Detecting Lag-One Autocorrelation in Interrupted Time Series Experiments with Small Datasets

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1 Journal of Modern Applied Saisical Mehods Volume 8 Issue Aricle Deecing Lag-One Auocorrelaion in Inerruped Time Series Experimens wih Small Daases Clare Riviello Universiy of Texas a Ausin, clareriviello@gmail.com S. aasha Berevas The Universiy of Texas a Ausin, asha.berevas@mail.uexas.edu Follow his and addiional works a: hp://digialcommons.wayne.edu/jmasm Par of he Applied Saisics Commons, Social and Behavioral Sciences Commons, and he Saisical Theory Commons Recommended Ciaion Riviello, Clare and Berevas, S. aasha (009) "Deecing Lag-One Auocorrelaion in Inerruped Time Series Experimens wih Small Daases," Journal of Modern Applied Saisical Mehods: Vol. 8 : Iss., Aricle. DOI: 0.37/jmasm/ Available a: hp://digialcommons.wayne.edu/jmasm/vol8/iss/ This Regular Aricle is brough o you for free and open access by he Open Access Journals a DigialCommons@WayneSae. I has been acceped for inclusion in Journal of Modern Applied Saisical Mehods by an auhorized edior of DigialCommons@WayneSae.

2 Journal of Modern Applied Saisical Mehods Copyrigh 009 JMASM, Inc. ovember 009, Vol. 8, o., /09/$95.00 Deecing Lag-One Auocorrelaion in Inerruped Time Series Experimens wih Small Daases Clare Riviello S. aasha Berevas Universiy of Texas a Ausin The power and ype I error raes of eigh indices for lag-one auocorrelaion deecion were assessed for inerruped ime series experimens (ITSEs) wih small numbers of daa poins. Performance of Huiema and McKean s (000) z saisic was modified and compared wih he z, five informaion crieria and he Durbin-Wason saisic. Key words: Auocorrelaion, informaion crieria, ype I error, power. Inroducion Educaional research conains many examples of single-subjec designs (Huiema, McKean, & McKnigh, 999). Single-subjec designs, also known as inerruped ime series experimens (ITSEs), are ypically used o assess a reamen s effec on special populaions such as children wih auism or developmenal disabiliies (Tawney & Gas, 984). The design consiss of repeaed measures on an oucome for an individual during baseline and reamen condiions (A and B phases, respecively). Use of repeaed measures on an individual is designed such ha he subjec acs as his/her own conrol; his also helps rule ou he possible influence of poenial hreas o validiy including hisory, pracice, and mauraion effecs. Wih ITSE daa, he paern of scores over ime is compared for he A (baseline) versus he B (reamen) phases. The comparison Clare Riviello is an Engineering Scienis a Applied Research Laboraories. clareriviello@gmail.com. aasha Berevas is an Associae Professor and chair of Quaniaive Mehods in he Deparmen of Educaional Psychology. Her ineress are in mulilevel and mea-analyic modeling echniques. asha.berevas@mail.uexas.edu. can lead o inferences abou he effec of inroducing he reamen on he rend in he oucome scores. To describe he change in rend, he effec on he level of he scores and on he possible growh paern mus be assessed. umerical descripors of hese rends are no well esimaed given he number of repeaed measures is as small as is commonly found in educaional single-case design research (Busk & Marascuilo, 988; Huiema, 985). One of he sources of hese esimaion problems is relaed o he auocorrelaed srucure inheren in such designs (Huiema & McKean, 99; Whie, 96; Kendall, 954; Marrio & Pope, 954). Several es saisics and indices recommended for idenifying poenial auocorrelaion exis. Unforunaely hese saisics are ypically recommended only for daases wih a larger numbers of daa poins han are ypically encounered wih ITSEs. Huiema and McKean (000) inroduced a es saisic, z, o idenify lag-one auocorrelaion in small daases. The Type I error rae of he z was wihin nominal levels and sufficien power was associaed wih his saisic. The curren sudy inroduces a modificaion of he z designed o enhance furher is saisical power. This sudy assesses he Type I error rae and power of boh versions of he z. The performance of he wo z saisics is also compared wih ha of oher es saisics and indices ha are commonly used o idenify auocorrelaed residuals for models used o summarize rends for small ITSE daases. 469

3 LAG-OE AUTOCORRELATIO DETECTIO Auocorrelaion One of he fundamenal assumpions when using ordinary leas squares esimaion for muliple regression is ha errors are independen. When he independence assumpion does no hold, his leads o inaccurae ess of he parial regression coefficiens (Huiema and McKean, 000). For daa consising of repeaed measures on an individual, i is likely ha a model can explain some bu no all of he auocorrelaion. In addiion, when he residuals in a regression model are auocorrelaed he model mus accoun for his o ensure accurae and precise esimaion of parameers and sandard errors. Thus, i is imporan o be able o deec auocorrelaion so ha he proper mehods for esimaing he regression model can be employed. This sudy is designed o focus solely on firs-order (lag-one) auocorrelaion. For a muliple regression model including k predicors, x i, of oucome y a ime using: y = β β x β x... β x ε 0 k k. () If here is a lag-one auocorrelaion, ρ, beween residuals, hen ε, he residual a ime, is relaed o ε, he residual a ime as follows: ε ρ ε a () = where ρ is he auocorrelaion beween residuals separaed by one ime period. I is assumed ha ε and ε have he same variance, and a is assumed o follow a sandard normal disribuion. Esimaing Lag-One Auocorrelaion Several formulas are available for esimaing he lag-one correlaion coefficien, ρ, for a ime series consising of daa poins. The convenional esimaor is calculaed as follows: = = r ( Y Y )( Y = ( Y Y ) Y ) (3) where Y is he simple average of he values of y. Unforunaely, as evidenced by is common usage, he bias of r is ofen ignored. The expeced value of a lag- auocorrelaion coefficien for a series consising of daa poins was analyically derived by Marrio and Pope (954) o be: E( r ) = ρ ( 3ρ) O( ). (4) I should be noed ha he expression in Equaion only covers erms o order [hus, he erm: O ( ) ]; here are addiional erms for higher orders of he inverse of. For large samples, hese higher order erms end owards zero. However, he ITSEs of ineres in his sudy end o involve shor series where is reasonably small and hese higher order erms are hus no as negligible. Bias clearly exiss in he esimaion of he auocorrelaion. Huiema and McKean (99) lised four addiional, fairly common esimaors designed o reduce he bias observed in r. However, each of hese is also highly biased for small daa ses. Huiema and McKean (99) suggesed correcing for he bias in r by using r = r (5) which, for smaller rue values of ρ incorporaes some of he noed bias eviden in Equaion. The auhors showed ha heir modified esimaor, r, is unbiased when ρ equaled zero even for sample sizes as small as = 6. Addiionally, he auhors found ha he bias was lower for posiive values of ρ bu higher for some negaive values. When esimaing he auocorrelaion, i is also necessary o calculae he error variance 470

4 RIVIELLO & BERETVAS of he esimaor because he esimaor and is variance can be combined o produce a saisic ha can be used o saisically es for he auocorrelaion. Barle (946) derived his variance formula for he variance of r : σ ρ =. (6) r by ignoring erms of order or higher. This formula is commonly reduced o: ˆ σ r = (7) under he assumpion of he null hypohesis ha ρ = 0 (Huiema & McKean, 99). Huiema and McKean (99) assered ha he commonly used Barle variance approximaion is no saisfacory for small sample sizes. Their simulaion sudy indicaed ha ˆ σ r (see Equaion 7) consisenly overesimaed he empirical variance. This overesimaion performed quie badly for values of of less han weny wih Barle s variance approximaion exceeding he empirical variance by 83% and 40% for = 6 and = 0, respecively. The auhors explored he performance of Moran s variance esimae: * ( ) ˆ σ r = (8) ( ) which, under he null hypohesis (ρ = 0), gives precise error variance esimaes. Afer looking a he performance of an auocorrelaion es * saisic using ˆ σ r as he error variance * esimaor, he auhors concluded ha ˆ σ r was no adequae for small sample sizes. In ess for posiive values of auocorrelaion, is resuls were oo conservaive excep for large values of. They recommended using: ( ) ˆ σ r = { [ E( r )] } (9) ( ) where E ( r ) ρ ( 3ρ). (0) (Marrio & Pope, 954) as follows from Equaion 4. Use of Equaion 9 yielded values close o he empirical values of he variance of ρ esimaes even for s as small as = 6. Deecing Auocorrelaion The main purpose of esimaing he correlaion coefficien and calculaing is error variance is o deec he presence of auocorrelaion in a daa se. If daa are known o be auocorrelaed, hen mehods oher han ordinary leas squares should be used o more accuraely esimae he regression coefficiens and heir sandard errors. One of he more commonly used ess for auocorrelaion in residuals is he Durbin-Wason es saisic: ( ε ε = d = ε = ) () where ε represens he residual a ime (see Equaion ). The procedure for carrying ou his es can be confusing, hus he sequence of seps for esing he non-direcional H 0 : ρ = 0 is explained here. Firs boh d and (4 d) should be compared wih he upper bound d u. If boh exceed his bound, hen he null hypohesis is reained; oherwise, boh d and (4 d) are compared wih he lower bound, d l. If eiher falls below d l, hen he null hypohesis is rejeced and a non-zero lag one auocorrelaion is inferred. If neiher d nor (4 d) falls below d l, he es is inconclusive. The concep of an inconclusive region is unseling and, alhough compuer mehods ha provide exac p-values are now becoming available, mos are slow or expensive (Huiema & McKean, 000). I is in his conex, ha Huiema and McKean (000) proposed an alernaive es saisic ha is simple o compue, 47

5 LAG-OE AUTOCORRELATIO DETECTIO approximaely normally disribued and does no have an inconclusive region. The es saisic was evaluaed for is use o es residuals from ITSE models ha have one o four phases. Huiema and McKean s es saisic is defined as: z = P r ( ) () ( ) where P is he number of parameers in he imeseries regression model and is he oal number of observaions in he ime series. The auhors found ha P r, P = r (3) provided an unbiased esimae of ρ and ha he denominaor of he es saisic (in Equaion ) approximaes he empirical variance of r,p (see Equaion 8). The z es saisic is a generalizaion of he es proposed in Huiema and McKean s (99) earlier work was designed for a singlephase model of ITSE daa. However, he auhors failed o implemen all of he suggesions from heir previous sudy. Specifically, he auhors did no use he correced error variance, σ, ˆr (see Equaion 9) ha hey had recommended. * Insead hey used ˆ σ r (see Equaion 8). Because { [ E ( r )] }, use of σ should ˆr lead o a smaller variance and hus a larger value of he es saisic and increased power over ˆ * σ r. Informaion Crieria As an alernaive o using es saisics o deec auocorrelaed residuals, i is also possible o esimae a model using ordinary leas squares regression, esimae he same model assuming auocorrelaed residuals, and hen compare he fi of he wo models. A pos-hoc evaluaion ha compares he wo models fi can be hen be conduced using an informaion crierion such as Akaike s Informaion Crierion (AIC): AIC = Log( L) k (4) where L is he value of he likelihood funcion evaluaed for he parameer esimaes and k is he number of esimaed parameers in a given model. The model wih he smalles informaion crierion value is considered he bes fiing model. As an alernaive o he asympoically efficien bu inconsisen AIC, several more consisen model fi saisics have been proposed (Bozdogan, 987; Hannon & Quinn, 979; Hurvich &Tsai, 989; Schwarz, 978). These include Swarz s (978) Bayesian crierion: SBC = Log( L) Log( ) k (5) where is he number of observaions, Hannon and Quinn s (979) informaion crierion HQIC = Log( L) klog( Log( )) ; (6) and Bozdogan s (987) consisen AIC CAIC = Log( L) ( Log( ) ) k. (7) In addiion, Hurvich and Tsai (989) developed a correced AIC specifically for small sample sizes, which deals wih AIC s endency o overfi models: k AICC = Log( L). (8) k For each of hese informaion crieria formulaions, he smaller he value, he beer he model fi. The AIC and SBC are supplied by defaul by mos saisical sofware. For example, when using SAS s PROC AUTOREG (SAS Insiue Inc., 003) o esimae an auoregressive model, he procedure also provides resuls under he assumpion of no 47

6 RIVIELLO & BERETVAS auocorrelaion in residuals (i.e., using ordinary leas squares, OLS, esimaion). The procedure auomaically provides he AIC and SBC for he OLS and auoregressive models o enable a comparison of he fi of he wo models. To dae, no sudies have been conduced o compare use of informaion crieria for idenificaion of auocorrelaed residuals for ITSE daa wih small sample sizes. Research Quesion This sudy is designed o inroduce and evaluae use of he variance correcion suggesed by Huiema and McKean (99) in a modified version of heir es saisic, z. Specifically, he correced es saisic being suggesed and evaluaed is: P r z = (9) ( ) { [ E( r )] } ( ) Idenificaion of lag-one auocorrelaion (of residuals) was compared for he z es saisics, he Durbin-Wason es saisic and he AIC, SBC, HQIC, CAIC, and AICC fi indices for condiions when ρ = 0 and when ρ 0. This sudy focused only on wo-phase ITSE daa. This design lies a he roo of commonly used single-subjec designs and provides an imporan saring poin for his invesigaion. Mehodology SAS code was used o generae daa, esimae models, and summarize resuls (Fan, Felsovalyi, Keenan, & Sivo, 00). Several design condiions were manipulaed o assess heir effec on he performance of he es saisics and fi indices. These condiions included he magniude of he reamen s effec on he level and linear growh, he degree of auocorrelaion and he overall sample size of he ITSE daa. Model and Assumpions The following wo-phase, ITSE model (Huiema & McKean, 99) was used o generae he daa: y = β β β d β ( n )] d ε 0 3 [ A (0) where n A is he number of daa poins in he firs phase (baseline phase A), d is he dummy variable coded wih a zero for daa poins in he baseline phase and wih a one for daa poins in he second phase, and [ ( na )] d is he cenered ineracion beween ime and reamen. The ineracion erm is cenered in his way o provide a coefficien, β 3, ha represens he reamen s effec on he slope (i.e., he difference in he linear growh beween ha prediced using he reamen phase daa and ha prediced using he baseline daa). The coefficien, β, represens he change in he inercep from he baseline o he reamen phase (specifically, he difference in he value of y, when = n A, prediced using reamen versus baseline phase daa). Thus, he β and β 3 coefficiens describe he effec of he reamen on he level and growh in y, respecively. The residuals (ε ) were generaed such ha ε = ρ ε a wih ρ being he rue lag-one auocorrelaion beween residuals separaed by one ime uni, and a was randomly and independenly seleced from a sandard normal disribuion. Because he focus in ITSE designs is on he effec of he inervenion, he β and β 3 coefficiens (see Equaion 0) are of mos ineres. Thus, when generaing he daa in his simulaion sudy, values of β 0 (baseline daa s inercep) and of β (baseline daa s linear growh) were no manipulaed bu were fixed such ha β 0 was se o zero and β was se o a value of 0. in all scenarios. This modeled daa wih an inercep of zero (i.e., y = 0 a = 0) and a sligh baseline rend. Values of β and β 3, however, were varied o invesigae heir effec on deecing auocorrelaion. Each parameer ook on values 0, 0., and 0.4 in his fully crossed design. In order o evaluae how he model selecion crieria performed over he range of possible values for ρ, is value was varied o range from 0.8 up o 0.8 in incremens of

7 LAG-OE AUTOCORRELATIO DETECTIO Finally, he number of daa poins,, in he wo phases for each scenario were varied o be, 0, 30, 50, or 00 wih he poins being divided equally beween he wo phases so ha n A = n B wih values for each of: 6, 0, 5, 5, or 50. The simulaion sudy hus enailed a fully crossed design consising of hree values of β crossed wih hree values of β 3, crossed wih nine values of ρ, crossed wih five values of for a oal of 405 combinaions of condiions. One housand daases were generaed for each of hese 405 scenarios. Analyses Afer each daase was generaed, he regression model in Equaion 0 was esimaed using SAS s PROC AUTOREG. This procedure esimaes he model using boh ordinary leas squares (OLS) (assuming ρ = 0) and auoregressive mehods (assuming ρ 0). The procedure provides values for he AIC and SBC for boh models. HQIC, CAIC, and AICC were hen calculaed (see Equaions 6, 7 and 8, respecively) using he log likelihood obained from he AIC value. For each informaion crierion, a ally was kep describing when he auoregressive model s informaion crierion was lower han ha of he OLS model. PROC AUTOREG addiionally provides he p-value for he Durbin-Wason es saisic. As wih he AIC and SBC, a ally was kep of he proporion of rials for which his p-value led o a rejecion of he null hypohesis ha ρ = 0 (p <.05). The z were also calculaed (see Equaion and 9, respecively) using he residuals from he OLS regression. The E(r ) in he denominaor of Equaion 9 was obained by subsiuing r,p for he unknown ρ in Equaion 6. Again, a ally was kep describing he proporion of rials for which he null hypohesis of no auocorrelaion was rejeced (p <.05). For condiions in which ρ 0, he ally by scenario for each of he eigh model selecion crieria provided he power o idenify he correc model. For condiions in which ρ = 0, he ally provided he ype I error rae. Resuls Type I Error Raes Table conains Type I error raes by condiion and crierion. Sample size appeared o have he sronges effec on ype I error raes. The ype I error rae was no grealy affeced by he values of β and β 3. Overall, he Type I error raes for z were he bes of he eigh crieria invesigaed. The raes were somewha conservaive for he smalles sample size condiions ( = ) wih values of 0.0 and for z, respecively. The z mainained ype I error raes a he nominal level across sample size condiions (wih a maximum value of 0.05). The raes for z were slighly elevaed (wih values of 0.059) alhough he saisic performed much beer han did he Durbin-Wason (DW) and he five informaion crieria (ICs) invesigaed. The Type I error raes of he five ICs (SBC, AIC, HQIC, CAIC and AICC) and for he DW saisic were generally inflaed across he ρ = 0 condiions examined wih he indices performing from wors o bes as follows: AIC, HQIC, SBC, AICC, DW, CAIC. The Type I error rae inflaion, however, decreased wih increasing sample size. Only in he scenarios wih he larges sample size ( = 00), were he CAIC and SBC s Type I error raes accepable if somewha conservaive. The CAIC s Type I error rae performance was also accepable (0.056) for condiions in which was 50. Power Table displays he power of he eigh crieria used o evaluae he presence of lag-one auocorrelaed residuals. In he presence of ype I error inflaion, he power of a crierion becomes somewha moo. Thus, i should be kep in mind ha he Type I error inflaion noed for he DW and he five ICs. As would be expeced, for all crieria he power was found o increase for larger sample sizes. Similarly, i was expeced and found ha as he magniude of ρ increased so did he power o deec he ρ of he ICs and es saisics. The z exhibied consisenly beer power levels han he SBC and DW for all posiive values of ρ. 474

8 RIVIELLO & BERETVAS Table : Type I Error Raes (False Deecion) of Lag-One Auocorrelaion by Crierion and Condiion Condiion Informaion Crierion Tes Saisics (p <.05) Parm* True Value SBC AIC HQIC CAIC AICC DW z ρ β β z *Parm. = Parameer Boh of hese es saisics had beer power han all oher indices when ρ 0.6. These resuls also suppored he heoreical conclusion menioned earlier ha z will always have more power han z. For negaive values of ρ, he ICs and DW saisic exhibied beer power han he z. And he ICs ha performed wors in erms of ype I error conrol performed bes in erms of power. The power was also unaffeced by he rue values of β and β 3. The power of z was quie low (0.089 and 0.33, respecively) for he = condiions bu he power levels become more comparable o hose of he oher crieria for larger. However, only z had exhibied accepable ype I error raes. Conclusion The resuls of he simulaion sudy suppor use of he z for idenificaion of lag-one auocorrelaion in small ITSE daases. Boh saisics mainain nominal raes of ype I error conrol alhough z s raes seemed slighly inflaed in he larger sample size condiions. Concomian wih he ype I error conrol were found somewha lower empirical power levels. However he ype I error inflaion of he five ICs and he DW prohibi heir use for deecion of auocorrelaion in he condiions examined here and especially wih ITSE daa consising of a small number of daa poins. A ype I error in he curren conex means ha an auoregressive model will be esimaed unnecessarily. While his should have minimal effec on he esimaion of he β coefficiens in Equaion 0, i will likely affec he sandard error (SE) esimaes used o es he 475

9 LAG-OE AUTOCORRELATIO DETECTIO Table : Power o Deec Lag-One Auocorrelaion by Crierion and Condiion Condiion Informaion Crierion Tes Saisic (p <.05) Parm* True Value SBC AIC HQIC CAIC AICC DW z z ρ β β *Parm. = Parameer saisical significance of hese coefficiens. The curren evaluaion could be exended furher by comparing esimaion of he OLS versus auoregressive model coefficiens and heir SEs for differen levels of auocorrelaion. This could help inform he curren sudy s ype I error and power resuls by indicaing he magniude of he effec of incorrec modeling of auocorrelaion. For example, if only a small degree of accuracy and precision is gained by modeling he auocorrelaion for a cerain value of ρ, hen i may no maer ha he model selecion crieria has low power a ha value. Similarly, if an insubsanial degree of accuracy and precision resuls from false idenificaion of auocorrelaion, hen he ype I error inflaion noed in his sudy migh be of minimal imporance. As wih mos simulaion sudies, resuls are limied by he condiions invesigaed: he values of he β and β 3 coefficiens (see Equaion 0) do no seem o have much effec on idenificaion of ρ, bu i should be invesigaed wheher his is really he case or wheher i jus appears ha way from he limied range of values of β and β 3 ha were chosen in his sudy. One of he main limiaions of his sudy is ha i considers only he wo-phase ITSE daa and only invesigaed firs-order auocorrelaion. Anoher imporan limiaion is ha performance was evaluaed only for a small subse of possible daa rends. All condiions included a sligh posiive linear rend in 476

10 RIVIELLO & BERETVAS baseline. In addiion, he only model misspecificaion assessed was wheher he residuals were auocorrelaed. Fuure research should invesigae use of he z for furher misspecified models including when a rue non-linear rend is ignored o mimic asympoic rends resuling from ceiling or floor effecs. The performance of hese saisics could also be assessed for ITSEs wih more han wo phases (e.g., for ABAB designs) as invesigaed by Huiema and McKean (000). This sudy also only invesigaed condiions in which he reamen and baseline phases had equal numbers of daa poins (n B = n A ). Single-subjec sudies frequenly enail unequal sample sizes per phase and he effec of uneven n should be invesigaed. Based on he resuls of his sudy, researchers ineresed in modeling linear growh in ITSE daa wih a small number of daa poins should use z or z o es for he presence of lag-one auocorrelaion. Researchers are cauioned agains using he Durbin-Wason es saisic and he various informaion crieria evaluaed here including he AIC, HQIC, SBC, AICC, DW and he CAIC for wo-phase ITSEs wih s less han 50. References Barle, M. S. (946). On he heoreical specificaion and sampling properies of auocorrelaed ime-series. Journal of he Royal Saisical Sociey, 8, 7-4. Bozdogan, H. (987). Model selecion and Akaike s informaion crierion (AIC): The general heory and is analyical exensions. Psychomerika, 5, Busk, P. L., & Marascuilo, L. A. (988). Auocorrelaion in single-subjec research: A counerargumen o he myh of no auocorrelaion. Behavioral Assessmen, 0, 9-4. Fan, X., Felsovalyi, A., Keenan, S. C., & Sivo, S. (00). SAS for Mone Carlo sudies: A guide for quaniaive researchers. Cary, C: SAS Insiue, Inc. Hannon, E. J., & Quinn, B. G. (979). The deerminaion of he order of an auoregression. Journal of he Royal Saisical Sociey, Series B, 4, Huiema, B. E. (985). Auocorrelaion in applied behavior analysis: A myh. Behavioral Assessmen, 7, 07-8 Huiema, B. E., & McKean, J. W. (99). Auocorrelaion esimaion and inference wih small samples. Psychological Bullein, 0, Huiema, B. E., & McKean, J. W. (000). A simple and powerful es for auocorrelaed errors in OLS inervenion models. Psychological Repors, 87, 3-0. Huiema, B. E., Mckean, J. W., & McKnigh S. (999). Auocorrelaion effecs on leas-squares inervenion analysis of shor ime series. Educaional and Psychological Measuremen, 59, Hurvich, C. M., & Tsai, C. L. (989). Regression and ime series model selecion in small samples. Biomerika, 76, Kendall, M. G. (954). oe on bias in he esimaion of auocorrelaion. Biomerika, 4, Marrio, F. H. C., & Pope, J. A. (954). Bias in he esimaion of auocorrelaions. Biomerika, 4, SAS Insiue, Inc. (003). SAS (Version 9.) [Compuer Sofware]. Cary, C: SAS Insiue, Inc. Schwarz, G. (978). Esimaing he dimension of a model. Annals of Saisics, 6, Tawney, L., & Gas, D. (984). Singlesubjec research in special educaion. Columbus, OR: Merrill. Whie, J. S. (96). Asympoic expansions for he mean and variance of he serial correlaion coefficien. Biomerika, 48,