# Quantifying uncertainty of regression models

I have built various different types of regression model (linear model, non-linear model, generalized linear model), and wish to determine the error/uncertainty of each one in order to compare them.

I have built the three models in R, and understand that I am able to use the predict function to obtain a confidence interval around each of the fitted values. A bit of background about my models. Each output gives a prediction of a households electricity usage when it is given a predefined continuous variable. The way in which these models will be used, is to sum the predicted values, in order to obtain a total electricity consumption for an area. Therefore, i would like to quantify the error surrounding this summed value to determine which model has the least uncertainty in it. The models will then be used in other areas for which we do not have the actual values, so the model with the smallest error will be desired.

My understanding, is that I can do the following, to work out the model error:

1. Compute the width of the confidence interval for each of the fitted values.
2. Divide the width by two, for each fitted value.
3. Each 'half-width' is equivalent to the error for that reading. Square each result.
4. Sum all of the squared errors, and square root the result.
5. This should give the error for the sum of the fitted values.

Firstly, is this correct? Does it give the error for the model?

Some R code (to demonstrate) is given at the end.

If this is correct, then my next question asks whether it is possible to use this same method for linear models which have manually computed coefficients?

Since I have created a linear model which uses the log of y and x, I have to take exponentials for my results to have meaning. Therefore, I have introduced Duan's Smearing factor, which applies a correction factor to the result. Therefore, my alpha (intercept) coefficient is different for one of my models, and i wish to determine the uncertainty of this type of model to compare.

R Code:

DF=data.frame(y=c(182.00455,606.27273,2401.03918,1597.27500,179.68846,76.82728,313.85000,1431.00000,16.43620,22887.81818,1010.00000,20.65909,184.17273,483.35000,21.45000,291.01500,359.10000,602.75000,253.18636,35.74286),
x=c(133.955464,1.913142,88.887131,95.512793,25.247257,11.938203,51.246909,96.265030,42.701863,9.082072,42.466148,86.741979,15.908011,55.756779,79.432585,61.395584,22.822762,22.853197,30.154734,96.494249))

fit <- lm(log(y) ~ log(x), data=DF)
summary(fit)

prediction=predict.lm(fit, interval="confidence")

prediction=data.frame(prediction)
prediction=exp(prediction) # Since the model was built using log, the values are exponentiated before use.
prediction$halfwidth=(prediction$upr-prediction$lwr)/2 prediction$error_squared=prediction$halfwidth^2 sqrt(sum(prediction$error_squared, na.rm=T))

sum(prediction$fit) ## uncertainty percentage: (sqrt(sum(prediction$error_squared, na.rm=T))/sum(prediction\$fit))*100
#  108.656%

• You need to define what you mean by "uncertainty for the model". Feb 27, 2017 at 14:01
• I suppose I mean model error. In other words, where could we expect the true value to lie when using the sum of all results. So, say the model produced a sum of 1000 when using the model on 100 sites, then what is the error around this value? Could we say it was 1000 +- x? Feb 27, 2017 at 14:07
• You may have to define that more clearly, it is not clear to me. E.g. I am not at all sure what "the model produces a sum" means. Do you mean the uncertainty around a predicted value that some how is a sum, do you mean the root mean squared error of the model on future predictions, do you mean something like R-squared on the present data you used to fit the model, do you mean the uncertainty around individual model coefficients, do you wish to compare how well different models fit the data relative to each other (not quite sure why you mention the several models otherwise), etc.? Feb 27, 2017 at 14:46
• I have added some background info to the question. Hopefully this clears up some of the confusion. Feb 27, 2017 at 15:02
• You have two fundamental problems to overcome. First, your models do not seem to accommodate the likely correlation among nearby houses. Second, you will be too optimistic about the predictive accuracy of the sum because you aren't actually comparing the predictions to anything: you are only computing an internal measure whose validity depends on unverified assumptions. For these reasons you can find a very large number of threads on this site that refer to cross validation, especially as part of the process of building predictive models.
– whuber
Feb 27, 2017 at 15:08