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The average rate of change for continuous time models.
1Department of Management, University of Notre Dame, Notre Dame, Indiana 46556, USA. kkelley@nd.edu
Behavior Research Methods
|April 14, 2009
Summary
The average rate of change (ARC) is often confused with the slope in linear models. This study clarifies the true mathematical definition of ARC in continuous time, showing they are not generally equal.
Area of Science:
- Longitudinal data analysis
- Statistical modeling
- Biostatistics
Background:
- The average rate of change (ARC) is frequently misinterpreted in applied longitudinal data analysis.
- The slope derived from a straight-line change model is often incorrectly equated with the true ARC.
Purpose of the Study:
- To mathematically define and clarify the concept of ARC for continuous time measurements.
- To demonstrate that the slope from a straight-line model is generally not equivalent to the ARC.
- To provide equations for quantifying the discrepancy between estimated and true ARC.
Main Methods:
- Mathematical derivation of ARC for continuous time data.
- Comparison of ARC with the slope from linear change models.
- Development of discrepancy measures for linear and nonlinear models.
Main Results:
- The slope from a straight-line change model is not generally equal to the ARC.
- General equations are presented to quantify the difference between the slope estimate and the true ARC.
- Discrepancy equations are provided for models linear and nonlinear in their parameters.
Conclusions:
- Accurate estimation of ARC in longitudinal studies requires distinguishing it from the linear model slope.
- The provided mathematical framework and discrepancy measures aid in correct interpretation of change over time.
- This work addresses a common misconception in statistical analysis of longitudinal data.
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