# Describing normal volume change with age: use a linear model and the standard deviation?

I would like to use a large dataset with tissue volume measures of 500 volunteers over a given ageframe to compare it to a small dataset with volunteers and patients (20-30 datapoints).

The tissue volumes correlate well with age and did a linear regression model for the large dataset and I thought I could also use the standard deviation to describe how the small data differs from the large dataset.

I explored the data and found that age is normally distributed around its mean and the Confidence Interval is slightly larger at the ends of the linear regression.

• How can I show this uncertainty in the standard deviation too?
• I am not sure how to test if it is feasible to assume the standard deviation is not changing with age (that's how it looks by eye).

My aim with this would be to (1) compare the volunteers from the small set to a large pool to volunteers and ideally show, that they are within a normal range and (2) to be able to compare individual patients with a larger dataset of "normal" and describe how far away they are from a "typical" volume measurement at their age. This could then maybe be used to subgroup the patients for further analysis.

You have 3 groups: the large original group of normals, the smaller newly recruited group of normals, and your patients. One simple way to address you question would be to include both that group membership (a 3-level categorical predictor) and an interaction of group with age as predictors in a regression model, with tissue volume as the outcome variable.

In usual treatment coding, you might use the original normal group as the reference level of group. Then the interaction coefficients for the other 2 group values with age represent how much their associations of volume with age differ from that of your original group.

Your hypothesis would then be that the interaction coefficient for the new normal group is 0 (same association with age as the original normals), while that for the patient group differs from 0. This approach has the advantage of evaluating and showing the differences (or lack of difference) based on all the data together, with a reasonably simple test of your hypothesis.

• Thank you for your answer @EdM. I think this could be a solution, however, the goal would be to detect for individual patients (also future ones) if they are within the "norm" that is based on this large dataset. How would I decide that with a model?
– CST
Jan 12 at 19:20
• @CST what you seek is the predicted value of volume for a given age and its standard error so you can see if an individual is far outside the expected distribution. That includes error both in the regression estimate itself and from the remaining mean-squared error. Standard software can do that, e.g. predict.lm() in R. See this Penn State page for formulas. But you should be very cautious about making individual decisions that way. 5 out of 100 individuals will normally be outside 95% limits, and you might risk bias by removing them.
– EdM
Jan 12 at 19:57
• thank you, I think this might be a solution. As I tried to describe in the text, the goal would be to make a decision for individual cases, but they would not be excluded, I would simply try to group them based on how extreme they are to obsereve if those cases have something in common.
– CST
Jan 12 at 20:02
• @CST that approach also might lead to bias. See this answer, provided in a different context, as an example of the dangers of focusing on extreme values simply because they are extreme. If you think that other variables might be leading to different age/volume associations, you would be better off including them in the regression model from the beginning.
– EdM
Jan 12 at 20:07