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| An alternative approach uses first and second derivatives of the scores on each individual over time to deduce trends in longitudinal behaviour across individuals although plenty of time points are required (preferable stretching daily over weeks). Further details of this method including the theory, examples of applications and R code (in the appendices) for implementation is available in Deboeck, PR, Montpetit, MA, Bergeman, CS and Boker, SM (2009) available for free download (as of January 2010) to CBSUers via sciencedirect on the [http://intranet.mrc-cbu.cam.ac.uk/library/ CBSU intranet library pages.] | An alternative approach compares trends in scores with a second measure. It uses first and second derivatives of the scores representing magnitude of change and rate of change on each individual over time. Summary measures of skew and/or kurtosis of these derivatives are then correlated with a second measure to see how change and rate of change over time relate to this second measure. Plenty of time points are required (preferable stretching daily over weeks) and a second measure which is thought to relate to the score measured over time. Further details of this method including the theory, examples of applications and R code (in the appendices) for implementation is available in Deboeck, PR, Montpetit, MA, Bergeman, CS and Boker, SM (2009) available for free download (as of January 2010) to CBSUers via sciencedirect on the [http://intranet.mrc-cbu.cam.ac.uk/library/ CBSU intranet library pages.] |
How do I compute slopes of linear trend for single cases?
It is often useful to produce [http://imaging.mrc-cbu.cam.ac.uk/statswiki/FAQ/Auc a summary measure of a variable measured over time] (e.g. to assess learning). This can be later used, for example to correlate with IQ to assess if rate of learning is related to IQ. SPSS and other packages only produce these for groups, rather than individuals.
The table below gives regression estimates of slopes corresponding to individual linear trends involving three up to six points. These may be computed from the data assuming each time point is entered in a separate column.
The formula is obtained by multiplying the k observed responses over time, $$y_text{1}$$ to $$y_text{k}$$, (in chronological order with $$y_text{1}$$ being the first response and $$y_text{k}$$ the last) by orthogonal polynomial coefficients and dividing by the sum of their squared coefficients (These may be found on page 678 of Howell DC (1997). It should, in fact, be found in the appendices of any of the five editions of this book. There are copies in the CBU library.
No. Points |
Slope |
|
3 |
($$y_text{3} - y_text{1}$$)/2 |
|
4 |
($$-3y_text{1} -3y_text{2} + y_text{3} + 3y_text{4}$$) / 20 |
|
5 |
($$-2y_text{1} -y_text{2} + y_text{4} + 2y_text{5}$$) / 10 |
|
6 |
($$-5y_text{1} -3y_text{2} - y_text{3} + y_text{4} + 3y_text{5} +5y_text{6}$$) / 70 |
|
An alternative approach compares trends in scores with a second measure. It uses first and second derivatives of the scores representing magnitude of change and rate of change on each individual over time. Summary measures of skew and/or kurtosis of these derivatives are then correlated with a second measure to see how change and rate of change over time relate to this second measure. Plenty of time points are required (preferable stretching daily over weeks) and a second measure which is thought to relate to the score measured over time. Further details of this method including the theory, examples of applications and R code (in the appendices) for implementation is available in Deboeck, PR, Montpetit, MA, Bergeman, CS and Boker, SM (2009) available for free download (as of January 2010) to CBSUers via sciencedirect on the [http://intranet.mrc-cbu.cam.ac.uk/library/ CBSU intranet library pages.]
Reference
Deboeck, PR, Montpetit, MA, Bergeman, CS and Boker, SM (2009) Using derivative estimates to describe intraindividual variability at multiple time scales. Psychological Mathods 14(4) 367-386.
Howell DC (1997) Statistical methods for psychologists. Fourth Edition. Duxbury Press:Belmont,CA
Lorch Jr RF and Myers JL (1990) Regression analyses of repeated measures data in cognitive research. Journal of Experimental Psychology: Learning, Memory and Cognition 16(1) 169-157.