# Doing regression only with correlation matrix, means, and SDs [duplicate]

I was wondering how mathematically is it possible to run a full regression analysis between 3 predictors (x1 x2 x3) and a dependent variable (y) by only knowing the: Means, Ns, SDs, and the Correlations between all these 4 variables (without the original data)?

I highly appreciate an R demonstration.

ns    <- c(273, 273, 273, 273)
means <- c(15.4, 7.1, 3.5, 6.2)
sds   <- c(3.4, 0.9, 1.5, 1.4)

r <- matrix( c(
1.0,  .57,  -.4,  .48,
.57,  1.0, -.61,  .66,
-.4, -.61,  1.0, -.68,
.48,  .66, -.68,   1.0), 4)

rownames(r) <- colnames(r) <- c('y', paste0('x', 1:3))

• You cannot run a truly "full" regression analysis with just these statistics, because you will not be able to construct residuals and perform regression diagnostics that depend on them. You are limited to making and testing parameter estimates.
– whuber
Dec 5, 2018 at 14:35
• Structural equation modeling is the most obvious answer in my mind, which is more or less the math provided in the answer below. Most SEM programs will be able to accept your inputs (all need to be able to estimate a covariance matrix in the end). Dec 10, 2018 at 2:29
• @whuber, do I need to first convert my correlation matrix into a var-covariance matrix using r_x_iy_i * sd_x_i*sd_y_i and then go from there? Dec 10, 2018 at 4:52
• That's one approach, because it reduces your problem to one with a known, explicit solution.
– whuber
Dec 10, 2018 at 15:13

$$\left(\begin{matrix} 1.00 & -0.61 & 0.66\\ -.61 & 1.00 & -0.68\\ .66 & -.68 & 1.00\end{matrix}\right)^{-1}\left(\begin{matrix} 0.57\\ -.40\\ .48\end{matrix}\right)$$