Richard Gonzalez1
Analysis of Dyad and Group Data
The intraclass correlation is an index comparing the variability within a group to the variability across groups. The intuition is that if there is an effect due to social interaction, the variability of those subjects who interacted should be different (when measured as a social group) than the variability of subjects who did not interact (such as across social groups).
There are several ways of measuring variability, and these lead to different measures for the intraclass correlation (for a general review of many measures of the intraclass correlation see Shrout & Fleiss, [1979]). One of the common ways to measure variability uses an ANOVA framework. Within this framework one compares the mean squared between groups (variability of the group means from the grand mean; denoted MSB) to the mean squared within groups (variability of the individual scores from their respective group means; denoted MSW). As stated above, the underlying intuition is that if there is a group effect, then the variability within groups should differ from the variability between groups. This is the same intuition that leads to the omnibus F test in an analysis of variance. The difference in our application is that we are interested in the intraclass correlation as a measure of nonindependence rather than the usual concern of the ANOVA, which is a statistical test of the equality of group means. Below I will present a different, more intuitive, way to think about the intraclass correlation.
The term ``group'' above can refer to any relevant collection. For example, a researcher might have husbands and wives rate their satisfaction in the marriage. In this context the intraclass compares the variability within a dyad to the variability across dyads. Another example occurs in the context of assessing reliability. Consider four judges who rate several videotapes. Note how this design differs from the previous example, which had different husband and wife pairs. In the present design the same judges are used to rate a standard set of videotapes. Regardless of these design differences, the intraclass correlation is still relevant because it compares the variability within judges (across several videotapes) to the variability across judges (on a single videotape). Thus, the intraclass correlation has many uses such as in reliability theory.
The intraclass correlation measures a special meaning of the word independence. The idea is that if there is no group influence, then the variability within groups should be the same as the variability between groups. That is, when s2g = 0 reflecting no variability across groups, then MSB = MSW and the intraclass correlation equals 0. If the variability within groups is less than the variability across groups, then this is evidence of some kind of ``convergence'' within the group. The intraclass correlation will be greater than 0. However, if the variability within the group is greater than across groups, then this is evidence of ``divergence'' within a group and the intraclass correlation will be negative. Negative intraclass correlations need to be interpreted with caution.
The intraclass correlation is used to test the convergence or homogeneity of responses within groups. It applies only to the case of one dependent variable. When you want to test the relationship between two or more dependent variables, then one must turn to more complicated techniques (which we will cover during Days 3 and 4 of this workshop).
A natural situation to use an intraclass correlation is when group members are indistinguishable from one another. However, it is also possible to do an intraclass correlation when the partners are distinguishable, but it requires a small twist in the computation as shown below.
Assume that each observation is as additive function of three components: the grand mean, an effect due to group, and random noise. These three terms are denoted below as m, a, and e, respectively.
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| Roommate | ||
| Roommate Pair | 1 | 2 |
| 1 | ||
| 2 | ||
| 3 | ||
| 4 | ||
| 5 | ||
This design yields an ANOVA source table conceptually organized as follows:
| Source | Sum of Squares | Degrees of Freedom | Mean Square Terms | F | p-value |
| Between Groups | SSB | dfb | MSB | F | p |
| Within Groups | SSW | dfw | MSW | ||
| Total | SST | dft |
As I will show below, this source table is very important for computing the intraclass correlation. Note that everything I did in this section is identical to a one-way ANOVA where roommate pair is the grouping code (in this example there was one factor with 5 levels) and the grouping code is treated as a random effect. This makes sense because presumably roommate pairs were randomly sampled by the investigator to participate in the design. Once a roommate pair was randomly selected, then the two roommates are automatically in the study.
This question is tricky because there are several different ways to calculate the intraclass correlation depending on which theory one uses (method of moments or maximum likelihood) and how one treats the relevant factors (e.g., random v. fixed).
The most common formula for the intraclass correlation is in this context
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The intraclass correlation examines the difference between 1) the deviations from the group mean and the grand mean (i.e., MSB) and 2) the deviations of the individual scores from the group mean (i.e., MSW). This difference is then normalized by a weighted combination of the two components. The numerator is the important piece of this equation-the numerator determines the sign of the intraclass correlation.
It depends on the design. If the group members are indistinguishable, then the design is identical to a one-way ANOVA with the groups as the factor (as I showed above). The MSB and MSW are taken straight from the ANOVA source table.
For distinguishable group members (e.g., male/female, president/vice president) a two factor ANOVA is computed with groups and people as factors. The variable that is used to distinguish the people in the group should be theoretically meaningful. The two factor ANOVA gives a MS for groups, a MS for people, and an MSW. The MSW from this two factor ANOVA will usually be less than the MSW from the one-way ANOVA treating individuals as indistinguishable. The model for this two factor model is
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Note that there is no interaction term (the reason is that in this formulation there is one observation per cell and it is impossible to extract an independent interaction effect).
ID Group Person Score 1 1 1 71 2 2 1 79 3 3 1 105 4 4 1 115 5 5 1 76 6 6 1 83 7 7 1 114 8 8 1 57 9 9 1 114 10 10 1 94 11 11 1 75 12 12 1 76 13 1 2 71 14 2 2 82 15 3 2 99 16 4 2 114 17 5 2 70 18 6 2 82 19 7 2 113 20 8 2 44 21 9 2 113 22 10 2 91 23 11 2 83 24 12 2 72
First, we consider the case when dyad members are indistinguishable (such as in homosexual couples or same-sex roommates). The ANOVA for treating the cases as indistinguishable has one grouping code as a factor.
summary(aov(Score ~ factor(Group)))
.
Df Sum of Sq Mean Sq F Value Pr(F)
Between 11 8990.46 817.314 57.1882 1.26299e-08
Within 12 171.50 14.292
The intraclass correlation for the indistinguishable case is
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Now we turn to the case when the group members are distinguishable (such as husbands and wives who can be distinguished on gender). Note this ANOVA has two factors and no interaction term. The design is equivalent to a randomized block design (one observation per cell).
summary(aov(Score ~ factor(Group) + factor(Person)))
.
Df Sum of Sq Mean Sq F Value Pr(F)
Between Group 11 8990.46 817.314 61.8078 0.00000
Between Person 1 26.04 26.042 1.9693 0.18811
Within 11 145.46 13.223
This yields an intraclass correlation of
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In this example the difference between the two versions is trivial but it won't always be. Below I show an example where the intraclass goes from 1 to .78 as one switches analyses from the distinguishable case to the indistinguishable case for the same data.
We must first understand the ANOVA model that is being used. The assumption in the exchangeable case is that a particular score Yij for the ith person in the jth group is composed of three parts: a grand mean m, a group effect aj, and noise eij. People from the same group have the same aj but have different eij.
The aj are assumed to be normally distributed with mean 0 and variance t2, and the eij are assumed to be normally distributed with mean 0 and variance s2e.
The original question can now be answered, What do the terms MSB and MSW mean? The term MSW is an unbiased sample estimate of s2e. This is the noise attributable to individuals regardless of group membership. The term MSB is the unbiased sample estimate of Ns2g + s2e (where N is the size of the group). This is the variability of the group means and consists of two parts: noise due to people and the true variability of the groups.
If you have distinguishable dyads, then it may occur to you that a Pearson correlation on the raw data (i.e., husband scores correlated with wife scores) may be possible. There are several problems with this. One problem is that the Pearson correlation on the raw data doesn't measure agreement in a strict sense. That is, if you add a 1 to all the husband's score, the Pearson correlation won't change, but there is a sense in which the agreement between husband and wife is different. Unlike the Pearson correlation, the intraclass correlation is sensitive to additive constants.
Another argument for the intraclass correlation on distinguishable dyads, however, is that the intraclass correlation provides an adjusted R2 (i.e., a better estimate of the population R2). If the group size is greater than two, then the intraclass makes more sense than a Pearson correlation because the latter can only be computed over two people.
The intraclass correlation is not what nonstatisticians usually mean by a correlation because an intraclass correlation can be used when the members are indistinguishable whereas the Pearson correlation cannot be computed for indistinguishable cases.
Further, a Pearson correlation assesses association whereas the intraclass correlation assesses agreement. Association is a different concept from agreement. Association is about how two scores ``go together'' in the sense of a change in one score (or for one group member) produces, or is attached with, a linear change in the other score. However, agreement means that a score in one variable produces, or is attached with, an exact score on the other variable. One can also make use of the underlying geometry and show that association is about the angle between two vectors whereas agreement is about the distance between two vectors.2 I'll explain during the workshop the analogy of angles and association on one hand with distance and agreement on the other.
Given all those differences it is still possible to put the intraclass correlation can be put into the mathematical form of a correlation. Recall that the correlation is defined as
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All we need to show is that the numerator of the intraclass is a covariance and the denominator of the intraclass is the product of two standard deviations to prove that it is a correlation.
Recall that the expected values of MSB and MSW are ks2a + s2e and s2e, respectively. With a little high school algebra, one can re-express the intraclass
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The term s2a is a covariance because two people from the same group share the same a. The variance of each persons' score is s2a + s2e. Thus, the correlation between two people from the same group is
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The intraclass correlation is a ``mathematical correlation'' but it should not be confused with the standard treatment of a Pearson correlation involving the usual scatterplots because the former is not sensitive to linear transformation of the data.
If you can compute a meaningful Pearson correlation, then that means you have distinguishable cases (otherwise, the Pearson correlation would not make any sense because you wouldn't know which individual to put in ``column X'' and which individual to put in ``column Y'').
Consider this example:
example . ID Group Person Score 1 1 1 3 2 1 2 3 3 2 1 4 4 2 2 4 5 3 1 5 6 3 2 5 7 4 1 6 8 4 2 6 9 5 1 7 10 5 2 7
In the example the scores for each dyad member are identical. The Pearson correlation equals one as does the intraclass correlation for distinguishable cases.
summary(aov(Score ~ factor(Group) + factor(Person)))
Df Sum of Sq Mean Sq F Value Pr(F)
factor(Group) 4 20 5 2.62827e+31 0.000000
factor(Person) 1 0 0 4.00000e+00 0.105113
Residuals 4 0 0
But now consider the case where one partner always scores one point more. The correlation will still be one. But the only reason we can do a correlation is because the partners within a dyad are distinguishable. Therefore, we must also do a distinguishable intraclass correlation.
Example2 (add one to one partner's score) . Group Person Score 1 1 1 3 2 1 2 4 3 2 1 4 4 2 2 5 5 3 1 5 6 3 2 6 7 4 1 6 8 4 2 7 9 5 1 7 10 5 2 8
summary(aov(Score ~ factor(Group) + factor(Person)))
.
Df Sum of Sq Mean Sq F Value Pr(F)
factor(Group) 4 20.0 5.0 1.165237e+31 0
factor(Person) 1 2.5 2.5 5.826186e+30 0
Residuals 4 0.0 0.0
The one point difference between the
two partners is reflected in the MS for person. The MSW (or mean square
residual) equals 0 because the partners within a dyad did not differ in
variability but did differ by a constant. So, the Pearson correlation
equals one as does the intraclass correlation.
Using the same numerical example, if the partners are indistinguishable, then the intraclass will not be 1 even when the only difference between the partners is that one partner has one point higher than the other partner (which, in the distinguishable case would lead to a perfect Pearson correlation).
summary(aov(Score ~ factor(Group)))
.
Df Sum of Sq Mean Sq F Value Pr(F)
factor(Group) 4 20.0 5.0 10 0.0132602
Residuals 5 2.5 0.5
Using the same data for the indistinguishable case the intraclass
correlation becomes
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The intraclass correlation will be negative whenever MSB < MSW. In other words, the intraclass correlation will be negative whenever the variability within groups exceeds the variability across groups. This means that scores in a group ``diverge'' relative to the noise present in the individuals. A negative intraclass correlation is tricky to interpret because the intraclass is bounded on the negative side.
Unfortunately, the intraclass correlation has a lower bound of
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NO! The formula everyone uses is biased.
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An unbiased estimate of the intraclass correlation is given by this formula
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The unbiased form shares the same lower bound as the biased form.
Because most people don't care about the issue of bias, I won't discuss it again and will continue using biased measures in this workshop. Note that the bias gets very, very small as the number of groups gets large. Usually when you have 20 or 30 groups the bias is so small it won't make any difference in your data analysis.
The intraclass correlation is used in reliability theory. Recall that in classic test theory reliability is defined as
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The intraclass correlation also appears in the repeated-measures ANOVA where the assumption is that the intraclass correlation across time periods (as well as time periods × the between-subjects variables) is a constant. This is known as the compound symmetry assumption.
A third place where the intraclass correlation appears is in the adjusted R2 in the context of a random effects ANOVA. This is analogous to w2 (for a fixed-effects model) and closely related to h2 and R2.
The framework we use is the pairwise correlation, which dates back to Pearson ([1901]; see also Fisher, [1925]). The idea is to build the nonindependence directly into the organization of the data matrix, and then compute standard estimators on the reorganized data matrix with appropriate corrections to the standard error. The pairwise correlation is so named because each possible within-group pair of scores is used to compute the correlation. For example, on dyads with individuals Adam and Amos in the first dyad, there are two possible pairings: Adam in column one and Amos in column two; or Amos in column one and Adam in column two. This coding is represented symbolically in Table 1. Thus with N = 3 dyads, each column contains 2N = 6 scores because each individual is represented in both columns. The two columns (i.e., variables X and X�) are then correlated using the usual product-moment correlation. This correlation is denoted rxx�, and is called the pairwise intraclass correlation. It is an estimate of the intraclass correlation of one person's score with his or her partner's score. The pairwise intraclass correlation is the maximum likelihood estimate of the intraclass correlation and therefore is endowed with the usual properties of maximum likelihood estimators (such as consistency)3. For theoretical development of the exchangeable case, formulas, and examples see Griffin and Gonzalez ([1995]).
| Variable | ||
| Dyad # | X | X� |
| 1 | X11 | X12 |
| X12 | X11 | |
| 2 | X21 | X22 |
| X22 | X21 | |
| 3 | X31 | X32 |
| X32 | X31 | |
| 4 | X41 | X42 |
| X42 | X41 | |
The correlation rxx� indexes the absolute similarity between two exchangeable partners in a dyad. This can be seen in a simple scatterplot of X against X�. On this plot each dyad is represented twice, once as the point (Xi, X�i) and once as the point (X�i, Xi). We draw line segments between points from the same dyad. These line segments will all have a slope of -1 and are bisected by the identity line. An aggregate measure of the squared length of these line segments is proportional to rxx�; it is in this sense that the pairwise intraclass is a measure of similarity between dyad members. Note that when the two individuals in the dyad perfectly agree, then the line segments will have length 0 (i.e., all points will be on the identity line), and the pairwise intraclass correlation rxx� will equal 1. An analogous plot was proposed in the context of calibration and resolution of judgment (Liberman & Tversky, [1993]).
Two examples of these plots appear in Figure 1. The data, from Stinson and Ickes ([1992]), are the frequency of smiles and laughter between dyad members, separately for dyads consisting of strangers and dyads consisting of friends. For dyads of strangers the pairwise intraclass rxx� was .72 whereas for dyads of friends rxx� was .40. The plot highlights an interesting difference in interaction between friends and strangers. It appears that the interaction pattern between strangers involves matching each other's frequency of smiling to a higher degree than interaction between friends. That is, for strangers both partners' frequency of smiling was more similar than the frequency of smiling between friends. For friends, the interaction pattern consisted of pairs where one partner smiled relatively much more than the other partner. This matching difference was independent from the mean level of smiling. The closed circle on the identity line represents the mean frequency of smiles. Dyads of friends had a higher frequency of smiles than dyads of strangers, yet dyads of strangers had a higher degree of matching (as indexed by the pairwise intraclass).
It is important to remember that the correlation rxx� is computed over 2N pairs. However, because the correlation rxx� is based on 2N pairs rather than on N dyads as in the usual case, the test of significance must be adjusted. The sample value rxx� can be tested against the null hypothesis that rxx� using the asymptotic test4
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The pairwise intraclass correlation indexes the similarity of individuals within dyads, and is closely related to other estimators of the intraclass correlation such as the ANOVA estimator (Fisher, [1925]; Haggard, [1958]). However, the pairwise method has several important advantages in the present situation. Most important, it is calculated in the same manner as the usual Pearson correlation: the two ``reverse-coded'' columns are correlated in the usual manner, thus offering ease of computation, flexibility in the use of existing computer packages, and an intuitive link to general correlational methods. It also has certain statistical properties that make it ideal to serve as the basis for more complicated statistics of interdependence (e.g., it is the maximum likelihood estimator of the intraclass correlation on groups of equal size). Moreover, the pairwise method used to compute the intraclass correlation within a single variable can be used to compute the ``cross intraclass correlation'' across different variables, an important index discussed below. Thus, the pairwise approach can extend to multivariate situations.
The previous example on dyads (Table 1) was defined implicitly on dyads where the members are ``exchangeable''; that is, there is no a priori way to classify an individual in a dyad. Examples of exchangeable dyads include gay couples, same-sex roommates, and identical twins. However, examples of distinguishable dyads (such as heterosexual couples where individuals within a dyad can be classified by sex) also occur. The calculation of the partial pairwise intraclass correlation in the distinguishable case follows the same general pattern. In the distinguishable case the pairwise correlation model requires one extra piece of information: a grouping code indexing the dyad member. This extra information is needed because each dyad member is distinguishable according to some theoretically meaningful variable. One simply computes the usual partial correlation between the two reversed columns, i.e., partialling out the variable of the grouping code. This partial correlation is the maximum likelihood estimator of the pairwise intraclass correlation for the distinguishable case. For the theoretical background underlying the distinguishable case, relevant formulae, computational examples, and extensions to a structural equations modelling framework see Gonzalez & Griffin ([in press]).
| Variable | |||
| Dyad # | C | X | X� |
| 1 | 1 | X11 | X12 |
| 2 | X12 | X11 | |
| 2 | 1 | X21 | X22 |
| 2 | X22 | X21 | |
| 3 | 1 | X31 | X32 |
| 2 | X32 | X31 | |
| 4 | 1 | X41 | X42 |
| 2 | X42 | X41 | |
The sample estimate of the partial pairwise intraclass correlation is simply the Pearson correlation between X and X� partialling out variable C. The partial pairwise intraclass correlation is denoted rxx� .c. This correlation can be computed with standard statistical packages (e.g., the partial correlation routine in either SAS or SPSS). For completeness we present the formula for the partial correlation
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Here we show how to extend the pairwise approach to situations where all groups are of size k. The direct extension is to perform the pairwise coding for all possible combinations of dyads. For instance, in a group of size three with members denoted A, B and C, the possible combinations are AB, AC, BA, BC, CA, and CB. For each of the six combinations, data from the person coded on the left (e.g., A in AB) is entered into column X and data from the person coded on the right (e.g., B in AB) is entered into column X�. Thus, columns X and X� will contain 6N data points, where N is the number of groups. The Pearson correlation between columns X and X� is the pairwise intraclass correlation for the exchangeable case.
Obviously, with large groups the pairwise framework becomes cumbersome because of the many combinations that need to be coded, but it still maintains it's interpretational simplicity. A computational shortcut to the pairwise framework for groups is given by using a traditional analysis of variance source table. Compute a one-way ANOVA using the grouping code as the single factor (e.g., if there are 20 groups of size 4, then there will be 20 cells in the ANOVA, each cell having four observations). Denote the sum of squares between groups as SSB, the sum of squares within groups as SSW, and the corresponding mean square terms as MSB and MSW, respectively. The exchangeable pairwise intraclass correlation is identical to
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Extensions of the intraclass correlation (either in pairwise or ANOVA-based formulations) is not straightforward for situations where the size of the groups vary within the same study. For example, a study on families may have some families of size 3, some of size 4, etc. For preliminary treatments of this problem see Karlin, Cameron, and Williams ([1981]) and Donner ([1986]).
There has been much discussion lately in the groups literature about hierarchical linear models. They are useful because they are general and can be extended to many other situations (unequal group sizes, missing data, nonnormal data, etc). Here I will show how the intraclass correlation for the exchangeable case can be handled through HLM and also how HLM helps unify the ANOVA intraclass and the pairwise intraclass that I presented today. For an introductory comparison between standard ANOVA models and their HLM generalizations see Raudenbush ([1993]).
The intuition of HLM is very simple. The idea is to run a regression within each group (level 1 regressions), take the b's from those separate regressions and then use those bs as variables in a subsequent regression (level 2 regression). Of course, it isn't computationally efficient (nor correct) to manually do the two step process I just described. Rather, computer programs do both steps simultaneously. There are many reasons why it is a good idea to run everything simultaneously. For one, regressions should be aware that they are making use of parameters estimated by other regressions so that error estimates and degrees of freedom can be adjusted accordingly.
We'll begin with the exchangeable case. Let's define the level 1 regression as
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The level 2 regression will decompose the bi0 into constituent parts. This is accomplished by the following regression that uses bi0 as the dependent variable.
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An HLM program will estimate s2e and t2, and then on use those to estimates to compute the intraclass correlation, i.e.,
| (8) |
Here is a simple example using SAS PROC MIXED (identical results will emerge from other HLM programs such as Mln, HLM, VARCLUS, etc). The SAS code to read in the data and run the model is5
*INFILE 'rg1';
DATA rg1;
INFILE 'simple.data';
INPUT subject group data;
run;
proc mixed method=reml asycorr covtest cl info data=rg1;
class group;
model data = / solution predicted corrb;
random intercept/sub=group solution;
run;
THE OUTPUT IS:.
Covariance Parameter Estimates (REML)
Cov Parm Subject Estimate Std Error Z Pr > |Z| Alpha
INTERCEPT GROUP 0.44522222 0.38738209 1.15 0.2504 0.05
Residual 1.01777778 0.21456640 4.74 0.0001 0.05
Covariance Parameter Estimates (REML)
Lower Upper
0.1357 8.2051
0.7002 1.6146
Asymptotic Correlation Matrix of Estimates
Cov Parm Row COVP1 COVP2
INTERCEPT 1 1.00000000 -0.05538883
Residual 2 -0.05538883 1.00000000
Model Fitting Information for DATA
Description Value
Observations 50.0000
Res Log Likelihood -75.2790
Akaike's Information Criterion -77.2790
Schwarz's Bayesian Criterion -79.1709
-2 Res Log Likelihood 150.5581
Solution for Fixed Effects
Effect Estimate Std Error DF t Pr > |t|
INTERCEPT 2.08000000 0.33075671 4 6.29 0.0033
Correlation Matrix for Fixed Effects
Effect Row COL1
INTERCEPT 1 1.00000000
Solution for Random Effects
Effect GROUP Estimate SE Pred DF t Pr > |t|
INTERCEPT 1 -0.55347552 0.39410253 45 -1.40 0.1671
INTERCEPT 2 0.26045907 0.39410253 45 0.66 0.5121
INTERCEPT 3 0.34185253 0.39410253 45 0.87 0.3903
INTERCEPT 4 0.66742637 0.39410253 45 1.69 0.0973
INTERCEPT 5 -0.71626244 0.39410253 45 -1.82 0.0758
Predicted Values
DATA Predicted SE Pred L95 U95 Residual
0.0000 1.5265 0.2943 0.9337 2.1193 -1.5265
1.0000 1.5265 0.2943 0.9337 2.1193 -0.5265
3.0000 1.5265 0.2943 0.9337 2.1193 1.4735
1.0000 1.5265 0.2943 0.9337 2.1193 -0.5265
1.0000 1.5265 0.2943 0.9337 2.1193 -0.5265
2.0000 1.5265 0.2943 0.9337 2.1193 0.4735
2.0000 1.5265 0.2943 0.9337 2.1193 0.4735
1.0000 1.5265 0.2943 0.9337 2.1193 -0.5265
1.0000 1.5265 0.2943 0.9337 2.1193 -0.5265
2.0000 1.5265 0.2943 0.9337 2.1193 0.4735
2.0000 2.3405 0.2943 1.7477 2.9333 -0.3405
3.0000 2.3405 0.2943 1.7477 2.9333 0.6595
4.0000 2.3405 0.2943 1.7477 2.9333 1.6595
2.0000 2.3405 0.2943 1.7477 2.9333 -0.3405
1.0000 2.3405 0.2943 1.7477 2.9333 -1.3405
1.0000 2.3405 0.2943 1.7477 2.9333 -1.3405
2.0000 2.3405 0.2943 1.7477 2.9333 -0.3405
2.0000 2.3405 0.2943 1.7477 2.9333 -0.3405
3.0000 2.3405 0.2943 1.7477 2.9333 0.6595
4.0000 2.3405 0.2943 1.7477 2.9333 1.6595
2.0000 2.4219 0.2943 1.8290 3.0147 -0.4219
3.0000 2.4219 0.2943 1.8290 3.0147 0.5781
4.0000 2.4219 0.2943 1.8290 3.0147 1.5781
4.0000 2.4219 0.2943 1.8290 3.0147 1.5781
2.0000 2.4219 0.2943 1.8290 3.0147 -0.4219
1.0000 2.4219 0.2943 1.8290 3.0147 -1.4219
2.0000 2.4219 0.2943 1.8290 3.0147 -0.4219
3.0000 2.4219 0.2943 1.8290 3.0147 0.5781
2.0000 2.4219 0.2943 1.8290 3.0147 -0.4219
2.0000 2.4219 0.2943 1.8290 3.0147 -0.4219
2.0000 2.7474 0.2943 2.1546 3.3402 -0.7474
4.0000 2.7474 0.2943 2.1546 3.3402 1.2526
5.0000 2.7474 0.2943 2.1546 3.3402 2.2526
3.0000 2.7474 0.2943 2.1546 3.3402 0.2526
2.0000 2.7474 0.2943 2.1546 3.3402 -0.7474
1.0000 2.7474 0.2943 2.1546 3.3402 -1.7474
3.0000 2.7474 0.2943 2.1546 3.3402 0.2526
3.0000 2.7474 0.2943 2.1546 3.3402 0.2526
2.0000 2.7474 0.2943 2.1546 3.3402 -0.7474
4.0000 2.7474 0.2943 2.1546 3.3402 1.2526
1.0000 1.3637 0.2943 0.7709 1.9565 -0.3637
0.0000 1.3637 0.2943 0.7709 1.9565 -1.3637
2.0000 1.3637 0.2943 0.7709 1.9565 0.6363
1.0000 1.3637 0.2943 0.7709 1.9565 -0.3637
1.0000 1.3637 0.2943 0.7709 1.9565 -0.3637
2.0000 1.3637 0.2943 0.7709 1.9565 0.6363
1.0000 1.3637 0.2943 0.7709 1.9565 -0.3637
0.0000 1.3637 0.2943 0.7709 1.9565 -1.3637
1.0000 1.3637 0.2943 0.7709 1.9565 -0.3637
3.0000 1.3637 0.2943 0.7709 1.9565 1.6363
The PROC MIXED has three critical lines of syntax. The first is that it defines the variable ``group'' as a class variable; that means group will be treated like a factor in an ANOVA. The second line states the dependent variable is the variable data. The third line states that there is a random effect called the intercept which is nested within the group variable. The other words such as ``solution'', ``predicted'', ``corrb'', ``cl'', etc. pretty much handle printing options. The last critical subcommand is the the ``method=reml'' given in the PROC MIXED line. This tells SAS to use a ``REstricted Maximum Likelihood'' estimation procedure. This just means to use the standard ANOVA estimators for MS terms.
The output labels the parameter t2 as ``INTERCEPT GROUP'' and the parameter s2e as ``RESIDUAL''. No where in the output does one find the intraclass correlation printed, so it must be computed by hand using Equation 8. For this example we have an intraclass of
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Now here is something very interesting. If you change the command ``method=reml'' to ``method=ml'' (and don't change anything else) you automatically get the pairwise intraclass correlation. ``ml'' stands for ``Maximum Likelihood'', which is a slightly different estimator than the REML estimator. Interesting that what separates the ANOVA intraclass and the pairwise intraclass is whether or not ``re'' is added to the ``method'' option in PROC MIXED.
proc mixed method=ml asycorr covtest cl info data=rg1;
class group;
model data = / solution predicted corrb;
random intercept/sub=group solution;
run;
THE OUTPUT IS:
Covariance Parameter Estimates (MLE)
Cov Parm Subject Estimate Std Error Z Pr > |Z| Alpha
INTERCEPT GROUP 0.33582222 0.27759303 1.21 0.2264 0.05
Residual 1.01777778 0.21456640 4.74 0.0001 0.05
Covariance Parameter Estimates (MLE)
Lower Upper
0.1067 4.9168
0.7002 1.6146
Asymptotic Correlation Matrix of Estimates
Cov Parm Row COVP1 COVP2
INTERCEPT 1 1.00000000 -0.07729531
Residual 2 -0.07729531 1.00000000
Model Fitting Information for DATA
Description Value
Observations 50.0000
Log Likelihood -75.0338
Akaike's Information Criterion -77.0338
Schwarz's Bayesian Criterion -78.9458
-2 Log Likelihood 150.0675
Solution for Fixed Effects
Solution for Fixed Effects
Effect Estimate Std Error DF t Pr > |t|
INTERCEPT 2.08000000 0.29583779 4 7.03 0.0022
Correlation Matrix for Fixed Effects
Effect Row COL1
INTERCEPT 1 1.00000000
Solution for Random Effects
Effect GROUP Estimate SE Pred DF t Pr > |t|
INTERCEPT 1 -0.52184440 0.36006853 45 -1.45 0.1542
INTERCEPT 2 0.24557384 0.36006853 45 0.68 0.4987
INTERCEPT 3 0.32231566 0.36006853 45 0.90 0.3755
INTERCEPT 4 0.62928296 0.36006853 45 1.75 0.0873
INTERCEPT 5 -0.67532805 0.36006853 45 -1.88 0.0672
Predicted Values
DATA Predicted SE Pred L95 U95 Residual
0.0000 1.5582 0.2878 0.9785 2.1379 -1.5582
1.0000 1.5582 0.2878 0.9785 2.1379 -0.5582
3.0000 1.5582 0.2878 0.9785 2.1379 1.4418
1.0000 1.5582 0.2878 0.9785 2.1379 -0.5582
1.0000 1.5582 0.2878 0.9785 2.1379 -0.5582
2.0000 1.5582 0.2878 0.9785 2.1379 0.4418
2.0000 1.5582 0.2878 0.9785 2.1379 0.4418
1.0000 1.5582 0.2878 0.9785 2.1379 -0.5582
1.0000 1.5582 0.2878 0.9785 2.1379 -0.5582
2.0000 1.5582 0.2878 0.9785 2.1379 0.4418
2.0000 2.3256 0.2878 1.7459 2.9053 -0.3256
3.0000 2.3256 0.2878 1.7459 2.9053 0.6744
4.0000 2.3256 0.2878 1.7459 2.9053 1.6744
2.0000 2.3256 0.2878 1.7459 2.9053 -0.3256
1.0000 2.3256 0.2878 1.7459 2.9053 -1.3256
1.0000 2.3256 0.2878 1.7459 2.9053 -1.3256
2.0000 2.3256 0.2878 1.7459 2.9053 -0.3256
2.0000 2.3256 0.2878 1.7459 2.9053 -0.3256
3.0000 2.3256 0.2878 1.7459 2.9053 0.6744
4.0000 2.3256 0.2878 1.7459 2.9053 1.6744
2.0000 2.4023 0.2878 1.8226 2.9820 -0.4023
3.0000 2.4023 0.2878 1.8226 2.9820 0.5977
4.0000 2.4023 0.2878 1.8226 2.9820 1.5977
4.0000 2.4023 0.2878 1.8226 2.9820 1.5977
2.0000 2.4023 0.2878 1.8226 2.9820 -0.4023
1.0000 2.4023 0.2878 1.8226 2.9820 -1.4023
2.0000 2.4023 0.2878 1.8226 2.9820 -0.4023
3.0000 2.4023 0.2878 1.8226 2.9820 0.5977
2.0000 2.4023 0.2878 1.8226 2.9820 -0.4023
2.0000 2.4023 0.2878 1.8226 2.9820 -0.4023
2.0000 2.7093 0.2878 2.1296 3.2890 -0.7093
4.0000 2.7093 0.2878 2.1296 3.2890 1.2907
5.0000 2.7093 0.2878 2.1296 3.2890 2.2907
3.0000 2.7093 0.2878 2.1296 3.2890 0.2907
2.0000 2.7093 0.2878 2.1296 3.2890 -0.7093
1.0000 2.7093 0.2878 2.1296 3.2890 -1.7093
3.0000 2.7093 0.2878 2.1296 3.2890 0.2907
3.0000 2.7093 0.2878 2.1296 3.2890 0.2907
2.0000 2.7093 0.2878 2.1296 3.2890 -0.7093
4.0000 2.7093 0.2878 2.1296 3.2890 1.2907
1.0000 1.4047 0.2878 0.8250 1.9844 -0.4047
0.0000 1.4047 0.2878 0.8250 1.9844 -1.4047
2.0000 1.4047 0.2878 0.8250 1.9844 0.5953
1.0000 1.4047 0.2878 0.8250 1.9844 -0.4047
1.0000 1.4047 0.2878 0.8250 1.9844 -0.4047
2.0000 1.4047 0.2878 0.8250 1.9844 0.5953
1.0000 1.4047 0.2878 0.8250 1.9844 -0.4047
0.0000 1.4047 0.2878 0.8250 1.9844 -1.4047
1.0000 1.4047 0.2878 0.8250 1.9844 -0.4047
3.0000 1.4047 0.2878 0.8250 1.9844 1.5953
The intraclass correlation for this analysis is also found by applying Equation 8
| |||||||||||
This is identical to the value you would get using the pairwise formula I presented earlier. If you run an ANOVA on these data you will find that SSB = 21.88, SSW = 45.80, and k = 10 because there are 10 subjects in each of the 5 groups. Plugging in these numbers into Equation 4 yields the identical result for the intraclass as the SAS output using ``method=ml''
| |||||||||||
HLM is easiest through the windows interface but that interface does not include all the available options. There is also a syntax command that allows you to use all the available options. I'll give both the syntax file and excerpts from the resulting output.
#WHLM CMD FILE FOR hlmex.ssm nonlin:n numit:50 stopval:0.0000010000 level1:DATA=INTRCPT1+RANDOM level2:INTRCPT1=INTRCPT2+random/ fixtau:3 lev1ols:10 accel:5 resfil:n resfil:n hypoth:n homvar:n CONSTRAIN:N HETEROL1VAR:N LAPLACE:N,50 LVR:N title:no title output:C:\Rich\dyad\hlm2.out mlf:y
*************************************************************
* *
* H H L M M 22 *
* H H L MM MM 2 2 *
* HHHHH L M M M 2 Version 4.40 *
* H H L M M 2 *
* H H LLLLL M M 2222 *
* *
*************************************************************
SPECIFICATIONS FOR THIS HLM RUN Tue Jun 22 00:35:36 1999
-------------------------------------------------------------------------------
Problem Title: NO TITLE
Weighting Specification
-----------------------
Weight
Variable
Weighting? Name Normalized?
Level 1 no no
Level 2 no no
The outcome variable is DATA
The model specified for the fixed effects was:
----------------------------------------------------
Level-1 Level-2
Coefficients Predictors
---------------------- ---------------
INTRCPT1, B0 INTRCPT2, G00
The model specified for the covariance components was:
---------------------------------------------------------
Sigma squared (constant across level-2 units)
Tau dimensions
INTRCPT1
Summary of the model specified (in equation format)
---------------------------------------------------
Level-1 Model
Y = B0 + R
Level-2 Model
B0 = G00 + U0
Level-1 OLS regressions
-----------------------
Level-2 Unit INTRCPT1
------------------------------------------------------------------------------
1 1.40000
2 2.40000
3 2.50000
4 2.90000
5 1.20000
The average OLS level-1 coefficient for INTRCPT1 = 2.08000
Least Squares Estimates
-----------------------
sigma_squared = 1.35360
The outcome variable is DATA
Least-squares estimates of fixed effects
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 2.080000 0.164536 12.642 4 0.000
----------------------------------------------------------------------------
The outcome variable is DATA
Least-squares estimates of fixed effects
(with robust standard errors)
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 2.080000 0.295838 7.031 4 0.000
----------------------------------------------------------------------------
The least-squares likelihood value = -78.516119
Deviance = 157.03224
Number of estimated parameters = 2
STARTING VALUES
---------------
sigma(0)_squared = 1.01778
Tau(0)
INTRCPT1 0.33582
Estimation of fixed effects
(Based on starting values of covariance components)
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 2.080000 0.295838 7.031 4 0.000
----------------------------------------------------------------------------
The value of the likelihood function at iteration 1 = -7.503375E+001
Iterations stopped due to small change in likelihood function
****** ITERATION 2 *******
Sigma_squared = 1.01778
Standard Error of Sigma_squared = 0.21457
Tau
INTRCPT1 0.33582
Standard Errors of Tau
INTRCPT1 0.27759
Tau (as correlations)
INTRCPT1 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.767
----------------------------------------------------
The value of the likelihood function at iteration 2 = -7.503375E+001
The outcome variable is DATA
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 2.080000 0.295838 7.031 4 0.000
----------------------------------------------------------------------------
The outcome variable is DATA
Final estimation of fixed effects
(with robust standard errors)
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 2.080000 0.295838 7.031 4 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.57950 0.33582 4 21.49782 0.000
level-1, R 1.00885 1.01778
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 150.06750
Number of estimated parameters = 3
\end{verbatimcmd}
}
\newpage
\section{Appendix 1}
Data for HLM example. I used the data structure below both for SAS and
as input into the {\it HLM} program as level 1.
{\scriptsize
\begin{verbatim}
SUBID GROUP DATA
1 1 0
2 1 1
3 1 3
4 1 1
5 1 1
6 1 2
7 1 2
8 1 1
9 1 1
10 1 2
11 2 2
12 2 3
13 2 4
14 2 2
15 2 1
16 2 1
17 2 2
18 2 2
19 2 3
20 2 4
21 3 2
22 3 3
23 3 4
24 3 4
25 3 2
26 3 1
27 3 2
28 3 3
29 3 2
30 3 2
31 4 2
32 4 4
33 4 5
34 4 3
35 4 2
36 4 1
37 4 3
38 4 3
39 4 2
40 4 4
41 5 1
42 5 0
43 5 2
44 5 1
45 5 1
46 5 2
47 5 1
48 5 0
49 5 1
50 5 3
For HLM I also had to input a level 2 file structured like this:
GROUP GROUPID 1 1 2 2 3 3 4 4 5 5
1 These notes take liberally from joint work with Dale Griffin. When I say ``our'' or ``we'' in these notes, I'm refer to Dale Griffin and myself.
2 In the case of a dyad with distinguishable members the X and Y columns each define a vector in an N-dimensional space. The cosine of the angle between the two vectors is the correlation coefficient.
3 There are additional uses of the intraclass correlation. For instance, the intraclass correlation appears in reliability theory and can be used where a measure of similarity between two scores is needed.
4 To simplify matters,we have chosen to present large sample asymptotic significance tests. We present a null hypothesis testing approach rather than a confidence interval approach but the latter will also be developed. Deriving analytic results for confidence intervals over correlations has not been an easy problem. Fortunately, there have been recent advances in the variance components literature for deriving confidence intervals that are applicable to the pairwise models (e.g., Donner & Eliasziw, [1988]).
5 There are other ways to write the PROC MIXED code that produce identical output but I've chosen this way because it makes generalizations to what I want to do during Day 3 and 4 much easier. For instance, the line ``random intercept/sub=group solution'' can be replaced with ``random group/solution'' and the output is the same.