Therapist and Patient Perceptions of Alliance and Progress
in Psychological Therapy for Women Diagnosed With Gynecological Cancers
Online Appendix: Estimating the Longitudinal One-With-Many Design
The one-with-many (OWM; Kenny, Kashy, & Cook, 2006; Marcus, Kashy, & Baldwin,
2009) design is a family of designs in which there is a focal individual (i.e. the “one”) who is
linked to multiple partners (i.e., the “many”). In a reciprocal OWM, both the focal individual
and each of the partners provide data. The OWM is a variance decomposition model that
separates therapist-level from patient/dyad-level variance. The longitudinal one-with-many
design (LOWM) is an extension of the OWM when multiple observations are gathered for each
therapist-partner pair over time (e.g., the two sessions at which alliance rated). A key advantage
of the LOWM is that it allows us to separate stable patient/dyad-level variance from timespecific patient/dyad-level variance (which includes error variance).
The LOWM design has three sampling units: time, patient, and therapist. However,
these are not hierarchically nested units because both patients and therapists report on the same
session. As a result, time is crossed with patient, and the time by patient interaction is nested
within therapist. The LOWM provides a more accurate view of the degree to which therapeutic
alliance and session outcomes depend on the therapist, the particular patient that the therapist is
seeing, and time-specific factors. Moreover, if the longitudinal aspect of the study includes more
than 2 waves, the LOWM can estimate whether there is significant variance in the patients’
trajectories over time, and whether patients and therapists agree in those trajectories.
In this appendix, we describe how to use multilevel modeling, as implemented in either
PROC MIXED in SAS or MIXED in SPSS, to estimate the LOWM. Table 1 shows the data
layout required for the analysis of a reciprocal LOWM with two waves of data (i.e., for WAI
ratings reported in the main paper), and Table 2 shows the data layout with six waves (i.e., for
session outcome ratings). In general, to conduct a reciprocal one-with-many analysis using
MLM, there would be two records for each focal person-partner dyad, one that contains the focal
person’s rating with a specific partner and one that contains the partner’s rating. With the
LOMN, there would be two such records for each wave of data collection. Thus, for WAI
ratings, the data set would have 4 records for each patient, and for session outcome the data set
would have 12 records for each patient.
To accomplish the LOWM analysis, each record needs to include a patient identification
variable (patientID) that repeats on every line. Each record also needs to include a therapist
identification variable (therapistID) that indicates which patients a particular therapist saw, as
well as a time identification variable (timeID) that indicates wave of data collection. Three
additional variables, which we call rater, therapist, and patient respectively, must be created.
These variables specify who generated the data — the therapist or the patient. The rater would
equal –1 if the data are from the therapist, and it would equal 1 if the data are from the patient;
the therapist variable would be 1 if the outcome was generated by the focal person, and 0 if it
was generated by the patient; and the patient variable would be 0 if the outcome was generated
by the therapist, and 1 if it was generated by the patient.
The OWM and LOWM are models that specifically focus on the random effects (i.e., the
variances and covariances). However, they can also include fixed effects, and in the overtime
context a natural fixed-effects model is a growth model that assesses average change in the
outcome over time (alternative specifications of the fixed effects are possible, but are not
considered in this appendix). To estimate a growth model, our data set also needs to include a
variable denoting assessment wave, and the way that we code such a variable has important
ramifications for both the fixed and random effects. A key consideration is how time = 0 is
defined. In the main article, we defined time zero as the midpoint of the study, which is
equivalent to a grand-mean centering approach. This then defines the intercept as the average
rating by either the therapist or patient across all the waves. Given that WAI was assessed at
sessions 2 and 6, but session outcome was assessed at every session, we defined TimeCentered
as -2, 2 for analyses of WAI and -2.5, -1.5, -.5, .5, 1.5, 2.5 for analyses of session outcome. This
approach results in time slopes that estimate the predicted change in WAI or session outcome
over a 1 week period.
The LOWM can be estimated using multilevel modeling in SAS or SPSS, although SAS
seems to derive solutions to these complex models more quickly than SPSS. The SAS syntax for
the two-wave LOWM for working alliance is:
PROC MIXED COVTEST ;
CLASS therapistID patientID timeID rater;
MODEL wai= therapist patient therapist*timecentered patient*timecentered /
NOINT S DDFM=SATTERTH;
RANDOM therapist patient / SUBJECT=therapistID TYPE=UNR ;
RANDOM therapist patient / SUBJECT = therapistID*patientID TYPE=UNR;
REPEATED rater / SUBJECT = therapistID*patientID*timeID TYPE=UNR;
run;
This syntax specifies the fixed effects components of the model in the MODEL statement, and
the results for this analysis are presented in Table 3. The estimate for the therapist variable is
the intercept for therapist ratings, which is the average of the therapist alliance ratings across
therapists, patients, and time, and the patient effect estimates the intercept for patient ratings
averaging across therapists, patients, and time. The therapist*timecentered effect estimates
the average weekly change in therapist-rated alliance across therapists and patients, and likewise,
the patient*timecentered effect estimates this change for the patient-rated alliance scores.
The LOWM variances and correlations are specified in the RANDOM and REPEATED
statements. The first RANDOM statement defines the therapist-level variances for both therapist
and patient ratings. Given the ordering of therapist and then patient in this syntax, the Var(1)
component in the SAS output estimates the degree to which therapists varied in their average
ratings of alliance (i.e., did some therapists report establishing strong alliance across all partners
and time whereas other therapists reported weaker alliance?). The Var(2) component estimates
whether patients who saw the same therapist tended to agree in their ratings of alliance (i.e., were
there consistently high patient ratings of alliance for some therapists and consistently low patient
ratings of alliance for others?). The corr(2,1) estimates the generalized reciprocity correlation.
The second RANDOM statement defines the patient-level variances for both therapist
and patient ratings. Here Var(1) estimates the degree to which therapists’ ratings of alliance
differ from patient to patient (i.e., do therapists report establishing stronger alliances with some
patients than with others), and Var(2) estimates patient-level variance in patient ratings (i.e., do
patients vary in their reports of alliance. Note that these two variances estimate the amount of
variance at the patient level that is consistent across the two waves. In addition the corr(2,1)
correlation estimates dyadic reciprocity – which is the extent to which patients and therapists
agree in their assessments of alliance (i.e., if a patient reports especially high alliance with the
therapist, higher than the other patients who see the same therapist, does the therapist also report
especially high alliance?). The final line of syntax specifies the residual or time-specific
component of the model, and of interest here is the corr(2,1) value, which estimates whether
therapists and patients agree on how well alliance was established at that particular session, over
and above their general levels of agreement.
The same LOWM estimates can be derived in SPSS using the following syntax:
MIXED wai WITH therapist patient timecentered
/FIXED therapist patient therapist*timecentered patient*TimeCentered | NOINT
/METHOD=REML
/PRINT=SOLUTION TESTCOV
/RANDOM=therapist patient | SUBJECT(therapistID) COVTYPE(UNR)
/RANDOM=therapist patient | SUBJECT(therapistID*patientID) COVTYPE(UNR)
/REPEATED = rater | SUBJECT(therapistID*patientID*timeID) COVTYPE(UNR).
Finally, if there are more than two waves of data in the longitudinal study, the random effects
can be expanded to include a random slope at the patient level. In SAS, this would be
accomplished by adding the following random statement to the syntax:
RANDOM therapist*timecentered patient*timecentered /
SUBJECT = therapistID*patientID TYPE=UNR;
For SPSS, the additional line of syntax would be:
/RANDOM= therapist*timecentered patient*timecentered |
SUBJECT(therapistID*patientID) COVTYPE(UNR)
Table 1
Example LOWM Data for Analyzing Working Alliance Assessed Twice From Both Patients and
Therapists
Patient
Therapist
Time
Time
WAI
Initial BDI
ID
ID
ID
Rater
Patient
Therapist
Centered
total
Centered
1102
2
1
-1
0
1
-2
197
-5.27
1102
2
2
1
1
0
-2
238
-5.27
1102
2
1
-1
0
1
2
215
-5.27
1102
2
2
1
1
0
2
237
-5.27
1117
4
1
-1
0
1
-2
188
-1.27
1117
4
2
1
1
0
-2
187
-1.27
1117
4
1
-1
0
1
2
214
-1.27
1117
4
2
1
1
0
2
221
-1.27
1140
1
1
-1
0
1
-2
228
11.73
1140
1
2
1
1
0
-2
238
11.73
1140
1
1
-1
0
1
2
220
11.73
1140
1
2
1
1
0
2
235
11.73
Note. Time is coded -2 at session 2 and 2 at session 6. Rater is coded -1 for therapist rating and
1 for patient rating. Patient is dummy coded 1 if the rating was from the patient, 0 otherwise.
Therapist is dummy coded 1 if the rating was from the therapist, 0 otherwise. WAI total is the
patient’s (or therapist’s) overall rating of working alliance for the session. Initial BDI Centered
is the patient’s initial BDI score prior to the beginning of therapy, centered on the grand mean for
the sample.
Table 2
Example LOWM Data for Analyzing Session Outcome Assessed Six Times From Both Patients
and Therapists
Patient
ID
Therapist
ID
Time
ID
Rater
Patient
Therapist
Time
Centered
Session
Outcome
1102
1102
1102
1102
1102
1102
1102
1102
1102
1102
1102
1102
1117
1117
1117
1117
1117
1117
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
1
1
2
2
3
3
4
4
5
5
6
6
1
1
2
2
3
3
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
-2.5
-2.5
-1.5
-1.5
-.5
-.5
.5
.5
1.5
1.5
2.5
2.5
-2.5
-2.5
-1.5
-1.5
-.5
-.5
15
14
11
9
11
7
9
6
12
5
11
5
10
11
13
9
10
6
Initial
BDI
Centered
-5.02
-5.02
-5.02
-5.02
-5.02
-5.02
-5.02
-5.02
-5.02
-5.02
-5.02
-5.02
-1.02
-1.02
-1.02
-1.02
-1.02
-1.02
Note. Rater is coded -1 for therapist rating and 1 for patient rating. Patient is dummy coded 1 if
the rating was from the patient, 0 otherwise. Therapist is dummy coded 1 if the rating was from
the therapist, 0 otherwise. WAI total is the patient’s (or therapist’s) overall rating of working
alliance for the session. Initial BDI Centered is the patient’s initial BDI score prior to the
beginning of therapy, centered on the grand mean for the sample.
Table 3
Partial SAS Output From the LOWM With Two Waves of WAI Data
Covariance Parameter Estimates
Cov Parm
Var(1)
Var(2)
Corr(2,1)
Var(1)
Var(2)
Corr(2,1)
Var(1)
Var(2)
Corr(2,1)
Subject
Estimate
Ther ID
Ther ID
Ther ID
Ther ID*Pat ID
Ther ID*Pat ID
Ther ID*Pat ID
Ther ID*Pat ID*time ID
Ther ID*Pat ID*time ID
Ther ID*Pat ID*time ID
Standard
Error
Z
Value
Pr Z
112.19
60.57
0.34
48.47
54.35
0.08
18.47
19.11
0.07
2.36
1.67
-0.48
7.04
7.13
5.63
8.97
8.85
3.43
0.0092
0.0475
0.6312
<.0001
<.0001
<.0001
<.0001
<.0001
0.0006
264.62
101.13
-0.16
341.12
387.70
0.46
165.67
169.09
0.26
Solution for Fixed Effects
Effect
patient
therapist
therapist*timecentered
patient*timecentered
Estimate
212.93
201.44
1.82
1.99
Note. Ther = Therapist; Pat = Patient.
Standard
Error
3.14
4.43
0.36
0.36
DF t Value
12.6
16.5
167
163
67.85
45.44
5.12
5.46
Pr > |t|
<.0001
<.0001
<.0001
<.0001