Latent Curve Modeling

In Aim 1a, I used latent curve modeling to determine if PA and SB patterns over time can best be described as linear or curvilinear (i.e., quadratic) trajectories. Latent curve modeling is a modern longitudinal analysis technique in the SEM framework to estimate patterns of change.

Newer analytic methods were necessary because the HEAL dataset has a complex data structure that is not appropriate for traditional methods such an ANOVA. For instance, PA and SB variables in the HEAL dataset violate ANOVA’s four main assumptions: 1) no missing data; 2) equally spaced time points; 3) normal distribution; and 4) homogeneity of variance and covariance over time. Additionally, growth modeling may be a better tool than traditional methods for examining individual-level change described in theoretical models of health behavior. For example, Lazarus and Folkman’s Transactional Model of Stress and Coping provides an explanatory model of individual-level change in percevied threat, resources, and coping behavior. ANOVA would not be an appropriate test of this individual-level change because it test differences in groups means over time and cannot parse out individual-level change.

Therefore, latent curve modeling (LCM) was chosen as the analytic method. LCM relaxes the assumption that all individuals are drawn from a single population, resulting in separate intercepts, slopes, and variance parameters for each participant (Roth, 1994). LCM is flexible and robust to the following challenges encountered in the HEAL dataset: partially missing data, unequally spaced time points, non-normally distributed repeated measures, heterogeneity of variance and covariance over time, complex trajectories, and time-varying covariates (see Roth, 1994; Cohen & Cohen, 1983).

LCM has two parts: a measurement model and a structural model. The measurement model for LCM is a multivariate regression model that describes the relationship between one or more measured indicators and one or more latent variables (Goodman, 1974; Nylund, Asparouhov, & Muthen, 2007). The structural model for LCM describes three types of relationships in one set of

multivariate regression equations (relationships among latent variables alone, outcome variables alone, and latent and outcome variables together) (Goodman, 1974).

In LCM, repeated measures for a given individual are expressed as an additive function of the underlying trajectory weighted by time. Most breast cancer survivors in HEAL had data available for five time points (pre-diagnosis, six months post-diagnosis, and two, five, and ten years post-diagnosis), and thus five time points appear in the equation. For each time point, the general equation for an observed trajectory was in the form:

Observed trajectory for individual = intercept + slope(weighted by time) + error

Therefore, an individual breast cancer survivor’s linear model equation with five time points is:

y

ti

= ή

1i

+ ή

2i

λ

1

+ ή

2i

λ

2

+ ή

2i

λ

3

+ ή

2i

λ

4

+ ή

2i

λ

5

+ ε

ti

(1)

where t=time, i=individual, ή1i = intercept for individual i, λ=time point, ή2i = slope for individual i, and ε=error.

The equation is expanded to include a third eta (ή) at each time point for a quadratic (i.e., curvilinear) model:

y

ti

= ή

1i

+ ή

2i

λ

1

+ ή

2i

λ

2

+ ή

2i

λ

3

+ ή

2i

λ

4

+ ή

2i

λ

5

+ ή

3i

λ

2 1

+ ή

3i

λ

2 2

+ ή

3i

λ

2 3

+ ή

3i

λ

2 4

+ ή

3i

λ

2 5

+ ε

ti

(2)

where t=time, i=individual, ή1i = intercept for individual i, λ=time point, λ2=quadratic time point, ή2i = slope for individual i, ή3i=quadratic term for individual i, and ε=error.

Next, the mean linear trajectory with a covariate was calculated. The measurement equation is:

y

i

= Λή

i

+ ε

i (3)

The corresponding structural equation for the mean linear trajectory with covariates for the intercept and slope, respectively, is:

ή

1i

= α + y

11

x

1i

+ ζ

i

ή

2i

= α + y

21

x

1i

+ ζ

i

(4)

where ή = factor score, i=individual, α = factor mean, y and x = covariates, and ζ = factor disturbance.

Thus, these 2 mean linear trajectory with covariates equations reduced to:

y

ti

= Λ

1

(α

1

+α

2

λ

t

+y

11

x

1i

+y

21

λ

t

x

1i

) + (ζ

1i

+ ζ

2i

λ

t

+ ε

ti

)

(5)

where i=individual, Λ = factor loading, α = factor mean, λ=time point, y and x = covariate, ζ = factor disturbance, and ε = item residual. The mean linear trajectory is represented by the following structural equation model:

Figure 3.1. Mean linear trajectory structural equation model for physical activity (PA).

}

}

y

i

= Λή

i

+ ε

i

ή

1i

= α + y

11

x

1i

+ ζ

i

For a mean quadratic trajectory, the measurement equation is identical to the linear equation with 1 exception: the covariance matrix was expanded to include a third eta representing the quadratic function. Therefore, the quadratic trajectory equation is:

y

ti

= Λ

1

(α

1

+α

2

λ

t

+ α

3

λ

t

+y

11

x

1i

+y

21

λ

t

x

1i

+y

31

λ

t

x

1i

)

+ (ζ

1i

+ ζ

2i

λ

t

+ ζ

3i

λ

t

+ ε

ti

)

(6)

where i=individual, Λ = factor loading, α = factor mean, λ=time point, y and x = covariate, ζ = factor disturbance, and ε = item residual.

The structural equation model for the mean quadratic trajectory is:

Figure 3.2. Mean quadratic trajectory model for physical activity (PA).

Finally, PA and SB were modeled simultaneously using a growth modeling procedure called, “pararell process.”

}

}

y

i

= Λή

i

+ ε

i

ή

1i

= α + y

11

x

1i

+ ζ

i

ή

2i

= α + y

21

x

1i

+ ζ

i

ή

3i

= α + y

31

x

1i

+ ζ

i

The equation for a parallel process linear model with a covariate is:

y

i

= Λ

1

(α

1

+α

2

λ

t

+ α

3

λ

t

+α

4

λ

t

+y

11

x

1i

+y

21

λ

t

x

1i

)

z

i

+ (ζ

1i

+ ζ

2i

λ

t

+ζ

3i

λ

t

+ζ

4i

λ

t

+ε

ti

)

(7)

where i=individual, Λ = factor loading, α = factor mean, λ=time point, y and x = covariates, ζ = factor disturbance, and ε = item residual.

The equation for a parallel process quadratic model with a covariate is:

y

i

= Λ

1

(α

1

+α

2

λ

t

+ α

3

λ

t

+ α

4

λ

t

+ α

5

λ

t

+α

6

λ

t

+y

11

x

1i

+y

21

λ

t

x

1i

+y

31

λ

t

x

1i

z

i

+ y

12

x

2i

+y

22

λ

t

x

2i

+y

32

λ

t

x

2i

)

+ (ζ

1i

+ζ

2i

λ

t

+ζ

3i

λ

t

+ζ

4i

λ

t

+ζ

5i

λ

t

+ζ

6i

λ

t

+ε

ti

)

(8)

where i=individual, Λ = factor loading, α = factor mean, λ=time point, y and x = covariate, ζ = factor disturbance, and ε = item residual.

Figure 3.3. Parallel process model diagram for physical activity (PA) and sedentary behavior (SB).

}

}

y

i

= Λή

i

+ ε

i

Λ

1

(α

1

+α

2

λ

t

+ α

3

λ

t

+

α

4

λ

t

+ α

5

λ

t

+α

6

λ

t

+

y

11

x

1i

+

y

21

λ

t

x

1i

+y

31

λ

t

x

1i

+y

12

x

2i

+ y

22

λt

x2i

+ y

32

λt

x2i

) +

(ζ

1i

+ζ

2i

λ

t

+ζ

3i

λ

t

+ζ

4i

λ

t

+ζ

5i

λ

t

+ ζ

6i

λ

t

)

}

y

i

= Λή

i

+ ε

i

Mean variances for each parameter (intercepts, slopes, and quadratic coefficients) were examined to determine if they were significantly different from zero. Variance was calculated by subtracting each woman’s PA and SB estimates from the group mean and squaring the difference (Bollen & Curran, 2006). These differences were then averaged. High intercept variance

indicates significant individual variation in PA and SB levels reported for pre-diagnosis. High slope variance indicates large differences across women in propensity to change over time. Taken together, significant variance in intercept and slope coefficients implies the likely

existence of subgroups of women following different trajectories. However, the methodology of latent curve modeling cannot determine the number of subgroups. Thus, a second methodology was needed.

In document Stover_unc_0153D_15608.pdf (Page 83-89)