Correlation are not in general reflected in the regression coefficients

Demonstration that positive (or negative) raw data correlations in general do not result in postive (or negative) linear regression coefficients

Generate data

y  <- c(1.25,2.14,3.02,3.53,4.22,1.02)
x1 <- c(1,2,3,4,5,2)
x2 <- c(1.73,2.66,3.58,4.59,5.86,4.14)
dat <- data.frame(y=y, x1=x1, x2=x2)

Correlation table

cor(dat)

##            y        x1        x2
## y  1.0000000 0.9398212 0.7246516
## x1 0.9398212 1.0000000 0.9151660
## x2 0.7246516 0.9151660 1.0000000

Linear regression

summary(fit <- lm(y ~ x1 + x2, data=dat))

## 
## Call:
## lm(formula = y ~ x1 + x2, data = dat)
## 
## Residuals:
##          1          2          3          4          5          6 
## -5.454e-02  3.735e-02  1.119e-01 -1.177e-01  2.295e-02 -3.564e-05 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.09163    0.13449   8.117 0.003909 ** 
## x1           1.47840    0.07695  19.212 0.000308 ***
## x2          -0.73149    0.07777  -9.406 0.002546 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1021 on 3 degrees of freedom
## Multiple R-squared:  0.9962, Adjusted R-squared:  0.9936 
## F-statistic: 390.3 on 2 and 3 DF,  p-value: 0.0002369

Variance inflation factors (should be < 10)

car::vif(fit)

##       x1       x2 
## 6.154935 6.154935

Get regression coefficients

cf.mod <- coef(fit)

Calculate z on a grid of x-y values

x1.seq <- seq(min(x1),max(x1),length.out=25)
x2.seq <- seq(min(x2),max(x2),length.out=25)
z <- t(outer(x1.seq, x2.seq, function(x,y) cf.mod[1]+cf.mod[2]*x+cf.mod[3]*y))

Draw the plane with “plot_ly” and add points with “add_trace”

https://stackoverflow.com/a/44324365

library(plotly)

## Loading required package: ggplot2

## 
## Attaching package: 'plotly'

## The following object is masked from 'package:ggplot2':
## 
##     last_plot

## The following object is masked from 'package:stats':
## 
##     filter

## The following object is masked from 'package:graphics':
## 
##     layout

cols <- c("#f5cb11", "#b31d83")

p <- plot_ly(x=~x1.seq, y=~x2.seq, z=~z,
             colors = c("#f5cb11", "#b31d83"),
             type="surface") %>%
    add_trace(data=dat, x=x1, y=x2, z=y, mode="markers", type="scatter3d",
              marker = list(color=cols, opacity=0.7, symbol=105)) %>%
    layout(scene = list(
               aspectmode = "manual", aspectratio = list(x=1, y=1, z=1),
               xaxis = list(title = "X1", range = c(0,6)),
               yaxis = list(title = "X2", range = c(0,6)),
               zaxis = list(title = "Y", range = pretty(z)[c(1,6)])))
p