One main issue with correlations of observed variable difference (change) scores is that these correlations may be strongly attenuated due to measurement error (unreliability). In terms of classical test theory, observed variable difference (change) scores often have low reliabilities because measurement error from both pretest and posttest affect the error variance component of the difference (change) score. To avoid the problem of low change score reliability, latent difference (change) scores can be used that are based on differences between true scores rather than observed scores. (True scores are by definition free of measurement error.) The correlations between "true difference scores" can be estimated using methods of structural equation modeling/longitudinal confirmatory factor analysis.

To mathematically identify latent difference score variables, you either need multiple (at least 2) observed variables ("indicators", measures) for each construct X, Y, and Z at each of the two time points or find a way to otherwise identify the error variance component of each observed variable X, Y, Z through appropriate constraints.

If you have appropriate estimates of the reliabilities of X, Y, and Z, you could derive (compute) the error variance components and specify them accordingly in a latent difference (change) score model with fixed error variances. Other options may be available for your design. You could check out the extensive literature on latent change score modeling to explore this option further and see if it works for your design, e.g.:

Article Modeling True Intraindividual Change: True Change as a Latent Variable

Chapter Modeling True Intraindividual Change in Structural Equation ...

Article Latent Variable Modeling of Differences and Changes with Lon...

Another option that may be applicable in your case is a relatively simple computational correction for attenuation, see, e.g.:

https://www.youtube.com/watch?v=jJ-qLImQYZs&t=391s

Again, this also requires that the reliability of the change scores be known.

See these links:

https://www.researchgate.net/post/How_do_I_analyse_the_correlation_between_two_variables

https://www.dummies.com/education/math/statistics/how-to-calculate-a-correlation/

Jia Jiet Lim

you may use Kendall's Tau.

Just a suggestion here. Start with a correlation matrix between the x values and scatter plot the x variables. If you haven't found something of interest then come back with a more detailed question. Best wishes David Booth

For the correlation, may you need two ordinal data, because nominal data will not correlate, so firstly set your data and go for statistical Pearson or Spearman Correlation test.

Thank you

Christian Geiser you perfectly understood my question and thank you for introducing the latent difference score and correction of attenuation. I don't think SPSS can perform a latent difference score? Are you able to recommend any models that can be performed in SPSS?

Also, the reliability estimate is not clear to me, do you have notes or videos showing how to calculate the reliability estimates?

Thanks!

Jia Jiet Lim To my knowledge, SPSS is not able to fit latent difference score models. You would need a program that is capable of structural equation modeling / confirmatory factor analysis such as AMOS (which belongs to SPSS), Mplus, or lavaan (included as a package in the free R software).

There are different methods of reliability estimation based on concepts of classical test theory (CTT). To learn more, a good resource is Raykov and Marcoulides' book "Introduction to Psychometric Theory":

https://www.routledge.com/Introduction-to-Psychometric-Theory/Raykov-Marcoulides/p/book/9780415878227

SPSS allows for rudimentary reliability analysis / statistics based on CTT under ANALYZE --> SCALE --> RELIABILITY. For example, you can fit a model of essentially or strictly parallel variables in SPSS based on which composite (Spearman Brown) reliability can be estimated. SPSS also allows you to compute Cronbach's alpha (a composite reliability index for essentially and strictly tau equivalent measures).

For rudimentary analyses of the type Christian mentioned, you may find this article by Gardner & Neufeld (1987) helpful:

Article Use of the Simple Change Score in Correlational Analyses'

Gardner, R. C., & Neufeld, R. W. (1987). Use of the simple change score in correlational analyses'. Educational and Psychological Measurement, 47(4), 849-864.

The correlation can be determined by representing the two values as x and y and subjected the two scores in to Pearson Product Moment Correlation formula and the result that will be obtained will be referred to as correlation coefficient.

You can start with a linear correlation, using Pearson coefficient R-squared. If R is low, there is no linear correlation.