Illustrative examples of serial dependence

In this section we illustrate the procedures introduced in Chapter 5. First, results of a simulation study are described to make comparisons of different methods with respect to their coverage probabilities and expected areas. Next, a biological data set is utilized to compare the estimated trends via MLE method and our procedure. Finally, the generalized p-values to test a set of simulated data are presented.

6.3.1 Simulated studies (Comparison of coverage probabilities)

In simulation studies, we generate the data sets with the common trend of (0 2) matrices among groups. For demonstration purposes, we compare five procedures with respect to their coverage probabilities and expected areas. These five methods are as follows:

(1) Diff: The generalized method with different covariance matrices proposed in Section 5.2.2.

(2) Equal: The generalized method with identical covariance matrix.

(3) GC: Growth-curve method described in Section 4.2.

(4) CHI: Classical Chi-square approximation described in Section 5.3.

(5) Hotell: Hotelling’s T –statistic² described in Section 5.3.

Based on 1,000 replicates in each combination and 5,000 runs in the generalized methods (1) and (2), the coverage probabilities of the five methods under different combinations are given in Table 6.8, and the corresponding estimated expected areas of 95% confidence region are given in Table 6.9 under different scenarios.

Table 6.8: Comparison of 95% coverage probabilities of β under σ₁=1, ρ₁=0.1 From Table 6.8 and Table 6.9, we can see that the coverage probabilities obtained by the classical Chi-square approximation were below the nominal level 0.95 in all cases although its expected areas were small. Similarly, the coverage probabilities, obtained by methods (2), (3) and (5) with the identical covariance matrix assumption, decrease when the heteroscedasticities increase. On the other hand, the method (1), the generalized method without the equal covariance matrix assumption, had good coverage probabilities in all cases even when the heteroscedasticities among groups were large.

Table 6.9: Expected areas of 95% confidence regions of β under σ₁=1, ρ₁=0.1

(n₁=15, n₂ =7) Diff Equal GC CHI Hotell

Hence, based on the overall comparisons, the generalized method without equal covariance matrix assumption is better than the other four methods with respect to their coverage probabilities and expected areas, especially when small sample sizes are associated with large variances.

6.3.2 Example 3: the dental data

The dental data for 11 girls and 16 boys at ages 8, 10, 12 and 14 years were first considered by Potthoff and Roy (1964) and later analyzed by Lee and Geisser (1975), Lee (1988) and many others. The design matrix is set to be 1 1 1 1

(5.13) and (5.16), the generalized p-values for testing the equality of the trends are about 4*10^-7 and 2*10^-8 for distinct covariance matrices and equal covariance matrices, respectively.

Table 6.10: Estimated trends, expected areas and hypotheses of the dental data set The generalized method Growth-curve method Group Estimated trend Expected area Estimated trend Expected area 11 girls (22.638 0.485 )′ 0.999 (22.639 0.485 )′ 0.924 16 boys (25.063 0.769 )′ 1.113 (25.027 0.773 )′ 0.929 15 boys (25.107 0.782 )′ 1.083 (25.092 0.782 )′ 0.963

Lee and Geisser had pointed out that individual 20, who is a boy, should be excluded. In this case, the generalized p-values are about 6*10^-6 and 5*10^-6 under distinct covariance matrices and equal covariance matrices, respectively. Hence we treated this dental data as arising from two different groups with distinct trends and

serial covariance matrices. We used 10,000 runs to apply the generalized method to estimate trends and the expected areas of the 95% confidence region for the trends of 11 girls, 16 boys and 15 boys. The results are summarized in Table 6.10.

From Table 6.10, we can see that the estimated trends obtained by the generalized method and the growth-curve method are quite similar; however, the expected areas via the generalized method are slight larger than those via the growth-curve method. In general, the larger the expected areas, the larger the coverage probabilities. The simulation study in Section 6.3.1 also shows this phenomenon.

6.3.3 Example 4: the simulated data (Testing equality of the trends)

In order to illustrate our procedures to test the equality of the trends, five sets of data were generated assuming serial dependence regression model (5.1) with the small sample sizes n_i = 1, ,58, i= . The values of the parameters are σ₁=1, σ₂ =1.5, data sets are presented in Table 6.11. The p-values for testing the equality of the trends are displayed in Table 6.12.

It is noted that the p-values in Table 6.12 are computed with 10,000 runs in each combination, p means the p-value under equal covariance matrices assumption by _e using formula (5.16) and p means the p-value without the assumption of equal _u covariance matrices by (5.13). The smaller the p-values, the stronger is the evidence to reject the null hypothesis. From Table 6.12, the numerical results showed that when groups are homogeneous, both p and _e p reached the same conclusion that there was _u not sufficient evidence to reject the null hypothesis. However, when heteroscedasticity is present, p usually fails to detect the differences between groups. On the other hand, _e p is more sensitive and is able to detect the differences between the distinct groups. u

Thus compared to p , _e p is more powerful than _u p under heteroscedasticity. _e

It is also noted that if we change the serial dependence into uniform covariance structure in program, the results with the procedure in Section 5.2 are very close to

Table 6.11: The simulated data set for 5 groups

i=4 3.357 -0.004 12.236 13.394 13.968

i=4 9.838 16.003 20.672 19.727 20.724

i=4 9.826 13.713 7.021 17.078 13.031

i=4 11.567 9.680 10.762 17.249 22.972

i=5 5.634 13.870 21.912 -4.802 22.294

i=5 2.952 21.233 -26.393 15.328 4.827

Table 6.12: The generalized p-values for the testing equality of the trends

Hypothesis p u p _e

H :01 β₁ =β₂ =β ₃ 0.083499 0.189677

H :02 β₁=β₂ =β₄ 0.000447^* 0.000001^*

H :03 β₁=β₂ =β ₅ 0.047343^* 0.082289

H :04 β₁=β₃ =β ₄ 0.000418^* 0.000002^*

H :05 β₁=β₃ =β ₅ 0.032971^* 0.093071

H :06 β₁=β₄ =β ₅ 0.000187^* 0.106036

H :07 β₂ =β₄ =β ₅ 0.000231^* 0.098681

H :08 β₃ =β₄ =β ₅ 0.000398^* 0.113409

H :09 β₂ =β₃ =β₄ =β ₅ 0.000365^* 0.057247

* significance under the nominal level 0.05

Table 6.13: The generalized p-values for the testing equality of growth curves with uniform covariance matrices

Chi and

Weerahandi(1998) Section 5.2 (uniform) Examples

p u p _e p _u p _e

(1) Serious heteroscedasticity 0.0236 0.0817 0.0245 0.0826 (2) Mild heteroscedasticity 0.0441 0.0113 0.0491 0.0112

We demonstrate the advantages of our proposed method when there are few subjects or few measurements taken over time in this section. The other traditional methods usually are restricted to specific conditions that are sometimes violated when the serial covariance matrices are quite different or the sample sizes are small. According to the numerical examples, our proposed method is superior since it does not require the assumption of equal covariance matrices and the generalized p-values are better able to detect the differences between the trends among the groups and for the single group case, the estimated trend is the same via the growth-curve method. Moreover, the coverage probabilities and the expected areas for this method are satisfactory while the other methods become worse when the heteroscedasticities increase.