4
$\begingroup$

I have been searching for the answer for a problem with confidence intervals for a long time now, so I hope someone can help!

The two psychological papers below performed the same analysis, regressing irregular word reading onto nonword reading for controls, and plotting these regressions with 90% confidence intervals, before superimposing dyslexic participants to find outliers. However, one paper has straight line intervals, the other curved intervals. Why? (I have provided links to the papers so you can observe the plots).

I have scoured both textbooks and online resources, and the only descriptions of confidence intervals I have found are for confidence intervals of the regression line, which estimate an area in which there is a given probability that the regression line will fall. This is not what the studies below used, however. They used what may, I think, be termed ‘prediction intervals’ and describe an area in which a given percentage of the data will fall (which are then, in this kind of research, used to denote an area of 'normal' performance on a task). Can anyone help with these distinctions, and/or provide a calculation for this type of interval?

The curious thing is that SPSS/PASW will plot these confidence intervals for me, but offers no explanation of where they come from, nor can I edit the provided plot.

Any help, no matter how small, will be greatly appreciated!

References: Manis, F. R., Seidenberg, M. S., Doi, L. M., McBride-Chang, C. & Petersen, A. (1996). On the bases of two subtypes of developmental dyslexia. Cognition: 58, 157-195. LINK: http://lcnl.wisc.edu/publications/archive/45.pdf Stanovich, K.E., Siegel, L.S. & Gottardo, A. (1997). Converging evidence for phonological and surface subtypes of dyslexia. Journal of Educational Psychology: 89, 114-127. LINK: http://web.mac.com/kstanovich/Site/Research_on_Reading_files/RdJEdPsy97.pdf

$\endgroup$

2 Answers 2

4
$\begingroup$

The formula for the prediction interval can be found in many textbooks on regression. Here is one set of notes on the web that goes into the mathematical details (other results from a web search show the concept, but not all show the math).

The basic idea is that you add the estimated variance for the y-values around the line to the estimated variance used for the confidence interval and use that for creating a new interval. You have 2 sources of uncertainty, the line itself (as seen in the confidence bands) and the variation of the points around the line. Plotting the prediction bands will generally show a curve, but depending on the data and the range plotted it may be difficult to tell the difference between the curve and a straight line. So the one plot may be showing curves that are nearly linear, or they could be using a different method that finds straight lines for the bounds.

Note that assumptions of normality and equal variances are much more important for prediction intervals than for the other inferences on regression models, so if these assumptions don't hold for your case then the prediction intervals are going to be less useful (or possibly even meaningless). Also note that if you are using these for outlier detection then you also have the problem of multiple comparisons.

$\endgroup$
1
  • $\begingroup$ +1 Good description and commentary. In the cases I have seen, the parallel straight lines are merely set one or two standard errors above and below the fitted line: crude and misleading, but better than nothing at all. $\endgroup$
    – whuber
    Sep 8, 2011 at 16:25
2
$\begingroup$

Greg Snow's description is good, but I would add that you should probably use a least products type of regression rather than the normal least squares because both axes are subject to (I assume) an equivalent degree of error. The standard least squares regression privileges the x axis variable and often leads to the regression line having a lower slope than it should, a problem that can be seen in the figure 1 of the second paper that you have linked.

John Ludbrook has written about this a couple of times: http://www.cytel.com/papers/Ludbrook-Special-2002.pdf

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.