NETWORK > P1
PURPOSE Fits the Holland and Leinhardt P1 model for binary networks.
DESCRIPTION All dyads (i,j) in a sociometric choice matrix X can be classified as mutual (xij = xji = 1), asymmetric (xij not equal to xji), or null (xij = xji = 0). The probabilities of each type of dyad are modeled as a function of three sets of substantive parameters: expansiveness of each actor, popularity of each actor, and reciprocity. The probabilities of mutual, asymmetric and null dyads, denoted mij, aij, and nij respectively, are modeled as follows:
mij = lijexp(r+2q+ai+aj+bi+bj)
aij = lijexp(q+ai+bj)
nij = lij
In the equations, the a parameters are interpreted as "productivity" or "expansiveness" measures for each node. The b parameters are interpreted as "attractiveness" or "popularity" measures. The r parameter is interpreted as a general measure of the tendency towards "reciprocity" or "mutuality" in the network. The q parameter is a function of the density of the network, reflecting the total number of arcs observed. Finally, the l parameters are normalizing constants used to insure that the modeled probabilities add to 1 for any given dyad.
Name of file that contains network to be analyzed. Data type: Valued graph.
(Output) Parameter dataset (Default = 'Alphabet')
Name of file to contain alpha and beta parameters.
(Output) Expected values (Default = 'P1Expect')
Name of file to contain P1 expected values.
Output residual values (Default= 'P1Resid')
Name of file to contain P1 residuals.
LOG FILE G-squared negative goodness-of-fit value with degrees of freedom. Probabilities are not printed because the theoretical distribution governing these values has not yet been established.
Values of q and r .
Expansiveness (a) and popularity (b) parameters for each actor.
An nxn matrix containing the P1 expected value between each pair of actors.
An nxn matrix of residuals (observed data minus expected) between each pair of actors.
A single-link hierarchical clustering of symmetrized residuals.
COMMENTS The model would be more useful if the distribution of G-squared were known: as it is, we cannot say for certain when the model fits and when it does not.
REFERENCES Holland P and Leinhardt J (1981). "An Exponential Family of Probability Distributions for Directed Graphs." Journal of the American Statistical Association 76:33-6