Appendix

In the body of the paper, we have focused on general results, and relied on the figures to present detail about the forms of the change trajectories. Here, we present information about the parameters that describe the curves, as well as the variance components that describe individual variation.Each HLM model estimates the change trajectory for some measure of caregiver speech using a regression equation that predicts the measure with an intercept and a slope that reflects change over time. We code time so that its value is zero at the first observation, one at the second, and so on, ending at four for the fifth observation. In this system, the intercept is the expected value for the speech measure at the time of the first observation (14 months), and the slope represents the expected change between adjacent time points. For the one variable (words per sentence) which exhibited a slight curve in the change trajectory, a more complex model was used in which a second slope reflected quadratic change over time. In addition to the model for time, we consider one time-varying covariate: the presence or absence of older siblings during the visit. The HLM analyses are roughly equivalent to estimating regression equations separately for each caregiver, and summarizing the values of growth parameters across people.

Next, we consider how those linear or quadratic models vary across individuals. We employ a dummy coding system for education, which permits each educational group to have a different mean growth trajectory. In this system, we estimate the intercept for one group (the group with the highest level of education) and express the intercepts of other groups as changes or differences from that reference group. In addition, we consider models in which education predicts the slope (and, in the case of Words per Sentence, the quadratic slope). We investigate these explanatory models for any growth parameter that exhibits significant variation across individual participants, regardless of whether the average value of the parameter is significantly different from zero. This allows us to investigate the possibility that different education groups exhibit different growth rates as well as different starting points, and guards against the possibility that the flat growth trajectories for some outcomes result from a situation in which differential growth for different groups averages to zero.

Although education explains some of the variations in the parameters (in most cases the intercept, but for proportion of multi-clause sentences, the slope), there are generally still unexplained individual differences. The amount of the unexplained variation is quantified by variance component estimates, which are the estimated variances of the individual differences (in intercept or slope) within educational groups. We report the square root of these variance components, so that results are in the more interpretable standard deviation metric.

Appendix Table A1 provides a summary of the results of our final analyses. Each column of the table provides results for a different language measure. For each of these measures, the first four rows of the table present the estimates of the average intercept for the highest educational group and the differences in mean intercept between that group and each of the other groups, along with the standard errors of these estimates (in parentheses). The fifth row of the table presents the effects of the presence of older siblings. The sixth and seventh rows of the table present the linear and quadratic slopes (if these are statistically significant) along with their standard errors (in parentheses). The last two rows of the table present the variation of individual differences in intercepts or slopes (when they are statistically significant) within educational groups, expressed as standard deviations (square roots of the variance components). Note that the differences in intercept associated with caregiver education are very large in comparison to the remaining individual differences. For the quantitative measures the difference in intercept between the highest and lowest educational groups is approximately five standard deviations (five times the variance component for the intercept). For the compositional measures the difference in intercept between the highest and lowest educational groups is approximately ten standard deviations (ten times the variance component for the intercept).

For some of the variables, transformations or other special handling can improve the validity of statistical inference. One reason this may arise is because the variable is a proportion (e.g., the proportion of complex sentences), in which case we employ a logistic regression model rather than a conventional regression. Another reason transformations may be necessary is that the distribution of errors of prediction is not constant across the range of model-predicted values. This was true for several of our variables. Accordingly, we analyze the square root of number of tokens, number of utterances, number of sentences, and diversity—number of word types and sentence types. Inferences for these for variables are conducted in the square-root metric. However, for the variables that were transformed by the square root, the parameters that we present are in the untransformed metric, as these untransformed models are more easily interpreted, and the transformed and untransformed models result in very similar predictions.

The process of fitting the models and exploring moderating variables involved trying a number of different possible models for each of our measures. In the interests of parsimony, we sought to reduce these to models in which only significant coefficients remained. Table A1 presents the parameter estimates for each of the measures that resulted from that process; these are the parameter values that were used to produce the figures depicting the best-fitting change trajectories.

Table A1

Growth curve parameters and standard errors for the eight language measures

Parameter / Word Tokens / Number of Utterances / Number of Sentences / Word Types / Sentence Types / Complex Sentences / Noun Phrases per Sentence / Words per Sentence
Intercepts
For highest level of education (advanced degree) / 4097.88
(342.21) / 1033.55
(79.36) / 713.01
(56.67) / 374.49
(23.56) / 7.39
(0.68) / -2.48
(0.05) / 1.46
(0.04) / 4.69
(0.13)
Change for bachelor’s degree / -607.06
(468.25) / -101.43
(109.67) / -101.19
(77.79) / -24.67
(32.08) / -1.34
(0.91) / -0.01
(0.03) / -0.04
(0.05) / -0.13
(0.17)
Change for some college / -870.38
(570.86) / -214.15
(133.13) / -118.97
(94.57) / -40.23
(38.90) / 0.09
(1.11) / -0.06
(0.04) / -0.15
(0.07) / -0.35
(0.20)
Change for high school only / -2307.71
(660.67) / -563.29
(153.45) / -374.44
(109.37) / -156.35
(44.82) / -2.98
(1.29) / -0.12
(0.04) / -0.25
(0.08) / -0.68
(0.24)
Change if older siblings present / -1009.50
(251.66) / -274.40
(56.91) / -190.44
(44.75) / -58.36
(14.92) / -1.69
(0.69) / -- / -- / --
Slopes
Linear change / -- / -- / -- / 15.91
(3.06) / 1.03
(0.14) / 0.19
(0.02) / 0.06
(0.01) / 0.08
(0.05)
Quadratic change / -- / -- / -- / -- / -- / -- / -- / 0.03
(0.01)
Amount of Unexplained Individual Differences
Square root of variance component for intercept / 1348.91 / 306.88 / 221.20 / 93.21 / 2.12 / 0.36 / 0.16 / 0.50
Square root of variance component for slope / -- / -- / -- / 15.11 / 0.53 / 0.10 / 0.04 / 0.23

Note: Standard Errors are in parentheses below each estimate. For complex sentences, education increments are adjustments to the slope, not to the intercept