# Can slopes in linear regressions be used as the independent or dependent variables in other regression models?

I have 100 patients and each patient have 10 longitudinal serum creatinine measurements. The estimated glomerular filtration rates (eGFR) were calculated from a MDRD formula comprising gender, age and serum creatinine. eGFR is the dependent variable and time is the independent variable in linear regression for each patient.

1. Do linear regressions violate the "independent X's" assumption and linear mixed models should be used instead?
2. Can eGFR slopes (which are estimates with uncertainties instead of measured numbers) estimated from each patient (in linear regressions for each patient or in random effects mixed models [how to estimate slopes for each individual patient in mixed models?]) be used as the independent or dependent variables in other regression models?

Thank you.

• If X's (independent variables) are measured variables, then it is a "random variable" instead of a "fixed value". According to Wikipedia (en.wikipedia.org/wiki/Random_variable): "In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process." Am I correct? – user4389 Apr 29 '11 at 4:59
• Please don't use replies to ask questions. You could continue this conversation in comments. Another option is to re-post this as a question and link back here. – whuber Apr 29 '11 at 16:32

In effect, you are proposing to use linear regression as a mathematical procedure to condense a 10-variate observation into a single variable (the slope). As such it's just another example of similar procedures like (say) using an average of repeated measurements as a regression variable or including principal components scores in a regression.

(1) Linear regression does not require the X's (independent variables) to be "independent." Indeed, in the standard formulation the concept of independence does not even apply because the X's are fixed values, not realizations of a random variable.

(2) Yes, you can use the slopes as dependent variables. It would help to establish that they might behave like the dependent variable in linear regression. For ordinary least squares this means that

a. Slopes may depend on some of the patient attributes.

b. The dependence is approximately linear, at least for the range of observed patient attributes.

c. Any variation between an observed slope and the hypothesized slope can be considered random.

d. This random variation is (i) independent from patient to patient and (ii) has approximately the same distribution from patient to patient.

e. As before, the independent variables are not viewed as random but as fixed and measured without appreciable error.

If all these conditions approximately hold, you should be ok. Violations of (d) or (e) can be cured by using generalizations of ordinary least squares.

(2'). Because the slopes will exhibit uncertainty (as measured in the regression used to estimate the slopes), they might not be good candidates for independent variables unless you are treating them as random in a mixed model or are using an errors-in-variables model.

You can also cope with this situation by means of a hierarchical Bayes model.

• I am sorry for the misnomer of "independent X's". What I meant was "independent Y's conditional on X's". Namely, the residuals must be independent. This assumption is violated in longitudinal observations. Reference: Modeling longitudinal data, II: standard regression models and extensions. Methods Mol Biol. 2009;473:61-94. – Jinn-Yuh Guh Apr 28 '11 at 15:39
• You can find out more about the 'generalizations of least squares' mentioned above from Lewis and Linzer 2005 – conjugateprior Apr 29 '11 at 15:59