Model for Full Sample

Given the test results from Section 2.3.6, we combine the samples and fit the models to the full data set. We use similar procedures to the procedures of Sections 2.3.1 and 2.3.2. First, we fit model (2.6) using EGLS, where the weights are the inverses of the estimated variances given by (2.13) and (2.16) for elements in the 2003-2004 sample and 2005-2006 sample, respectively.

Second, we fit the variance model (2.12) using EGLS and the estimates and residuals from the initial model fit of (2.6), where the weights are the inverses of the estimated variances given by (2.13) and (2.16). Third, we refit model (2.12) using EGLS, where the weights are the inverses of the estimated values from the initial model fit. Fourth, we fit model (2.6) using EGLS with the estimated variances from the second fitting of model (2.12). The estimated regression coefficients and standard errors are given in Table 2.12. The estimates are similar to the estimates from Table 2.11, but have smaller standard errors given that the model is fit with a larger sample. Using the test described in Section 2.3.4, we test to see if the survey weights are significant in the regression estimation for the full sample. The F test statistic is 1.51 on (7, 3077) degrees of freedom with a p-value of 0.16. Hence, the evidence suggests that the survey weights have little influence on the expected value of the estimates.

To evaluate the EGLS model fit of (2.6) to the full sample, we look at group means and standard deviations of the standardized residuals. Let ˆy∗hij = x⁰_∗hijβˆ_EGLS^∗ and ˆe∗hij = y∗hij− ˆ

y∗hij for individual hij in the full sample, where x∗hij, y∗hij, and ˆβ_EGLS^∗ are defined in (2.8).

We sorted {ˆy∗hij, ˆe∗hij} by ˆy∗hij, divided the data into ten groups of approximately equal size, and computed group means of ˆy∗hij and ˆe∗hij and group standard deviations of ˆe∗hij. The first group in each of the samples is restricted to be the set of {ˆy∗hij, ˆe∗hij} values, where ˆy∗hij is equal to zero so that the group mean of ˆy∗hij is zero by default. Based on our model, individuals

Table 2.12 Estimated regression coefficients for models (2.6) and (2.12) fit to the full NHANES sample using EGLS

Model (2.6) Model (2.12)

Variable Est (SE) Coefficient Est (SE) trans 1.552 (0.191) α0 1.579 (3.484)

mod 1.184 (0.129) α₁ 12.417 (3.548) vig 1.692 (0.218) α₂ 1.203 (0.135) age2 0.104 (0.012)

mexblackmod -1.033 (0.170) mexvig -0.335 (0.497) mexblackage 0.035 (0.020)

older than 75 who report zero activity will have ˆy∗hij values of zero. Plots of the standardized group means and standard deviations are given in Figure 2.2. The plot on the left compares group means of the estimates to group means of the residuals. The plot on the right compares group means of the estimates to group standard deviations of the residuals. In the plot on the right, the standard deviation of the residuals for the first group is much lower than the standard deviations of residuals for the other nine groups. This occurs because a large majority of the residuals are zero when ˆy∗hij is zero. That is, the vast majority of individuals are estimated to have zero MVPA from the fitted model also measure zero accelerometer MVPA.

Figure 2.2 Plot of standardized group means and standard deviations

We consider an additional model adjustment to account for estimation of zero accelerometer MVPA. In the full sample there are 219 individuals with ˆy∗hij = 0 from the model fit. The sample mean and variance of measured accelerometer-based activity is 0.093 (min/day) and 0.612 (min/day)², respectively, for the same 219 individuals. We use these estimates as the intercepts in our mean and variance models. To implement the restrictions, we refit models (2.6) and (2.12) and fix the intercept in model (2.6) at 0.093 and the intercept in the variance model (2.12) at 0.612. The restricted models are

y_hij = 0.093 + β₁trans_hij+ β₂mod_hij+ β₃vig_hij+ β₄age2_hij+ β₅mexblackmod_hij + β6mexvighij+ β7mexblackagehij+ ehij

= 0.093 + x⁰_hijβ + ehij (2.17)

and

v_hij = 0.612 + α1(ˆy_hij)^α2, (2.18) where ˆyhij is the estimated value from model (2.17) for yhij.

To get estimates for the new means model we first regress y_hij−0.093 on the model variables (x_hij) in (2.17) using EGLS, where we use the estimated variances from the fitted variance model given in Table 2.12. The final set of estimates are computed using estimated variances from the restricted variance model (see below). We fit model (2.17) using only the data with positive ˆy∗hij values from the model fit of (2.6) given in Table 2.12. Let ˆβres denote the vector of fitted regression coefficients from model (2.17). The estimated variance matrix of ˆβ_res is

(X⁰X)⁻¹X⁰[I ˆσ_e²+ J J⁰σˆ_e0² ]X(X⁰X)⁻¹, (2.19) where X is the matrix of x∗hij variables from the model, I is an identity matrix, J J⁰ is a matrix of 1’s, ˆσ_e² is the estimated error variance from the model fit for the residuals with positive estimated MVPA, and ˆσ_e0² is the estimated variance of the mean of individuals with zero estimated MVPA.

Next, we fit model (2.18) using the squared residuals and ˆy_hij values from the model fit of (2.6), where the response variable is ˆe²_hij− 0.612. We fit this model with EGLS, using the

estimated variances from the variance model given in Table 2.12. We do the fitting using only the data with positive ˆy∗hij values from the model fit of (2.6) given in Table 2.12. The variance model is refit using the inverse of the estimated values from the first fit as the weights in the second fit to account for heterogeneity in the errors. Let ˆαres denote the vector of fitted regression coefficients from model (2.18). The estimated variance matrix of ˆαres is approximated using the form given by (2.19), where the rows in X are the partial derivatives of α1(ˆy_hij)^α2 with respect to α1and α2evaluated at ˆαres. The estimated regression coefficients and standard errors for models (2.17) and (2.18) are given in Table 2.13. Plots of the group means and standard deviations of the standardized residuals for the restricted model are given in Figure 2.3. The groups used to construct the plots in Figure 2.2 were also used to construct the plots in Figure 2.3.

Table 2.13 Estimates and standard errors for models (2.17) and (2.18) fit using EGLS

Model (2.17) Model (2.18)

Variable Est (SE) Coefficient Est (SE)

Intercept (restricted) 0.093 (0.002) Intercept (restricted) 0.612 (0.285)

trans 1.546 (0.221) α1 13.681 (3.510)

mod 1.181 (0.171) α2 1.167 (0.090)

vig 1.791 (0.245)

age2 0.095 (0.017) mexblackmod -0.984 (0.194) mexvig -0.587 (0.500) mexblackage 0.050 (0.021)