# Full text of "ERIC ED204360: Models of Continuous Growth and Their Implications for the Analysis of Nonequivalent Control Group Designs."

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ED 20** 360 DOCOHEHT RESOHE TH 810 3flO AOTHOR TITLE POB DATE NOTE EDRS PPICE DESCPIPTORS TDfNTIFIEPS Bl-utnbe??^^ Carol Joyce Kodels of CDTitiDUoas for the Analysis of H Designs. Apr B1 ' ' . 27p,: Paper presented American Educational Anaeles# CA, April 13 MP01/PC02 Plus Postag ♦Analysis of Co^varian ♦Mathematical Models: ♦Qaasiexperiiaental De ♦Change Scores: Pan S Control Groups: ♦Pesi : Porter, Andrew C. Gtbwth and Their Implications oneqaivalent Control Gtoup at the Annual Meeting of the Pefeearch Association (6 5th, Los -17^ 1981) , ce: ♦Analysis of Variance: Pretests Posttests: sign pread Hypothesis; Hone^uivalent dua 1 Scores ABSTRACT Analysis strategies are discussed for the nonequivalent control group design when three aodels of continuous n^turgil arowth are known. For Model I type natural growth it was shown tha* the fan spread hypothesis always holds# and Analysis of Cbvarlance (AHCOVA), Analysis of Variajice <AHOVAJ of fiesidualized Gain Scores^f and AHOVA of Sttindardized Change Scores all are potentially correct analysis strategies. For Model II and Model fll tvpe natural growth it was shown that the fan spread hypothesis l$olds atid the AHCOVA, A.HOVA of Pesiduall2ed> Gain Scores, and AHOVi of ^ Standardized^Ch^nge scores are>*potentiaLly correct analysis strateaies oTily when Model II and Kodel III type natur^al growl;h reduce to Model 1 type natural growth. Further, it was shown that aiveh any natural growth situation, there is a value of K for which ftNOVfr,of Index of Response is a piJtentially correct analysis strateay. In order that an index of response works, the exact form of natural arowth must be known. Most growth models, and in particular most linear growth models, do not' conform to the fan spread hypothesis nor are the usual analysis strategies correct fp^ these models, (Authpr/PLl , . «*«*««***«««*«««4t*1£******************4>***********jjc ******* *********i4c**** ♦ Reproductions supplied by POPS are the best that can be made * * from the original document, ♦ **********}|c*************^**«« *««4c*««««**«« *««««*«** ************ RIC o O rg CD DCPARTKCNT OF CIH>CATKM« KATiorfAi r^STrruTE of E&UCATIOIV SOUCATKMJAL HSSOUfiCSS WFOflMAliOM ^ This docufiMnL t«an rvproducM » /fom th* person <h of^anirdtion ongtnvung it . ' J -Mwiot chan^ hwe b«ftn mKte rc> m^ovir Models of Continuous Growth aiM their Implications for the Analysis of Nonequivalent Control Group Designs J _ ' Caro^ Joyce Blumberg ' . ' Andrew C- Porter ■ PERMISSION TO REPRODUCE THIS Michigan State UniV^rsityi*!^ material has been granted by April, 1981 TO THE EDUCAHONALRESOURCES INFORMATION CENTER rERiG* ' Paper presented at the 1981 Annual meeting of the' American Educational Research Association, Los Angeles, April, 1981. ERJC 2 t \ The authors wish to thank Dts^^^James.Stapleton^ Robert\Floden , William Schmidt* an<r Joseph Gardiner for their comments on earlier versions of^th^s p.aper* We also wish to thank Ms* Linila Stiles for her excellent typing bf the manuscript* [ ■ ABSTRACT The paper ^iscassee analysis strategies fpr the noiiequivalent , control group design when models ot natural growth are known* Several models of continuous growt'h are shown to satisfy the fan spread hypothesis* For tliese growth models it is also shqwn that Analysis of Covariance^ ANOVA of Kesidualized Gain Scores, and ANOVA of Standardized Change Scores yield correctly defined treatment" effects* On the one hand, these models are not restricted to linear growthj as past literature woul^ suggest* Onr the other hand^ these" are the only- models of growth shown to fit the fan ^spread hypothesis r and for which the thre^ analysis strategies yiel^^ correctly defined effects. Consequently, mist growth modelsj and in particular most linear growth models , do not conform to tlie fan spread hypotl^sis nor are the usual analysis strategies correct for these models/ 4 ■ In many educatioijai settings, a true-^perimental ciesign is not possible when a researcher wants to evaluate the effects of c3if£erent treatments. 'Thus, quasi-eXperimental ciesigns are employed. C>ne of the more commonly used quasi-exjS^imental^iKiesigtis is the nonequivalent control group design (Campbell S Stanley, 1966V pp* A7-50, 55-57, 6iT 6A; Campbell, 1969)- For this/ design, pre'^ and post-obsijervations on ^ the sam^ measure ^re avail*able for "feubjects in two non-randomly ^ created compar^ison groups^ ^ The tvo grou^Js ma^ be elt(ier a treatment L group and a control ^roup' or two different trea.tmeijft groups* While the desigti^a-llows for several pretest aTi<J po^ttesuf dbservations on each individual , in this pipex consiiderat^n' is ^^^rictjeti to designs with a single pretest and 'posVtes t ; ' i There has been much <Jiscus5lori in the- literature of -the analysis t ' ' ' * * * / strategies that are appropriate for use in\ connection with nonr equivalent',Qontrol^ group designs* The basio problem is to identify *-* ^ ^ / . / analysis strategies which will T)r6vide'!^^ased estimates of the treatment effects* This problem has come to be known as the prj5blem of measuring change. Unless some assumptions are made, there is no knowably correct analysis strategy for use with any particular application of a nonequivalent control group design. A short and excellent discussion of this is given by Lord (1967), One of the possible approaches is to make some assumptions about the^data that^ , would result under natural growth* It is oiAy within the context of ^a particular set of assumptions that th^ appropriateness and non- appropriateness of particular analysis 3t^ategies can be discussed. One assumption frequently made In the literature is that data conform to the fan spread hypothesis (see €i-g-» Campbell, 1571; Kenny '&^Cohen, 1980)- The fan spread hypothesis states that the ratio of' the difference of group*means to the standard deviation common to. the ♦ ■ * * two populations ^frqm which the two groups are drawn) Is constant over t±me< Without^ loss of ' generality , throughout this paper it will be assumed that the pretest, X, Ig given at time t ^ 0* The pogttest, Y(t), is given At some tiine,j^t > 0, Symbolically, then, the fan ^spVead hypothesis can be expressed ^ ' ^ f . 2 = 1 ^ (1) * where u ^..population mean for groups i on the pretest; i = 1,2 y„(t) = population mean for group i on tfie postte^t at time t;^ 1 = 1,2 Oy^ = . standaigpd^ devidflon common to both populations on the pretest^ and OyCt) = standard deviation coTimon to both populations on the post^ , test at tfeie t. Assuming' the fan spreStii— hypothesis for natural growth', the appropriateness of some analysis strategies can be discussed. There Is, however, a major problem with discussing analysis strategies for r^onetjulvalent control §roup designs only In terms of the fan spread hypothesis: As' Wtll>e shown later In this paper. It is only In rare cases^ that 'data conform to the fan spread hypothesis. Hence, the ■'fajjj ^spread hypothes-is'should not be the focus of attention when* dlsqusslng the t>robleia of measuring change* ■ ' ,*'Ft>116wln^ the lead oi Bryk and Welsberg (1977) and In contrast to. £he fan spread hypothesis, this paper considers assumptions ^(mcjdel'i) about cpntinuous natural growth. Even though making fx* ^ H as^umptlODg about contrtrruous growth Is further removed than *.the fan spread hypothesis from commonly employed analysis strategle^s, It Is closer (to the usual ways of thinking about growth a?Wso should facilitate judgement about the reasonableness of the assumptions for actual data. Another reason for considering continuous growth 'models is that they reflect growth as ,a dynamic process changing over f^e in a yhtinuous manner. By employing assumptions such as the fan spread hypothesis, much of the literature on the problem of measuring change , has i*gnored this dynamic nature of growth^ As will be shown later in this paper, an analysis strategy may be appropriate at certain time points for th€ posttest but *not at other time points. Thus, depen- dencies on time mu^t also, be considered, . ^ . The approach to investigating the problen: of measuring change taken in this paper is to first posit\ a particular model of natural growth. The models of natural growth posited will be representativa of gj^owth models that have been suggested for varl6us, types of academic and/or physical - growth. Given a particular mdel of natural' growthV' a description can then be given in terms of parameters estimable .from a-particular nonequivalent control group design. Against these parameters, the approprlatene&6 of analysis strategies cafn then be ' investigated. * - r ' * ^ , - : '■ ^■■f Before proceeding further, it is convenient; £o inf,rodijee;efie various analysis strategies to be considered iji this, pape&^^nd to ( ^ ■'•■..->. specify how treatment effects are defined under /isacn. Throughout 1 - ^ this paperj it is assumed that treatment 6f Teats are additive That is,^ that a treatment causes an 'increase or decrease of the same amount for everyone from their scores under natural growth* ^ The null and alternate hypotheses for each analysis strajtegy can / then be stated / ' ^ 0 analysis strategy ^ ^ 1 analysis strategy / — ' where a . . defines the treatment effect undet> a analysis strategy ' ^ , particular analysis model. The primary question investigated in this paper is which a^ialysis strategies yield null treatment ef fefltft'under each natural growth model considered. An, analysis strategy is to be considered a potentially correct analysis strategy if under natural growth it gives null treatment effects/ The reason for the word potential is' that there remain questions of distributional assumptions and precision, , " Analysis of Covariance / The linear model for anCC^A can be stated as (Winer, 1971) where X^^ and Y^^t)^4re the pr^^^pst and posttest scores ^ * , respectively ''for individual j in group i;'i * 1^2 Uyr and v^(t) are the population grand means ^r X and X V(t) respectively ^(t)'X slope of the regression line of Y(t) 0n' X for each group ^ ' ' h£ d^^enotes analysis of covariance ./ 8 and is the treatment effect )for group * 1,2 error term for An individual. AC ij ^ (2) In this paper, all designs considered are two group designs. For two group des|.gns, -(a^^)^' = - ^^aC^2 ^^^^^^^ ^ ^^AC^ ° ^ ' For later convenience, let a^^ symbolize the quantity' 2*^^^)]^ ■ since (a^^)^= -(a^^)^ [u^(t) - u-(t) - 6Y(,).,-(i-,^ - _ by (2) Hence Analysts of Variance of Index of Response ANOVA of Index of Response (Cox, -19^) Is actually a set of analysis strategies. ANOVS of Index of Response will fijrst be (liscussed in general anj^ then some specific cases will be discOssed in further detail. Let K be some r^I constant. An index of response is then defined by Z At) , where Z (t) « ^..(t) - K*X . An ANOVA ij 1] . ij 1] ^ ^ [ of Index of Response is nothing more than- an ANOVA petformed on the Z^j(t)'s, The linear model for ANOVa of Index of Resjjonse Is then as for ANOVA ERIC 6 where ' U-(t) " population grand, mean for Z ' IR denotes Index of Response ^^Ir\^ " u^Ct) - w^(t) - K(Ujj - Ujj) ,is the treatment' effect for group i; i « 1,2 an^i ^^IR-^tj error term for an indJ.vj^|Aal * Let a- = 2(a >■ - By a derivation analogous to that for XK IK L , ^ a,- it can be shown that AC ' * IR 1 2 Assuming that i , notice that a__ * 0 if and only if K * . Here, and throughout the rest of this paper, \~\ . \ . the cases where ij„ = v„ will be ignored. So, a proper index of ^1 ^2 ' (t) - Uy (t) 1 2 response, namely Z .(t) = ^ ' ~ ' - always exists ^1 ^2 unless = u„ * The problem i^, of course, that the value of ^1 ^^2 (t) - (t) . ■ • 1 * 2 2 is unknown .in most situations^ It should also be . (t) - y (t) 12'. noted that is a function of tipie, hence the indices ^1 ^2 - ) of response will be differ^t for different values of t. 10 ' Some specific values of K which are of interest are ^ , (1) K = I, When K - 1, the analysis strategy^ is more commonly known as ANOVA of Gain Scores, In this case, equation (A) reduces to^ ap = p (t) - p Ct) - Cu* - u ) , ' (5) ^ (2) K = ' J ' , When K = the analysis strat^egy is. sometimes called ANOVA of Standardized Change ^Scores (Kenny S Cohen, 1980), In thls'ease, equation '(A) irelluces to 1 2 1 2 . , . ' (3) K = \^^yy^ • When K = \(^-^^y^ the analysis ^strategy is sometimes called ANOVA of Reslduallzed Gain Scores. In ^t his case, .equation (A) i^educes to Sometimes ^Y(t) -X estimated when setting a valu^ far K,- In ♦ the. literature on the problem of measuring change, the distinction between knowing versus estimating 6Y(t)''X an.ANOVA'Ofe Index of Response Is nbt made clear. As will be seen, the distinction is not Important for this paper which foetus on null treatment effects defined by e'aqh of several analysis strategies. Nevertheless, there are important differences between these two procedures when* the topics of hypothesis testing or conf idence Interval estimation .are con- sldered, (Olejnlk & Porter^ 1981) , , , Some Particular Natural Growth Models Three models of Individual natural growth will be cpnsldered. I ^ 11 ERIC MODEL 1: Y^j(t) = g(t)*X^^ + h(t) '-where g(t) and 'h(t) are continuous functions in t, gCO) = .1 and h(OV«=" 0\ Further X >0 and g(t) > 0 for 0 < t < t where 13 ' * o - t^ is a time past which.no observati6ns -on YCt) will be*takep. ■ MODEL II: Y^'.Ct) = gCt)X^j + h^(t)' ' , ' : ' and -' ^ Y2jCt) = gCt)k2j + h2(t) •-■ where g(t), h^'(t) , and h2(t) are- continuous functions in, t, ^(0) = 1; h^(0) = h2(0) = 0, X^^.> 0, and gCt) > 0 for 0 < t < t* ; i = 1,2 .T . . MODEL 111:. Y^jCt) = g^Ct)X^j + h^(^v\ and ^ ' v. where Z^(^^/^2^^^^ h^(t) y a^ h^(t) are continuous functions in^ty . g^CO) ^ g2C0) =-1, h-^(O) = h2(0) = 0>.\j >-0> and g^Cr) i 0 ' . ■ for 0 < t < t ; i ^ 1,2 , f o ♦ -Model 111 is the functional representation of those growth ^ models where there is a perfect correlation between the pretest and the posttest within .each group- Models 1 and 11 are special inIt>ortant sub-models .of Model 111, In Mod^i 1, if gCt) = 0 for some time t, then the correlation is undefined for that point in *t±me- If g(iy) is negative then the correlation is -1. ^ Negative correlationsf, and in particular cotr^T^^^s of -1^ between the j>retest and posttest are not l^^kely in actual situations, H^ence, the restriction ■g(t) > 0 was m^e. The restriction X^^ > 0 wa^ made 12 solely for converrience. 'When g(t) < O.dr when 0 for one or more individuals the theory becomes much-more complicated, "It was : ' • ■ . • . ^ • ■ decided that these complications were beyond the scope of this paper* The^ reason for the restrictions g(0) = 1 and h(0) = 0 in Mod^l I Is * ■ consistency. By definition 7, , (0) = X,, / But under'Model I, 7..(0) , Ij ■ . ' ^ g(0-)X,. + h(0) . Hence, X =,g(0)X + h(0) for each individual. / . i ■ ■ ■ ^ Consequently/ g(0) 1 an<^ h(0) - 0. The restrictions for MoMels II ♦ and III w^re plaQ^ed ^there for analogous- reasons . ' - ' Model I Results Examples of Model I Type Growth (1) Parallel growth/ Parallel growth is defined by g(t) i 1 That is, "Y^^(t) = + h(t), where h(t) is any conti'huous function\ tee Figure^l for a pictorial representation ojf parallel growth. ,Y„(t), Person ? Person 2 Person 1 Figure 1; An Example of Parallel Growth (2) Differential linear growth* Differential linear" growth is defined by g(t) =*'m*t.+ 1 and h(t) i 0, where m is some nonz^tsro * 13 cons'tant. Thaf is, Y,,(t) ^ (m-t + 1)X,, . See Figures 2 and 0 for plc^torial representations of differential linear growth. The solid portions of each curve in these figures aTid in all the remaining figures Indicate that part of the natural groVth curves under con-^ sideration in \:his paper . , ^ * Figure 2: Differential Linear Growth under >Sodel 1 when m > 6 Person 1 14 11 if t = ^ Persorr,:> m V \ \ Pel Figure 3; Differential Linear Growth ^ under Model I when tn < 0 (3) Exponential type growth. Exponential . type growth Is defined by g(t) = a-b^ + c and h(t) ,i 0, where a, ^nd c .^e constants with b > 0, The reason for the Restriction b > 0 Is , t t ' to make b real "valued, Foi: .values of t) < 0, b takes on cot|ple:f values: Hence, ^^j^t) = (a*b^ + ^-^'^Ij' '^^^ Figures 4 and 5 - for pit;torlcal representations of exponential type growth- 15 t ^ 4. Y = ' c-X . P ers on _3 I = c'X Persoii'2 Y = c-^ Person 1 t = logj^(^)^" Figure 5: Exponential Type Growth when a < 0, b > 1» c > 1» and .a + c =^ r ERIC i7 ■14 The inclusion of monotone,decreasing functions . (sep e/g.,. Figures 3 and 5) as representatives of natural growth may aeem strange^ One topic'-which has received ,attentiort from learning theorists ts ^ y forgetting' curves Forgetting curves ate » of course, ,d^c/reasing ^.^ ' \ * / - functions and hence the decision was made to include mo^iotbne decreasing functions as representatives of natural growth ' ' * ^ \ ' ' Some other particular examples of Model I type natural" growth are: '(4) Logarithmic type growth = [log^ Cb^t +a)]^x:^ ^, / where a and b are constants with* a > 0 ^d a ?t \ i ■ C5) Cumulative nonnal (Nonnal'' Ogive) type growth t Y CtK- 2 il e^p (V)dv].£^ -» /27I ' / ij C6) Logistic type growth-^ .t (1 + c)^d 1 + c*d where c and d are constants with^c > 0 and d > 1 Fan Spread Hypothesis Under Model I Recall that Model I type growth js defined in general by ^ '^ij^V " gU)^X^j + h(t) ^ ' ■ ^ First, notice that ^ and Vy(0 - g(t) ^A(t) + • 1 h C8) / ^ / 15 'Secohd, notice that ol it) = g^(t)ol (t) -and ' ' ' . (9) oht) = g^(t)0^(t) . 2 ^ 2 where .^d are'the variances on the pretest for group 1 and group 2, respectively 2 2 - ■ ■ an4 Oy (t) and (t) are the variances on the posttest for group 1 and group 2, respectively* Third, notice that while Tf.,(t) and t can take any one of an infinity / , . . of possl±>le relationships, some of which have just bejsn illustrated, for any time t, X and Y have ^ lljiear relationship* Hence, P^^^j ^ ^ ■ . for each of the two populations. ^ For the remainder of tKe Model I results section it will be. assumed that there are equal vsfriances across the two populations on the pretest and cotjsequently , by equation (9), on, the pQsttest. Hence , * - [g(t) y„ + h(t)] - [g(t) u + h(t)] ' h;y equations (8) and (9) , 1 ■ ^ -2 g(t)*(ji^ - My ) "^1 ^2 ERIC * ■ / .19 Vyit) - IJL(t) - y 1 - 2 ^ _ti 22^ (lb) ^^-^nce the fan spread hypothesis holds when ^^.Ct) = g(t) X'., + h(t) ■ under natural growth. Equation (10) can be rewritten in c'h^ form tj (t) This form is mote convenient when, discussing potentially correct analysis strategies . I ' . - Analysis of covariance . A treatment effect und^r ANCX)VA is given by e^^^tion (3) V= - u^U)) - fi^^^^j.j^ (u - u ) . .Oy(t) und6r Model I type growth because So^ . / ■ H^nce, by comparing equations (11) and 042) > ANCOVA is alwajrs a po.tetltially correcif analysis strategy/undeiT Model 1 type natural growth. • AHOVA of index of Response / A treatment effect under ANOVA of Xndfix of Response is given by equation {^)/^ \ ^2 -^1 ^2 20 Hence for ANOVA of Indent of Rei&poii^a^ to. be' a poteyitially correct / ' ' ; ' * ' ■ analysis strategy it /Is necessary^tp^^.have \inder .n'atural growth' J But, by equatloi/ (11), for ModVjL' I type natural growth 0 i (u„(t.). vCt» - — Cm„ - Uy ) . (11> ^1- - ^2,. ■ "'X ^1 ^2 * Hence, -by /oraparing equations (13) and (n) , ANOVA of Index of Respons&^ls a potentially ccrtrrect analysis strategy if and only if K - For ANbVA of Standardized Change Scores, K For DVA of Residualized Gain Scores, K = ^(t) iCYCt) Butj p . . = 1 under Model I type growth. Hence, K = for A1?0VA of Residualized Gain Scores. Consequently, ANOVA of Standard^ized ' Change Scores and ANOVA of Residualized Gain Scores are both potentially correct analysis strategies ttx all Model I type natural growth situations with the additional assumption that a*, = cr , V - ^ ^1 ^2 For ANOVA. of Gain Scores, K = 1. ^ Hence,- ANOVA of Gain Scores is a, potentially correct analysis strategy only for any < time > t, when — - 1 . Recall that under parallel growth, g(t) = 1^ AI*so,/iy X ^ - ' / ) / equation (9), c^^Ct) = ,g(t) for all .Model I type natural growth , ^ V , ' Hence, under parallel growth, a (t) = q * So, ANOVA of Gain Scores **** ' / . is always a potentially correct analysis strategy under parallel growths 18 . ' / Model II Results ^ Recall that Model II type natural growth was defined by ,Y^^(t) - g(t) X^^ + h^(t) ' rr and / Y2j(t) = g(t) X^^ + h2(t) " , . whete g(t), h-(t) and h-(t) are continuous functions in t> g(0) = 1> h,(0) h ('O) = 0, X,, >. 0, and g(t) > 0 f«x^O < t '< t . Hence> 1 2 o anA • (14) / iJ^ (t) = git) -u^- -h^it) i further, ^ . 1 / ■ and ' (15) 0 Ct) = gCt)--<j - . . ■ ' " As was done In the Model I results section, assume *f or the remainder .of the Model II results section that o„ = 'o = , Hence, , , ^1 ^^2 ^ by the set of equations (15) » o„ (t) = "o (t) = ,o (t) . Further, 1 ^2 ^ ' ^XY(t) " ^ each group. So, Si,Y(t)'X sajae for both groups t ari4 this common value is — r * Thus, by the same derivations as given In the Model I results section, ANCOVA, ANOVA of .Reslduallzed- Gain £^cores, and AKOVA of Standardized Change Scores are all potentially correct analysis strategies and the fan spread^iiypothesls holds when 19 Pliigging equations (14) and (15) into equation (16) yields 1 * 2 X ^ 1 2 . This equation simplifies, to « 0 = h^(t) - h2(t) , ' Hence^ uader Mod^l II type. natural growth A^ICOVA,* ANOVA' of" Residuajrf-Ned Gain Scores, and ANOVA o£ Standardized Change , Scores are all potentially correct analysis strategies and the fan spread hypothesis holds if and only if h (t) = h^(t) . Consequently, the three analysis strategies are potentially cotrett and the fan spread hypothesis holds under Model l^^pe natural growth only^when it reduces to Model 1 type natural growth* ^ ^ Model III Results ^ Recall that Model 111 type natural growth was defined by ' V^j(t> 8i<t) X^^ + h^(t) and > * • ^ where g^(t), g2(t), hj^(t) and h2(t) are contiYiuous functions in ^> gl(0) = ■g2(0) ^ 1, h^(0) » h^iO) - 0, X > 0, and ^^(t) 0 for ' 0 -^^t < t *; i 1,2 • All of tMe specij^^c classes of natural growth * curves fisted ui^der Model 1 (e*g,, linear, exponential^ and logarithmic^, are also possible under Model 111, but under Model III each group*s ^ growth may be defined by a dif^ferent set^ of values itor the constants* 20 Even more generally, each groof/s natural growth maybe from^a. " , r different class of growth curves. .Fpr example., grouf> 1 may follow logarithmic type growth while group 2 follpws exp^Dnential type growth • . . I As was done under the Model I results section* assume for the remainder of this section that a ^ * Then ^1 ^2 ' '\ and ' ' / (17) As seen earlier when "PjfY(t) ^ ^' . • oM IV (t) y (t) - — (U^ - u^ ) ^1, ■ ^2 ■'x • ^ * -^2 is the expression for a treatment effect for ANCOVA* ANOVA of Standardized Cairi Stores, and ANOVA of Standardized Change, Scores • - Notice that the presence of o^(t} in. the expressidn above implies^that; a cotfimon variance is assumed for the two populations / which is, in general, not the ca^e for Model HI type natural growth (see the set of equations (17))^^ Nevetvth^less ^ any one of the /three. anal7sis strategies can- be used with data from Model III and so yield estimated treatment effects* The question can then be raised as to whether. these estimated effects have an expected value of zero under^Model 'III type natural growth. This question is*beyond the scope of the present paper. Thus* we stop with the conclusion that Model III type growth is inconsistent with the parametric definition for each of the- -three atrategies. The fan spread hypothesis also makes the assumption of 24 ^ . 21. ■ ; , , ^ ■ ' - * , L ' ' * ■ s'^ common variances for' both groups at* any time t* Hence, /the fail spread"^ hypothesis is also inconsistent with Model ill type growth, ^ Differential linear Growth It seems that several authors have either believed or else have by omission led their readers to believe that all (or at leasts mo^t) differential linear growth is equivalent to the fan spread hypothesis (e.g., Bryk & Weisberg, 1977; Ketmy, 1975; Olejnik, 1977). The concepts of differential linear growth and of the 'fan spread hypothesis are^ however, distinct conceptfs,- in the Model I results section it / was shown that many forms of natur^al growth other tfian differential linear growth conform to the fan spread hypothesis* licence, differen- tial linear growth that conforms to the fan spread hypothesis is a subset of all natural growth that conforpis to the fan spread hypothesis* Further, as will be shQwnbeldw, differential linear *■ growth conforms to the fan spread hypothesis only in rare cases. Differential linear growth under Model m is defined by Y^j(t) = (b^-t + 1)-X^j . That is, differential linear growth undei^ Model IH i% Model III type natural. growth with , , * ■ g^(t) = b^-t / ^ and h,(t) = h^(t) = 0 . In the previous section it was shown thaJ: Mojiel III type natural' ■ / ' , ' ■ ■ growth is inconsistent with the fan spread hypothesis unless it reduces to M del I type g/rowth. Differential linear growth under Model III ERIC 25 .22' * : reduces to Model' I type growths if and'only if ~ h^. Hence, differential linear growth undei: Model Itl conforms to the fan ^spread hypothesis only in those rare cases wherfe b'^ = This same argument also shows that'ANO^VA, ANOVA of Residuallzed Gain Scored,, and ANOVA of Standardized Change Scores are potentially' correct analysis stxateglefi under Model' III type* differential linear growth ^ if and only if - h^* {■ - ■ ' f ' ^Summary - - . , ' In this paper three models of continuous natural ^growth were considered. For Model I type natural growth It was shown that the ^■ fan spread hypothesis always holds* ^Additionally, foj:*Mc)del I type growth, it was shown that ANCOVA, ANOVA of Residualized Gain Scores, and ANOVA "of .Standardized* Change Scores all are potentially correct < analysis^strategies. For Model II andModel III type natural growth it was shown that the fan spread hypothesis holds and that ANCOVA, ANOVA of Residualized Cain Scotes, and ANOVA of Standardised piange Scores are potentially correct analyjsis strategies only wh-en Model II and Mod^l I^II type natural growth reduce to Model I type natural growth* Further, it was shown that given any*natural grovth situation, there is a value of K foi" which ANOVA* of Index of Response 15 a potentially correct analysis strategy. But the efficacy of this strategy is ' more apparent than real* In order that an index of -response works, the exact form of natural growth must be laiown* ^This is, of course, rarely the case for ei^iri-cal research. Be Referehces Bryk, A* S*, & Weisberg, H. I» Use of the,nonequivaient control group desi^ when subjects are giiQwing* Psychological Bulletin ^ 1977> 84, 950-962. * /" . ^ Capipbellf D* T» Refotms as experiments*. Ameylc^an Psychologist > 1969, Campbell^ D* T* Temporal changes in'treatment-ef feet correlations; A , quasi-experimental model for institutional record^ and longitudinal studies* *In V* GlassCEd,) Proceedings of the 1^70 invitational conference on testing problems: the t^romise and perils of educational information systems . New York; Educational Testing Service,' 1971. Campbell, D>T*, & Stanley J» C*. Experimental and quasi-experimental designs for research . Chicago: Ra^d McHally, 1966* Cox> D» R. Planning of e^cperiiflents ^ New York: John Wiiey ^ Sons> Inc*> 1958* ■ Kenny> D* A quasi-experimental approach to assessing treatment effects in the non^equivalent ^control group design* * Psychological 6ulletin , 1975> 82, 345-362* ^ ^ , ' . Kenny, D* A*, & Cohen, S* H* A reexamination of selection and growth processes. in the nonequivalent control group design. In iC. Schuesaler(Ed*) , Sociological Methodology 1980 * Jossey-Bass, 1980* Lord,fF*%MT A paradox* in the interpretation of *group comparisons* Psychological Bulletin , 1967, 68_, 304-305* Olejnik, S* F* Data analysis strategies^ for quasi-experimental studies where differential group and individual growth rates are assumed * Unpublished Ph*D* dissertation, East Lansing: Mlchig^an ^^t;ate ^. "University, 1977* , ' . . ' Olcjnik,. S*F. , ^ Porter* A*C* Bias . and mean square errors of estimators as criteria ^or evaluating competing analysis strategies* in quasi- experiments* Journal pf Educational Statistics , 1981, 6> 33-53* Winer> B* J* StatistjLcal principles in fexperlmental design (2nd ed*). ♦ New York: McGraw-Hill, l971\ . - , . ■ • ' ■ 27