Linear Model 3: return of the yi

good models will have a large signal-to-noise ratio
- the signal (what the model can explain) will be big
- the noise (what the model can’t explain) will be small
the bigger the signal the more noise it can tolerate and still be good
the F-statistic value depends on the measurement of units and is unbounded so can be very big or small
a ratio >1 usually means our model can explain more variance than it can’t explain
The F-statistic - a measure of how good a model is (like R2) but is based around a comparison to the simplest model and the mount of error in the model
- both R2 and the F-statistic are based upon our sample data rather than trying to estimate something in the population
we get an associated p-value for the F-statistic
- the p-value represents how likely we are to find an F-statistic as large as the one we’ve observed if the null hypothesis is true

the linear model should fit the data better than the simplest possible model
- the simplest model is the null model - where there is no relationship between the predictor(s) and outcome
- a linear model with no predictors is just the outcome predicted by the mean of the outcome

example mean plot

the lines from each point to the mean represent the distance between the score of each participant and the mean score for the sample (aka residuals)
- large residuals - scores further away from the mean (the mean was a poor predictor of their score)
- for good models we want a larger difference between the mean and the line of best fit (so smaller residuals)
adding a predictor creates a model that takes into account the impact of that predictor on the outcome
- added precision → improvement on the null model
- adding a predictor does not always reduce error and improve fit

some error in the model can be seen with the line of best fit and the residuals
a good model would have a small amount of error
- the line of best fit is not the best judge of that as identical lines can still have different residuals