by Roland
Last Updated April 23, 2018 11:19 AM

What I have:

- a linear model $y=a_0+a_1x$ with given parameter estimates,
- the number of values used for fitting the model,
- the Pearson R² value.

I need to estimate errors of prediction. I don't see a way to calculate it, but is there a way to at least get a rough estimate?

Okay, I'm sure the folks who know more than I do will correct me if I am off base here. Heck, maybe I'm misinterpreting what you mean when you say "errors of prediction". My interpretation is that you are asking if you can estimate the errors of the slope and of the intercept.

My intuition is that depending on how rough you are willing to accept... it isn't quite hopeless.

General stuff: $\sqrt{R^2}$ gives us the correlation between our predicted values $\hat{y}$ and $y$ and in fact (in the single predictor case) is synonymous with $\beta_{a_1}$. The estimation of the intercept (and intercept error) does not affect this value/correlation.

Solution 1: We know the standard error of a pearson product moment correlation transformed into a Fisher $Z_r$ is $\frac{1}{\sqrt{N-3}}$, so we can find the larger of those distances when we transform back to the scale of r and treat that as the standard error of $\beta$. Knowing the nature of whatever system $x$ is as well as the nature of system $y$ you might be able to speculate regarding the standard deviations and extrapolate a likely scenario for what the naturally scaled error of $a_1$ is.

Solution 2: One worst case scenario is that *all* of the rest of the variance is in the estimate of the slope. I'm admittedly stumped and this seems like a complex topic, but shesh it really does seem that you should be able to come up with an estimate of the standard error of the standardized slope given these constraints.

If you know the model was fit by least squares, then we have:

$$\hat{a}_0=\overline{y}-\hat{a}_1\overline{x}\;\;\;\;\;var(\hat{a}_0)=\hat{\sigma}^2\frac{s_x^2+\overline{x}^2}{ns_x^2}$$ $$\hat{a}_1=r_{xy}\frac{s_y}{s_x}\;\;\;\;\;var(\hat{a}_1)=\hat{\sigma}^2\frac{1}{ns_x^2}$$ $$cov(\hat{a}_0,\hat{a}_1)=-\hat{\sigma}^2\frac{\overline{x}}{ns_x^2}$$ $$R^2=r_{xy}^2=1-\frac{(n-2)\hat{\sigma}^2}{ns_y^2}$$

Where $\overline{x}=\frac{1}{n}\sum_ix_i$ and $s_x^2=\frac{1}{n}\sum_i(x_i-\overline{x})^2$ and similarly for $\overline{y},s^2_y$ and $r_{xy}$ is the correlation between $x$ and $y$.

Now the forumal for the prediction error is:

$$mse(\hat{y})=\hat{\sigma}^2(1+\frac{1+z^2}{n})$$

Where $z=\frac{x_p-\overline{x}}{s_x}$ and $x_p$ is the predictor used. Hence you need to know $\hat{\sigma}^2,n,\overline{x},s_x$.

However, you need $s_y^2$ in order to rescale $R^2$ properly. One way to get around this, is to note that:

$$\hat{\sigma}^2=\frac{n}{n-2}s_y^2(1-R^2)=\frac{n}{n-2}\frac{\hat{a}_1^2s_x^2}{R^2}(1-R^2)$$

One rough approximation is to use $\hat{y}^2$ in place of $s_y^2$ to get $\hat{\sigma}^2\approx \frac{n}{n-2}\hat{y}^2(1-R^2)$. We can safely approximate $\hat{z}^2= 4$ provided $x_p$ is "typical" of the units used in the model fitting.

An alternative (better?) approach is to estimate $\hat{\overline{x}}$ and $\hat{s}_x^2$ from the predictors that you have available, say as $\hat{\overline{x}}=\frac{1}{n_p}\sum_{j}x_{pj}$ and $\hat{s}_x^2=\frac{1}{n_p}\sum_j(x_{pj}-\hat{\overline{x}})^2$ where $n_p$ is the number of observations you are producing predictions for with $x_{pj}$ being the predictor for the "jth" observation. Then you replace $\hat{z}_j=\frac{x_{pj}-\hat{\overline{x}}}{\hat{s}_x}$ and $\hat{\sigma}^2\approx \frac{n}{n-2}\hat{a}_1^2\hat{s}_x^2\frac{1-R^2}{R^2}$.

Note that if you add $\overline{x}$ and $s_x^2$ to your available information, then you have everything you need to know about the regression fit.

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