The following data are from Pothoff and Roy (1964) and consist of growth measurements for 11 girls and 16 boys at ages 8, 10, 12, and 14. Some of the observations are suspect (for example, the third observation for person 20); however, all of the data are used here for comparison purposes.

The analysis strategy employs a linear growth curve model for the boys and girls as well as a variance-covariance model that incorporates correlations for all of the observations arising from the same person. The data are assumed to be Gaussian, and their likelihood is maximized to estimate the model parameters. See Jennrich and Schluchter (1986), Louis (1988), Crowder and Hand (1990), Diggle, Liang, and Zeger (1994), and Everitt (1995) for overviews of this approach to repeated measures. Jennrich and Schluchter present results for the Pothoff and Roy data from various covariance structures. The PROC MIXED statements to fit an unstructured variance matrix (their Model 2) are as follows:

data pr; input Person Gender $ y1 y2 y3 y4; y=y1; Age=8; output; y=y2; Age=10; output; y=y3; Age=12; output; y=y4; Age=14; output; drop y1-y4; datalines; 1 F 21.0 20.0 21.5 23.0 2 F 21.0 21.5 24.0 25.5 3 F 20.5 24.0 24.5 26.0 4 F 23.5 24.5 25.0 26.5 5 F 21.5 23.0 22.5 23.5 6 F 20.0 21.0 21.0 22.5 7 F 21.5 22.5 23.0 25.0 8 F 23.0 23.0 23.5 24.0 9 F 20.0 21.0 22.0 21.5 10 F 16.5 19.0 19.0 19.5 11 F 24.5 25.0 28.0 28.0 12 M 26.0 25.0 29.0 31.0 13 M 21.5 22.5 23.0 26.5 14 M 23.0 22.5 24.0 27.5 15 M 25.5 27.5 26.5 27.0 16 M 20.0 23.5 22.5 26.0 17 M 24.5 25.5 27.0 28.5 18 M 22.0 22.0 24.5 26.5 19 M 24.0 21.5 24.5 25.5 20 M 23.0 20.5 31.0 26.0 21 M 27.5 28.0 31.0 31.5 22 M 23.0 23.0 23.5 25.0 23 M 21.5 23.5 24.0 28.0 24 M 17.0 24.5 26.0 29.5 25 M 22.5 25.5 25.5 26.0 26 M 23.0 24.5 26.0 30.0 27 M 22.0 21.5 23.5 25.0 ;

proc mixed data=pr method=ml covtest; class Person Gender; model y = Gender Age Gender*Age / s; repeated / type=un subject=Person r; run;

To follow Jennrich and Schluchter, this example uses maximum likelihood (METHOD=ML) instead of the default REML to estimate the unknown covariance parameters. The COVTEST option requests asymptotic tests of all the covariance parameters.

The MODEL statement first lists the dependent variable Y. The fixed effects are then listed after the equal sign. The variable Gender requests a different intercept for the girls and boys, Age models an overall linear growth trend, and Gender*Age makes the slopes different over time. It is actually not necessary to specify Age separately, but doing so enables PROC MIXED to carry out a test for heterogeneous slopes. The SOLUTION option requests the display of the fixed-effects solution vector.

The REPEATED statement contains no effects, taking advantage of the default assumption that the observations are ordered similarly for each subject. The TYPE=UN option requests an unstructured block for each SUBJECT=Person. The matrix is, therefore, block diagonal with 27 blocks, each block consisting of identical 44 unstructured matrices. The 10 parameters of these unstructured blocks make up the covariance parameters estimated by maximum likelihood. The R option requests that the first block of be displayed.

The results from this analysis are shown in Output 58.2.1–Output 58.2.9.

The Mixed Procedure

Model Information | |
---|---|

Data Set | WORK.PR |

Dependent Variable | y |

Covariance Structure | Unstructured |

Subject Effect | Person |

Estimation Method | ML |

Residual Variance Method | None |

Fixed Effects SE Method | Model-Based |

Degrees of Freedom Method | Between-Within |

In Output 58.2.1, the covariance structure is listed as "Unstructured," and no residual variance is used with this structure. The default degrees-of-freedom method here is "Between-Within."

Class Level Information | ||
---|---|---|

Class | Levels | Values |

Person | 27 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 |

Gender | 2 | F M |

In Output 58.2.2, note that Person has 27 levels and Gender has 2.

Dimensions | |
---|---|

Covariance Parameters | 10 |

Columns in X | 6 |

Columns in Z | 0 |

Subjects | 27 |

Max Obs Per Subject | 4 |

In Output 58.2.3, the 10 covariance parameters result from the unstructured blocks of . There is no matrix for this model, and each of the 27 subjects has a maximum of 4 observations.

Number of Observations | |
---|---|

Number of Observations Read | 108 |

Number of Observations Used | 108 |

Number of Observations Not Used | 0 |

Iteration History | |||
---|---|---|---|

Iteration | Evaluations | -2 Log Like | Criterion |

0 | 1 | 478.24175986 | |

1 | 2 | 419.47721707 | 0.00000152 |

2 | 1 | 419.47704812 | 0.00000000 |

Convergence criteria met. |

Three Newton-Raphson iterations are required to find the maximum likelihood estimates (Output 58.2.4). The default relative Hessian criterion has a final value less than 1E8, indicating the convergence of the Newton-Raphson algorithm and the attainment of an optimum.

Estimated R Matrix for Person 1 | ||||
---|---|---|---|---|

Row | Col1 | Col2 | Col3 | Col4 |

1 | 5.1192 | 2.4409 | 3.6105 | 2.5222 |

2 | 2.4409 | 3.9279 | 2.7175 | 3.0624 |

3 | 3.6105 | 2.7175 | 5.9798 | 3.8235 |

4 | 2.5222 | 3.0624 | 3.8235 | 4.6180 |

The 44 matrix in Output 58.2.5 is the estimated unstructured covariance matrix. It is the estimate of the first block of , and the other 26 blocks all have the same estimate.

Covariance Parameter Estimates | |||||
---|---|---|---|---|---|

Cov Parm | Subject | Estimate | Standard Error | Z Value | Pr Z |

UN(1,1) | Person | 5.1192 | 1.4169 | 3.61 | 0.0002 |

UN(2,1) | Person | 2.4409 | 0.9835 | 2.48 | 0.0131 |

UN(2,2) | Person | 3.9279 | 1.0824 | 3.63 | 0.0001 |

UN(3,1) | Person | 3.6105 | 1.2767 | 2.83 | 0.0047 |

UN(3,2) | Person | 2.7175 | 1.0740 | 2.53 | 0.0114 |

UN(3,3) | Person | 5.9798 | 1.6279 | 3.67 | 0.0001 |

UN(4,1) | Person | 2.5222 | 1.0649 | 2.37 | 0.0179 |

UN(4,2) | Person | 3.0624 | 1.0135 | 3.02 | 0.0025 |

UN(4,3) | Person | 3.8235 | 1.2508 | 3.06 | 0.0022 |

UN(4,4) | Person | 4.6180 | 1.2573 | 3.67 | 0.0001 |

The "Covariance Parameter Estimates" table in Output 58.2.6 lists the 10 estimated covariance parameters in order; note their correspondence to the first block of displayed in Output 58.2.5. The parameter estimates are labeled according to their location in the block in the Cov Parm column, and all of these estimates are associated with Person as the subject effect. The Std Error column lists approximate standard errors of the covariance parameters obtained from the inverse Hessian matrix. These standard errors lead to approximate Wald *Z* statistics, which are compared with the standard normal distribution The results of these tests indicate that all the parameters are significantly different from 0; however, the Wald test can be unreliable in small samples.

To carry out Wald tests of various linear combinations of these parameters, use the following procedure. First, run the statements again, adding the ASYCOV option and an ODS statement:

ods output CovParms=cp AsyCov=asy; proc mixed data=pr method=ml covtest asycov; class Person Gender; model y = Gender Age Gender*Age / s; repeated / type=un subject=Person r; run;

This creates two data sets, cp and asy, which contain the covariance parameter estimates and their asymptotic variance covariance matrix, respectively. Then read these data sets into the SAS/IML matrix programming language as follows:

proc iml; use cp; read all var {Estimate} into est; use asy; read all var ('CovP1':'CovP10') into asy;

You can then construct your desired linear combinations and corresponding quadratic forms with the asy matrix.

Fit Statistics | |
---|---|

-2 Log Likelihood | 419.5 |

AIC (smaller is better) | 447.5 |

AICC (smaller is better) | 452.0 |

BIC (smaller is better) | 465.6 |

Null Model Likelihood Ratio Test | ||
---|---|---|

DF | Chi-Square | Pr > ChiSq |

9 | 58.76 | <.0001 |

The null model likelihood ratio test (LRT) in Output 58.2.7 is highly significant for this model, indicating that the unstructured covariance matrix is preferred to the diagonal matrix of the ordinary least squares null model. The degrees of freedom for this test is 9, which is the difference between 10 and the 1 parameter for the null model’s diagonal matrix.

Solution for Fixed Effects | ||||||
---|---|---|---|---|---|---|

Effect | Gender | Estimate | Standard Error | DF | t Value | Pr > |t| |

Intercept | 15.8423 | 0.9356 | 25 | 16.93 | <.0001 | |

Gender | F | 1.5831 | 1.4658 | 25 | 1.08 | 0.2904 |

Gender | M | 0 | . | . | . | . |

Age | 0.8268 | 0.07911 | 25 | 10.45 | <.0001 | |

Age*Gender | F | -0.3504 | 0.1239 | 25 | -2.83 | 0.0091 |

Age*Gender | M | 0 | . | . | . | . |

The "Solution for Fixed Effects" table in Output 58.2.8 lists the solution vector for the fixed effects. The estimate of the boys’ intercept is , while that for the girls is . Similarly, the estimate for the boys’ slope is , while that for the girls is . Thus the girls’ starting point is larger than that for the boys, but their growth rate is about half that of the boys.

Note that two of the estimates equal 0; this is a result of the overparameterized model used by PROC MIXED. You can obtain a full-rank parameterization by using the following MODEL statement:

model y = Gender Gender*Age / noint s;

Here, the NOINT option causes the different intercepts to be fit directly as the two levels of Gender. However, this alternative specification results in different tests for these effects.

Type 3 Tests of Fixed Effects | ||||
---|---|---|---|---|

Effect | Num DF | Den DF | F Value | Pr > F |

Gender | 1 | 25 | 1.17 | 0.2904 |

Age | 1 | 25 | 110.54 | <.0001 |

Age*Gender | 1 | 25 | 7.99 | 0.0091 |

The "Type 3 Tests of Fixed Effects" table in Output 58.2.9 displays Type 3 tests for all of the fixed effects. These tests are partial in the sense that they account for all of the other fixed effects in the model. In addition, you can use the HTYPE= option in the MODEL statement to obtain Type 1 (sequential) or Type 2 (also partial) tests of effects.

It is usually best to consider higher-order terms first, and in this case the Age*Gender test reveals a difference between the slopes that is statistically significant at the 1% level. Note that the -value for this test () is the same as the -value in the "Age*Gender F" row in the "Solution for Fixed Effects" table (Output 58.2.8) and that the *F* statistic () is the square of the *t* statistic (), ignoring rounding error. Similar connections are evident among the other rows in these two tables.

The Age test is one for an overall growth curve accounting for possible heterogeneous slopes, and it is highly significant. Finally, the Gender row tests the null hypothesis of a common intercept, and this hypothesis cannot be rejected from these data.

As an alternative to the *F* tests shown here, you can carry out likelihood ratio tests of various hypotheses by fitting the reduced models, subtracting 2 log likelihoods, and comparing the resulting statistics with distributions.

Since the different levels of the repeated effect represent different years, it is natural to try fitting a time series model to the data within each subject. To obtain time series structures in , you can replace TYPE=UN with TYPE=AR(1) or TYPE=TOEP to obtain the first- or *n*th-order autoregressive covariance matrices, respectively. For example, the statements to fit an AR(1) structure are as follows:

proc mixed data=pr method=ml; class Person Gender; model y = Gender Age Gender*Age / s; repeated / type=ar(1) sub=Person r; run;

To fit a random coefficients model, use the following statements:

proc mixed data=pr method=ml; class Person Gender; model y = Gender Age Gender*Age / s; random intercept Age / type=un sub=Person g; run;

This specifies an unstructured covariance matrix for the random intercept and slope. In mixed model notation, is block diagonal with identical 22 unstructured blocks for each person. By default, becomes . See Example 58.5 for further information about this model.

Finally, you can fit a compound symmetry structure by using TYPE=CS, as follows:

proc mixed data=pr method=ml covtest; class Person Gender; model y = Gender Age Gender*Age / s; repeated / type=cs subject=Person r; run;

The results from this analysis are shown in Output 58.2.10–Output 58.2.17.

The "Model Information" table in Output 58.2.10 is the same as before except for the change in "Covariance Structure."

The Mixed Procedure

Model Information | |
---|---|

Data Set | WORK.PR |

Dependent Variable | y |

Covariance Structure | Compound Symmetry |

Subject Effect | Person |

Estimation Method | ML |

Residual Variance Method | Profile |

Fixed Effects SE Method | Model-Based |

Degrees of Freedom Method | Between-Within |

The "Dimensions" table in Output 58.2.11 shows that there are only two covariance parameters in the compound symmetry model; this covariance structure has common variance and common covariance.

Class Level Information | ||
---|---|---|

Class | Levels | Values |

Person | 27 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 |

Gender | 2 | F M |

Dimensions | |
---|---|

Covariance Parameters | 2 |

Columns in X | 6 |

Columns in Z | 0 |

Subjects | 27 |

Max Obs Per Subject | 4 |

Number of Observations | |
---|---|

Number of Observations Read | 108 |

Number of Observations Used | 108 |

Number of Observations Not Used | 0 |

Since the data are balanced, only one step is required to find the estimates (Output 58.2.12).

Iteration History | |||
---|---|---|---|

Iteration | Evaluations | -2 Log Like | Criterion |

0 | 1 | 478.24175986 | |

1 | 1 | 428.63905802 | 0.00000000 |

Convergence criteria met. |

Output 58.2.13 displays the estimated matrix for the first subject. Note the compound symmetry structure here, which consists of a common covariance with a diagonal enhancement.

Estimated R Matrix for Person 1 | ||||
---|---|---|---|---|

Row | Col1 | Col2 | Col3 | Col4 |

1 | 4.9052 | 3.0306 | 3.0306 | 3.0306 |

2 | 3.0306 | 4.9052 | 3.0306 | 3.0306 |

3 | 3.0306 | 3.0306 | 4.9052 | 3.0306 |

4 | 3.0306 | 3.0306 | 3.0306 | 4.9052 |

The common covariance is estimated to be , as listed in the CS row of the "Covariance Parameter Estimates" table in Output 58.2.14, and the residual variance is estimated to be , as listed in the Residual row. You can use these two numbers to estimate the intraclass correlation coefficient (ICC) for this model. Here, the ICC estimate equals . You can also obtain this number by adding the RCORR option to the REPEATED statement.

Covariance Parameter Estimates | |||||
---|---|---|---|---|---|

Cov Parm | Subject | Estimate | Standard Error | Z Value | Pr Z |

CS | Person | 3.0306 | 0.9552 | 3.17 | 0.0015 |

Residual | 1.8746 | 0.2946 | 6.36 | <.0001 |

In the case shown in Output 58.2.15, the null model LRT has only one degree of freedom, corresponding to the common covariance parameter. The test indicates that modeling this extra covariance is superior to fitting the simple null model.

Fit Statistics | |
---|---|

-2 Log Likelihood | 428.6 |

AIC (smaller is better) | 440.6 |

AICC (smaller is better) | 441.5 |

BIC (smaller is better) | 448.4 |

Null Model Likelihood Ratio Test | ||
---|---|---|

DF | Chi-Square | Pr > ChiSq |

1 | 49.60 | <.0001 |

Note that the fixed-effects estimates and their standard errors (Output 58.2.16) are not very different from those in the preceding unstructured example (Output 58.2.8).

Solution for Fixed Effects | ||||||
---|---|---|---|---|---|---|

Effect | Gender | Estimate | Standard Error | DF | t Value | Pr > |t| |

Intercept | 16.3406 | 0.9631 | 25 | 16.97 | <.0001 | |

Gender | F | 1.0321 | 1.5089 | 25 | 0.68 | 0.5003 |

Gender | M | 0 | . | . | . | . |

Age | 0.7844 | 0.07654 | 79 | 10.25 | <.0001 | |

Age*Gender | F | -0.3048 | 0.1199 | 79 | -2.54 | 0.0130 |

Age*Gender | M | 0 | . | . | . | . |

The *F* tests shown in Output 58.2.17 are also similar to those from the preceding unstructured example (Output 58.2.9). Again, the slopes are significantly different but the intercepts are not.

Type 3 Tests of Fixed Effects | ||||
---|---|---|---|---|

Effect | Num DF | Den DF | F Value | Pr > F |

Gender | 1 | 25 | 0.47 | 0.5003 |

Age | 1 | 79 | 111.10 | <.0001 |

Age*Gender | 1 | 79 | 6.46 | 0.0130 |

You can fit the same compound symmetry model with the following specification by using the RANDOM statement:

proc mixed data=pr method=ml; class Person Gender; model y = Gender Age Gender*Age / s; random Person; run;

Compound symmetry is the structure that Jennrich and Schluchter deemed best among the ones they fit. To carry the analysis one step further, you can use the GROUP= option as follows to specify heterogeneity of this structure across girls and boys:

proc mixed data=pr method=ml; class Person Gender; model y = Gender Age Gender*Age / s; repeated / type=cs subject=Person group=Gender; run;

The results from this analysis are shown in Output 58.2.18–Output 58.2.24. Note that in Output 58.2.18 Gender is listed as a "Group Effect."

The Mixed Procedure

Model Information | |
---|---|

Data Set | WORK.PR |

Dependent Variable | y |

Covariance Structure | Compound Symmetry |

Subject Effect | Person |

Group Effect | Gender |

Estimation Method | ML |

Residual Variance Method | None |

Fixed Effects SE Method | Model-Based |

Degrees of Freedom Method | Between-Within |

The four covariance parameters listed in Output 58.2.19 result from the two compound symmetry structures corresponding to the two levels of Gender.

Class Level Information | ||
---|---|---|

Class | Levels | Values |

Person | 27 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 |

Gender | 2 | F M |

Dimensions | |
---|---|

Covariance Parameters | 4 |

Columns in X | 6 |

Columns in Z | 0 |

Subjects | 27 |

Max Obs Per Subject | 4 |

Number of Observations | |
---|---|

Number of Observations Read | 108 |

Number of Observations Used | 108 |

Number of Observations Not Used | 0 |

As Output 58.2.20 shows, even with the heterogeneity, only one iteration is required for convergence.

Iteration History | |||
---|---|---|---|

Iteration | Evaluations | -2 Log Like | Criterion |

0 | 1 | 478.24175986 | |

1 | 1 | 408.81297228 | 0.00000000 |

Convergence criteria met. |

The "Covariance Parameter Estimates" table in Output 58.2.21 lists the heterogeneous estimates. Note that both the common covariance and the diagonal enhancement differ between girls and boys.

Covariance Parameter Estimates | |||
---|---|---|---|

Cov Parm | Subject | Group | Estimate |

Variance | Person | Gender F | 0.5900 |

CS | Person | Gender F | 3.8804 |

Variance | Person | Gender M | 2.7577 |

CS | Person | Gender M | 2.4463 |

As Output 58.2.22 shows, both Akaike’s information criterion (424.8) and Schwarz’s Bayesian information criterion (435.2) are smaller for this model than for the homogeneous compound symmetry model (440.6 and 448.4, respectively). This indicates that the heterogeneous model is more appropriate. To construct the likelihood ratio test between the two models, subtract the 2 log likelihood values: . Comparing this value with the distribution with two degrees of freedom yields a *p*-value less than 0.0001, again favoring the heterogeneous model.

Fit Statistics | |
---|---|

-2 Log Likelihood | 408.8 |

AIC (smaller is better) | 424.8 |

AICC (smaller is better) | 426.3 |

BIC (smaller is better) | 435.2 |

Null Model Likelihood Ratio Test | ||
---|---|---|

DF | Chi-Square | Pr > ChiSq |

3 | 69.43 | <.0001 |

Note that the fixed-effects estimates shown in Output 58.2.23 are the same as in the homogeneous case, but the standard errors are different.

Solution for Fixed Effects | ||||||
---|---|---|---|---|---|---|

Effect | Gender | Estimate | Standard Error | DF | t Value | Pr > |t| |

Intercept | 16.3406 | 1.1130 | 25 | 14.68 | <.0001 | |

Gender | F | 1.0321 | 1.3890 | 25 | 0.74 | 0.4644 |

Gender | M | 0 | . | . | . | . |

Age | 0.7844 | 0.09283 | 79 | 8.45 | <.0001 | |

Age*Gender | F | -0.3048 | 0.1063 | 79 | -2.87 | 0.0053 |

Age*Gender | M | 0 | . | . | . | . |

The fixed-effects tests shown in Output 58.2.24 are similar to those from previous models, although the *p*-values do change as a result of specifying a different covariance structure. It is important for you to select a reasonable covariance structure in order to obtain valid inferences for your fixed effects.

Type 3 Tests of Fixed Effects | ||||
---|---|---|---|---|

Effect | Num DF | Den DF | F Value | Pr > F |

Gender | 1 | 25 | 0.55 | 0.4644 |

Age | 1 | 79 | 141.37 | <.0001 |

Age*Gender | 1 | 79 | 8.22 | 0.0053 |