When fitting the random-intercept vs. random slope mixed-effect regression model
Hai Nguyen
Last compiled on 18 March, 2021
Question
Why should we fit the random slope mixed-effect regression model (MRM) instead of the random intercept?
- In practice, we often care of producing the estimates from MRM. That’s good enough! Since that, we avoid the complicated problem by choosing to fit the random intercept.
- The situation was raised when I was suggesting that we should fit the random intercept and slope MRM, not the random-intercept MRM instead when we do the analysis of the group effect (Exparel vs. Epidural) across 6 time points (12h, 24h, 36h, 48h, 60h and 72h) on the pain scale (0-10). The question was asked by an PI assistant “Why should we do that?”
Review the concepts
Random intercept model
\[ y = Time + Grp + (Grp \times Time) + Subject + Error, \]
\[ y_{ij} = β_0 + β_1t_{ij} + β_2G_i + β_3(G_i \times T_{ij}) + \nu_{0i} + \epsilon_{ij}, \]
\[ \nu_{0i} \sim N(0, \sigma^2_{\nu}) \ \ \ \ \epsilon_{ij} \sim N(0, \sigma^2) \]
where represents the influence of individual \(i\) on the patient’s repeated observations.
As mentioned, individuals deviate from the regression of \(y\) on \(t\) and \(grp\) in a parallel manner. It is referred to as a random-intercept model, with each \(\nu_{0i}\) indicating how individual \(i\) deviates from the population trend.
Random intercept model for longitudinal data is often too simplistic for reasons:
- it is unlikely that the rate of change across time is the same for all individuals, and more likely that individuals differ in their time trends; not everyone changes at the same rate.
- The compound symmetry assumption of the random intercept model is usually untenable for most longitudinal data
- In general, measurements at points close in time tend to be more highly correlated than measurements further separated in time.
- Also, in many studies, subjects are more similar at baseline, and they grow at different rates across time. Thus, it is natural to expect that variability will increase over time.
- it is unlikely that the rate of change across time is the same for all individuals, and more likely that individuals differ in their time trends; not everyone changes at the same rate.
\(\Longrightarrow\) Come up with the random intercept and slope (trend) mixed-effect regression model
Random intercept and slope (trend) mixed-effect regression model
where
* \(\beta_0\) the overall population intercept,
* \(\beta_1\) is the overall population slope,
* \(\nu_{0i}\): the intercept deviation for subject \(i\),
* \(\nu_{1i}\): the slope deviation for subject \(i\)
- The intercept parameters indicate the starting point, and the slope parameters indicate the degree of change over time. The population intercept and slope parameters represent the overall (population) trend, while the individual parameters express how subjects deviate from the population trend (so we did not see the parallel of individuals to the population trend anymore).
Answer
\(\Longrightarrow\) To sum up, model with the intercept and trend (slope) reflex more exact the behave of each individual and the observations of individual comparing to using just intercept MRM.
Intuitively, we can image the complex of the depart and on-going path of the pain progression overtime on the Spaghetti plot. This is the visualized evidence we should cook up with the random slope model.
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