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 |
|
Reference
Howell DC (1997) Statistical methods for psychologists. Fourth Edition. Duxbury Press:Belmont,CA
Matthews, JNS, Altman, DG, Campbell, MJ and Royston, P (1990) Analysis of serial measurements in medical research. 'British Medical Journal' 300 230-235.