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Recently I encountered a situation where each subject has a continuous response variable at 3 time points ($t_0$, $t_1$, $t_2$). There are $3$ treatment groups ($A,B,C$), and some covariates.

Consider the follow variables:

  • $y_i$: continuous response at time $t_i$

  • $x_1$: group (categorical, $A,B,C$)

  • $x2,x3,...$: covariates (e.g.) age, gender

Measurements of $y$ are taken at the $3$ time points. A special note is that $t_0$ is the baseline and no actual treatment is performed at $t_0$.

Goal: To assess whether the rate of change in $y$ differ among the $3$ groups from $t_0$ to $t_1$, and $t_1$ to $t_2$.

There seems to be many possible analysis method as below to achieve the above goal, and I am not sure if I understand the difference or equivalence among them. Might someone be willing to explain or suggest a viable way to analyze the data?

Options

(1) Paired t-test

Assuming normality, we only consider $y_0$ and $y_1$ for each group. Then a difference:

$D = y_1 - y_0$

can be computed, and we can test whether mean of $D=0$. If significant, we can conclude change of $y$ from $t_0$ to $t_1$ is significant for that group. We then repeat the same for $t_1$ and $t_2$.

(2) ANCOVA

Assuming normality, we only consider $y_0$ and $y_1$ for all subjects. ANCOVA can be performed to see if $y$ is significantly different between time points or among groups. Covariates can also be considered. We then repeat the same for $t_1$ and $t_2$.

(3) Linear Mixed Model

Assuming normality, we simply consider all data in long format where each subject has $3$ rows of data representing the $3$ time points. A linear mixed effect model can be fitted by considering group as the fixed effect, and time point as the random effect. Covariates can also be considered in the model.

I guess the part I am confused the most is that we are interested in the "rate of change in $y$" instead of whether y is different between time points or among groups. In method (1) I can 'see' rate of change is assessed because of how D is computed. I am not sure about (2) and (3). Just a short add-on question, what if $y$ is not normally distributed even after transformation, what other viable method is available for the same goal?

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    $\begingroup$ "a continuous response variable at 2 time points (t_0, t_1, t_2)" ... that looks like three times rather than two $\endgroup$
    – Glen_b
    Mar 16 at 6:50
  • $\begingroup$ The first thing is to clarify what you want. When you say "rate of change" do you want to account for the covariates? Also, do you want to look at the two periods separately or together? That is are you interested in 1) T_0 vs. T_1 and T_1 vs. T_2 but not overall change/ 2) Whether the rate of change changed? 3) Whether the rate of change is different in the two groups? Or 4) Something else? $\endgroup$
    – Peter Flom
    Mar 16 at 10:05
  • $\begingroup$ Thank you both. It's 3 time points. Yes I would like to account for covariates, but also like to see the numerical results without covariates. (1) is right, we are interested in T0 to T1, and then T1 to T2. (2) and (3) are of interest as well. $\endgroup$ Mar 17 at 6:21

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Don't bother with the difference scores, the paired t-test, or the ANCOVA. Difference scores are somewhat controversial to use, and both statistical tests are highly inflexible, in that they require a bunch of assumptions, promote what Winter calls the "cookbook approach" to testing hypotheses, and require more assumptions than a mixed model would. I will also add that you don't need normally distributed data for a mixed model, nor a normally distributed residual term, so long as you specify which residuals you feel are most appropriate.

The mixed model has the most straightforward way of estimating this, but even a regular regression or GLM would model this fine given the number of time points you have (typically mixed models are better for lots of clusters of time variables). Depending on how you consider the main effects and interactions, you can very straightforwardly model:

  • The main effects of time, group, and the covariates.
  • What I assume are the important interactions between time and group (since you label the other variables as somewhat secondary by the language you use).
  • The residual term that best specifies this model.

As an example of a normal regression, this could look like:

$$ y = \beta_0 + + \beta_1 \text{Time} + \beta_2 \text{Group} + \beta_3 (\text{Time} \times \text{Group}) + \cdot \cdot \cdot x_k + \epsilon \\ \epsilon \sim \text{Normal}(0,1) $$

Where $\beta$ is the coefficient and $\epsilon$ is the error term, which in the typical case would have errors which are normally distributed with mean zero and an SD of $1$. For non-normal error terms, this would be directly specified in a Generalized Linear Model (GLM) explicitly (for example binary data is usually modeled with a logit link function rather than the typical OLS method).

To make the answer here more clear, the above model in a standard regression looks at the main effects of time and group, their interaction, and the additional control variables you mention. None of this explicitly requires a linear mixed model and seem to be accomplishable in a typical regression instead.

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  • $\begingroup$ Thank you Shawn. What specific statistics tests would you consider for addressing the goal (Goal: To assess whether the rate of change in y differ among the 3 groups from t0 to t1, and t1 to t2.) in that model? $\endgroup$ Mar 18 at 4:40
  • $\begingroup$ If you mean inferential tests, then every coefficient in a regression has an associated $p$ value, from the main effects down to the interactions. Anything that involves pairwise comparisons can be similarly looked at in a regression. $\endgroup$ Mar 18 at 6:02

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