What would be a "reasonable" minimal number of observations to look for a trend over time with a linear regression? what about fitting a quadratic model?

I work with composite indices of inequality in health (SII,RII), and have only 4 waves of the survey, so 4 points (1997,2001,2004,2008).

I am not statistician, but I have the intuitive impression 4 points are not sufficient. Do you have an answer, and/or references ?

Thanks a lot,

Françoise

## Best Answer

Peters rule of thumb of 10 per covariate is a reasonable rule. A straight line can be fit perfectly with any two points regardless of the amount of noise in the response values and a quadratic can be fit perfectly with just 3 points. So clearly in almost any circumstance it would be proper to say that 4 points is insufficient. However, like most rules of thumb it does not cover every situation. Cases where the noise term in the model has a large variance will require more samples than a similar case where the error variance is small.

The required number of sample points does depend on objects. If you are doing exploratory analysis just to see if one model (say linear in a covariate) looks better than another (say a quadratic function of the covariate) less than 10 points may be enough. But if you want very accurate estimates of the correlation and regression coefficients for the covariates you could need more than 10 per covariate. An accuracy of prediction criterion could require even more samples than accurate parameter estimates. Note that the variance of the estimates and prediction all involve the variance of the models error term.