Why does standardizing the DV in multilevel modeling change the pattern of results?

Question

I have time series data and am modeling the outcome using multilevel modeling. When I within-person standardize the IV and the DV I get a different pattern of results than when the DV is unstandardized and the IV is centered (but not standardized). This seems to be due to the within-person standardized DV, not the IV (note: I realize the coefficient will change with standardization, but not why the pattern of results changes).

By within-person standardization, I mean:

• $y_{it} = (y_{it} - y_{i}) / s_{yi}$ where $y_{it}$ is the DV for individual $i$ at time $t$, $y_{i}$ is the mean of the DV for individual $i$, and $s_{yi}$ is the SD for person $i$ ($s_{yi} = \sqrt{\Sigma_{t}(y_{it}-y_{i})^2/(T_{i}-1)}$).
• $x_{it} = (x_{it} - x_{i}) / s_{xi}$ where $x_{it}$ is the IV for individual $i$ at time $t$, $x_{i}$ is the mean of the IV for individual $i$, and $s_{xi}$ is the SD for person $i$ ($s_{xi} = \sqrt{\Sigma_{t}(x_{it}-x_{i})^2/(T_{i}-1)}$)

In Model 1 I have:

• DV: not centered, not standardized
• IV: within-person centered (not standardized)
• IV: grand-mean centered (not standardized)

In Model 2 I have:

• DV: within-person standardized
• IV: within-person standardized
• IV: grand-mean centered (not standardized)

In Model 1 the within-person standardized IV not associated with the DV, and the grand-mean centered IV is negatively associated. In Model 2 the within-person standardized IV is negatively associated with the (within-person standardized) DV, and the grand-mean centered IV is not associated with it.

Why would standardizing the DV change the pattern of the results in this way? Is there a different interpretation of the coefficients in Model 1 vs. Model 2?

Notes:

• 'IV' above refers to the same variable, just transformed as described
• My actual model has more than 1 IV included

Answer 1

In the multilevel or mixed effects model, there is a latent decomposition of the outcome across the levels, so your centering of the dependent variable is unnecessary and likely causing problems. I understand that you wish to standardize the outcome within cluster, but in the case of these models, that is somewhat counter-productive.

Imagining that you have 2 levels, repeated observations within clusters (e.g., persons, firms, etc.), the model decomposes the outcome into a fixed mean and two random errors (residuals):

At level 1 (within cluster): $y_{ij} = \beta_{0j} + e_{ij}, e_{ij}\sim N(0, \sigma_e^2)$

and at level 2 (between cluster): $\beta_{0j} = \gamma_{00} + u_{0j}, u_{0j}\sim N(0, \sigma_u^2)$

combined model: $y_{ij} = \gamma_{00} + u_{0j} + e_{ij}$

In the longitudinal context, $\gamma_{00}$ is the grand mean estimated from all the observed data points - the average value of the outcome y. If not all clusters have the same number of repeated measures, then this is becomes a weighted mean of the outcome. The two random error components, $u_{0j}$ and $e_{ij}$ are latent variables that decompose the outcome further.

The level 2 residual error, $u_{0j}$, is a cluster deviation from $\gamma_{00}$. The level 1 error, $e_{ij}$, is the deviation of a given observation from the cluster mean ($\gamma_{00} + u_{0j}$).

The multilevel model does this person mean centering implicitly for the outcome and it does so using latent variables, which are superior to any observed mean you can compute. Why? The latent decomposition corrects for sampling error in the aggregation of time-varying observed scores when forming the unique cluster means ($u_{0j}$)*. See the paper by Marsh et al. in 2009 in Multivariate Behavioral Research for an in depth discussion of this topic.

However, predictors are not similarly decomposed, as pointed out in that paper. Thus, it is perfectly logical to decompose the predictor into within- and between-cluster components. I am not familiar with the centering approach you take, specifically dividing by the within-person standard deviation. But that does not mean it is invalid. Typically, I center each value of the time-varying predictor around the person's mean across occasions, which is sometimes referred to as centering within cluster.

*This is particularly useful when you have fewer data points/observations. As the number of data points/observations increases, then the latent mean and observed mean tend to converge. See this article.