Let's assume we fit a linear regression to our data $(X_i,Y_i)_{old}$, we lose the old data points but we still have access to the coefficients and the statistics of our linear regression. New data $(X_i,Y_i)_{new}$ comes in. We want to fit a linear regression model again. What is the best method to incorporate the statistics from fit for $(X_i,Y_i)_{old}$ into our new regression?
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$\begingroup$ Are you open to using a bayesian method with the priors specified by the old coefficients and standard errors? $\endgroup$– David Luke ThiessenJun 30, 2021 at 15:09
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2$\begingroup$ If you have the statistics from the old regression for $n, \sum x_i, \sum x_i^2, \sum y_i, \sum y_i^2, \sum x_i y_i$ (or their equivalents of number, means, variances and covariance) then you can incorporate them into the new regression $\endgroup$– HenryJun 30, 2021 at 15:10
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$\begingroup$ @DavidLukeThiessen If it works why not. $\endgroup$– user127776Jun 30, 2021 at 15:11
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1$\begingroup$ The simple linear regression coefficients are related to the means, variances and covariance and these statistics allow you to calculate these for the combined regression $\endgroup$– HenryJun 30, 2021 at 15:14
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2$\begingroup$ Specific solutions along the lines indicated by @Henry appear here on CV: see stats.stackexchange.com/questions/6920 for instance. Also look for threads about updating means and covariances (or moments generally) with new data -- stats.stackexchange.com/questions/51622 will do -- and apply them to regression, using threads that describe how to do regression using only those moments -- such as stats.stackexchange.com/questions/107597. $\endgroup$– whuber ♦Jun 30, 2021 at 15:53
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