# correlation vs lm with two predictors

I'm trying to find an equivalent result between cor.test and lm with two predictors.

dat <- data.frame(y=c(1.4, 6.3, 3.2, 1.6, 4.3, 4.5, 8.4, 2.2, 4.2, 6.3, 8.3,
2.2, 1.1, 5.3, 2.2, 1.8, 7.5, 1.4),
x=c(22.2, 44.3, 13.3, 11.4, 57.3, 54.8, 78.5, 22.6, 45.6,
65.4, 14.5, 78.9, 14.4, 67.4, 11.1, 66.8, 91.4, 39.6),
d=c(1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0))


x and y are correlated when d == 1

cor.test(dat$$y[dat$$d == 1], dat$$x[dat$$d == 1])
t = 4.5776, df = 7, p-value = 0.002551


How can I reach the equivalent conclusion with lm?

mod <- lm(y ~ x * d, data=dat)

• A correlation coefficient is not equivalent to a linear model beta coefficients, so why are you expecting to get the same result? Oct 28, 2021 at 11:28
• Why do you have $d$ as a feature in your regression? You totally ignore the $d=0$ data when you do your correlation test.
– Dave
Oct 28, 2021 at 11:28
• @user2974951 I don't expect the same result, just a similar conclusion that there is dependency between the two predictors when d==1. Oct 28, 2021 at 11:29
• @Dave yes, the correlation was just to show that there is a dependency between x and y (when d==1). I was hoping to extract some information from mod that could provide me with a similar conclusion that x and y are related when d==1. Oct 28, 2021 at 11:32

I wonder whether you really mean that d is continuous. If you take d as categorical with two levels you could do:

library(data.table)
library(ggplot2)

dat <- data.table(y=c(1.4, 6.3, 3.2, 1.6, 4.3, 4.5, 8.4, 2.2, 4.2, 6.3, 8.3, 2.2, 1.1, 5.3, 2.2, 1.8, 7.5,1.4),
x=c(22.2,44.3,13.3,11.4,57.3,54.8,78.5,22.6,45.6,65.4,14.5,78.9,14.4,67.4,11.1,66.8,91.4,39.6),
d=c(1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0))

# Make d categorical, set '1' as reference level:
dat[, d := as.factor(d)]
dat[, d := relevel(d, ref= '1')]


Now the output of the regression recapitulates the correlation:

mod <- lm(y ~ x * d, data= dat)
summary(mod)

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  0.66304    1.54848   0.428   0.6750
x            0.08609    0.03482   2.473   0.0268 *
d0           2.27474    2.15567   1.055   0.3092
x:d0        -0.06460    0.04345  -1.487   0.1592

• (Intercept) is the value of y when x == 0 and d == 1
• x is the change in y for 1 unit change in x when d == 1 (i.e. the slope of the regression line). The slope is different from 0 at the significance level 0.0268 (compared to 0.002 when regressing using only the data at d == 1).
• d0 is the change in intercept of the regression line between d == 1 and d == 0. I.e. the regression line when d == 0 has intercept 0.66304 + 2.27474
• x:d0 Is the difference in slopes between the regression line with d == 1 and d == 0 (effect of interaction between x and d). I.e. the regression line when d == 0 has slope 0.08609 - 0.06460

You can visualize this with:

ggplot(data= dat, aes(x, y, colour= d)) +
geom_point() +
geom_smooth(se= FALSE, method= 'lm') +
xlim(c(0, NA)) +
theme_light() • I was going to post this exact answer, +1 Oct 28, 2021 at 13:02
• @Firebug this may be a separate question: I cannot quite explain why the estimate of x (slope when d == 1) has p-value ~0.03 as opposed to ~0.002 when subsetting first and then regressing Oct 28, 2021 at 13:18
• @dariober, thanks for your answer, exactly what I was looking for. And yes d should be categorical, not continuous. Oct 28, 2021 at 13:22
• @dariober Parameters estimates in multiple regression are correlated. The estimates of the variance/standard errors are not independent of the estimated covariances. Also, the degrees of freedom for the t-distribution are different (but that has a smaller effect if the they are not extremely small). Oct 28, 2021 at 14:16
• Basically what Roland said. If you look how the t-statistics are derived, you have to use the estimated expectation and standard error of coefficients. See stats.stackexchange.com/questions/27916/… Oct 28, 2021 at 14:42

As Dave pointed out, the current regression includes d as a regressor and also specifies an interaction term, cf.

> summary(mod)

Call:
lm(formula = y ~ x * d, data = dat)

Residuals:
Min      1Q  Median      3Q     Max
-2.5733 -1.2657 -0.3988  1.2886  5.0506

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  2.93777    1.49971   1.959   0.0704 .
x            0.02149    0.02599   0.827   0.4222
d           -2.27474    2.15567  -1.055   0.3092
x:d          0.06460    0.04345   1.487   0.1592


In cor.test you however drop the observations for which d==0. To achieve something similar in a linear regression, try

dat2 <- dat[dat\$d==1,1:2]
mod2 <- lm(y ~ x, data=dat2)
> summary(mod2)

Call:
lm(formula = y ~ x, data = dat2)

Residuals:
Min      1Q  Median      3Q     Max
-1.2962 -0.8810 -0.3889  0.9786  1.8230

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  0.66304    0.83643   0.793  0.45398
x            0.08609    0.01881   4.578  0.00255 **


Now, the coefficient on x is significant (where the coefficient, as was also mentioned in the comments, is related to, but not the same as the correlation).

The null hypothesis you are testing against with the correlation test is: $$E[x \cdot y |d=1] = E[x|d=1] \cdot E[y|d=1]$$

Your model though does not directly check for the hypothesis above, but you test for a general effect of x on y plus an additional effect of x on y given d=1.

In order to test the hypothesis as in your correlation test, you can do this:

mod <- lm(y ~ x:d, data = dat %>% dplyr::filter(d == 1))
summary(mod)


This yields the same t-statistic (and therefore also the same p-value) as in your correlation-test.