Analysis of Variance and Design of Experiments-II

Transcription

1 Analyss of Varance and Desgn of Experments-II MODULE VI LECTURE - 4 SPLIT-PLOT AND STRIP-PLOT DESIGNS Dr. Shalabh Department of Mathematcs & Statstcs Indan Insttute of Technology Kanpur

2 An example to motvate the use of splt plot desgn s as follows. Suppose we wsh to study two factors, say methods of cultvaton and varetes of wheat. Suppose, the frst factor has t levels and the second factor conssts of s varetes. The frst factor requres the use of a large complex equpment and consequently, relatvely large plots of land are needed. Ths wll requre hgher cost and puts a restrcton on the number of plots to be used. Because of the nature of the equpment used for plantng the wheat, the second factor can be accommodated n much smaller plots. To acheve ths, the large plots are splt nto smaller plots at the plantng stage. Ths means that snce the plots are close together, so less varablty s expected among the plots and n turn, more plots and less varablty among plots s expected, whch mples that the contrasts wll have more nformaton n terms of smaller standard errors. Ths suggests that the experment can be conducted wth two strata. The whole-plot stratum conssts of large plots n whch the plots can be assgned as per any standard desgn, e.g. CRD, RBD, or Latn square desgn. Next stratum s the splt-plot stratum whch conssts of the splt-plots. There are the smaller plots that are obtaned by splttng each of the large plots nto s parts. The treatments assgned to the large whole-plots are replcated r tmes, and treatments assgned to the splt-plots are replcated rt tmes. Now much more nformaton on the splt-plot factor s avalable because of the extra replcaton, and n turn, a smaller spltplot-to-splt-plot varance s expected. The nteracton contrasts between whole- and splt-plot treatment also fall nto the splt-plot stratum and beneft due to smaller varance.

3 3 There are two dstnct randomzatons n the splt plot desgns:. The frst randomzaton taes place n stratum 1, when the levels of the whole-plot treatment are randomly assgned to the whole plots.. The second randomzaton taes place n stratum where the levels n the splt-plot treatment are randomly assgned n the splt-plot. Many splt-plot plans can easly be modfed to become strp-plot experments. These have ther own advantages and dsadvantages. Examples Followng examples have been opted from Gesbrecht and Gumpertz (004). Consder a hypothetcal cae bang study n an ndustry. Assume that there are r recpes and c bang condtons are to be studed. A smple splt- plot experment wth the recpes as a whole-plot factor and the bang condton as a splt-plot factor can be set up f cae batters are made up usng recpes n a random order. Each batch of batter s then splt nto c portons. The porton sare then baed under the c condtons. A new random bang order s selected for each batter. Replcaton s provded by repeatng recpes. Another opton s to mae up enough batter to mae one cae from each of the r recpes. All caes based on r recpes are then baed at one tme n an oven at one of the c condtons. Now we have an experment wth bang condtons as a whole-plot factor and recpe as a splt-plot factor.

4 4 In case of a strp-plot desgn, the expermenter would mae up batches of each of the batters large enough. Then partton each batch nto c caes and then bae the caes n sets, wth one cae of each recpe n each set. In terms of row-column structure of desgn, the rows represent recpes and the columns represent bang condtons. The advantage here s that n the absence of replcaton, only r batches of batter need be mxed and the oven need only be set up c tmes. In another example of a splt-plot experment n ndustral qualty research s as follows: The object of the project s to develop a pacagng materal that would gve a better seal under the wde range of possble sealng process condtons used by potental customers. The pacage manufacturer dentfes a number of factors whch can affect the qualty of the seal. In the whole-plot part of the experment, the sample lots of eght dfferent pacagng materals are produced. These lots of materal are then sent to a customer s plant, where each of them s subdvded nto sx subplots. The subplots are used n sx dfferent sealng processes. Ths consttutes the splt-plot part of the study.

5 5 Statstcal analyss of splt-plot experments Splt-plot experment wth whole-plots n a CRD Statstcal Model The statstcal model for a splt-plot conssts of the two randomzaton steps n the splt-plot experment, one n each stratum. So t s a model wth two terms. We consder an experment wth whole-plots arranged n a CRD. Suppose W represents the whole-plot treatment and S represents the splt-plot treatment, then the lnear statstcal model s wrtten as y µ + w + ε(1) + s + ( w s) + ε(), j j j where ε (1) j ' s and ε '() j are dentcally and ndependently dstrbuted random errors, each wth mean 0 but dfferent varances σ and σ respectvely, 1,,..., t ; j 1,,..., s and 1,,..., s. Moreover, ε (1) ' s and are 1, mutually ndependent. j ε () ' s j The whole-plot stratum of the model contans the whole-plot treatment effects µ w and the whole-plot error terms ε (1). If we nclude the mean, ths part of the model s smlar to the case n one way model for CRD. The splt-plot stratum contans the splt-plot treatment effects s, the nteracton effect of w and s as ( w s) and the expermental error assocated wth ndvdual splt-plots. All the terms on the rght-hand sde of the model (except ) are assumed to have observatons measured as the devaton from the respectve mean. ε µ () j j

6 6 Analyss of varance The analyss of varance for the splt-plot experment n the CRD s le an extended analyss for the CRD. Ths can be consdered as two separate analyss of varance for each of two strata wth two separate error terms. Ths s llustrated n the followng table. ANOVA table for a splt-plot experment wth whole-plots arranged n a CRD Source Degrees of freedom Sum of squares Mean squares E( MS ) F - rato W t 1 t ( oo ooo) MSW rs y y σ + sσ + rsφ 1 w MSW MSE(1) Error(1) tr ( 1) t r ( jo oo) MSE (1) j s y y σ + sσ 1 S s 1 s ( oo ooo) MSS rt y y σ + rtφ s MSS MSE() W S ( t1)( s 1) t s ( o oo oo + ooo) MS( W S) r y y y y σ + rsφ w s Error() tr ( 1)( s 1) t r s ( yj yjo yo + yoo ) MSE () j σ MS( W S) MSE() Total (corrected) rts 1 t r s j ( y y ) j ooo

7 t s Note that the sum of squares due to W x S s ry rsty SSW SSS, where SSW s the whole-plot treatment sum of squares and SSS s the splt-plot treatment sum of squares. The error() sum of squares s obtaned by subtracton. The mean squares are obtaned by dvdng the sum of squares entres by respectve degrees of freedom. o ooo 7 The quanttes,, and represent quadratc forms as follows: φ w φ s φ w s j j t 1 s s 1 φw φs φ w s j w ( W S) ( t1)( s1) j. These quadratc forms wll be zero under the approprate null hypotheses. It s clear from the expected mean squares that - error (1) s used to test the hypothess of no whole-plot treatment effect and error and - error () s used to test the hypotheses of no nteracton or splt plot treatment effects. The test for nteracton s performed frst, otherwse other tests of hypothess are doubtful. Note that snce all the levels of each factor are tested n combnaton wth every level of the other factor, so the analyss n the whole-plot stratum and the splt-plot stratum are orthogonal. The estmates of nteractons between whole-and splt-unt factors are the contrasts that are orthogonal to both whole-plot and splt-plot treatment contrasts.

8 8 Standard errors of man-effect contrasts The standard errors n the splt-plot are more complex than n other desgns. We frst consder the contrasts among levels of the splt-plot treatment. We wrte the general form of a splt-plot contrast as c y Then and Snce E cy ce y [ ] oo oo cs Var cyoo Var c ε() j / rt j [ ] σ E MSE(), c σ / rt. oo. t follows that the estmated standard error (s.e) of a splt-plot treatment contrast s of the form c MSE() se. c yoo. rt For the case c 1 and c -1, the contrast s y y where '. It follows that MSE() se.( yoo yoo '). rt oo oo ' The confdence ntervals computed for the contrasts are based on t(r 1)(s 1) degrees of freedom.

9 9 The general form of a whole-plot treatment contrast taes the form E cy ce y cw. [ ] oo oo ε(1) j ε() j j j Var c yoo Var c + c r rs c ( σ + sσ1). rs c y oo. We have Snce s [ (1) ] E MSE σ + sσ, t follows that the estmated standard error of a whole-plot treatment contrast of the form 1 c MSE(1) se. cy oo. rs For the whole-plot treatment dfference, the estmate of standard error s MSE(1) se. ( yoo y ' oo ). rs