What to set "intercept factor" and "slope factor" as in model?

Question

I was struggling a bit with understanding what exactly the intercept factor and slope factor is in a model. I need to estimate these in an a priori power analysis I am trying to do.

My understanding is that a "high" intercept would be 1, with the slope being the rate of change (either positive or negative). I noticed you can set the intercept above 1, and wondered when this would be the case. What is the difference between an intercept factor of 1 and 2? Is an intercept factor of 1 with a slope of -0.1 the same as an intercept factor of 2 with a slope of -0.2, for example.

How does the intercept change depending on the observed measure, if at all?

Added R code below. Latent variable is depression, with observed measure of PHQ-9 scores collected at 7 timepoints. I plan to analyse this with a latent growth curve model.

library(semPower)
powerLGCM <- semPower.powerLGCM(
  # define type of power analysis
  type = 'a-priori', alpha = .05, power = .80,
  # define hypothesis 
  nWaves = 7,
  means = c(1, -0.15),     # i, s
  variances = c(.8, .7),  # i, s
  covariances = .5,
  nullEffect = 'sMean = 0',
  # define measurement model
  nIndicator = rep(1, 7), loadM = .7
)

Answer 1

I would strongly suggest that you read a bit more about latent growth curve (LGC) models before you run your power analysis. This will help you understand what the parameters of an LGC model are, how they relate to your observed variables, and how to set the parameters appropriately for your power analysis. There is a lot of good introductory literature available on this topic, for example:

Bollen, K. A., & Curran, P. J. (2006). Latent curve models: A structural equation perspective. New York: Wiley.

Duncan, T. E., Duncan, S. C., & Strycker, L. A. (2013). An introduction to latent variable growth curve modeling: Concepts, issues, and application. Routledge.

Little, T. D. (2013). Longitudinal structural equation modeling. New York: Guilford.

Newsom, J. T. (2015). Longitudinal structural equation modeling: A comprehensive introduction. Routledge.

Preacher, K. J. (2008). Latent growth curve modeling (No. 157). Sage.

https://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=1ACF96E33449B682E740E025BEAF408E?doi=10.1.1.138.4268&rep=rep1&type=pdf

https://www.youtube.com/watch?v=f4_FE9Q5LCo

In a nutshell, an LGC model, in the simple linear growth case consists of an intercept factor (initial or Time-1 latent variable) that represents the depression true (error-free) scores at Time 1, a linear slope factor (latent variable) that represents true inter-individual differences in linear change across time, and time-specific error variables that represent random measurement error and situation-specific variance.

The intercept and slope factors each have a mean and a variance as free parameters as indicated in your syntax. The intercept factor mean represents the average depression true (error-free) scores at Time 1. The variance of the intercept factor represents true (error-free) individual differences in depression at Time 1. The slope factor can be interpreted as a latent difference or latent change score variable that represents linear change in true scores across time. The mean of the slope factor represents the average rate of linear change in true scores across time. The variance of the slope factor represents the extent of inter-individual differences in linear change across time.

There is also a covariance parameter between intercept and slope that represents the linear association between the true initial status and linear change over time. Lastly, there are error variance parameters to be set for the observed variables. The error variances reflect the degree of unreliability and/or situation-specificity in the observed variables.

It may be difficult for you to guess the "right" (or at least plausible) parameter values for your simulation. (This is probably the hardest part of any kind of power analysis!) You might want to check out whether there is any prior literature that has reported an LGC model application to the same depression scale. Perhaps you can derive approximate parameter values from such an application. In any case, I would try out multiple different sets of parameter values to see how that affects estimated power, especially if you are not sure about what the actual parameters might look like in the population.