0
$\begingroup$

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!

Improve this question
$\endgroup$
2
  • $\begingroup$ 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
    Commented Jul 2, 2018 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
    Commented Jul 2, 2018 at 13:31

1 Answer 1

Reset to default
0
$\begingroup$

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

Improve this answer
$\endgroup$

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.