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where the ys is a latent slope score that is constant over time, and the and are coef cients describing the change This second expression (Equation 1834) is more general because we can add restrictions ( = , = ) and obtain the rst expression (Equation 1833) We refer to this as a dual change score (DCS) model because it permits both a systematic constant change ( ) and a systematic proportional change ( ) over time, and no stochastic residual is added (ie, z[t]; see McArdle, 2001) This is an interesting linear model because the expectations lead to a mixed-effects model trajectory with a distinct nonlinear form (eg, A[t] in Equation 1822), but the corresponding accumulation of differences (Equation 1832) remains unchanged One advantage of this approach is that this dynamic model can tted using standard structural modeling software.

The structural path diagram in Figure 1812 illustrates how the latent change score model (Equations 1830 1834) can be directly represented using standard longitudinal structural equation models This set of equations is drawn in Figure 1812 by using (a) unit-valued regression weights among variables by xed nonzero constraints (as in McArdle & Nesselroade, 1994), (b) a constant time lag by using additional latent variables as placeholders (as in Horn & McArdle, 1980), (c) each latent change score as the focal outcome variable, and (d) a repetition (by equality constraints) of the and structural coef cients Following the standard linear growth models, we assume the unobserved initial-level component (y0) has a mean and variance ie, 0 and 2 , while the error of measurement has mean zero, has 0 constant variance 2 > 0), and is uncorrelated with every e other component.

FIGURE 24-24

As in the linear change model of Figure 187, the constant change component (ys) has a nonzero mean (ie, s , the average of the latent change scores), a nonzero variance (ie, 2 , the variability of the latent change scores), s and a nonzero correlation with the latent initial levels (ie, 0s ) As in other latent growth models, the numerical values of the parameters and can now be combined to form.

To begin your journey into the world of color, Photoshop provides four color controls in the Toolbox, as shown in Figure 4.1. These icons work as follows: n Foreground color: The foreground color icon indicates the color you apply when you use the Paint Bucket, Pencil, Brush, or one of the type tools, and also if you Alt-drag (Option-drag on the Mac) with the Smudge tool. The foreground color also begins any gradation created with the Gradient tool in its default state. Photoshop uses the current foreground color to fill any shape you create with the shape tools, and if you make a selection and then click inside it with the Paint Bucket or use the Edit Fill command, the foreground color fills it up. Remember that you also can apply the foreground color to a standard selection by pressing Alt+Backspace (Option-Delete on the Mac). To change the foreground color, click the foreground color icon to display the Color Picker dialog box. Then select a new color in the Color palette, or click anywhere in an open image window with the Eyedropper tool. You also can set the foreground color by clicking a swatch in the Swatches palette. (All are explained later in this chapter.)