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Joint Statistical Meetings
New York City, August 11-15, 2002
A Bayesian Model for the Study of Accuracy, Reciprocity and
Congruence in Interpersonal Perception
Paramjit Gill
Okanagan University College, Kelowna, BC, Canada
[email protected]
Abstract
A fully Bayesian approach is proposed for the analysis of accuracy and mutuality
in interpersonal perceptions. The Bayesian analysis is based on social relations
model (SRM) formulation. Inference is straightforward using Markov chain Monte
Carlo (MCMC) methods as implemented in the software package WinBugs. An
example is provided to highlight the use of Bayesian analysis of interpersonal
attraction data.
1. Introduction
Accuracy in interpersonal perception is a fundamental and one of the oldest
topics in social and personal psychology. Are people’s perceptions of others
valid? This is the most obvious question in the field of interpersonal
perception, yet, surprisingly, the most difficult to study (Kenny 1994, Chapter
7). In the late 1940's and early 1950's, the study of individual differences in
the accuracy of social perception became a dominant area of research but
Cronbach (1955) and others argued that a comprehensive understanding of
accuracy requires more sophisticated statistical and computational procedures
than those available at that time.
The “second wave of accuracy research” promised to provide a satisfactory
solution. Kenny & Albright (1987) argued that the accuracy research must be
nomothetic, interpersonal, and compartmental. They proposed the use of
social relations model (SRM) as an appropriate tool to do so. Accuracy is
thus defined by the links between various components of the SRM. Although
the SRM provides a methodological framework for the analysis, actual
practical usage is hampered by lack of available computational machinery.
Purpose of this communication is to present a computationally tractable,
fully Bayesian approach to the analysis of accuracy in interpersonal
perceptions. The vehicle for doing this is modern Bayesian computation
made accessible in the software package WinBugs (Spiegelhalter et al.,
2000). The Bayesian approach is based on SRM formulation which
partitions a response into various components. Accuracy is measured by
interrelationships among these components.
2. Social Relations Model
We follow Kenny (1994, Chapter 7) where social relations model is
proposed to for the study of accuracy of personal perceptions. We assume
that the design used in the study is round robin or reciprocal. That is, each
subject serves as judge and target and each subject interacts with all other
subjects.
For each dyad (pair) of subjects i and j, we have four measurements on the
level of a trait yij, yji, xij and xji. Here yij represents the response (impression)
of subject i as an actor (judge) towards subject j as a partner (target) and xji
represents a postdiction (perception) by partner j of the impression yij. In yji
and xij the roles are reversed. The SRM partitions the responses into
population-specific, actor-specific, partner-specific and dyadic components in
an additive fashion
2.1 Statistical Assumptions
2.2 Bayesian Formulation
We note that SRM is a random effects model. The subjects
involved in the study are assumed to be a random sample from a
population and we are interested in generalizing beyond the
particular persons involved in the study. Subject-specific and dyadspecific effects are assumed Normal random random variables with
The population parameters {s2a1,s2a2,s2b1,s2b2 ,s2g1 ,s2g2, r1, r2, r3,
r4, r5, f1, f2, f3 }are called the variance-covariance parameters and are
of primary interest. They are parts of matrices
E(a1i) = E(a2i) = E(b1i) = E(b2i) = E(g1ij) = E(g2ij) = 0
var(a1i) = s2a1 ; var(a2i) = s2a2 ; var(b1i) = s2b1
var(b2i) = s2b2 ; var(g1ij) = s2g1 ; var(g2ij) = s2g2
Correlations among subject-specific effects represent various kinds
of accuracy and reciprocity as follows.
If one does not have strong prior opinion, diffuse prior distributions
for parameters and hyper-parameters can be used. Following
conventional Bayesian protocol, we assume
m1 ~ N[qm1, s2m1]; qm1 ~ N[0,10000]; s2m1 ~ IG[0.0001,0.0001]
m2 ~ N[qm2, s2m2]; qm2 ~ N[0,10000]; s2m2 ~ IG[0.0001,0.0001]
 corr(a1i ,b1i) = r1 measures individual-level reciprocity of
impression. A positive value means that people who are seen by
others as possessing a given trait also see others as possessing the
same trait.
We use independent inverse-Wisharts as priors for S1 and S2 :
 corr(a1i ,a2i) = r2 measures assumed (or perceived) individual
reciprocity. A positive value indicates that people who think of
others possessing a given trait also perceive that others think
similarly about them. We would expect this correlation to be higher
than r1 which measures actual reciprocity.
Having specified the Bayesian model; the model assumptions and data
induce a posterior distribution in accordance with the Bayesian
paradigm. The posterior distribution is the distribution of the parameters
conditional on the data and is the final product from which inference
proceeds. Typically however, one is interested in the average value and
variation of some of the parameters. If we repeatedly generate values of
a parameter from the posterior distribution, average those values and
calculate their standard deviation, we will then have obtained estimates
of the posterior mean and posterior standard deviation.
 corr(a1i ,b2i) = r3 measures perceiver accuracy. A positive value
means that perceiver’s average response (perception) well
corresponds to the average impression of his interaction partners.
 corr(a2i ,b1i) = r4 measures individual-level accuracy. A positive
value means that people have a reasonable understanding of how
they are generally viewed by others as a whole.
 corr(b1i ,b2i) = r5 measures assumed individual-level accuracy.
When people see a subject A possessing a trait (say friendly), they
assume that A knows that other see him friendly. This correlation is
higher than r4 which measures actual accuracy.
Correlations among dyad-specific effects measure dyadic accuracy,
mutuality and congruence as follows (see Figure 1).
 corr(g1ij ,g1ji) = f1 indicates mutuality or dyadic reciprocity in the
sense that if subject A treats subject B in an especially friendly
manner, does B treat A in an especially friendly manner in return?
 corr(g1ij ,g2ij) = f2 measures dyadic congruence or assumed
dyadic reciprocity in the sense that subject A likes subject B because
A thinks that B likes A.
 corr(g1ij ,g2ji) = f3 measures dyadic accuracy of a perceiver to
predict his partner’s behavior towards the perceiver. That is, if
subject A sees subject B as especially friendly, does B act especially
friendly with A?
-1
S1 ~ Wishart4
[(4I)-1,4]
-1
; S2 ~ Wishart4
[(4I)-1,4]
Methods of Markov chain Monte Carlo (MCMC) provide an iterative
approach to variate generation from posterior distributions. Gibbs
sampling algorithm (as implemented in WinBugs) is used to simulate
from the marginal posterior distributions of the parameters of interest.
3. Curry & Emerson Data Example
Curry & Emerson (1970) conducted a study on previously
unacquainted students who lived together in a residence-hall at the
University of Washington. Six 8-person round robin groups of students
reported their attraction toward their group members on a 100-point
scale at weeks 1, 2, 4, 6, and 8. The subjects also provided perception of
attraction ratings towards them by other subjects.
For simplicity, we consider five time points as replicates. A more
realistic analysis would consider longitudinal profiling of variancecovariance components. Table 1 shows means, standard deviations,
2.5% and 97.5% quantiles of the marginal posterior distributions of the
some key parameters for the attraction data.
We see that the perception bias m2-m1 is positive but small. It means
that subjects, on the average, have a pretty good idea when estimating
the level of attraction they command from others.
Individual level reciprocity (Mean r1 = 0.12) and its perception (Mean
r2 = 0.85) make an interesting comparison. Low reciprocity means that
people who are seen by others as attractive, do not see others as
attractive. But people who think others as attractive assume that others
think similarly about them.
yij = m1 + a1i + b1j + g1ij
xji = m2 + a2j + b2i + g2ji
yji = m1 + a1j + b1i + g1ji
xij = m2 + a2i + b2j + g2ij
Individual level accuracy (Mean r4 = 0.34) is lower than the assumed
individual level accuracy (Mean r5 = 0.85). This tells us that people
have a poor understanding of how they are generally viewed by others.
On the other hand, people assume that others have an almost perfect
notion of how they are seen by others.
The model parameters are divided into three groups
Population-specific:
m1 = population average impression
m2 = population average perception
m2 - m1 = Perception bias the subjects have in estimating their own trait level
Both the dyadic level reciprocity (Mean f1 = 0.39) and accuracy
(Mean f3 = 0.33) are rather low. It is possible that these values increase
with time which could be confirmed with a detailed longitudinal
analysis. Not surprisingly, the dyadic congruence (Mean f2 = 0.65) is
high which tells us that subjects have a tendency to like specific others
because they think that those specific others like them. When compared
with mean f1 = 0.39, it means that subjects believed that their unique
impressions of specific partners were reciprocated more than they really
were reciprocated.
Subject-specific:
a1i = average impression that subject i has about others
a2i = average perception that subject i has about what others think of him
b1i = average impression others have of subject i
b2i = average perception others have of subject i’s impression of others
Dyad-specific:
g1ij = dyadic interaction between subjects i and j as reported by i as a judge. It
is the special relative impression of i toward j, subtracting out the actor and
partner effects.
g2ji = perception by subject j of the dyadic interaction g1ij
S1 = Cov(a1i,b1i,a2i,b2i) and S2 = Cov(g1ij,g1ji, g2ij,g2ji)
Figure 1. Mutuality, congruence, and accuracy triangle (Kenny & Albright, 1987)
Table 1. Some summary results from the Bayesian analysis
of attraction data
Parameter
Mean
SD
2.5%
97.5%
m2-m1
1.65
0.48
0.67
2.52
r1
0.12
0.08
-0.04
0.28
r2
0.85
0.03
0.80
0.90
r3
0.12
0.09
-0.06
0.29
r4
0.34
0.07
0.19
0.47
r5
0.85
0.03
0.79
0.90
f1
0.39
0.03
0.32
0.46
f2
0.65
0.02
0.62
0.68
f3
0.33
0.03
0.27
0.39
s2a1
66
8.7
51
85
s2b1
69
9.0
53
88
s2a2
82
9.0
66
101
s2b2
21
3.2
16
28
s2g1
152
6.8
140
166
s2g2
76
3.2
70
82
Among the variance components, we find that actor and partner
variations in attraction levels are very similar (Mean s2a1 = 66, Mean
s2b1 = 69). It is, however, interesting to note that there is substantial
actor variation in perception (Mean s2a2 = 82). It tells us that some
people believe that they are more attractive and others believe that they
are not that attractive. On the other hand, partner variation in perception
is relatively much lower (Mean s2b2 = 21). That is, there is a slight
tendency for some people to be seen as harsh judges and others to be
seen as lenient ones.
The dyadic variance in the reported attraction level (Mean s2g1 = 152)
is almost double than the dyadic variance in the perceived attraction
(Mean s2g2 = 76). It tell us that subjects were not capable of fully
realizing the extent of variability in dyadic interactions.
4. Future Research
The relationship between persons develops over time and therefore,
the model should accommodate the longitudinal nature of data. It
means that the model parameters are assumed to be occasion-specific.
It would be of interest to include covariates (such as sex) in the model.
For example, in the Curry-Emerson study, the students lived in the
residences as room-mate pairs. Work on measuring the effect of
physical proximity on the degree of accuracy, reciprocity and
congruence is under progress.
5. References
1. Cronbach, J. L (1955). Psychological Bulletin, 52, 177-193.
2. Curry, T.J. & Emerson, R.M.(1970). Sociometry, 33, 216-238.
3. Kenny, D.A.(1994). Interpersonal Perceptions. Guilford Press: New York
4. Kenny, D.A. & Albright, L.(1987). Psychological Bulletin, 102, 390-402.
5. Spiegelhalter, D., Thomas, A. & Best, N.(2000). WinBUGS User Manual. MRC
Biostatistics Unit: Cambridge.
6. Acknowledgements
This research is being supported by a grant from the Natural Sciences and
Engineering Research Council (NSERC) of Canada and is a part of ongoing joint
work with Professor C. F. Bond Jr. of Texas Christian University, Fort Worth.