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I am currently running an analysis to quantify temporal linear trends across repeated measurements acquired during a physical ergonomics experiment (i.e., evaluating posture metrics collected over several cycles during a physical movement task). For this particular analysis, I am interested in describing the linear temporal trend by simultaneously combining the linear slope with the fit (for example, R squared) into a single metric (computed using linear regression and a weighted least squares algorithm).

I have discussed this matter with my research advisor, who suggested to compute the ratio of standard error of the slope over the slope estimate. I have also considered computing the ratio of the slope estimate over the range of the lower and upper bounds of the 95% confidence interval of the slope estimate. However, we are not certain on an appropriate strategy that is statistically valid.

Does anyone have any suggestions on a method to simultaneously combine the linear slope and error (i.e., model fit) outputs computed using linear regression? Thank you for your time and help!

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  • $\begingroup$ Let's say that you are able to come up with such a metric (call it M). How would you interpret it? If you combine information on the trend (slope) and fit (R-squared, say), what would it mean that M is small/large? Would you trust a large slope from a model with low R squared more than a large slope from a model with high R squared? $\endgroup$ – Isabella Ghement Jul 2 '18 at 13:29
  • $\begingroup$ Is the metric M intended to facilitate comparisons of slopes and R squared values across different studies (or experiments)? Then why not consider instead multivariate meta-analysis as an option? Then you compare (i)slopes with slopes across studies and (ii) R-squared values with R-squares values, while allowing for the within-study correlation between slope and R-squared. $\endgroup$ – Isabella Ghement Jul 2 '18 at 13:31
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The slope is basically the change in modelled value per unit change in input value, so is a normalisation of explained variance in input vs output.

The error is the unexplained variance not accounted for by the model.

$R^2$ is one minus the ratio of unexplained variance to unexplained variance: $$R^2 = 1-SS_{res}/SS_{tot}$$ so is related to your desired parameter.

$SS_{res}$ is the sum of squares of the residual after the model has been applied, i.e. the square of the error. $SS_{tot}$ is the total sum of squares, i.e. the sum of the square of (error plus explained variance). The slope is the explained variance per unit of the input axis.

so $R^2$ so what you are looking for

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