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ED 20** 360 


TH 810 3flO 





Bl-utnbe??^^ Carol Joyce 
Kodels of CDTitiDUoas 
for the Analysis of H 

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: 

pread Hypothesis; Hone^uivalent 
dua 1 Scores 


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 , . 

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KATiorfAi r^STrruTE of E&UCATIOIV 

^ 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 


Michigan State UniV^rsityi*!^ material has been granted by 

April, 1981 


Paper presented at the 1981 Annual meeting of the' American Educational 
Research Association, Los Angeles, April, 1981. 





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* 

[ ■ 


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 


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 


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 




is the treatment effect )for group * 1,2 

error term for An individual. 
AC ij ^ 


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) 


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 



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) 

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. 


^ 11 


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 observati6ns -on YCt) will be*takep. 

■ MODEL II: Y^'.Ct) = gCt)X^j + h^(t)' ' , ' : ' 


-' ^ 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 


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. 


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 * 


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 




t = 

^ Persorr,:> 

m V 




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- 


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 




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 


Y CtK- 2 il e^p (V)dv].£^ 

-» /27I ' / ij 

C6) Logistic type growth-^ 


(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 ^ 


Vy(0 - g(t) ^A(t) + 

• 1 h 


/ ^ 


'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 ) . 


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 

• AHOVA of index of Response / A treatment effect under ANOVA of 
Xndfix of Response is given by equation {^)/^ 

\ ^2 -^1 ^2 


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' 


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 


DVA of Residualized Gain Scores, K = ^(t) 



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 


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 


^ . 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 


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) 


> * • ^ 

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* 


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 


^ . 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 



.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. 



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\ . - , . 

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