Multiple Regression Analysis of Reading Performance Data from Twin Pairs with Reading Difficulties and Non-twin Siblings: The Augmented Model

Abstract

The augmented multiple regression model for the analysis of data from selected twin pairs was extended to facilitate analyses of data from twin pairs and non-twin siblings. Fitting this extended model to data from both selected twin pairs and siblings yields direct estimates of heritability (h²) and the difference between environmental influences shared by members of twin pairs and those of sib or twin/sib pairs [i.e., c²_(t) − c²_(s)]. When this model was fitted to reading performance data from 293 MZ and 436 DZ pairs selected for reading difficulties, and 291 of their non-twin siblings, h² = .48 ± .22, p = .03, and c²_(t) − c²_(s) = .22 ± .12, p = .06. Although the test for differential shared environmental influences is only marginally significant, the results of this analysis suggest that environmental influences on reading performance that are shared by members of twin pairs (.36) may be substantially greater than those for less contemporaneous twin/sibling pairs (.14).

The basic and augmented multiple regression models that DeFries and Fulker (1985, 1988) advocated for analyzing data from selected twin pairs have become standards of behavioral genetic analysis. The basic model is appropriate for assessing the heritable nature of group deficits whereas the augmented model is used to assess heritability and shared environmentality of individual differences within the selected group. The basic and augmented models are as follows:

$$C=B_{1}P+B_{2}R+A,$$ (1)

$$C=B_{3}P+B_{4}R+B_{5}PR+A,$$ (2)

where C is the co-twin’s score, P is the proband’s score, R is the coefficient of relationship [1.0 and 0.5 for identical (MZ) and fraternal (DZ) twin pairs, respectively], and A is the regression constant. From the basic model, when the data are transformed by expressing each score as a deviation from the unselected population mean and dividing by the difference between the proband mean and the mean of the unselected population (i.e., the selection differential), B₂ = h²_g, an estimate of the extent to which the deviant scores of probands are due to heritable influences. From the augmented model, B₅ = h², a measure of the extent to which individual differences within the selected group are heritable, B₃ = c²_(t), a corresponding estimate of the extent to which environmental influences are shared by members of twin pairs, and B₄ = h²_g − h².

Recently, Astrom et al. (2011, submitted) extended the basic model to facilitate analyses of data from selected twin pairs and non-twin siblings as follows:

$$C=B_{1}P+B_{2}R+B_{6}S+A,$$ (3)

where C is the co-twin’s or co-sib’s score, P is the proband’s score, R is the coefficient of relationship (1.0 for MZ pairs, and 0.5 for DZ pairs and sib or twin/sib pairs), and S is a dummy code for pair type (+.5 for both MZ twin and DZ twin pairs, and −.5 for sib or twin/sib pairs). When this model is fitted to transformed twin/sib data, B₂ estimates h²_g from the twin data (i.e., twice the difference between the MZ and DZ cotwin means), and B₆ estimates the difference between the DZ co-twin and co-sib means, thereby providing a test of the difference between environmental influences shared by members of DZ twin pairs and those of sib or twin/sib pairs, i.e. c²_g(t) − c²_g(s).

For example, Astrom et al. (submitted) recently analyzed discriminant function data (a composite measure of reading performance) from 254 MZ and 420 DZ twin pairs in which at least one twin met proband criteria, and from 303 of their non-twin siblings. Results obtained from analyses of the twin-only data yielded estimates of h²_g = .667 and c²_g(t) = .252. When Equation 3 was fitted to the combined twin and co-sib/twin data, B₂ = h²_g = .667 and B₆ = c²_g(t) − c²_g(s) = .082. Although the B₆ coefficient was relatively small, it was significant (p = .02), suggesting that environmental influences shared by members of twin pairs (.252) are somewhat greater than those for less contemporaneous twin/non-twin sibling pairs (.252 − .082 = .170) for this measure.

In a corresponding manner, Equation 2 may also be easily extended to assess the etiologies of individual differences in selected twin pairs and non-twin siblings. Thus, the primary objective of this brief note is to present this extended augmented model and illustrate its application.

Materials and Methods

Subjects and Measures

The subjects participated in either the Colorado Reading Project (DeFries et al., 1991) or the ongoing Colorado Learning Disabilities Research Center (DeFries et al., 1997). In order to minimize the possibility of ascertainment bias, the sample was systematically obtained through 27 school districts in the state of Colorado. School administrators identified twin pairs in each school without regard to reading status, and then permission was sought from parents to review the school records of their children for evidence of reading problems (e.g., low reading achievement test scores or referral to resource rooms because of low reading performance). Parents were also asked whether either twin had difficulties learning to read. Twin pairs in which at least one member had a history of reading problems were invited to complete extensive test batteries of reading-related and cognitive measures in laboratories at both the University of Colorado, Boulder, and the University of Denver. Data from the Peabody Individual Achievement Test (Dunn and Markwardt, 1970) Reading Recognition, Reading Comprehension and Spelling subtests were then used to compute a discriminant function score for each child, employing weights estimated from an analysis of data from an independent sample tested during an earlier phase of the project (DeFries, 1985). Twin pairs with a positive history of reading problems were included in the proband sample if at least one member of the pair was also classified as affected by the discriminant score and had a Verbal or Performance IQ of at least 85, no serious neurological, emotional, or behavioral problems, and no uncorrected visual or auditory acuity deficits. Control twins were matched to probands on the basis of age, gender, and school district and had a negative history for reading problems.

Selected items from the Nichols and Bilbro (1966) twin questionnaire were used to classify zygosity. In doubtful cases, zygosity was confirmed by genotyping blood or buccal samples. Twin pairs were reared in English-speaking, primarily middle-class homes, and ranged in age from 8 to 20 years of age. Prior to testing, the session was fully described to family members, and informed consent was obtained from subjects 18 years of age or older. For children who were younger than 18, assent was obtained from the children and permission was obtained from their parents. The resulting sample included 293 MZ and 436 DZ pairs in which at least one member of each pair met proband criteria and 291 of their non-twin siblings. In addition, 728 control twin pairs were also tested.

Analyses

Table 1 presents the three expected co-relative/proband regressions, viz., co-twins on MZ probands (B_MZ), co-twins on DZ probands (B_DZ) and co-sibs on twin or sib probands (B_S), where c²_(t) is the proportion of observed variance due to environmental influences shared by twin pairs, and c²_(s) is that shared by probands and non-twin siblings. It can be seen that h² equals twice the difference between B_MZ and B_DZ. Further, the difference between B_DZ and B_S is a simple function of the difference between shared environmental influences in twin pairs versus those in sibling pairs.

Table 1.

Expected and Observed Twin and Co-Sib Regressions for Reading Performance^a

BMZ = h2 + c2(t) = .839

BDZ = .5 h2 + c2(t) = .599

BS = .5 h2 + c2(s) = .378

^aB_MZ is the regression of co-twin scores on MZ proband scores, B_DZ is the regression of co-twin scores on DZ proband scores, and B_S is the regression of co-sib scores on twin or sib proband scores.

Therefore, to test for differential c² between twin pairs and sibling pairs, the following simple extension of the augmented model can be fitted simultaneously to data from probands, their co-twins and co-sibs:

$$C=B_{3}P+B_{4}R+B_{5}PR+B_{6}S+B_{7}PS+A,$$ (4)

where PS is the product of the proband’s score and the dummy code (+.5 for twin pairs and −.5 for sib or twin/sib pairs). When Equation 4 is fitted to the data and the five partial regression coefficients are estimated simultaneously, B₇ = B_DZ − B_S and, thereby, estimates the extent to which individual differences that are due to shared environmental influences differ in twin pairs versus sib pairs, i.e., c²_(t) − c²_(s). Likewise, B₅ = 2(B_MZ − B_DZ), again estimating h², but only from the twin data.

Because truncate selection was employed (DeFries and Gillis, 1991), pairs in which both members met criteria for reading difficulties were double-entered for regression analyses. Standard error estimates and significance tests were adjusted accordingly. All regression models were fit using SPSS for Unix 17.0.

Results and Discussion

The observed regression coefficients for the discriminant function scores (a composite measure of reading performance) estimated from data of MZ twin pairs, DZ pairs and twin/sib pairs are also presented in Table 1. Doubling the difference between B_MZ and B_DZ indicates that h² = .480. Also, B_MZ − h² = c²_(t) = .359, and B_DZ − B_S = c²_(t) − c²_(s) = .221.

Results of fitting Equation 3 to the twin data only and Equation 4 to the twin/sibling data are presented in Table 2. When Equation 3 was fitted to data from twins only, the B₃ and B₅ estimates are c²_(t) = .359 and h² = .480 as expected. Similarly, when Equation 4 was fitted to the data from both twin pairs and twin/sib pairs, B₅ was again .480 and B₇ = .221. Although B₇ is only marginally significant (p = .056), this result suggests that shared environmental influences for members of twin pairs (.359) are substantially greater than those for less contemporaneous twin/sib pairs (359 − .221 = .138).

Table 2.

Results of fitting the Augmented and Extended-Augmented Models to Twin Data Only and to Data from Both Selected Twin Pairs and Non-twin Siblings

Twins only

B3 = c2(t) = .359 ± .148, t = 2.42, p = .016

B5 = h2 = .480 ± .197, t = 2.43, p = .015

Twins and non-twin co-sibs

B5 = h2 = .480 ±.219, t = 2.19, p = .028

B7 = c2(t) − c2(s) = .221 ±.116, t = 1.91, p = .056

As illustrated, the results obtained by fitting the augmented twin/co-sib model to the data may also be obtained by comparing the MZ, DZ, and co-sib/twin regressions presented in Table 1. However, fitting Equation 4 to the data obviously provides a more comprehensive analysis of combined twin and twin/co-sib data, as well as direct estimates and tests of significance for h² and c²_(t) − c²_(s). The multiple regression analysis of twin and sibling data is also more versatile. For example, Equations 1–4 may be easily extended to include various covariates and tests of significance for differential etiology as a function of either dichotomous or continuous variables (Wadsworth, Olson & DeFries, 2010).

Although fitting Equation 4 to the twin and co-sib data facilitates a more comprehensive analysis of the combined data set, it should be noted that in Table 2 the significance of the h² estimate is slightly less (p = .028) than that when Equation 2 is fitted to the twin-only data ( p = .015). As recently discussed by Astrom et al. (submitted), the inclusion of data from siblings in twin studies may result in a reduction in power, e.g., when shared environmental influences in twin pairs are substantially greater than those in twin/sib pairs. In such cases, Equation 2 should be fitted to the twin data to estimate h² and c²_(t), and then Equation 2 should be fitted to the combined twin/sibling data to test for the difference between shared environmental influences in twin pairs vs. twin/sib pairs. However, when shared environmental influences in twin pairs are not substantially greater than those for sib pairs, power may be increased by the inclusion of data from siblings in twin studies, especially for those with relatively small sample sizes (cf., Astrom et al., in press).

It should also be noted that Zieleniewski et al. (1987) previously extended Equation 2 to analyze data from both twin pairs and sibling pairs. However, their model included two dummy variables (E_T and E_S) to index shared environmental influences for twins and sibling pairs, respectively, rather than the single dummy variable (S) used in Equation 4. Moreover, fitting their extended model to combined twin and sibling data yields separate estimates of c²_(t) and c²_(s), but no direct test of significance for their difference. Although Zieleniewski et al. (1987) fitted subsequent models that equated E_T and E_S to obtain combined estimates of c², and then used changes in R² for tests of significance, fitting the more parsimonious Equation 4 to combined twin and sib-pair data facilitates direct estimates of c²_(t) − c²_(s) and corresponding significance tests.