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I have a longitudinal outcome of two time points(2018 and 2020), the outcome is a quality of life score generated from a validated instrument, the score ranges from -0.158 to 1, a value of 1 indicate perfect health state, a values of 0 indicate a health state equal to death, negative values indicate a health state worse than death. I have a sample size of 457 longitudinal profile, the distribution of this outcome is heavily skewed to the left and multimodal, 55% of the scores lies in [0.8 ; 1 ]. Histogram of data

I tried the linear mixed model using some covariates such as age gender region, the plot of the residuals looked like this : residuals

It was clearly that the linear mixed model would not fit well, the log or square root transformations are not applicable since I have negative values, I tried this transformation :

  • transform into scale of 0-1 by a normal linear transformation.
  • apply the logit and transform to the whole real line.

then the distribution looked like this hist

and when i fitted a linear mixed model the residuals were like this :

res

both raw variable and the transformed one are not suitable for modelling. How to handle this type of data?

A social preference valuations set for EQ-5D health states in Flanders, Belgium Irina Cleemput

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    $\begingroup$ FWIW, these are left skewed data. Negate them and apply methods to analyze right skewed data! (But that's not your real problem: the residuals belie a need to include a categorical variable that can discriminate among the three regimes.) $\endgroup$
    – whuber
    Commented May 24, 2021 at 17:04
  • $\begingroup$ Since the outcome is at the subject level and the outcome is a quality of life measure then you will probably need some explanatory variables that are related to quality of life. $\endgroup$ Commented May 24, 2021 at 17:27
  • $\begingroup$ Thank you for your reply Robert Long, but I think the question here is not the explanatory variables, what I need is a modeling technique to handle the skewness because my data does not come from a known distribution. $\endgroup$
    – ilyas
    Commented May 24, 2021 at 18:04
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    $\begingroup$ No, the residual plots indicate that there is an explanatory variable missing. The skewness of the outcome is not relevant - it is the residuals that matter, and the residual plots tell us that there is a missing explanatory variable, as @whuber also noted. $\endgroup$ Commented May 25, 2021 at 8:03
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    $\begingroup$ The distribution of the outcome is never an assumption of any regression model: only the conditional distribution is relevant. If you successfully account for the three categories then you will have a (much) better sense of what the conditional distribution might be. $\endgroup$
    – whuber
    Commented May 25, 2021 at 11:11

2 Answers 2

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It is important to understand that the distribution of the outcome/response is not important. Depending on what the purpose of the model is, it is the conditional distribution that matters. One way to write a linear mixed model is:

$y = X \beta + Zu + \epsilon$

where $\epsilon \sim \text{N}(0, \sigma^2_\epsilon)$ and $X \beta + Zu$ is the linear predictor.

This can also be written as:

$y|X \sim \mathcal{N}(X \beta + Zu, \sigma^2_\epsilon)$

that is, the distribution of $y$, conditional on $X$, is normally distributed with a mean of the predicted values and variance $\sigma^2_\epsilon$.

So this is why we care about the residuals, and not the distribution of the outcome/response. The outcome distribution can have all kinds of weird and wonderful shapes including skewness and multi-modality.

It is also important to realise that often the conditional distribution need not be normal, and as mentioned already this is dictated by what you want to use the model for, and in particular what properties you would like the estimates to have. There are a number of excellent threads on this topic:
Assumptions of linear models and what to do if the residuals are not normally distributed
What is a complete list of the usual assumptions for linear regression?
Where do the assumptions for linear regression come from?

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  • $\begingroup$ Does this answer your question ? If so please consider marking it as the accepted answer. If not, please let us know why. Also, if you haven't already, please consider upvoting it. $\endgroup$ Commented Jun 26, 2021 at 12:25
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The residuals plots may also suggest that you have a bounded outcome. You may consider a Beta distribution instead. This option is available in the R packages GLMMadaptive and glmmTMB. Also, for such outcomes standard residuals diagnostics that are primarily aimed at models with normal error terms may not give you a very clear picture. You could consider instead the simulated residuals from the DHARMa package.

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    $\begingroup$ (+1) Dimitris I should have mentioned that too :D $\endgroup$ Commented May 27, 2021 at 19:04

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