Mathematics of the SRM (David A. Kenny)
Mathematics of the SRM
(click colored terms for definitions)

The basic model for the Social Relations Model (SRM) is a two-way random effects model. The random effects are actor and partner, and the relationship effect is the interaction of actor and partner. A random effect is assumed to be sampled from a population of effects. So the actors and partners are assumed to sampled from the population of persons. In random effects models, the focus is on the variance of the effects and not the specific level of the effect. So the interest is not in a particular actor's effect but rather in the variance across actors of the effect.

As in any analysis of variance, effect estimates can be computed. The estimation of effects for the round robin design is complicated by the missing data in the diagonal. However, the formulas have been developed for the estimation of effects (see Appendix B of Interpersonal Perception: A Social Relations Analysis). However, the variance of the effects themselves do not provide an very good estimate of the true variance of the effects. Consider the following data from a round-robin design of six persons:

```

Partner

A  B  C  D  E  F

A    A  -  4  5  6  7  5

c    B  2  -  1  3  2  4

t    C  5  4  -  3  2  6

o    D  4  5  4  -  2  5

r    E  1  4  4  3  -  3

F  5  7  5  3  9  -

```

Using specialized formulas, the estimates of the actor effects are 1.208, -1.625, -.167, -.208, -1.083, and 1.875, respectively for the six persons. The variance of these six estimates is 1.772. Though not obvious, it can be shown that this estimate of actor variance depends on the number of partners. If, using the very same data, we computed the estimate of actor variance from just five of the six persons, the actor variance would increase to 1.894. This value would be obtained if we re-analyzed the data thinking we had six groups, each group containing five of the members of the original group. Further reducing the group size leads to even lower estimates of variance. For instance, using four actors results in a even larger estimate of actor variance of 2.112. Generally, the greater the number of partners, the smaller the actor variance. It is undesirable that the estimate of actor variance depends on the number of persons sampled.

One solution to this problem is to use the data to forecast what the actor variance would be if there were an infinite number of partners. Of course, there are not and can never be an infinite number of partners, but it is possible to use statistical methods to make such a prediction (see Appendix B of Interpersonal Perception: A Social Relations Analysis). What is done is to subtract from the estimated actor variance the relationship variance (and relationship covariance which is described below) times a weighting factor that is based on the group size. For the example, with six persons the correction factor is .466 and so the estimate of the partner variance if there were an infinite number of partners is 1.306 (1.772 - .466). Note that 1.306 is less than the variance of the estimates. Also when the estimated variances are corrected for groups of size 4 and 5, the same value of 1.306 is obtained. Thus, although the estimated actor variance depends on the number of partners, the corrected actor variance does not.

Because there is the correction term that is used to estimate actor and partner variances, it is possible that the estimated variance can be negative. Although negative variances are impossible using the standard formula for a variance, they are possible in SRM analyses. These negative variances occur because the true variance is very near zero, and so about half the time it is estimated as zero. The usual practice is to treat these negative estimates as zero.

In Social Relations modeling, correlations can also be out of range. That is, the estimated correlation might be larger than one or smaller than minus one. Recall that estimates of actor and partner variances are estimates of what the variances would be if there were a large number of persons. In the same way, the correlations are forecasts of what the correlation would be if there were an infinite number of persons in the study.

The definitional formula for a correlation is the covariance divided by the product of the standard deviations of the two variables that are correlated. Recall that the standard deviations of both actor and partner variance are adjusted down by the relationship variance. Thus, the estimated SRM correlation tends to be larger because the denominator is smaller. One reasonable approach is not to compute correlations between actor or partner effects unless they explain at least ten percent of the total variance.

In principle, negative variances and out-of-range correlations could be avoided if maximum likelihood estimation, not least squares, were used. However, currently maximum likelihood estimation is not been used for the SRM due to its great computational complexity. However, it seems likely that eventually maximum likelihood will be the estimation method for the model.

When there are two variables there are six possible SRM correlations. There are four at the individual level -- actor-actor, actor-partner, partner-actor, and partner-partner -- where the first term represents one variable and the second the other. There are two dyadic correlations: relationship intrapersonal and relationship interpersonal. So if there are two variables, say gaze and liking, the six correlations would be as follows:

```

actor-actor:  Does a person who gazes more at others like others more?

actor-partner:  Is a person who gazes more at others liked by others more?

partner-actor:  Does looks a person who is gazed at more by others like

others more?

partner-partner:  Does looks a person who is gazed at more by others  liked by others more?

intrapersonal:  If a person especially gazes at another person, does that person especially like the other?

interpersonal:  If a person especially gazes at another person, is that

person especially liked by the other?

```

There is a further complication in the statistical estimation of Social Relations terms: nonindependence between terms. In the analysis of a single variable, there are two sources of nonindependence: the actor-partner correlation and dyadic correlation. The actor-partner correlation represents the correlation between a person's actor effect with his or her partner effect. The dyadic correlation is between pairs of relationship effects. If liking is measured, the dyadic correlation would assess whether if A particularly liked B, does B particularly like A? The actor-partner correlation would assess if someone tended to like others, was that person liked by others?

The last estimation complication to be discussed is the separation of error from relationship variance. Most of the error variance will be present in the relationship variance. To separate error from relationship variance, there must be multiple measures of the construct. Either the same construct is measured in two different ways or the same variable is measured at two points in time. If there is correlation between these multiple measures of the same theoretical construct, the correlation can be treated as relationship variance. More technically, for actor, partner, and relationship effect, variance is partitioned into stable and unstable components. Stable refers to variance that correlates across measures of the same construct and unstable refers to variance that does not correlate across measures.

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