# Reporting results of simple linear regression: what information to include?

I have just performed some (very) simple linear regression in Genstat and would like to include a succinct and meaningful summary of the output in my report. I'm not sure exactly what or how much of the information I should be including.

The main bits of my Genstat output look like this:

Summary of analysis
Source      d.f.    s.s.       m.s.       v.r.    F pr.
Regression    1   8128935.   8128935.    814.41   <.001
Residual     53    529015.      9981.
Total        54   8657950.    160332.

Percentage variance accounted for 93.8
Standard error of observations is estimated to be 99.9.

Estimates of parameters
Parameter    estimate    s.e.     t(53)   t pr.
Constant      41.5      30.7       1.35   0.182
UKHR_Ref       0.8659    0.0303   28.54   <.001


I was intending to report this simply as:

Adjusted R2 = 0.94 (slope = 0.87, p < 0.001; intercept not significantly different from 0).

but a colleague has suggested that I should also include at least the root mean squared error (which I believe in this case is equal to the standard error of the observations i.e. 99.9?).

Does including the RMSE provide additional useful information, or is the goodness of fit already adequately explained by the adjusted-R2 value?

Are there hard-and-fast rules for how much information to report, or is it fairly subjective?

• "Are there hard-and-fast rules for how much information to report" - it really depends on what you want to do after the regression. One might be happy with just the correlation coefficient; one might need the Durbin-Watson value on top of that, and still another one might want to see the diagonal of the hat matrix... it really depends. Commented Oct 28, 2011 at 11:46
• Some organizations do have rules. See the APA guidelines for instance.
– whuber
Commented Oct 28, 2011 at 14:14

For a simple linear regression, I would always produce a plot of the x variable against the y variable, with the regression line super-imposed on the plot (always plot your data whenever its feasible!). This will tell you very easily how well your model fits, and is easy to read for 1 variable regression. Adding that to what you've already got would probably be sufficient, although you may want to include some diagnostic plots (leverage, cooks distance, residuals, etc.). It depends on how good that x-y plot is, and on your intended audience, and any protocols that your audience expect.

$R^2$ vs RMSE

$R^2$ is a relative measure, whereas the RMSE is more of an absolute measure, as you would expect most observations to be within $\pm$RMSE from the fitted line, and nearly all to be within $\pm 2$RMSE. If you want to convey "explanatory power" $R^2$ is probably better, and if you want to convey "predictive power", the RMSE is probably better.

• Thanks @probabilityislogic. I have included a plot in my report, and it sounds like that plus my original sentence should be sufficient in this case. I think that including diagnostic plots is probably unnecessary for this audience, though I've obviously checked them myself and they look reasonable. Thanks also for the explanation of R2 vs. RMSE - it's very helpful. Commented Oct 31, 2011 at 9:56
• What about the t-value(s) and d.f.(s)? When should these be included? Does it only make sense to include both or neither? Commented Jun 5, 2013 at 18:46

I use to report the β coefficient plus the 95% CI, the p value and adjusted Rsquared. Ex:

(β = 1.46, 95% CI [1.19, 1.8], p = 0.001 **, adjusted R2 = 0.48)

If reporting a multiple regression or a regression with factor variables, I report the coefficient, the 95% CI, the p values and then separately the F(degres of freedom) statistics, the adjusted R2 and the p value of the model.