R squared and goodness of fit in linear regression

I've been teaching a modelling course recently, and have been reading and thinking about the notion of goodness of fit. R squared, the proportion of variation in the outcome Y, explained by the covariates X, is commonly described as a measure of goodness of fit. This of course seems very reasonable, since R squared measures how close the observed Y values are to the predicted (fitted) values from the model.

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R squared and adjusted R squared

One quantity people often report when fitting linear regression models is the R squared value. This measures what proportion of the variation in the outcome Y can be explained by the covariates/predictors. If R squared is close to 1 (unusual in my line of work), it means that the covariates can jointly explain the variation in the outcome Y. This means Y can be accurately predicted (in some sense) using the covariates. Conversely, a low R squared means Y is poorly predicted by the covariates. Of course, an effect can be substantively important but not necessarily explain a large amount of variance - blood pressure affects the risk of cardiovascular disease, but it is not a strong enough predictor to explain a large amount of variation in outcomes. Put another way, knowing someone's blood pressure can't tell you with much certainty whether a particular individual will suffer from cardiovascular disease.

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The robust sandwich variance estimator for linear regression (theory)

In a previous post we looked at the properties of the ordinary least squares linear regression estimator when the covariates, as well as the outcome, are considered as random variables. In this post we'll look at the theory sandwich (sometimes called robust) variance estimator for linear regression. See this post for details on how to use the sandwich variance estimator in R.

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The t-test and robustness to non-normality

The t-test is one of the most commonly used tests in statistics. The two-sample t-test allows us to test the null hypothesis that the population means of two groups are equal, based on samples from each of the two groups. In its simplest form, it assumes that in the population, the variable/quantity of interest X follows a normal distribution N(\mu_{1},\sigma^{2}) in the first group and is N(\mu_{2},\sigma^{2}) in the second group. That is, the variance is assumed to be the same in both groups, and the variable is normally distributed around the group mean. The null hypothesis is then that \mu_{1}=\mu_{2}.

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Linear regression with random regressors, part 2

Previously I wrote about how when linear regression is introduced and derived, it is almost always done assuming the covariates/regressors/independent variables are fixed quantities. As I wrote, in many studies such an assumption does not match reality, in that both the regressors and outcome in the regression are realised values of random variables. I showed that the usual ordinary least squares (OLS) estimators are unbiased with random covariates, and that the usual standard error estimator, derived assuming fixed covariates, is unbiased with random covariates. This gives us some understand of the behaviour of these estimators in the random covariate setting.

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Regression inference assuming predictors are fixed

Linear regression is one the work horses of statistical analysis, permitting us to model how the expectation of an outcome Y depends on one or more predictors (or covariates, regressors, independent variables) X. Previously I wrote about the assumptions required for validity of ordinary linear regression estimates and their inferential procedures (tests, confidence intervals) assuming (as we often do) that the residuals are normally distributed with constant variance.

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