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I came across some statements saying that if the within-subjects and between-subjects standard deviations are similar, the observations can be considered independent even if repeated measures exist (e.g., from a longitudinal survey). Therefore, linear regression can be used instead of a multilevel model that models the dependency in the data.

I believe this statement is unfounded, but I have not found any reference against it. Any thoughts about it?

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  • $\begingroup$ Can you be more specific? Do you mean in an anova with one factor A the Between Mean Square and the within or Residual Mean Square are equal, so that the F statistic for testing the null hypothesis equals 1? $\endgroup$
    – BenP
    Commented May 7 at 16:17
  • $\begingroup$ In general this is untrue, as a trivial counterexample consider a sequence of non-Gaussian RVs with identical variance and dependence in the higher moments $\endgroup$
    – Randy Savage
    Commented May 7 at 16:43
  • $\begingroup$ @BenP the statement was about longitudinal survey. $\endgroup$
    – fever88
    Commented May 7 at 16:51
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    $\begingroup$ @fever88 I got that. But the whole setting is unclear. Without more details the question will be closed soon I suspect. $\endgroup$
    – BenP
    Commented May 7 at 19:00
  • $\begingroup$ The standard, lowest-order measure of "dependency in the data" is a serial correlation coefficient. Its estimate is nearly zero when the within and between variances are nearly equal. That might provide some insight into the implicit assumptions behind the recommendation you refer to. $\endgroup$
    – whuber ♦
    Commented May 7 at 19:10

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In a multilevel framework the question is more clear. Suppose you have data $y_{tj}$ of individuals $j$ measured at a number of time points $t$. A simple model with a random intercept across subjects (individuals) only and no independent variables is:

$y_{tj}=b_{0j} + e_{tj}$

The variance of the random intercept, $\sigma_{b_{0j}}^2$ gives the variance between subjects. The variance $\sigma_e^2$ gives the variance within subjects. The total variance in the data across all individuals and time points equals $\sigma_{b_{0j}}^2 + \sigma_e^2$. The proportion between subjects variance of the total variance would be equal to 0.5 if the within and between variance would be equal. However, this proportion 0.5 also stands for the correlation between two randomly drawn $y$ values of a randomly drawn individual. So, equal between and within subject variance leads to correlated data, seen from this simple multilevel model perspective. Only if the between subjects variance would be zero, the data at any pair of time points would be independent.

It's relevant to add that a "simple" multilevel model which assumes one and the same variance at each time point and one and the same correlation between any pair of time points may be inappropriate for the data at hand. E.g. adding a fixed or random time effect, linear, quadratic etc. or splines could make a multilevel model more flexible. Further, other models than multilevel models could be used and be more appropriate for longitudinal data. Hence, I think it is difficult or impossible to give a general rule for the relation of between/within subject variance and dependence of data. See e.g. the comment of whuber to this question about serial correlation.

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