Mixed Models Random Coefficients

Transcription

1 NCSS Saisical Sofware Chaper Mixed Models Random Coefficiens Inroducion This specialized Mixed Models procedure analyzes random coefficien regression models. In his case, he regression coefficiens (he inerceps and slopes) are unique o each subjec. Since he subjecs are a random sample from a populaion of subjecs, his echnique is called random coefficiens. The echnique is also known as mulilevel modeling or hierarchical modeling. This procedure uses he sandard mixed model calculaion engine o perform all calculaions. However, he userinerface has been simplified o make specifying he random coefficiens analysis much easier. Random Coefficiens Models I is ofen imporan in a sudy o deermine he relaionship beween he response and ime. This is ofen done by including he measuremen ime as a covariae in he model, wih a corresponding slope, say β. I is plausible and likely ha he slope will vary wih subjec, so i migh be useful o model a separae inercep and slope for each subjec in he sudy. This is done by fiing he subjec variable as he inercep and he subjec*ime ineracion as he slope for each paien. These wo erms could reasonably be assumed o arise a random from a disribuion and, hus, would be specified as random effecs. This gives rise o wha is called a random coefficiens model. A random coefficiens model is one in which he subjec erm and a subjec*ime ineracion erm are boh included as random effecs in he model. This ype of model is differen from an ordinary random effecs model because when we fi a sraigh line, he esimaes of he slope and inercep are no independen. Thus, he subjec and subjec*ime effecs in he model are correlaed. The random effecs model mus be adaped o his siuaion o allow for correlaion among hese random effecs. This is done using he bivariae normal disribuion. The bivariae random effec becomes where subjec subjec, subjec* ime G =. subjec, subjec* ime subjec* ime subjeck ( subjec * ime) k ~ N(, G), The random coefficiens model is ofen used if he relaionship wih ime is of ineres or if he repeaed measuremens do no occur a fixed inervals. - NCSS, LLC. All Righs Reserved.

2 NCSS Saisical Sofware Mixed Models Random Coefficiens Types of Facors I is imporan o undersand beween-subjec facors and wihin-subjec facors. Beween-Subjec Facors Each subjec is assigned o only one caegory of a each beween-subjec facor. For example, if subjecs are randomly assigned o hree reamen groups (four subjecs per group), reamen is a beween-subjec facor. Wihin-Subjec Facors Wihin-subjec facors are hose in which he subjec s response is measured a several ime poins. Wihin-subjec facors are hose facors for which muliple levels of he facor are measured on he same subjec. If each subjec is measured a he low, medium, and high level of he reamen, reamen is a wihin-subjec facor. Deermining he Correc Model of he Variance-Covariance of Y Akaike Informaion Crierion (AIC) for Model Assessmen Akaike informaion crierion (AIC) is ool for assessing model fi (Akaike, 97, 97). The formula is AIC = L + p where L is he (ML or REML) log-likelihood and p depends on he ype of likelihood seleced. If he ML mehod is used, p is he oal number of parameers. If he REML mehod is used, p is he number of variance componen parameers. The formula is designed so ha a smaller AIC value indicaes a beer model. AIC penalizes models wih larger numbers of parameers. Tha is, if a model wih a much larger number of parameers produces only a sligh improvemen in likelihood, he values of AIC for he wo models will sugges ha he more parsimonious (limied) model is sill he beer model. As an example, suppose a researcher would like o deermine he appropriae variance-covariance srucure for a longiudinal model wih four equal ime poins. The researcher uses REML as he likelihood ype. The analysis is run five imes, each wih a differen covariance paern, and he AIC values are recorded as follows. Paern Number of Parameers - log-likelihood AIC Diagonal. 6. Compound Symmery AR() Toepliz Unsrucured The recommended variance-covariance srucure among hese five is he Toepliz paern, since i resuls in he smalles AIC value. - NCSS, LLC. All Righs Reserved.

3 NCSS Saisical Sofware Mixed Models Random Coefficiens Wha o Do When You Encouner a Variance Esimae ha is Equal o Zero I is possible ha a mixed models daa analysis resuls in a variance componen esimae ha is negaive or equal o zero. This is paricularly rue in he case of random coefficiens models. When his happens, he componen ha has a variance esimae equal o zero should be removed from he random facors model saemen (or, if possible, he repeaed paern should be simplified o diagonal ), and he analysis should be rerun. Fixed Effecs A fixed effec (or facor) is a variable for which levels in he sudy represen all levels of ineres, or a leas all levels ha are imporan for inference (e.g., reamen, dose, ec.). The fixed effecs in he model include hose facor for which means, sandard errors, and confidence inervals will be esimaed and ess of hypoheses will be performed. Oher variables for which he model is o be adjused (ha are no imporan for esimaion or hypohesis esing) may also be included in he model as fixed facors. Fixed facors may be discree variables or coninuous covariaes. The correc model for fixed effecs depends on he number of fixed facors, he quesions o be answered by he analysis, and he amoun of daa available for he analysis. When more han one fixed facor may influence he response, i is common o include hose facors in he model, along wih heir ineracions (wo-way, hree-way, ec.). Difficulies arise when here are no sufficien daa o model he higher-order ineracions. In his case, some ineracions mus be omied from he model. I is usually suggesed ha if you include an ineracion in he model, you should also include he main effecs (i.e. individual facors) involved in he ineracion even if he hypohesis es for he main effecs in no significan. Covariaes Covariaes are coninuous measuremens ha may no be of primary ineres in he sudy, bu poenially have an influence on he response. Two ypes of covariaes ypically arise in mixed models designs: subjec covariaes and wihin-subjec covariaes. Time as a Fixed Effecs Facor vs. Time as a Covariae Time is an essenial measuremen in many mixed model designs. In some analyses, ime may be considered a fixed facor, while in ohers i is covariae. In random coefficien models, ime is considered o be a covariae. Muliple Comparisons of Fixed Effec Levels If here is evidence ha a fixed facor of a mixed model has difference responses among is levels, i is usually of ineres o perform pos-hoc pair-wise comparisons of he leas-squares means o furher clarify hose differences. I is well-known ha p-value adjusmens need o be made when muliple ess are performed (see Hochberg and Tamhane, 987, or Hsu, 996, for general discussion and deails of he need for mulipliciy adjusmen). Such adjusmens are usually made o preserve he family-wise error rae (FWER), also called he experimen-wise error rae, of he group of ess. FWER is he probabiliy of incorrecly rejecing a leas one of he pair-wise ess. We refer you o he Mixed Models chaper for more deails on muliple comparisons. - NCSS, LLC. All Righs Reserved.

4 NCSS Saisical Sofware Mixed Models Random Coefficiens Speciying he Wihin-Subjecs Variance-Covariance Marix The R Marix The R marix is he variance-covariance marix for errors, ε. When he R marix is used o specify he variancecovariance srucure of y, he G sub marix is no used. The full R marix is made up of N symmeric R sub-marices, R R = R where R, R, R,, R N are all of he same srucure, bu, unlike he G sub marices, differ according o he number of repeaed measuremens on each subjec. When he R marix is specified in NCSS, i is assumed ha here is a fixed, known se of repeaed measuremen imes. (If he repeaed measuremen imes are random, specificaion of he G sub marix wih R = I should be used insead for specifying covariance srucure.) Thus, he differences in he dimensions of he R sub-marices occur only when some measuremens for a subjec are missing. As an example, suppose an R sub-marix is of he form R R N R = Sub, 5 where here are five ime poins a which each subjec is inended o be measured: hour, hours, 5 hours, hours, and hours. If he firs subjec has measuremens a all five ime poins, hen n = 5, and he sub-marix is idenical o R Sub above, and R = R Sub. Suppose he second subjec is measured a hour, 5 hours, and hours, bu misses he -hour and -hour measuremens. The R marix for his subjec is R =. 5 For his subjec, n =. Tha is, for he case when he ime poins are fixed, insead of having missing values in he R sub-marices, he marix is collapsed o accommodae he number of realized measuremens. Srucures of R There are many possible srucures for he sub-marices ha make up he R marix. The R Sub srucures ha can be specified in NCSS are shown below. - NCSS, LLC. All Righs Reserved.

5 NCSS Saisical Sofware Mixed Models Random Coefficiens -5 NCSS, LLC. All Righs Reserved. Diagonal Homogeneous Heerogeneous Correlaion Compound Symmery Homogeneous Heerogeneous Correlaion AR() Homogeneous Heerogeneous Correlaion Toepliz Homogeneous Heerogeneous Correlaion

6 NCSS Saisical Sofware Mixed Models Random Coefficiens -6 NCSS, LLC. All Righs Reserved. Toepliz() Homogeneous Heerogeneous Correlaion Noe: This is he same as Banded(). Toepliz() Homogeneous Heerogeneous Correlaion Toepliz() and Toepliz(5) Toepliz() and Toepliz(5) follow he same paern as Toepliz() and Toepliz(), bu wih he corresponding numbers of bands. Banded() Homogeneous Heerogeneous Correlaion Noe: This is he same as Toepliz().

7 NCSS Saisical Sofware Mixed Models Random Coefficiens Banded() Homogeneous Heerogeneous Correlaion Banded() and Banded (5) Banded() and Banded(5) follow he same paern as Banded() and Banded(), bu wih he corresponding numbers of bands. Unsrucured Homogeneous Heerogeneous Correlaion Pariioning he Variance-Covariance Srucure wih Groups In he case where i is expeced ha he variance-covariance parameers are differen across group levels of he daa, i may be useful o specify a differen se of R parameers for each level of a group variable. This produces a se of variance-covariance parameers ha is differen for each level of he chosen group variable, bu each se has he same srucure as he oher groups. Pariioning he R Marix Parameers Suppose he srucure of R in a sudy wih four ime poins is specified o be Toepliz: R =. -7 NCSS, LLC. All Righs Reserved.

8 NCSS Saisical Sofware Mixed Models Random Coefficiens If here are sixeen subjecs hen R R = The oal number of variance-covariance parameers is four:,,, and. R Suppose now ha here are wo groups of eigh subjecs, and i is believed ha he four variance parameers of he firs group are differen from he four variance parameers of he second group. We now have R R 6. and R,, R8 =, R 9,, R6 =. The oal number of variance-covariance parameers is now eigh. I is easy o see how quickly he number of variance-covariance parameers increases when R or G is pariioned by groups. Procedure Opions This secion describes he opions available in his procedure. Variables Response Variable This variable conains he numeric responses (measuremens) for each of he subjecs. There is one measuremen per subjec per ime poin. Hence, all responses are in a single column (variable) of he spreadshee. Subjec Variable This variable conains an idenificaion value for each subjec. Each subjec mus have a unique idenificaion number (or name). In a random coefficien design, several measuremens are made on each subjec. Regressions This opion specifies he polynomial order of he regression fi for each subjec. Usually, linear regression is used, bu quadraic and cubic regressions are also possible. -8 NCSS, LLC. All Righs Reserved.

9 NCSS Saisical Sofware Mixed Models Random Coefficiens Covariances If his box is checked, he G-marix (covariance marix) will include he covariance of each pair of variance componens (diagonal elemen of he G-marix). If he box is no checked, all off-diagonal elemens will be se o zero (he G-marix will be diagonal). This opion is commonly checked when fiing a random coefficien model. Beween and Wihin Fixed Facors This secion les you specify all fixed facors wheher hey are beween or wihin. Number Ener he number of facors (up o 6) ha you wan o use. This opion conrols how many facor variable enry boxes are displayed and used. Noe ha if you selec facor variables in he boxes below, and hen reduce his number so hose boxes are no longer visible, he hidden facors will no be used. Fixed Facor Variables Selec a fixed facor (caegorical or class) variable here. Capializaion is ignored when deermining unique ex values. A caegorical variable has only a few unique values (ex or numeric) which are used o idenify he caegories (groups) ino which he subjec falls. ² (Unequal Group Variance) One facor variable can have a differen variance in each group. Check his box o indicae ha his facor should have unequal variances. Oher facors will have equal variances. This panel is used o specify muliple comparisons or cusom conrass for facor variables. Comparison This opion specifies he se of muliple comparisons ha will be compued for his facor. Several predefined ses are available or you can specify up o wo of your own in he Cusom (-) opions. For ineracions, hese comparisons are run for each caegory of he second facor. Possible choices are: Firs versus Each The muliple comparisons are each caegory esed agains he firs caegory. This opion would be used when he firs caegory is he conrol (sandard) caegory. Noe: he firs is deermined alphabeically. nd versus Each The muliple comparisons are each caegory esed agains he second caegory. This opion would be used when he second caegory is he conrol (sandard) caegory. rd versus Each The muliple comparisons are each caegory esed agains he hird caegory. This opion would be used when he hird caegory is he conrol (sandard) caegory. Las versus Each The muliple comparisons are each caegory esed agains he las caegory. This opion would be used when he las caegory is he conrol (sandard) caegory. -9 NCSS, LLC. All Righs Reserved.

10 NCSS Saisical Sofware Mixed Models Random Coefficiens Baseline versus Each The muliple comparisons are each caegory esed agains he baseline caegory. This opion would be used when he baseline caegory is he conrol (sandard) caegory. The baseline caegory is enered o he righ. Ave versus Each The muliple comparisons are each caegory esed agains he average of he oher caegories. All Pairs The muliple comparisons are each caegory esed agains every oher caegory. Cusom The muliple comparisons are deermined by he coefficiens enered in he wo Cusom boxes o he righ. Baseline Ener he level of he corresponding Facor Variable o which comparisons will be made. The Baseline is used only when Comparison is se o Baseline vs Each. The value enered here mus be one of he levels of he Facor Variable. The enry is no case sensiive and values should no be enered wih quoes. Cusom - This opion specifies he weighs of a comparison. I is used when he Comparison is se o Cusom. There are no numerical resricions on hese coefficiens. They do no even have o sum o zero. However, his is recommended. If he coefficiens do sum o zero, he comparison is called a CONTRAST. The significance ess anicipae ha only one or wo of hese comparisons are run. If you run several, you should make some ype of Bonferroni adjusmen o your alpha value. Specifying he Coefficiens When you pu in your own conrass, you mus be careful ha you specify he appropriae number of coefficiens. For example, if he facor has four levels, four coefficiens mus be specified, separaed by blanks or commas. Exra coefficiens are ignored. If oo few coefficiens are specified, he missing coefficiens are assumed o be zero. These comparison coefficiens designae weighed averages of he level-means ha are o be saisically esed. The null hypohesis is ha he weighed average is zero. The alernaive hypohesis is ha he weighed average is nonzero. The coefficiens are specified here in his box. As an example, suppose you wan o compare he average of he firs wo levels wih he average of he las wo levels in a six-level facor. You would ener - -. As a second example, suppose you wan o compare he average of he firs wo levels wih he average of he las hree levels in a six-level facor. The cusom conras would be - -. Noe ha in each example, coefficiens were used ha sum o zero. Ones were no used in he second example because he resul would no sum o zero. Covariaes Number Ener he number of covariaes (up o six) ha you wan o use. This opion conrols how many covariae variable enry boxes are displayed and used. Since you have o specify a variable conaining a ime covariae, he minimum number of covariaes is one. Noe ha if you selec covariae variables in he boxes below, and hen reduce his number so hose boxes are no longer visible, he hidden covariaes will no be used. - NCSS, LLC. All Righs Reserved.

11 NCSS Saisical Sofware Mixed Models Random Coefficiens - NCSS, LLC. All Righs Reserved. Covariae Variables Specify each covariae variable here. The values in his variable mus be numeric and should be a leas ordinal. The firs covariae becomes he ime variable. I indicaes he ime values. We have found i useful o scale he ime values beween - and. Doing his will avoid serious rounding errors ha may occur oherwise. Compue Means a hese Values Specify one or more values a which means and comparisons are o be calculaed. A separae repor is calculaed for each unique se of covariae values. Mean Ener Mean o indicae ha he covariae mean should be used as he value of he covariae in he various repors. Wihin-Subjec Variance-Covariance Marix The repeaed componen is used o specify he R marix in he mixed model. Usually, he diagonal paern is used. Paern Specify he ype of R (error covariance) marix o be generaed. This represens he relaionship beween observaions from he same subjec. The R srucures ha can be specified in NCSS are shown below. The usual ype is he 'Diagonal' marix. The opions are: Diagonal Homogeneous Heerogeneous Compound Symmery Homogeneous Heerogeneous AR() Homogeneous Heerogeneous

12 NCSS Saisical Sofware Mixed Models Random Coefficiens - NCSS, LLC. All Righs Reserved. AR(Time Diff) Homogeneous Heerogeneous Toepliz (All) Homogeneous Heerogeneous Toepliz() Homogeneous Heerogeneous Noe: This is he same as Banded(). Toepliz() Homogeneous Heerogeneous Toepliz() and Toepliz(5) Toepliz() and Toepliz(5) follow he same paern as Toepliz() and Toepliz(), bu wih he corresponding numbers of bands.

13 NCSS Saisical Sofware Mixed Models Random Coefficiens Banded() Homogeneous Noe: This is he same as Toepliz(). Heerogeneous Banded() Homogeneous Heerogeneous Banded() and Banded (5) Banded() and Banded(5) follow he same paern as Banded() and Banded(), bu wih he corresponding numbers of bands. Unsrucured Homogeneous Heerogeneous Force Posiive Correlaions When checked, his opion forces all covariances (off-diagonal elemens of he R marix) o be non-negaive. When his opion is no checked, covariances can be negaive. Usually, negaive covariances are okay and should be allowed. However, some Repeaed paerns such as Compound Symmery assume ha covariances (correlaions) are posiive. Checking his box will help hose runs o converge. Model (Fixed Terms) Terms This opion specifies which erms (erms, powers, cross-producs, and ineracions) are included in he fixed porion of he mixed model. The opions are -Way All covariaes and facors are included in he model. No ineracion or power erms are included. Use his opion when you jus wan o use he variables you have specified. - NCSS, LLC. All Righs Reserved.

14 NCSS Saisical Sofware Mixed Models Random Coefficiens Up o -Way All individual variables, wo-way ineracions, cross-producs, and squared covariaes are included. For example, if you are analyzing four facors named A, B, C, and D, his opion would generae he model as: A + B + C + D + AB + AC + AD + BC + BD + CD. Up o -Way All individual variables, wo-way ineracions, hree-way ineracions, squared covariaes, cross-producs, and cubed covariaes are included in he model. For example, if you are analyzing wo covariaes called X and X and a facor named A, his opion would generae he model as: X + X + C + X*X + X*C + X*C + X*X + X*X + X*X*C + X*X*C + X*X*C + X*X*X + X*X*X + X*X*X + X*X*X. Up o -Way All individual variables, wo-way ineracions, hree-way ineracions, and four-way ineracions, along wih he squares, cubics, quarics, and cross-producs of covariaes and heir ineracions are included in he model. Ineracion Model All individual variables and heir ineracions are included. No powers of covariaes are included. This requires a daase in which all combinaions of facor variables are presen. Cusom The model specified in he Cusom box is used. Maximum Exponen of Covariaes This opion specifies he maximum exponen of each covariae in he erms of he model. This maximum is furher consrained by he seing of he Terms opion. Cusom Model Specify a cusom saisical model for fixed effecs here. Saisical hypohesis ess will be generaed for each erm in his model. Variables for which hypohesis ess are o be performed should be included in his model saemen. You may also include variables in his model ha are solely o be used for adjusmen and no imporan for inference or hypohesis esing. For caegorical facors, each erm represens a se of indicaor variables in he expanded design marix. The componens of his model come from he variables lised in he facor and covariae variables. If you wan o use hem, hey mus be lised here. Synax In he examples ha follow each synax descripion, A, B, C, and X represen variable names. Assume ha A, B, and C are facors, and X is a covariae.. Specify main effecs by specifying heir variable names, separaed by blanks or he '+' (plus) sign. A+B A B C A B X Main effecs for A and B only Main effecs for A, B, and C only Main effecs for A and B, plus he covariae effec of X. Specify ineracions and cross producs using an aserisk (*) beween variable names, such as Frui*Nus or A*B*C. When an ineracion beween a facor and a covariae is specified, a cross-produc is generaed for each value of he facor. For covariaes, higher order (e.g. squared, cubic) erms may be added by repeaing he covariae name. If X is a covariae, X*X represens he covariae squared, and X*X*X represens he covariae cubed, ec. Only covariaes can be repeaed. Facors canno be squared or cubed. Tha is, if A is a facor, you would no include A*A nor A*A*A in your model. - NCSS, LLC. All Righs Reserved.

15 NCSS Saisical Sofware Mixed Models Random Coefficiens A+B+A*B A+B+C+A*B+A*C+B*C+A*B*C A+B+C+A*X A+X+X*X A+B*B Main effecs for A and B plus he AB ineracion Full model for facors A, B, and C Main effecs for A, B, and C plus he ineracion of A wih he covariae X Main Effec for A plus X and he square of X No valid since B is caegorical and canno be squared. Use he ' ' (bar) symbol as a shorhand echnique for specifying large models quickly. A B = A+B+A*B A B C = A+B+C+A*B+A*C+B*C+A*B*C A B C X*X = A+B+A*B+C+X*X A B C X = A+B+A*B+C+X+C*X. You can use parenheses for muliplicaion. (A+B)*(C+X) = A*C+A*X+B*C+B*X (A+B) C = A+B+C+(A+B)*C = A+B+C+A*C+B*C 5. Use he '@' (a) symbol o limi he order of ineracion and cross-produc erms in he model. A B = A+B+C+A*B+A*C+B*C A B X X (@) = A+B+X+A*B+A*X+B*X+X*X Inercep Check his box o include he inercep in he model. Under mos circumsances, you should include he inercep erm in your model. Maximizaion Tab This ab conrols he Newon-Raphson, Fisher-Scoring, and Differenial Evoluion likelihood-maximizaion algorihms. Opions Likelihood Type Specify he ype of likelihood equaion o be solved. The opions are: MLE The 'Maximum Likelihood' soluion has become less popular. REML (recommended) The 'Resriced Maximum Likelihood' soluion is recommended. I is he defaul in oher sofware programs (such as SAS). Soluion Mehod Specify he mehod o be used o solve he likelihood equaions. The opions are: Newon-Raphson This is an implemenaion of he popular 'gradien search' procedure for maximizing he likelihood equaions. Whenever possible, we recommend ha you use his mehod. -5 NCSS, LLC. All Righs Reserved.

16 NCSS Saisical Sofware Mixed Models Random Coefficiens Fisher-Scoring This is an inermediae sep in he Newon-Raphson procedure. However, when he Newon-Raphson fails o converge, you may wan o sop wih his procedure. MIVQUE This non-ieraive mehod is used o provide saring values for he Newon-Raphson mehod. For large problems, you may wan o invesigae he model using his mehod since i is much faser. Differenial Evoluion This grid search echnique will ofen find a soluion when he oher mehods fail o converge. However, i is painfully slow--ofen requiring hours o converge--and so should only be used as a las resor. Read in from a Variable Use his opion when you wan o use a soluion from a previous run or from anoher source. The soluion is read in from he variable seleced in he 'Read Soluion From' variable. Read Soluion From (Variable) This opional variable conains he variance-covariance parameer values of a soluion ha has been found previously. The order of he parameer values is he same as on he parameer repors. This opion is useful when problem requires a grea deal of ime o solve. Once you have achieved a soluion, you can reuse i by enering his variable here and seing he 'Soluion Mehod' opion o 'Read in from a Variable'. Wrie Soluion To (Variable) Selec an empy variable ino which he soluion is auomaically sored. Noe ha any previous informaion in his variable will be desroyed. This opion is useful when problem requires a grea deal of ime o solve. Once you have achieved a soluion, you can hen reuse i by enering his variable in he 'Read Soluion From' variable box and seing he 'Soluion Mehod' opion o 'Read in from a Variable'. Newon-Raphson / Fisher-Scoring Opions Max Fisher Scoring Ieraions This is he maximum number of Fisher Scoring ieraions ha occur in he maximum likelihood finding process. When Soluion Mehod (Variables ab) is se o 'Newon-Raphson', up o his number of Fisher Scoring ieraions occur before beginning Newon-Raphson ieraions. Max Newon-Raphson Ieraions This is he maximum number of Newon-Raphson ieraions ha occur in he maximum likelihood finding process. When Soluion Mehod (Variables ab) is se o 'Newon-Raphson', Fisher-scoring ieraions occur before beginning Newon-Raphson ieraions. Lambda Each parameer's change is muliplied by his value a each ieraion. Usually, his value can be se o one. However, i may be necessary o se his value o.5 o implemen sep-halving: a process ha is necessary when he Newon-Raphson diverges. Noe: his parameer only used by he Fisher-Scoring and Newon-Raphson mehods. -6 NCSS, LLC. All Righs Reserved.

17 NCSS Saisical Sofware Mixed Models Random Coefficiens Convergence Crierion This procedure uses relaive Hessian convergence (or he Relaive Offse Orhogonaliy Convergence Crierion) as described by Baes and Was (98). Recommended: The defaul value, E-8, will be adequae for many problems. When he rouine fails o converge, ry increasing he value o E-6. Differenial Evoluion Opions Crossover Rae This value conrols he amoun of movemen of he differenial evoluion algorihm oward he curren bes. Larger values accelerae movemen oward he curren bes, bu reduce he chance of locaing he global maximum. Smaller values improve he chances of finding he global, raher han a local, soluion, bu increase he number of ieraions unil convergence. RANGE: Usually, a value beween.5 and. is used. RECOMMENDED:.9. Muaion Rae This value ses he muaion rae of he search algorihm. This is he probabiliy ha a parameer is se o a random value wihin he parameer space. I keeps he algorihm from salling on a local maximum. RANGE: Values beween and are allowed. RECOMMENDED:.9 for random coefficiens (complex) models or.5 for random effecs (simple) models. Minimum Relaive Change This parameer conrols he convergence of he likelihood maximizer. When he relaive change in he likelihoods from one generaion o he nex is less han his amoun, he algorihm concludes ha i has converged. The relaive change is L(g+) - L(g) / L(g) where L(g) is absolue value of he likelihood a generaion 'g'. Noe ha he algorihm also erminaes if he Maximum Generaions are reached or if he number of individuals ha are replaced in a generaion is zero. The value. (en zeros) seems o work well in pracice. Se his value o zero o ignore his convergence crierion. Soluions/Ieraion This is he number of rial poins (soluion ses) ha are used by he differenial evoluion algorihm during each ieraion. In he erminology of differenial evoluion, his is he populaion size. RECOMMENDED: A value beween 5 and 5 is recommended. More poins may dramaically increase he running ime. Fewer poins may no allow he algorihm o converge. Max Ieraions Specify he maximum number of differenial evoluion ieraions used by he differenial evoluion algorihm. A value beween and is usually adequae. For large daases, i.e., number of rows greaer han, you may wan o reduce his number. Oher Opions Zero (Algorihm Rounding) This cuoff value is used by he leas-squares algorihm o lessen he influence of rounding error. Values lower han his are rese o zero. If unexpeced resuls are obained, ry using a smaller value, such as E-. Noe ha E-5 is an abbreviaion for he number.. RECOMMENDED: E- or E-. RANGE: E- o E-. -7 NCSS, LLC. All Righs Reserved.

18 NCSS Saisical Sofware Mixed Models Random Coefficiens Variance Zero When an esimaed variance componen (diagonal elemen) is less han his value, he variance is assumed o be zero and all reporing is erminaed since he algorihm has no converged properly. To correc his problem, remove he corresponding erm from he Random Facors Model or simplify he Repeaed Variance Paern. Since he parameer is zero, why would you wan o keep i? RECOMMENDED: E-6 or E-8. RANGE: E- o E-. Correlaion Zero When an esimaed correlaion (off-diagonal elemen) is less han his value, he correlaion is assumed o be zero and all reporing is erminaed since he algorihm has no converged properly. To correc his problem, remove he corresponding erm from he Random Facors Model or simplify he Repeaed Variance Paern. Since he parameer is zero, why would you wan o keep i? RECOMMENDED: E-6 or E-8. RANGE: E- o E-. Max Reries Specify he maximum number of reries o occur. During he maximum likelihood search process, he search may lead o an impossible combinaion of variance-covariance parameers (as defined by a marix of variancecovariance parameers ha is no posiive definie). When such a combinaion arises, he search algorihm will begin again. Max Reries is he maximum number of imes he process will re-sar o avoid such combinaions. Repors Tab The following opions conrol which repors are displayed. Selec Repors Run Summary Repor Check his box o obain a summary of he likelihood ype, he model, he ieraions, he resuling likelihood/aic, and run ime. Variance Esimaes Repor Check his box o obain esimaes of random and repeaed componens of he model. Hypohesis Tess Repor Check his box o obain F-Tess for all erms in he Fixed (Means) Specificaion (see Variables ab). L-Marices Terms Repor Check his box o obain L marices for each erm in he model. Each L marix describes he linear combinaion of he beas ha is used o es he corresponding erm in he model. Cauion: Selecing his opion can generae a very large amoun of oupu, as he L marices can be very numerous and lenghy. -8 NCSS, LLC. All Righs Reserved.

19 NCSS Saisical Sofware Mixed Models Random Coefficiens Comparisons by Fixed Effecs Repor Check his box o obain planned comparison ess, comparing levels of he fixed effecs. Deails of he comparisons o be made are specified under he Comparisons and Covariaes abs. When more han one covariae value is specified under he Covariaes ab, he comparisons are grouped such ha for each fixed effec, comparisons for all covariae(s) values are displayed. Compare o Comparisons by Covariae Values. Comparisons by Covariae Values Repor Check his box o obain planned comparison ess, comparing levels of he fixed effecs. Deails of he comparisons o be made are specified under he Comparisons and Covariaes abs. When more han one covariae value is specified under he Covariaes ab, he comparisons are grouped such ha for each value of he covariae(s), a new se of comparisons is displayed. Compare o Comparisons by Fixed Effecs. L-Marices Comparisons Repor Check his box o obain L marices for each planned comparison. Each L marix describes he linear combinaion of he beas ha is used o es he corresponding comparison. Cauion: Selecing his opion can generae a very large amoun of oupu, as he L marices can be very numerous and lenghy. Means by Fixed Effecs Repor Check his box o obain means and confidence limis for each fixed effec level. When more han one covariae value is specified under he Covariaes ab, he means are grouped such ha for each fixed effec, means for all covariae(s) values are displayed. Compare o Means by Covariae Values. Means by Covariae Values Repor Check his box o obain means and confidence limis for each fixed effec level. When more han one covariae value is specified under he Covariaes ab, he means are grouped such ha for each value of he covariae(s), a new se of means is displayed. Compare o Means by Fixed Effecs. L-Marices LS Means Repor Check his box o obain L marices for each leas squares mean (of he fixed effecs). Each L marix describes he linear combinaion of he beas ha is used o generae he leas squares mean. Cauion: Selecing his opion can generae a very large amoun of oupu, as he L marices can be very numerous and lenghy. Fixed Effecs Soluion Repor Check his box o obain esimaes, P-values and confidence limis of he fixed effecs and covariaes (beas). Asympoic VC Marix Repor Check his box o obain he asympoic variance-covariance marix of he random (and repeaed) componens of he model.. Vi Marices (s Subjecs) Repor Check his box o display he Vi marices of he firs hree subjecs. Hessian Marix Repor Check his box o obain he Hessian marix. The Hessian marix is direcly associaed wih he variancecovariance marix of he random (and repeaed) componens of he model. -9 NCSS, LLC. All Righs Reserved.

20 NCSS Saisical Sofware Mixed Models Random Coefficiens Show Repor Definiions Indicae wheher o show he definiions a he end of repors. Alhough hese definiions are helpful a firs, hey may end o cluer he oupu. This opion les you omi hem. Selec Repors Alpha Alpha Specify he alpha value (significance level) used for F-ess, T-ess, and confidence inervals. Alpha is he probabiliy of rejecing he null hypohesis of equal means when i is acually rue. Usually, an alpha of.5 is used. Typical choices for alpha range from. o.. Repor Opions Sagger label and oupu of label lengh is When wriing a row of informaion o a repor, some variable names/labels may be oo long o fi in he space allocaed. If he name (or label) conains more characers han enered here, he res of he oupu for ha line is moved down o he nex line. Mos repors are designed o hold a label of up o 5 characers. Ener when you always wan each row's oupu o be prined on wo lines. Ener when you wan each row prined on only one line. Noe ha his may cause some columns o be mis-aligned. Repor Opions Decimal Places Precision Specifies wheher unformaed numbers are displayed as single (7-digi) or double (-digi) precision numbers in he oupu. All calculaions are performed in double precision regardless of he Precision seleced here. Single Unformaed numbers are displayed wih 7-digis. This is he defaul seing. All repors have been formaed for single precision. Double Unformaed numbers are displayed wih -digis. This opion is mos ofen used when he exremely accurae resuls are needed for furher calculaion. For example, double precision migh be used when you are going o use he Muliple Regression model in a ransformaion. Double Precision Forma Misalignmen Double precision numbers require more space han is available in he oupu columns, causing column alignmen problems. The double precision opion is for hose insances when accuracy is more imporan han forma alignmen. Effecs/Beas... DF Denominaor Specify he number of digis afer he decimal poin o display on he oupu of values of his ype. Noe ha his opion in no way influences he accuracy wih which he calculaions are done. Ener 'General' o display all digis available. The number of digis displayed by his opion is conrolled by wheher he PRECISION opion is SINGLE or DOUBLE. - NCSS, LLC. All Righs Reserved.

21 NCSS Saisical Sofware Mixed Models Random Coefficiens Plos Tab These opions specify he plos of group means and subjecs. Click he plo forma buon o change he plo seings. Selec Plos Means Plos Check his box o obain plos of means for each fixed effecs erm of he model. Deails of he appearance of he plos are specified under he Means Plos and Symbols abs. Subjec Plos Check his box o obain plos of he repeaed values for each subjec. Plos comparing main effecs for each subjec are also given. The repeaed values for each subjec are ordered according o he order he values appear in he daa se. Deails of he appearance of he plos are specified under he Subjec Plos and Symbols abs. Y-Axis Scaling Specify he mehod for calculaing he minimum and maximum along he verical axis. Separae means ha each plo is scaled independenly. Uniform means ha all plos use he overall minimum and maximum of he daa. This opion is ignored if a minimum or maximum is specified. These opions specify he subjec plos. - NCSS, LLC. All Righs Reserved.

22 NCSS Saisical Sofware Mixed Models Random Coefficiens Example Random Coefficiens Model wih a Beween Subjecs Facor and Two, Wihin-Subjecs Covariaes This example should acquain he reader wih he oupu for all oupu opions. I presens an analysis of a longiudinal design in which here is one beween-subjecs facor (Drug) a ime variable (Time), and a covariae (Cov). The response is a measure of pain (Pain). Two drugs (Kerlosin and Laposec) are compared o a placebo for heir effeciveness in reducing pain following a surgical eye procedure. A sandard pain measuremen for each paien is measured a minue inervals following surgery and adminisraion of he drug (or placebo). Six measuremens, wih he las a Time = hours, are made for each of he paiens (7 per group). A blood pressure measuremen of each individual a he ime of pain measuremen is measured as a covariae. The researchers wish o compare he drugs a he ime value of. and he Cov value of. Pain Daase Drug Paien Time Cov Pain Kerlosin Kerlosin Kerlosin Kerlosin 5 57 Kerlosin.5 8 Kerlosin 5 7 Kerlosin Kerlosin 5 68 Kerlosin Kerlosin 7 Kerlosin Kerlosin Placebo 9 7 Placebo Placebo 58 7 The following plo shows he relaionship among all variables excep he covariae. - NCSS, LLC. All Righs Reserved.

23 NCSS Saisical Sofware Mixed Models Random Coefficiens To run he analysis using he Mixed Models Random Coefficiens procedure, you can ener he values according o he insrucions below (beginning wih Sep ) or load he compleed emplae Example by clicking on Open Example Templae from he File menu of he Mixed Models Random Coefficiens window. Open he Pain daase. From he File menu of he NCSS Daa window, selec Open Example Daa. Click on he file Pain.NCSS. Click Open. Open he Mixed Models Random Coefficiens window. Using he Analysis menu or he Procedure Navigaor, find and selec he Mixed Models Random Coefficiens procedure. On he menus, selec File, hen New Templae. This will fill he procedure wih he defaul emplae. Specify he variables. On he Mixed Models Random Coefficiens window, selec he Variables ab. Double-click in he Response Variable ex box. This will bring up he variable selecion window. Selec Pain from he lis of variables and hen click Ok. Pain will appear in he Response Variable box. Double-click in he Subjec Variable ex box. This will bring up he variable selecion window. Selec Paien from he lis of variables and hen click Ok. Paien will appear in he Subjec Variable box. Se he Regressions opion o Linear. Check he Covariances box. Se he number of Fixed Facors o. Se he Variable o Drug. Se he Comparison o All Pairs. Se he number of Covariaes o. Se he firs covariae o Time. Se he Compue Means a hese Values o. Se he second covariae o Cov. Se he Compue Means a hese Values o. Se he Terms o Up o -Way. Se he Maximum Exponen of Covariaes o. Specify he repors. On he Mixed Models Random Coefficiens window, selec he Repors ab. Check all repor and plo checkboxes excep L Marices Comparisons and L Marices LS Means. 5 Run he procedure. From he Run menu, selec Run Procedure. Alernaively, jus click he green Run buon. - NCSS, LLC. All Righs Reserved.

24 NCSS Saisical Sofware Run Summary Secion Mixed Models Random Coefficiens Parameer Likelihood Type Fixed Model Random Model Repeaed Paern Value Resriced Maximum Likelihood COV+DRUG+TIME+COV*DRUG+COV*TIME+DRUG*TIME PATIENT+PATIENT*TIME Diagonal Number of Rows 6 Number of Subjecs Soluion Type Newon-Raphson Fisher Ieraions 5 of a possible 5 Newon Ieraions of a possible Max Reries Lambda Log Likelihood Log Likelihood AIC (Smaller Beer) Convergence Normal Run Time (Seconds). This secion provides a summary of he model and he ieraions oward he maximum log likelihood. Likelihood Type This value indicaes ha resriced maximum likelihood was used raher han maximum likelihood. Fixed Model The fixed model specified for his run. I includes fixed erms and covariaes. Random Model The random model as specified by he Regressions seing. Repeaed Model The paern seleced for he wihin-subjecs variance-covariance marix. Number of Rows The number of rows processed from he daabase. Number of Subjecs The number of unique subjecs from he daabase. Soluion Type The soluion ype is mehod used for finding he maximum (resriced) maximum likelihood soluion. Newon- Raphson is he recommended mehod. Fisher Ieraions Some Fisher-Scoring ieraions are used as par of he Newon-Raphson algorihm. The 5 of a possible 5 means five Fisher-Scoring ieraions were used, while five was he maximum ha were allowed (as specified on he Maximizaion ab). Newon Ieraions The of a possible means ha all fory Newon-Raphson ieraions were used. You may wan o increase his value and rerun so he algorihm has a chance o converge. - NCSS, LLC. All Righs Reserved.

25 NCSS Saisical Sofware Mixed Models Random Coefficiens Max Reries The maximum number of imes ha lambda was changed and new variance-covariance parameers found during an ieraion was en. If he values of he parameers resul in a negaive variance, lambda is divided by wo and new parameers are generaed. This process coninues unil a posiive variance occurs or unil Max Reries is reached. Lambda Lambda is a parameer used in he Newon-Raphson process o specify he amoun of change in parameer esimaes beween ieraions. One is generally an appropriae selecion. When convergence problems occur, rese his o.5. If he values of he parameers resul in a negaive variance, lambda is divided by wo and new parameers are generaed. This process coninues unil a posiive variance occurs or unil Max Reries is reached. Log Likelihood This is he log of he likelihood of he daa given he variance-covariance parameer esimaes. When a maximum is reached, he algorihm converges. - Log Likelihood This is minus imes he log of he likelihood. When a minimum is reached, he algorihm converges. AIC The Akaike Informaion Crierion is used for comparing covariance srucures in models. I gives a penaly for increasing he number of covariance parameers in he model. Convergence Normal convergence indicaes ha convergence was reached before he limi. Run Time (Seconds) The run ime is he amoun of ime used o solve he problem and generae he oupu. Random Componen Parameer Esimaes (G Marix) Random Componen Parameer Esimaes (G Marix) Parameer Esimaed Model Number Value Term 9.57 Paien.77 Paien*Time -.68 Paien, Paien*Time This secion gives he random componen esimaes. Parameer Number When he random componen model resuls in more han one parameer for he componen, he parameer number idenifies parameers wihin he componen. Esimaed Value The esimaed values of he hree variance componens. Model Term The name of he random erm being presened on his line. -5 NCSS, LLC. All Righs Reserved.

26 NCSS Saisical Sofware Mixed Models Random Coefficiens Repeaed Componen Parameer Esimaes (R Marix) Repeaed Componen Parameer Esimaes (R Marix) Parameer Esimaed Parameer Number Value Type 7.68 Diagonal (Variance) This secion gives he repeaed componen esimaes according o he Repeaed Variance Paern specificaions of he Variables ab. Parameer Number When he repeaed paern resuls in more han one parameer for he componen, he parameer number idenifies parameers wihin he componen. Esimaed Value The esimaed value 7.68 is he esimaed residual (error) variance. Parameer Type The parameer ype describes he srucure of he R marix. Term-by-Term Hypohesis Tes Resuls Term-by-Term Hypohesis Tes Resuls Model Num Denom Prob Term F-Value DF DF Level Cov Drug Time Cov*Drug Cov*Time Drug*Time This secion conains an F-es for each fixed erm in he model according o he mehods described by Kenward and Roger (997). Model Term This is he name of he erm in he model. F-Value The F-Value corresponds o he L marix used for esing his erm in he model. The F-Value is based on he F approximaion described in Kenward and Roger (997). Num DF This is he numeraor degrees of freedom for he corresponding erm. Denom DF This is he approximae denominaor degrees of freedom for his comparison as described in Kenward and Roger (997). Prob Level The Probabiliy Level (or P-value) gives he srengh of evidence (smaller Prob Level implies more evidence) ha a erm in he model has differences among is levels, or a slope differen from zero in he case of covariae. I is he probabiliy of obaining he corresponding F-Value (or greaer) if he null hypohesis of equal means (or no slope) is rue. -6 NCSS, LLC. All Righs Reserved.

27 NCSS Saisical Sofware Mixed Models Random Coefficiens Individual Comparison Hypohesis Tes Resuls Individual Comparison Hypohesis Tes Resuls Covariaes: Cov=., Time=. Comparison Raw Bonferroni Comparison/ Mean Num Denom Prob Prob Covariae(s) Difference F-Value DF DF Level Level Drug.8.8. Drug: Kerlosin - Laposec [] Drug: Kerlosin - Placebo [] Drug: Laposec - Placebo [] This secion shows he F-ess for comparisons of he levels of he fixed erms of he model according o he mehods described by Kenward and Roger (997). The individual comparisons are grouped ino subses of he fixed model erms. Comparison/Covariae(s) This is he comparison being made. The firs line is Drug. On his line, he levels of drug are compared when he covariae is equal o. The second line is Drug: Placebo Kerlosin. On his line, Kerlosin is compared o Placebo when he covariae is equal o. Comparison Mean Difference This is he difference in he leas squares means for each comparison. F-Value The F-Value corresponds o he L marix used for esing his comparison. The F-Value is based on he F approximaion described in Kenward and Roger (997). Num DF This is he numeraor degrees of freedom for his comparison. Denom DF This is he approximae denominaor degrees of freedom for his comparison as described in Kenward and Roger (997). Raw Prob Level The Raw Probabiliy Level (or Raw P-value) gives he srengh of evidence for a single comparison, unadjused for muliple esing. I is he single es probabiliy of obaining he corresponding difference if he null hypohesis of equal means is rue. Bonferroni Prob Level The Bonferroni Prob Level is adjused for muliple ess. The number of ess adjused for is enclosed in brackes following each Bonferroni Prob Level. For example,.766 [] signifies ha he probabiliy he means are equal, given he daa, is.766, afer adjusing for ess. -7 NCSS, LLC. All Righs Reserved.

28 NCSS Saisical Sofware Mixed Models Random Coefficiens Leas Squares (Adjused) Means Leas Squares (Adjused) Means Covariaes: Cov=., Time=. 95.% 95.% Sandard Lower Upper Error Conf. Limi Conf. Limi Name Mean of Mean for Mean for Mean DF Inercep Drug Kerlosin Laposec Placebo This secion gives he adjused means for he levels of each fixed facor when Cov = and Time =. Name This is he level of he fixed erm ha is esimaed on he line. Mean The mean is he esimaed leas squares (adjused or marginal) mean a he specified value of he covariae. Sandard Error of Mean This is he sandard error of he mean. 95.% Lower (Upper) Conf. Limi for Mean These limis give a 95% confidence inerval for he mean. DF The degrees of freedom used for he confidence limis are calculaed using he mehod of Kenward and Roger (997). -8 NCSS, LLC. All Righs Reserved.

29 NCSS Saisical Sofware Means Plos Mixed Models Random Coefficiens Means Plos These plos show he means broken up ino he caegories of he fixed effecs of he model. Some general rends ha can be seen are hose of pain decreasing wih ime and lower pain for he wo drugs afer wo hours. -9 NCSS, LLC. All Righs Reserved.

30 NCSS Saisical Sofware Subjec Plos Mixed Models Random Coefficiens Subjec Plos Each se of conneced dos of he Subjec plos show he response rajecory of paricular subjec. Soluion for Fixed Effecs Soluion for Fixed Effecs 95.% 95.% Effec Effec Lower Upper Effec Esimae Sandard Prob Conf. Limi Conf. Limi Effec Name (Bea) Error Level of Bea of Bea DF No. Inercep Cov (Drug="Kerlosin") (Drug="Laposec") (Drug="Placebo").. 5 Time Cov*(Drug="Kerlosin") Cov*(Drug="Laposec") Cov*(Drug="Placebo").. 9 Cov*Time (Drug="Kerlosin")*Time (Drug="Laposec")*Time (Drug="Placebo")*Time.. This secion shows he model esimaes for all he model erms (beas). Effec Name The Effec Name is he level of he fixed effec ha is examine on he line. - NCSS, LLC. All Righs Reserved.