Supplemental Materials
Latent Growth Modeling With Domain-Specific Outcomes Comprised of Mixed Response
Types in Intervention Studies
By T. A. Whittaker et al., 2014, Journal of Consulting and Clinical Psychology
http://dx.doi.org/10.1037/a0036664.supp
Supplemental Material A
Equations for a Latent Growth Model With a Single Outcome Measure
In a latent growth model with a single outcome repeatedly measured, an observed
variable y is measured for individual i on T different occasions such that yi  [ yi1 , yi 2 ,, yiT ]
where t = 1, 2, … , T. The measurement portion of the model, which is comparable to a firstorder common factor model, with an incorporated mean structure is given as
y i  ν  Λη i  ε i ,
(SA1)
where y i is a T  1 vector containing raw scores of the observed variable y at each time t for
individual i; ν is a T  1 vector containing intercepts for each observed variable y at each time t;
Λ is a T  2 matrix containing intercept and slope factor loadings (  ) at each time t; ηi is a 2 
1 vector containing factor scores for the intercept factor ( αi ) and the slope factor ( βi ); and ε i is
a T  1 vector containing normal, random errors for the observed variable y at each time t. The
observed variable intercepts in ν are generally set equal to zero for model identification
purposes. Also, model identification problems will be encountered if data for participants are not
collected during a minimum of three measurement waves.
The intercept factor loadings in Λ are all fixed to values of 1.0. While different temporal
reference points may be modeled, a common reference point in latent growth models is the initial
amount of the manifest variable, which is accomplished by fixing the first factor loading in Λ on
the slope factor to a value of zero. While various growth patterns may be modeled in a latent
growth model, linear growth across time is the simplest trajectory to model. Linear growth is
modeled by fixing the slope factor loadings in Λ equal to values representing the difference
between the measurement occasions. Latent means of the intercept and slope factors are
estimated by regressing the factors on a constant value of 1.0 (typically represented in diagrams
as a triangle in which a value of 1.0 is enclosed) and are given by
Supplemental Materials
Latent Growth Modeling With Domain-Specific Outcomes Comprised of Mixed Response
Types in Intervention Studies
By T. A. Whittaker et al., 2014, Journal of Consulting and Clinical Psychology
http://dx.doi.org/10.1037/a0036664.supp
ηi  μ  ζ i ,
(SA2)
where ηi is a 2  1 vector containing intercept and slope parameters for individual i; μ is a 2  1
vector containing latent means for the intercept,  a , and the slope,   , across individuals; and
ζ i is a 2  1 vector containing the disturbances associated with the intercept and slope factors,
  and   , respectively.
i
i
The covariance matrix of raw scores (  yy ) on each observed variable y at each
measurement occasion t is given as

yy
 ΛΨΛ  Θ ε ,
(SA3)
where Λ is a T  2 matrix containing intercept and slope factor loadings at each time t; Ψ is a 2 
2 covariance matrix for the intercept and slope factors, and Θ ε is a T  T covariance matrix for
the observed y variables’ residuals at each time t. The variances of the intercept and slope as well
as the covariance between the intercept and slope constructs are estimated via their disturbances.
Variances of and covariances between the disturbances are given in matrix Ψ :
  2
i
Ψ

  i  i
    
,
 2 
i
i
(SA4)
i
where  2 i is the variance of the intercept factor disturbance,  2i is the variance of the slope
factor disturbance, and    i  i (=   i   i ) is the covariance between the intercept and slope factor
disturbances.
Supplemental Material B
Equations for a Latent Growth Model With Ordered Categorical Data
Supplemental Materials
Latent Growth Modeling With Domain-Specific Outcomes Comprised of Mixed Response
Types in Intervention Studies
By T. A. Whittaker et al., 2014, Journal of Consulting and Clinical Psychology
http://dx.doi.org/10.1037/a0036664.supp
With ordered categorical data, the latent growth model in Equation 1 may be given as
y *i  ν  Λη i  ε i ,
(SB1)
where y *i is a T  1 vector containing the underlying continuous latent item responses at each
time t for individual i. The observed, ordered categorical variable y with c response options
measured at each time t for individual i is related to the underlying continuous latent item
response y* at each time t for individual i, respectively, by the use of j threshold parameters, τj (
j  1...c  1), per outcome measure at each time t:
y = c if  j  y *   ( j 1) .
(SB2)
Again, the intercepts in ν are set equal to zero for model identification purposes.
Supplemental Material C
Mplus Code for the Simultaneous Conditional MDLGM
TITLE:
CONDITIONAL MDLGM WITH MIXED RESPONSE TYPESASSESSING TREATMENT EFFECTS IN INTERCEPT AND SLOPE FACTORS
DATA: FILE IS C:\SENIORWISE\MDLGM.DAT;
Supplemental Materials
Latent Growth Modeling With Domain-Specific Outcomes Comprised of Mixed Response
Types in Intervention Studies
By T. A. Whittaker et al., 2014, Journal of Consulting and Clinical Psychology
http://dx.doi.org/10.1037/a0036664.supp
!INDICATING LOCATION OF THE DATASET;
VARIABLE:
NAMES ARE GROUP SF1-SF5 MC1-MC5;
MISSING ARE ALL (999);
!INDICATING HOW MISSING DATA ARE CODED IN THE DATASET;
CATEGORICAL ARE MC1-MC5;
!MEMORY COMPLAINTS IS AN ORDERED CATEGORICAL VARIABLE;
MODEL:
SF_I SF_S | SF1@-26 SF2@-14 SF3@-6 SF4@-2 SF5@0;
!LINEAR MODEL FOR SOCIAL FUNCTIONING;
!SF_I = SOCIAL FUNCTIONING INTERCEPT;
!SF_S = SOCIAL FUNCTIONING SLOPE;
MC_I MC_S | MC1@-26 MC2@-14 MC3@-6 MC4@-2 MC5@0;
!LINEAR MODEL FOR MEMORY COMPLAINTS;
!MC_I = MEMORY COMPLAINT INTERCEPT;
!MC_S = MEMORY COMPLAINTS SLOPE;
MC_S@0;
!SETTING MEMORY COMPLAINTS SLOPE VARIANCE EQUAL TO ZERO;
SF_I SF_S MC_I MC_S ON GROUP;
!REGRESSING INTERCEPT & SLOPE FACTORS ON GROUP VARIABLE;
!ASSESSING TREATMENT EFFECTS;
OUTPUT: STDYX;
!REQUESTING STANDARDIZED OUTPUT IN MPLUS;
SAVEDATA: DIFFTEST IS DERIV.DAT;
!SAVING THE DERIVATIVES IN DERIV.DAT FILE;
!THIS IS USED FOR THE CHI-SQUARE DIFFERENCE TEST;
Supplemental Material D
Mplus Code for the Simultaneous Unconditional MDLGM
Supplemental Materials
Latent Growth Modeling With Domain-Specific Outcomes Comprised of Mixed Response
Types in Intervention Studies
By T. A. Whittaker et al., 2014, Journal of Consulting and Clinical Psychology
http://dx.doi.org/10.1037/a0036664.supp
TITLE:
CONDITIONAL MDLGM WITH MIXED RESPONSE TYPESASSESSING TREATMENT EFFECTS IN INTERCEPT AND SLOPE FACTORS
DATA: FILE IS C:\SENIORWISE\MDLGM.DAT;
!INDICATING LOCATION OF THE DATASET;
VARIABLE:
NAMES ARE GROUP SF1-SF5 MC1-MC5;
MISSING ARE ALL (999);
!INDICATING HOW MISSING DATA ARE CODED IN THE DATASET;
CATEGORICAL ARE MC1-MC5;
!MEMORY COMPLAINTS IS AN ORDERED CATEGORICAL VARIABLE;
ANALYSIS: DIFFTEST IS DERIV.DAT;
!USING THE SAVED DERIVATIVES FROM THE CONDITIONAL MDLGM;
!TO CALCULATE THE CHI-SQUARE DIFFERENCE TEST;
MODEL:
SF_I SF_S | SF1@-26 SF2@-14 SF3@-6 SF4@-2 SF5@0;
!LINEAR MODEL FOR SOCIAL FUNCTIONING;
!SF_I = SOCIAL FUNCTIONING INTERCEPT;
!SF_S = SOCIAL FUNCTIONING SLOPE;
MC_I MC_S | MC1@-26 MC2@-14 MC3@-6 MC4@-2 MC5@0;
!LINEAR MODEL FOR MEMORY COMPLAINTS;
!MC_I = MEMORY COMPLAINT INTERCEPT;
!MC_S = MEMORY COMPLAINTS SLOPE;
MC_S@0;
!SETTING MEMORY COMPLAINTS SLOPE VARIANCE EQUAL TO ZERO;
SF_I SF_S MC_I MC_S ON GROUP;
!REGRESSING INTERCEPT & SLOPE FACTORS ON GROUP VARIABLE;
!ASSESSING TREATMENT EFFECTS;
OUTPUT: STDYX;
!REQUESTING STANDARDIZED OUTPUT IN MPLUS;
Supplemental Materials
Latent Growth Modeling With Domain-Specific Outcomes Comprised of Mixed Response
Types in Intervention Studies
By T. A. Whittaker et al., 2014, Journal of Consulting and Clinical Psychology
http://dx.doi.org/10.1037/a0036664.supp
Supplemental Material E
Mplus Code for the Simultaneous Conditional MDLGM with the TWOLEVEL Approach
Incorporated
TITLE:
CONDITIONAL MULTILEVEL MDLGM WITH MIXED RESPONSE TYPESASSESSING TREATMENT EFFECTS IN INTERCEPT AND SLOPE FACTORS
DATA: FILE IS C:\SENIORWISE\MDLGM_CLUSTER.DAT;
!INDICATING LOCATION OF THE DATASET;
VARIABLE:
NAMES ARE GROUP CLUSTER SF1-SF5 MC1-MC5;
MISSING ARE ALL (999);
!INDICATING HOW MISSING DATA ARE CODED IN THE DATASET;
CATEGORICAL ARE MC1-MC5;
!MEMORY COMPLAINTS IS AN ORDERED CATEGORICAL VARIABLE;
CLUSTER = CLUSTER;
!INDICATING WHICH VARIABLE CONTAINS INFORMATION ABOUT
CLUSTERING DEPENDENCY;
BETWEEN = GROUP;
!INDICATING THAT THE GROUP VARIABLE WILL BE USED AS
A COVARIATE FOR THE BETWEEN-LEVEL MODEL;
MODEL:
%WITHIN%
!WITHIN-LEVEL MODEL;
SF_IW SF_SW | SF1@-26 SF2@-14 SF3@-6 SF4@-2 SF5@0;
!LINEAR MODEL FOR SOCIAL FUNCTIONING;
!SF_IW = SOCIAL FUNCTIONING INTERCEPT WITHIN-LEVEL;
!SF_SW = SOCIAL FUNCTIONING SLOPE WITHIN-LEVEL;
MC_IW MC_SW | MC1@-26 MC2@-14 MC3@-6 MC4@-2 MC5@0;
!LINEAR MODEL FOR MEMORY COMPLAINTS;
!MC_IW = MEMORY COMPLAINT INTERCEPT WITHIN-LEVEL;
!MC_SW = MEMORY COMPLAINTS SLOPE WITHIN-LEVEL;
MC_SW@0;
!SETTING MEMORY COMPLAINTS SLOPE VARIANCE WITHIN-LEVEL
EQUAL TO ZERO;
Supplemental Materials
Latent Growth Modeling With Domain-Specific Outcomes Comprised of Mixed Response
Types in Intervention Studies
By T. A. Whittaker et al., 2014, Journal of Consulting and Clinical Psychology
http://dx.doi.org/10.1037/a0036664.supp
%BETWEEN%
!BETWEEN-LEVEL MODEL;
SF_IB SF_SB | SF1@-26 SF2@-14 SF3@-6 SF4@-2 SF5@0;
!LINEAR MODEL FOR SOCIAL FUNCTIONING;
!SF_IB = SOCIAL FUNCTIONING INTERCEPT BETWEEN-LEVEL;
!SF_SB = SOCIAL FUNCTIONING SLOPE BETWEEN-LEVEL;
MC_IB MC_SB | MC1@-26 MC2@-14 MC3@-6 MC4@-2 MC5@0;
!LINEAR MODEL FOR MEMORY COMPLAINTS;
!MC_IB = MEMORY COMPLAINT INTERCEPT BETWEEN-LEVEL;
!MC_SB = MEMORY COMPLAINTS SLOPE BETWEEN-LEVEL;
MC_SB@0;
!SETTING MEMORY COMPLAINTS SLOPE VARIANCE BETWEEN-LEVEL
EQUAL TO ZERO;
SF1-SF5@0;
!SETTING RESIDUAL VARIANCES OF SOCIAL FUNCTIONING
OUTCOMES AT EACH TIME EQUAL TO ZERO DUE TO
ESTIMATION PROBLEMS AT BETWEEN-LEVEL;
!NEGATIVE RESIDUAL VARIANCES OCCURRED DURING ESTIMATION;
MC1-MC5@0;
!SETTING RESIDUAL VARIANCES OF MEMORY COMPLAINTS
OUTCOMES AT EACH TIME EQUAL TO ZERO DUE TO
ESTIMATION PROBLEMS AT BETWEEN-LEVEL;
!NEGATIVE RESIDUAL VARIANCES OCCURRED DURING ESTIMATION;
SF_IB SF_SB MC_IB MC_SB ON GROUP;
!REGRESSING INTERCEPT & SLOPE FACTORS BETWEEN-LEVEL
ON GROUP VARIABLE;
!ASSESSING TREATMENT EFFECTS;
OUTPUT: STDYX;
!REQUESTING STANDARDIZED OUTPUT IN MPLUS;