A Bayesian Model for the Study of Accuracy, Reciprocity and Congruence in Interpersonal Perception Paramjit Gill Okanagan University College, Kelowna, BC, Canada [e-mail]

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A Bayesian Model for the Study of Accuracy,
Reciprocity and Congruence in Interpersonal
Perception Paramjit GillOkanagan University
College, Kelowna, BC, Canadapgill_at_ouc.bc.ca
Joint Statistical Meetings New York City, August
11-15, 2002

• 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 peoples 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
• yij m1 a1i b1j g1ij

2.2 Bayesian Formulation 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 S1
Cov(a1i,b1i,a2i,b2i) and S2 Cov(g1ij,g1ji,
g2ij,g2ji) 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 Nqm1, s2m1 qm1 N0,10000 s2m1
IG0.0001,0.0001 m2 Nqm2, s2m2 qm2
N0,10000 s2m2 IG0.0001,0.0001 We use
independent inverse-Wisharts as priors for S1 and
S2 S1-1 Wishart4(4I)-1,4 S2-1
Wishart4(4I)-1,4 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. 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
variance-covariance 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. 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.
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.
2.1 Statistical Assumptions 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
dyad-specific effects are assumed Normal random
random variables with 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. ? 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. ? 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. ? corr(a1i
,b2i) r3 measures perceiver accuracy. A
positive value means that perceivers 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 partners behavior
towards the perceiver. That is, if subject A sees
subject B as especially friendly, does B act
especially friendly with A?
Table 1. Some summary results from the Bayesian
analysis of attraction data  
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.
 

Figure 1. Mutuality, congruence, and accuracy
triangle (Kenny Albright, 1987)