# Longitudinal data analysis where meaning and metric of response variable varies over time

Determining what factors predict change over time is a topic of investigation in many fields and there are a variety of readily implemented methods for analysing repeated measures in the same metric.

A very simple scenario might involve examining the differences between individual outcomes at two time points $Y_{t2} - Y_{t1}$ as a linear function of various predictors/covariates $X_1 ... X_n$:

$$Y_{t2_i} - Y_{t1_i} = \beta_0 + \beta_1 X_{1_i} +\ ...+\beta_n X_{n_i} + \varepsilon_i$$

Here, parameter estimates $\beta_1 ... \beta_n$ provide information about the relationship between a given predictor and change in the outcome variable, above and beyond the contributions of the other predictors.

A more complex situation arises when we have measures of the same outcome over several time points. Techniques like latent growth curve modeling/linear mixed-effects models allow us to examine the relationships between our predictors and individuals' trajectories over time:

$$Y_{it} = \beta_{0i} + \beta_{1i} \lambda_{it} + \varepsilon_{it}$$ $$\beta_{0i} = \alpha_{00} + \alpha_{01} X_{1_i} +\ ...+\alpha_{0n} X_{n_i} + \mu_{0i}$$ $$\beta_{1i} = \alpha_{10} + \alpha_{11} X_{1_i} +\ ...+\alpha_{1n} X_{n_i} + \mu_{01i}$$

Here, we are able to examine our predictors' contributions to individuals' slopes (i.e., effect of time on outcome) and intercepts.

Whereas the latter paradigm is flexible (e.g., doesn't assume homogeneity of variance, relatively robust to missing data), both these techniques are typically applied to the same outcome measured repeatedly over time. But how do you approach longitudinal data where the outcome measure differs qualitatively over time?

A toy example helps to motivate this question: disruptive/deviant behaviors in adults are measured by different instruments then are similar behaviors in children. For example, a questionnaire administered to a child measures behaviors present in delinquent adolescents (e.g., involvement in physical altercations, shoplifting) whereas the adult version might probe related, but different, behaviors (e.g., armed robbery). Let's assume previous research provides a strong empirical and theoretical rationale that these behaviors, though different between children and adults, reflect the same underlying trait (i.e., disregard for societal conventions or aggression).

What then is the appropriate framework for examining what effects changes in this trait over the transition from adolescence to adulthood? Say we have three time points of data: at $t_1$ everybody is a child and is assessed using the child scale $C_{it_1}$ and we also measure predictors of interest $X_{1i} ... X_{ni}$; at $t_2$ some people are still adolescents and measured as such ($C_{it_2}$) and others are now adults and are measured with the adult scale $A_{it_2}$; finally, at $t_3$, everyone is measured as an adult ($A_{it_3}$). How do we examine the association between $X_{1i} ... X_{ni}$ and changes in our dependent variable(s) over time?

• (+1) As you correctly recognise because the instrument of examination (eg. the test used) changes over time you are having two distinct dependent variables. Lamping them together is wrong because you might have completely different rationale as well as error structures behind two. Can you please give more information about you $X$? Also I think your later example is a bit off because you do not have longitudinal data. In three time-points $t_1$, $t_2$ and $t_3$ three distinct variables $Y_1$, $Y_2$ and $Y_3$ are measured. Apologies if I misinterpret your question. – usεr11852 Apr 26 '15 at 19:27
• @usεr11852 $X_1 ... X_n$ could be a variety of pseudo-continuous (e.g., socio-economic status rated from 1-6) and continuous (e.g., age) variables correlated with our outcomes. I'm not sure it would be correct to say three distinct variables are measured but this is likely a matter of perspective: for some subjects we might have recorded two scores on the child measure at $t_1$ and $t_2$ and one score on the adult measure at $t_3$. For others, one child score and two adult scores. Cheers. – Richard Border Apr 26 '15 at 20:01
• Is it important that you only have 3 time points for the purpose of your question? If not, it seems like you could estimate a step function that allows for essentially a different line of best fit starting at adult hood for each child. Although the assumptions about errors are relaxed with linear mixed effects models, they have their own set of assumptions and you would want to ensure that you were not violating them. With 3 time points, I don't think my approach would work though. – wools May 4 '15 at 22:48
• @Buckminster: Well... Assume that in $t_1$ and $t_2$ you are measuring the cognitive abilities of a child using the Wechsler intelligence scale for Children and then at $t_3$ you are using the "standard" IQ for adults as the child matured. Yeah, sure you are measuring the same thing IN THEORY (eg. cognitive ability). Nevertheless to generally assume any notion of homoskedasticy or even correlation between the two tests is over-simplifying. – usεr11852 May 5 '15 at 2:58
• @wools Unfortunately, yes, the three time point issue is central to the dilemma. – Richard Border May 5 '15 at 15:37