Social Networks
North-Holland
359
12 (1990) 359-373
TESTING FOR CHANGE IN A DIGRAPH AT TWO TIME POINTS
Tom A.B. SNIJDERS
University of Groningen,
*
University
of Utrecht
A method is presented for testing change of digraphs (representing
some binary relation) observed
at two points in time, labeled I and II. The test is conditional
on the entire digraph at time I, the
numbers of new arcs to and from each actor, and the numbers of disappeared
arcs to and from
each actor. A new arc is defined as an arc existing at time II but not at time I; a disappeared
arc is
an arc existing at time I but not at time II. In particular,
tests are conditional
simultaneously
on
in-degrees
and out-degrees
at times I and II. The elements of the dyad transition
matrix,
indicating the numbers of dyads of some particular type (mutual, asymmetric,
or null) at time I,
and of some (same or other) type at time II, are proposed as possible test statistics.
1. Testing change in networks
Methods for carrying out exact, tests of the null hypothesis
that a
random directed graph (digraph) has a conditionally
uniform distribution, given the in- and out-degrees of the various vertices, have recently
been developed
and practically
elaborated.
They are presented
in
Snijders (1991). The same principle is used in the present paper for a
conceptually rather simple approach to testing change in networks. The
null hypotheses tested here can be viewed as random digraph models
where the digraphs are uniformly distributed
conditional
on certain
relevant statistics. Consider a network observed at two time points,
denoted “time I” and “time II”. Actors and vertices will be treated as
synonyms. The sets of actors at both time points are identical, but the
arcs between the actors may have changed. There are many ways in
which one could specify the notion of randomness for the change in the
network from time I to time II. In the approach proposed in this paper,
the focus is on one hand on the newly created arcs, defined as the arcs
* Department
of Statistics and Measurement
Theory, University of Groningen,
Grote
2/l, 9712 TS Groningen,
The Netherlands;
and Department
of Empirical Theoretical
University of Utrecht, Heidelberglaan
2, 3584 CS Utrecht, The Netherlands.
0378-8733/90/$3.50
0 1990 - Elsevier Science Publishers
B.V. (North-Holland)
Kruisstraat
Sociology,
360
T.A.B. Sni~ders / Testing
for change
present at time II but not at time I; and on the other hand on the
disappeared
arcs: those present at time I but not at time II. The null
hypothesis of random change can be loosely described in the following
way; a formal definition is given in Section 2. For new arcs the null
model is that these are created at random, subject to the following
restrictions:
(1) new directed arcs from actor i to actor j can be created
those i and j where an arc from i to j does not yet exist
(2) the number of new arcs from each actor i is considered to
(3) the number of new arcs to each actor j is considered to
only for
at time I;
be given;
be given.
The null hypothesis for disappeared arcs is the mirror image of this; it
is formally defined in the next section. Combining these two, the null
hypothesis for the entire digruph at time II is the conditionally
uniform
distribution
given the following statistics: the entire digraph at time I,
the number of new arcs from each actor, the number of new arcs to
each actor, the number of disappeared
arcs from each actor, and the
number of disappeared
arcs to each actor. This null hypothesis
is
related to Holland and Leinhardt’s
(1981) p1 distribution
and to
Wasserman’s (1987) models for comparing two digraphs; this relation
is discussed in Section 6 of the present paper.
The number of new arcs from an actor is an indication of the activity
of this actor; the number of new arcs to an actor is an indication of his
attractiveness.
The same holds, in an inverse sense, for the numbers of
disappeared
arcs to and from each actor. While recognizing the fact
that activity and attractiveness
may be influenced by network (or other
social) properties, we take the position here that these statistics primarily reflect actor properties and that in the statistical analysis actor
properties
should be separated
from network properties.
Network
change therefore is tested while controlling simultaneously
for differential activity and for differential attractiveness
of individual actors.
The test statistic for testing this null hypothesis can be any statistic
expressing the theory that is being considered
as an alternative
hypothesis. For instance, the test statistic can be a function of the dyad or
the triad counts at times I and II. In Section 3, the dyad transition
matrix is proposed as a relevant set of test statistics. In Sections 6 and
7, some contrasts with other approaches
to statistical methods for
digraphs observed at two time points are briefly discussed.
361
T.A. B. Snijders / Testing for change
2. Formulation of the null hypothesis
Some notation must be defined for a more detailed treatment. Consider
an inter-actor relation on a set G of g actors, observed at time points I
and II. This relation will be called “choice”. Data pertaining
to the
relation are represented by a directed graph (digraph) or, equivalently,
by an incidence matrix X = (X;,). Presence of the relationship
for an
ordered pair (i, j) of actors in G is represented by an arc from i to j
in the digraph, and by Xjj = 1 in the incidence matrix; absence of the
relationship
is represented
by absence of the arc and by X,, = 0. The
relationship is only between (not within) actors, so that Xiii= 0 for all i.
The incidence matrices for the two time points are denoted
X’ and
Xii, respectively. For each of the two time points t = I, II the out-degrees (numbers of choices made) are given by
x,: = c jxi;.
and the in-degrees
(numbers
of choices received)
by
X:j = C,Xl).
The vectors of in- and out-degrees are denoted Xl, = (Xi,, . . . , Xi,)
and X:+=(X:+,...,
X,‘,), respectively.
Dyads (unordered
pairs of
actors with the arcs that may exist between them) can be classified into
mutual ( Xii = Xji = 1: reciprocal choices), asymmetric
( Xij + Xjj = l),
and null ( Xii = Xii = 0) dyads. The number of mutuals is
M’=
~,,x;>x,:,
the number
N’= CICJ(l
of null dyads is
- X:)(1
and the number
A’=
I,,(
- J$),
of asymmetric
X;J(l -x,:)
dyads is
+ X,::(l - xl>)) = g(g - Q/2
-M’-
N’*
362
T.A.B. Snijders / Testrng
The total number
for change
of arcs is
,Xjj = 2M’ + A’.
T’ = c;
The change in the total number of arcs from T’ to T” may reflect the
observation design (e.g., actors may have been asked to make a given
number of choices) or development
processes in the group as a whole,
while changes in in- and out-degrees may, in addition, reflect characteristics of actors. The present paper does not discuss how to analyse the
changes in the total number of arcs or those in the in- and out-degrees,
but regards the in- and out-degrees
as given values; we discuss the
question how to analyse changes in the inter-actor relation given the inand out-degrees at both time points.
A statistical test is constituted by a test statistic, chosen in view of
the theory tested, together with a null hypothesis,
or a model of
randomness.
The null model is used as a gauge to indicate which
outcomes of the test statistic are sufficiently unusual to conclude that
something interesting
is going on. In this view of testing, the null
hypothesis is a straw man; rejecting the hypothesis amounts to crossing
the barrier beyond which the researcher is allowed to make a substantive interpretation.
In analogy to what is proposed by Holland and
Leinhardt (1975) for the analysis of digraphs observed at one point in
time, the test statistic will have to be derived from theoretical considerations about the process of change leading from X’ to X”. Null
models are formulated here separately for new arcs and for disappeared
arcs. New arcs correspond to entries equal to 1 in the incidence matrix
X” defined by
Xi7 =
i
1
if X:,! = 1
0
otherwise
more compactly,
X”
=
x”
and Xl: = 0 .
>
X” can be defined
* (1 - XI),
as
(1)
where * denotes the elementwise
(Hadamard)
product
and 1 the
matrix with all elements unity. It is evident that X” must satisfy
xI;=l’xj’j=o.
(2)
T.A. B. Snijders / Tesiing for change
363
The row sum Xi: denotes the number of new choices made by actor i;
the column sum XT, is the number of new choices received by actor j.
The null hypothesis for new arcs is defined as follows:
the matrix of new arcs X” has the conditionally
uniform distribution,
given the network X’ at time I, the vector X,“, of new choices made by
each actor, and the vector X:, of new choices received by each actor.
In order to understand the notion of a conditionally
uniform distribution, recall that the uniform distribution for a discrete outcome space is
defined by the property
that all possible outcomes
have the same
probability. The conditionally
uniform distribution for a digraph, given
certain statistics, is the distribution
with the properties
that every
digraph for which these statistics have the given values, is equally
probable, and that every other digraph has probability 0. An equivalent
formulation
of the null hypothesis for new arcs is therefore that every
matrix X” satisfying (2) and having the required row and column
sums, has the same probability.
The interpretation
is that, apart from
the information
possible contained in the “statistics”
Xi, X,“, and
X”+*, all other structure in X” is random noise.
This null model refers only to Xi and X”, and says nothing yet
about the disappeared
arcs. A similar model can be formulated
for
disappeared arcs. The matrix Xd of disappeared arcs is given by
Xd=X’*
(1-x”);
(3)
it must satisfy
(4
For the disappeared
arcs, the corresponding
null model is
the matrix of disappeared
arcs X d has the conditionally
uniform
distribution,
given the network Xi at time I, the vector X,“, of arcs to
each actor that disappeared,
and the vector Xi+ of arcs from each
actor that disappeared.
The incidence
matrix
p
-
= y
+ X”
p,
X” can be expressed
as
(5)
364
T.A. B. Snbders
/ Testing
for change
which demonstrates
that X’, X”, and Xd jointly determine
X” completely. The matrices X” and Xd refer to two complementary
sets of
directed pairs: for every directed pair (i, j), X14 and Xl”. cannot both
be equal to 1. This is implied by (2) and (4). In some cases it will be
relevant to consider separately the new and the disappeared
arcs, and
test these two null models separately. It can also be relevant to consider
these null models jointly, and combine them as follows in a null
hypothesis for XT’:
the network X” at time II has the conditionally
given the network Xi at time I, and the vectors
uniform distribution,
Xc+, X:,,
Xi+, and
Xdt*.
In other words, all outcomes of X” that agree with the given outcomes
all strucof Xi, X2+, X:*X$+, and Xd+* have the same probability;
ture in X” that is not a function of these statistics is random noise. In
the suggestive notation from Holland and Leinhardt
(1975), the digraph distribution corresponding
to this null model could be called the
Note that formulae (l), (3)
u I X’? X*“+, x:*,
X2+, X”, * distribution.
and (5) imply that, for a given value of X’, there is a one-to-one
correspondence
between X” and the pair (X”, Xd). This implies that
this model for X” is equivalent to X” and Xd being conditionally
independent
and uniformly distributed. From (5) it follows that under
this model the in-degrees and out-degrees of X” are fixed.
The three null models proposed in this section are models expressing
randomness
of change in the structure of the network, which control
for possible changes in actor-related
characteristics.
The next two
sections explain how these null models can be tested.
3. Test statistics; the dyad transition matrix
As stated in
theory tested.
are unstable,
to turn either
tendencies is
SC+&
=
Section 1, the test statistic should reflect the substantive
For example, theory might say that asymmetric relations
so that there should be a tendency for asymmetric dyads
into null or into mutual dyads. Whether either of these
present will be reflected in the &ad transition matrix
‘0,
\C nm
‘%
Cna
T. A. B. Snijders
365
/ Testing for change
also considered in Wasserman (1980; see Table 5). The entries of the
dyad transition matrix are defined as follows. (Strictly speaking it is
not a matrix, as the (2, 2) element consists of two entries instead of
one.) For dyad types u and u (mutual, asymmetric, or null abbreviated
to m, a, and n), S,, denotes the number of dyads which stayed the
Same, being of type u at times I and II; C,, denotes the number of
dyads which Changed from type u to type u. For asymmetric dyads,
there are two possibilities for a dyad to remain asymmetric:
the single
unreciprocated
arc can remain the same, but it can also change direction while remaining asymmetric. The associated numbers are
S,=
C,,=
#((i,
#((i,
j)Ii<j, X,:+X:l=Xj;+X):=l,
j)Ii<j, X;.+Xj=Xiy+X$i=l,
Xi:=X:j)
Xi:#X;).
The row sums of SCdyads are M’, A’ and N’. The column sums (both
S, and C,, are counted in column 2) are MI’, An, and N”. Under the
the row sums of SCdyads are
u I X1, X*“+, x:*3 Xi+, Xd+* distribution,
fixed while the entries, including the column sums, are random. The
dyad count at time II and the entries of the dyad transition matrix are
the simplest relevant statistics which can be used to test randomness of
change as modeled by the U 1X’, Xt+, X:,,
Xi+, Xd+* distribution.
A class of slightly more complicated
statistics can be based on the
triad count for X”, and on transition
counts of triad types. For
example, one could consider a theory stating that there is a tendency
towards transitivity;
see, e.g., Holland and Leinhardt (1975). This can
be expressed for times t = I, II in the number of transitive relations in
triads given by
T,; =
c
x;jx;kx;k
i, j,k
and in the number
of intransitive
Tin,= C xI)X:k(l
- xl:).
relations
in triads,
i, j.k
A relevant
intransitive
test statistic
to transitive
would here be the number of changes from
relations: i.e., the number of ordered triples
366
T.A.B. Snijders / Testing
(i, j, k) where at time I relations i +j
time II a new arc i -+ k has been
remain:
ci, =
c
x;jx:,(l - x;gx,:lx;;x,:
=
for change
+ k exist but not i + k, while at
created and relations
i -j
-+ k
c x$x;kx,;(l - x$)(1 - x;q.
i. j.k
r.J.k
Other test statistics are T,:’ - T,: and Ti$ - Ti,!,,; or, alternatively,
“normalized”
combination
of these two given by
The choice between
considerations.
these test statistics
has to depend
the
on subject matter
4. Null distribution of test statistics
Explicit formulas for probabilities,
expected values, etc., of test statistics under the U ) Xi, Xt+, X: *, Xz+, Xd+* distribution
are presumably impossible to derive. However, the techniques
given in Snijders
(1991) for enumeration
and Monte Carlo simulation for O-l matrices
with given marginals and with a set of structural zeros are immediately
applicable to the I/ ( X’, Xz+, XT,, X$+, Xd+* distribution.
Since X”
and Xd are independent
under this distribution,
enumeration
or simulation can be carried out separately for X” and Xd. In view of the
definitions of the conditionally
uniform distributions
given in Section
2, the question here is the enumeration
or simulation of the conditionally uniform distribution
of a O-l matrix with structurally
zero values
in a prespecified set of cells, and with given row and column sums. As
enumeration
is too time-consuming
in most cases, it is not pursued any
further here. As to simulation,
Snijders (1991) argues that it is not
feasible to efficiently simulate a random sample from this conditionally
uniform
distribution.
However,
it is feasible to sample incidence
matrices with unequal probabilities
(hence, not according to the uniby
form distribution)
and to correct for these “ incorrect” probabilities
the estimator used. An efficient method for taking an unequal probability sample of incidence matrices is given in Snijders (1991) and is here
T.A.B. Snijders / Testing for change
367
taken for granted. Denote by q( X”) the probability
function of the
sampled distribution.
Suppose that N random incidence matrices are
independently
generated from this distribution
and denote them by X,
for h= l,..., N. Then for any function +(X), the expected value p of
G(X) under the U 1X1, Xc+, X:,,
Xz+, X:* distribution
can be
consistently estimated by the ratio estimator
More details can be found in Snijders (1991), were also a formula for
the standard error is given.
Using this method, p-values can be calculated for any test statistic in
the following way. Denote the test statistic by t( X”) and its observed
value by t,, and take the function + as the indicator function of the
event { t( X”) 2 to}; the p-value can then be estimated by (6), and the
standard error of (6) can be given an arbitrarily small positive value by
taking N sufficiently large.
This simulation procedure
is comprised in the little PC software
package Z - 0 of enumeration
and simulation methods for Zero-One
matrices, written in Turbo Pascal, and obtainable from the author.
5. Application to Newcomb’s fraternity data
The data of a college fraternity studied by Newcomb (1961) is used as
an application,
purely as an example of the testing method. This
fraternity
consisted of 17 men without prior acquaintance
of one
another. The particular data used for this example are obtained from
rankings they were asked to give of their fellow members on the basis
of positive feeling. These rankings were asked for each of 15 weeks; we
use the first and last preference rankings, referring to week 0 and week
15. The data are included in UCINET. Rankings were dichotomized
by
defining a directed arc from i to j being present if j was among i’s
top 4 choices. This cut-off point was motivated by Wasserman (1980),
who also studied these data, by the consideration
that the top 4 may
surely be considered
to be “positive”
choices. Thus two incidence
matrices are obtained with out-degree all equal to 4. In-degrees ranged
from 0 to to 8 at time I and from 0 to 9 at time II. The in-degrees had a
368
T.A.B. Snijders / Testing for change
considerably
higher variance at time II than at time I: the normalized
heterogeneity
index H based on the in-degree variance (Snijders 1981)
had values H’ = 0.19 and H” = 0.33. This points towards increased
differences in popularity.
For the matrix X” of new arcs, the out-degrees
(number of new
choices made) were
22214012322333412
and the in-degrees
(new choices received)
34052323402312012.
For the matrix Xd of disappeared
arcs, the out-degrees
(number of
choices withdrawn) are identical to those of X”, as the out-degrees of
the digraphs at times I and II are identical. The in-degrees of Xd
(number of choices lost) were
00313341148300411.
The dyad counts at both times were identical:
M’ = MT’ = 17, A’ = A” = 34, N’ = NIT = 85.
To what extent can this stability be explained
from the notion of
“random change, conditionally
on the out- and in-degrees of X” and
Xd” as defined in Section 3? The dyad transition matrix is
6
=dyads
=
i
6
5
5
8
(7)
17
Note that the row and column marginals both are equal to the dyad
count 17, 34, 85. A simulation of 10,000 runs was carried out to give
Monte Carlo estimates of parameters
under the U 1X’, X2+, X: *,
distribution
of
the
entries
of
the dyad transition
matrix,
X*+3 Xd+*
using the method of Section 4. The estimated means (upper numbers)
and standard deviations (lower numbers) of the dyad transition counts
T.A. B. Snijders / Testing for change
under this null distribution
5.6
0.5
2.4
1.1
369
are
6.0
0.8
11.3
1.2
3.0
1.4
5.3
0.5
17
0
17.3
1.4
34
0
85
0
17.3
1.4
26.6
2.7
55.9
1.9
10.5
1.6
46.9
3.2
78.6
2.5
All means in this table have standard errors less than 0.06.
Estimated one-sided p-values for the observed dyad transition counts
in (7) calculated by the method of Section 4, are given in the following
table. The one-sided p-values are given either with a - sign, indicating
that it is a left-sided p-value (“too low outcome”);
or with a + sign,
indicating that it is a right-sided p-value (“too high outcome”).
0.65 +
o.oo+
0.06 -
ooo-
0.00 -
0.33+
0.29 0.00 +
0.00 -
0.00 +
o.33-
0.36 +
x
X
X
I
0.00 +
A brief summary of this table of p-values is as follows. The column
marginals show that, under random formation
and disappearance
of
arcs as in the U 1X’, X2+, XT,, Xi+, Xd+* model, it is extremely
unlikely that the dyad count would have remained stable at 17, 34, 85:
the number of asymmetric dyads would have decreased and the numbers of mutuals and null dyads would have increased. So the constancy
of the dyad count points to the activity of social processes favoring
symmetry of choices. The bottom line within the table (referring to null
dyads at time I) shows that null dyads are relatively stable: they remain
null much more frequently
than can be explained by chance. The
middle line (asymmetric dyads at time I) shows that asymmetric dyads
are relatively unstable:
they change into mutual dyads more, and
remain unchanged less frequently,
than can be explained by chance.
The top line does not include any significant p-values, so the change
pattern from mutual dyads is compatible
with the null model of
370
T.A.B. Snijders / Testing
for change
random change. In conclusion, the observed pattern seems to confirm
the theoretical notion of instability of asymmetric dyads.
6. Relation with loglinear families of distributions
The model for the null hypothesis
formulated
in Section 2 can be
related to the models for conformity
of two sociometric
relations
treated by Wasserman
(1987). His model (11) is a loglinear model
which, when applied to our matrices X1 and X1’, can be defined as
follows:
log P{X’=x’,
X”=x”)
+
c,p,‘x:j + ~jp:lxI:, + p’M’ + p”M”
+
s*cx:,x;;
+p*Cx,:x;,I + A.
i ,j
(8)
1.1
The greek letters denote parameters of which the interpretation
is given
in Wasserman (1987). Parameter 0 * is called by Wasserman
a multiplexity parameter;
in our case of observations
of the “same” network
at two time points, it can be called a steadiness parameter.
p* can be
called a delayed mutuality parameter.
There is a well known relation between tests of hypotheses
in
loglinear models and tests of conditionally
uniform distributions
(Lehmann, 1986): an exact test of a null hypothesis in a loglinear model
must be carried out as a test of the uniform distribution
that is
conditional on the complete sufficient statistics for the null hypothesis.
Our null hypothesis is conditional on X’, Xc+, XT,, Xz+, and Xd+.+;
to which null hypothesis
in the loglinear approach
does this correspond? Note that the entire digraph X1 is among the conditioning
statistics, so only X” is effectively random. Denoting the Hadamard
product of X1 and X” by
x* =x1 * x”,
T.A.B. Sni/ders / Testing for change
371
it can be seen that Xi:= X$- Xji*,, X,!+= X,5-- X,:, and similarly for
the column sums of X” and Xd. This shows that sufficient statistics for
the null hypothesis
in the loglinear approach,
that corresponds
to
u I X1, X*“+, X:*9 X*+3 Xd+* are, in addition to X1: all marginal sums
X,‘:, Xyj, Xj(i*,,and XT;. In comparison with (8), this means that the
parameters
for the null model are, beside parameters
for X’: the
actor-specific activity and attractiveness parameters for time II, CX~’
and
(3,: and 0.7 which are not only actor#q’, and steadiness parameters
specific but also direction- (“in” and “out”) specific. The resulting null
model is
log P {
=
x” = XI11x1 = x1}
#Ix11
+
+++
C,a,“x;:+
pyx’:,
+
~,e,l”x,*++ C,e.;x;j + X(x’).
p”M’I
(9)
An actor with a ,high value for 0,: has a high tendency of maintaining
choices made; an actor with a high value of 0,; has a high tendency of
retaining choices received. The parameters in (8) that do not occur in
our null model (9) are the mutuality parameter
p” and the delayed
mutuality parameter p*. This interpretation
in a loglinear framework is
in accordance
with the description
of our null model in Section 2:
individual actor characteristics
are maximally controlled for; mutuality
is seen as an aspect of social structure, and is therefore absent from the
null model.
7. Discussion
The approach to testing change in digraphs presented here is conceptually rather simple. Is such an approach based on conditionally
uniform
distributions
not a step backward, however, in view of some existing
modeling approaches
to change in networks: the approach based on
continuous-time
Markov models proposed by Holland and Leinhardt
(1977) and Wasserman (1980), and the loglinear modeling approach of
Wasserman (1987) and Wasserman
and Iacobucci (1988)? It will be
argued that both the approach presented in the present paper and the
existing modeling approaches have their own virtues.
312
T.A. B. Smjders / Testing for change
The approach by Holland, Leinhardt, and Wasserman is based on a
continuous-time
Markov chain model for the “network process”. Some
of its characteristics
are the following.
(a) Change is not only tested but also modeled, involving parameters
expressing change properties of the network. This is obviously a strong
point in favour of this approach. Still, the models “are purposefully
simple so that.. . they can be treated as null models, against which data
with more complicated
structure can be tested” (Wasserman
1980, p.
281).
(b) The assumption
of a continuous-time
process underlying
the
observations
is theoretically
attractive.
However, a price is paid in
terms of difficult mathematics
and the invocation
of unobservable
entities: after all, observation is necessarily restricted to some discrete
time points.
(c) The Markov assumption is a strong one, and will not always be
tenable.
(d) The likelihood calculations used to estimate the parameters
are
complex; see Wasserman (1980).
(e) Although Holland and Leinhardt (1977, p. 10) start by formulating a very general model for the dependence
of change intensities on
the network structure, all practical elaborations
of this model are much
more restrictive, with change intensities depending on reciprocity only,
or on in-degrees only, or on out-degrees only (see Wasserman, 1980).
Summarizing, the virtue of the Holland-Leinhardt-Wasserman
approach resides in the presence of parameters for change, which are
estimated within an elegant mathematical
model. The associated assumptions, however, will be too restrictive for some applications;
in
such applications,
the results may be deceiving. Of course, this holds
often for advanced modeling. The virtue of the approach proposed in
the present paper is its conceptual
simplicity; moreover, it is easily
applicable for those who have at their disposal the (publicly available)
Z-O software package.
The loglinear modeling approach of Wasserman (1987) and Wasserman and Iacobucci (1988) may be conceptually
somewhat less elegant
than the continuous-time
Markov approach; it is, however, much more
easily applicable, using standard loglinear modeling packages. If one
wishes to keep actor differences into account in the loglinear approach,
one might use models with incidental parameters (one or more parameters per actor). This leads, however, to statistical problems in parameter
T.A.B. Snijders
/ Testing for change
373
estimation. In order not to get caught in such problems, Wassermann
and Iacobucci (1988) divide the set of points (actors) into S subgroups,
the points in each subgroup having common loglinear model parameters. In the present paper, the focus is more on testing, whereas in
Wasserman
and Iacobucci (1988) the focus is more on estimation.
Testing in the loglinear modeling approach relies on the asymptotic
distribution
of likelihood ratio tests. In the case of network modeling,
little is known about requirements
on network size and number of
model parameters in order to make these approximations
sufficiently
precise. Moreover, it may be difficult to find a relevant division of
points into subgroups. This implies that the present approach will be
preferable if one wants to focus on testing randomness of change, while
the number of points in the network is too low for the loglinear
approach, or the division of points into homogeneous
subgroups is not
feasible; whereas if one focuses on parameter
estimation,
or if the
requirements
for validity of likelihood
ratio tests in the loglinear
modeling approach are met, the latter approach will be the preferred
one.
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