mssng data To ad n the nterpretaton of epermental results Some specfc applcatons of the covarance technque n controllng epermental error and n adustng treatment means are descrbed below: a. Sol Heterogenety The covarance technque s effectve n controllng epermental error caused by sol heterogenety when the pattern of sol heterogenety s spotty or unknown varablty between plots n the same blocks remans large despte blockng Use of ANCOVA n such cases nvolves the measurement, from ndvdual epermental plots, of a covarate that can dstngush dfferences n the natve sol fertlty between plots and, at the same tme, s lnearly related to the character of prmary nterest. Two

3 Analyss of Covarance types of covarate that are commonly used for controllng epermental error due to sol heterogenety are unformty tral data and crop performance data pror to treatment. ) Unformty tral data: A unform sol cropped unformly gves unform crop performance; sol heterogenety s measured as the dfferences n crop performance from one plot to another. Thus, n usng unformty tral data as the covarate, the two varables nvolved are: The prmary varable Y, recorded from epermental plots after the treatments are appled The covarate X, recorded from a unformty tral n the same area but before the eperment s conducted Unformty tral data s an deal covarate to correct for varablty due to sol heterogenety. It clearly satsfes the requrement that t s not affected by the treatments snce measurements are made before the start of the eperment and, thus, before the treatments are appled. ) Data collected pror to treatment mplementaton: In eperments where there s a tme lag between crop establshment and treatment applcaton, some crop characterstcs can be measured before treatments are appled. Such data represent the nherent varaton between epermental plots. In such cases, one or more plant characters that are closely related to crop growth, such as plant heght and tller number, may be measured for each epermental plot ust before the treatments are appled. Because all plots are managed unformly before treatment, any dfferences n crop performance between plots at ths stage can be attrbuted prmarly to sol heterogenety. Crop performance data s clearly easer and cheaper to gather than unformty tral data because crop performance data s obtaned from the crop used for the eperment. However, such data are only avalable when the treatments are appled late n the lfe cycle of the epermental plants. b. Resdual effects of Prevous Trals In cases where sol heterogenety caused by resdual effects of prevous trals s epected to be large, the feld should be left n fallow or green manure for a perod of tme between eperments. Ths practce, however, s costly and s not commonly used where epermental feld areas are lmted. When a fallow perod or a green manure crop cannot be used, the epected resdual effects can be corrected by blockng, or by a covarance technque, or by both. Wth the covarance technque, the covarance could be plot yeld n the prevous tral or an nde representng the epected resdual effects of the prevous treatments. c. Stand Irregulartes Varaton n the number of plants per plot often becomes an mportant source of varaton n feld eperments. Ths s especally so n the annual crops where populaton densty s hgh and accdental loss of one or more plants s qute common. Among several methods, 3

4 Analyss of Covarance ANCOVA wth stand count as the covarate s one of the best alternatves, provded that stand rregulartes are not caused by treatments. Such a condton ests when: The treatments are appled after the crop s already well establshed. The plants are lost due to mechancal errors on plantng or damage durng cultvaton. The plants are damaged n a random fashon from causes such as rat nfestaton, cattle grazng. In such cases covarance analyss may be appled but ts purpose s totally dfferent. Such usage of the covarance technque s to ad n the nterpretaton of results rather than to control epermental error and to adust treatment means. d. Non unformty n pest ncdence The dstrbuton of pest damage s usually spotty and the pattern of occurrence s dffcult to predct. Consequently, blockng s usually neffectve and covarance analyss, wth pest damage as the covarate, s an deal alternatve. e. Non unformty n envronmental stress To select varetes that are tolerant to envronmental stress, genotypes are usually tested at a specfed level of stress n a partcular envronment. Some eamples are nsect and dsease ncdence, drought or water loggng, salnty, ron tocty and low fertlty. It s usually dffcult to mantan a unform level of stress for all test varetes. For eamples, n feld screenng for varetals resstance to an nsect nfestaton, t s often mpossble to ensure a unform nsect eposure for all test varetes. To montor the varaton n the stress condton over an epermental area, a commonly used feld plot technque s to plant, at regular ntervals, a check varety whose reacton to the stress of nterest s well establshed. A susceptble varety s commonly used for ths purpose. The reacton of the susceptble check nearest to, or surroundng, each test varety can be used as the covarate to adust for varablty n the stress levels between the test plots..1 Error Control and Adustment of Treatment Means It s known that blockng can reduce epermental error by mamzng the dfferences between blocks and thus mnmzng dfferences wthn blocks. Blockng, however, cannot cope wth certan type of varablty such as spotty sol heterogenety and unpredctable nsect ncdence. In both nstances, heterogenety between epermental plots does not follow a defnte pattern, whch causes dffculty n gettng mamum dfferences between blocks. Blockng s neffectve n the case of non unform nsect ncdences because blockng must be done before the ncdence occurs. Federer and Schlottfeldt (1954) used covarance as a substtute for the use of block to control gradents n epermental materal. Outhwate and Rutherford (1955) etended Federer and Schlottfeldt s results. Furthermore, even though t s true that a researcher may have some nformaton on the probable path and drecton movement, unless the drecton of nsect movement concde wth the fertlty gradent, the choce of whether sol heterogenety or nsect ncdence should be the crteron for blockng s dffcult. The choce s especally dffcult f both sources of varaton have about the same mportance. 4

5 Analyss of Covarance Use of ANCOVA should be consdered n eperments n whch blockng cannot adequately reduce the epermental error. By measurng an addtonal varable (.e., covarate X) whch s known to be lnearly related to the prmary varable Y, the source of varaton assocated wth the covarate can be deducted from epermental error. Model: Lnear model s * Υ Χ θ + e, (.1) where, n number of observatons Y n 1 vector of observatons X * n r order of ncdence matr θ r 1 vector of parameters e n 1 vector of errors Covarance analyss s acheved by parttonng θ nto two parts: a, of order p1 for general mean μ and the effects correspondng to the levels of factors and ther nteractons, and b of order k1 for the coeffcents of covarates. Smlarly X * s parttoned nto two parts wth X of order n p for the dummy varables and of order n k for the values of covarates. In ths way model can be wrtten as: Υ Χa + Ζb + e, (.) E (e), Var (e) σ Ι Assumptons: error term has zero mean and constant varance σ error s ndependently dstrbuted the covarate, s unaffected by the treatments (t can t be nfluenced by the factor under study) y and have a lnear relatonshp errors follow a normal dstrbuton Followng Searle (1971), t s assumed that X does not necessarly have full column rank whereas does. Furthermore, t s assumed that the columns of are lnearly ndependent of those of X. Let a and b be the least squares estmates of a and b. The normal equaton for a and b are obtaned as: X X X X a b X Y Y Thus, a ( X X) ( X Y X b ) ( X X) X Y ( X X) * a ( X X) X b X b 5

6 Analyss of Covarance where a ( X X) X Y and * b b P Ι { } [ Ι X( X X) X ] [ Ι X( X X) X ] 1 ( P) PY X( X X) X Y Let the model nvolvng the covarates, be defned as: Y Pb+ e (.3) Then, b ( P) PY 1 Let P R. Thus (.3) can be wrtten as Y R b + e Thus, R R mn Y Y * * ( Y X θ)( Y X θ) a [ Y X Y ] b Let R (a, b) be the reducton n sum of squares due to parameters a and b.e. R ( a,b) Y Xa Y X X X Y X X X Y X X X + Y b ( 1 ) X Y Y X( X X) X ( P) PY + Y ( P) ( 1 ) X Y + Y P( P) PY ( 1 ) X Y + Y R ( R R ) R Y 1 PY Ths s the sum of two reductons due to fttng of a and b. When only a s consdered n the model: Υ Χa + e R a Y X X X * ( ) ( ) X Y Y Xa, when only b s consdered n the model: Y R b + e SSRb Y R R ( a, b) R( a) + SSRb 1 ( R R ) R Y Y b Thus reducton n sum of squares due to parameter b s n the presence of a s R(b/a) R(a,b) + R(a) SSRb b R z Y 6

7 Analyss of Covarance Thus SSRb s the reducton n sum of squares attrbutable to fttng the covarates, havng already ftted the factors-and-nteractons part of the model. 3. ANCOVA for Randomzed Block Desgn The model for the analyss of covarance for two-way classfed data wth k treatments n r blocks of the randomzed block desgn (Das and Gr, 1986) s y μ + t + b + b + e, where y s the observaton correspondng to the th treatment n the th block; 1,,,k, 1,,,r; μ s the fed effect of general mean; t s the fed effect of th treatment; b s the fed effect of th block; s the observaton on the covarate correspondng to y ; b s the regresson coeffcent of y on ; e s are the error components whch are assumed to be ndependently and dentcally dstrbuted wth zero mean and a constant varance σ ; Now, y T, y B, Let T(), B (), T G, T() G. where T s the th treatment total, B s th block total, T () s th treatment total for covarate, B () th block total for covarate, G grand total and G grand total of covarate. Applyng the technque of least squares, the estmates of the fed effects along wth the regresson coeffcent are as shown below: μ T bt t r B bb b k b y b y () () μ, μ, TT G bg r T r () () B B B, k () k () GG + G + E E y, The error sum of squares (E) adusted for the varaton of s gven by E (y μ t b b) T B G y bey. r k 7

8 Analyss of Covarance The degrees of freedom of the adusted error sum of squares s (r-1)(k-1)-1. Ths has been reduced by one degree of freedom from that n the randomzed block desgn because the estmaton of error varance here s subect to one more restrcton mposed for estmatng b. For obtanng the adusted treatment sum of squares we make the hypothess that all the treatment effects are zero. The error sum of squares (E 1 ) for ths hypothetcal model comes out as below: Adusted error sum of squares on the hypothess B E1 y b E y k where E y B B () b, E y y E k B () E, k The adusted treatment sum of squares s now E 1 E wth k-1 degrees of freedom. The estmates and the varance of dfferent treatment contrasts are obtaned as below. t for 1,,, k. T T G () G b( ) r r y y b( ),(say) Estmate of any contrast V( l t l t ) σ σ because V(b). E l (y b ) lt s then gven by ( l + r l ), E ( m) In partcular, V(t t m) σ [ + ] and σ s estmated by the error mean square r E obtanable from E m E (r 1)(k 1) 1 8

9 Analyss of Covarance It s seen that the varance of the estmate of the dfference between any par of treatments s not constant, as we get n ANOVA of ths desgn. Sometmes to make the crtcal dfference (C.D.) a constant for each par of treatments, ( n the above varance, T m) s replaced by, the treatment mean square for the -varate. (k 1) Varance covarance table showng the sum of square and sum of products Sources of varaton Blocks Treatments d.f. (r-1) (k-1) SS due to - covarate S B () k G T() G T r Error (r-1)(k-1) By subtracton(e ) Total -1 G Sum of products S y B B SS due to y- varate S yy () GG B G k k T() T r GG T y T r G T By subtracton(e y ) By subtracton(e yy ) y GG y G yy Adusted error and treatment sums of squares Sources of varaton d.f. S S y S yy Adusted SS d.f. Treatments k-1 T T y yy T E k-1 E 1 E y Error (r-1)(k-1) E E y E yy E yy E (r-1)(k-1) -1 E Treatment +error r(k-1) T + E E E y y + Ey E y T yy + E yy E yy E yy E 1 E T r(k-1)-1 Illustraton 3.1: A varetal tral was conducted at Vvekanand laboratory, Almora wth 8 varetes of hybrd maze. The tral was lad put n a randomzed block desgn wth 4 replcatons. At the tme of harvest the number of plants per plot of sze 48 1 was also recorded along wth the plot yeld. The data of gran yeld (Y) n lbs per plot and the number of plants (X) are presented below. 9

10 Analyss of Covarance Rep I Rep II Rep III Rep IV S.No. Varetes X Y X Y X Y X Y 1 V.S N.C Ind 816A V.L V.L V.L V.L T Source: Gomez and Gomez (1984) The ANOVA of RBD for the gven data set wthout consderng the covarate s as gven: ANOVA Source Sum of Squares df Mean Square F Sg. Varety Replcaton Error Total ANCOVA was performed n the gven data set wth X as the covarate and the results are as follows: ANCOVA Source Sum of Squares df Mean Square F Sg. Varety Replcaton X Error Total

11 Analyss of Covarance ANCOVA Usng Two Covarates Wth two or more ndependent varable there s no change n the theory beyond the addton of etra term n X. Illustraton 3.: The method s llustrated for a one way classfcaton by the average daly gan of pg weghts gven n Snedecor and Cochran (198) and reproceduced n the followng table. Intal age (X1), ntal weght (X) and weght gan(y) of 4 pgs Treatment 1 Treatment X1 X Y X1 X Y Treatment3 Treatment 4 X1 X Y X1 X Y The ANOVA of CRD for the data set wthout consderng the covarate s as gven: Error Source.845 Sum of Squares 36 df.3 Mean Square F Sg. Total treatment

12 Analyss of Covarance ANCOVA was performed for the data set wth covarates and the results are as follows: Source Type III Sum of Squares df Mean Square F Sg. treatment covarate covarate Error Total The above table shows that there s a reducton n the error sum of square when two covarates were used. As one of the covarates, vz. ntal weght (X1) s not sgnfcant, the reducton n error SS s not very large. The adustment of treatment means accomplshes two mportant mprovements: 1. The treatment mean s adusted to a value that t would have had, had there been no dfferences n the values of covarate.. The epermental error s reduced to half and thus the precson for comparng treatment means s ncreased. Although blockng and covarance technque are both used to reduce epermental error, the dfferences between the two technques are such that they are usually not nterchangeable. The ANCOVA, for eample, can be used only when the covarate representng the heterogenety between epermental unts can be measured quanttatvely. However, that s not a necessary condton for blockng, In addton, because blockng s done before the start of eperment, t can be used only to cope wth the sources of varaton that are known or predctable. ANCOVA, on the other hand, can take care of unepected sources of varaton that occur durng the eperment. Thus ANCOVA s useful as a supplementary procedure to take care of sources of varaton that can not be accounted for by blockng. 4. Estmaton of Mssng data ANCOVA offers an alternatve to the mssng data formula technque. It s applcable to any number of mssng data. One covarate s assgned to each mssng observaton. The technque prescrbes an approprate set of values for each covarate. The only dfference, between the use of ANCOVA for error control and that for estmaton of mssng data, s the manner n whch the values of the covarate are assgned. When covarance analyss s used to control error and to adust treatment means, the covarate s measured along wth the Y varable for each epermental unt. But when covarance analyss s used to estmate mssng data, the covarate s not measured but s assgned, one each, to a mssng observaton. The case of only one mssng observaton s dscussed here. The rules for the applcaton of ANCOVA to a data set wth one mssng observaton are: 1

13 Analyss of Covarance For the mssng observaton, set Y. Assgn the values of covarate as X 1 for the epermental unt wth the mssng observaton, and X otherwse. Wth the complete set of data for the Y varable and the X varable as assgned above, perform the ANCOVA followng the standard procedures. Illustraton 4.1: The data from the RBD nvolvng s rates of seedng s gven where the observaton of the fourth treatment (1 kg seed/ha)n the second replcaton s mssng. Treatment Kg seed/ha Data of gran yeld (Y) and covarate (X) for one mssng data Rep I Rep II Rep III Rep IV Treatment total X Y X Y X Y X Y X Y Rep total Grand total Varous sums of squares for the Y varable followng the standard analyss of covarance procedure for a RBD are computed. The results are shown n followng table: Sources of varaton d.f. Sum of cross-products Y adusted for X XX XY YY d.f. SS MS F Replcaton Treatment Error Treatment +Error Treatment adusted Total

14 Analyss of Covarance The varous sum of squares for the X varable are: 1 1 Total SS rt Replcaton SS.15 t rt Treatment SS.83 r rt 4 4 Error SS Total SS Replcaton SS Treatment SS The varous sum of cross products are computed as: G C.F y (r)(t) (4)(6) Total SCP - (C.F.) B Replcaton SCP y 6543 C.F t 6 T Treatment SCP y 1456 C.F r 4 Error SCP Total SCP Replcaton SCP Treatment SCP ( )-( ) where B y s the replcaton total for the Y varable, of the replcaton n whch the mssng data occurred, and T y s the treatment total, for the Y varable, correspondng to the treatment wth the mssng data. The estmate of the mssng data s computed as: ErrorSCP Estmate of mssng data - b y. - ErrorSSX ( ) 565 kg/ha.65 Estmate of the mssng observaton of fourth treatment n replcaton II s 565 kg/ha. 5. Conclusons Analyss of covarance (ANCOVA) utlzes the nformaton on the concomtant varables whch are hghly correlated wth the response varable and are not affected by the treatments. It s a statstcal method for mprovng the precson of estmates of contrasts n treatment effects by reducng the epermental error. ANCOVA technque has also been advantageously used for analyzng the data whch have mssng observatons. References Das, M.N. and Gr, N.C. (1986). Desgn and analyss of eperments. New age nternatonal (P) lmted. 14

15 Analyss of Covarance Federer,W.T. and Schlottfeldt, C.S. (1954). The use of covarance to control gradents n eperments, Bometrcs, 1, 8-9. Forester, H.C. (1937). Desgn of agronomc eperments for plots dfferentated n fertlty by past treatments, Iowa Agr. Ep.Sta.Res.Bull. 6. Gomez, K.A. and Gomez, A.A. (1984) Statstcal procedures for agrcultural research, John wley and sons, Inc. Outhwate, A.D. and Rutherford, A. (1955). Covarance analyss an alternatve to stratfcaton n the control of gradents, Bometrcs, 11, Searle, S.R. (1971). Lnear models, John wley and sons, Inc. Snedecor, G.W. and Cochran, W.G.(198). Statstcal methods, Iowa State Unversty Press Ames, Iowa. 15