set.seed(1)
x1 <- runif(100)
x2 <- 0.5 * x1 + rnorm(100) / 10
y <- 2 + 2 * x1 + 0.3 * x2 + rnorm(100)
set.seed(1)
x1 <- runif(100)
x2 <- 0.5 * x1 + rnorm(100) / 10
y <- 2 + 2 * x1 + 0.3 * x2 + rnorm(100)
cor(x1, x2)
## [1] 0.8351212
plot(x1, x2)
summary(lm(y ~ x1 + x2))
##
## Call:
## lm(formula = y ~ x1 + x2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.8311 -0.7273 -0.0537 0.6338 2.3359
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.1305 0.2319 9.188 7.61e-15 ***
## x1 1.4396 0.7212 1.996 0.0487 *
## x2 1.0097 1.1337 0.891 0.3754
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.056 on 97 degrees of freedom
## Multiple R-squared: 0.2088, Adjusted R-squared: 0.1925
## F-statistic: 12.8 on 2 and 97 DF, p-value: 1.164e-05
summary(lm(y ~ x1))
##
## Call:
## lm(formula = y ~ x1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.89495 -0.66874 -0.07785 0.59221 2.45560
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.1124 0.2307 9.155 8.27e-15 ***
## x1 1.9759 0.3963 4.986 2.66e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.055 on 98 degrees of freedom
## Multiple R-squared: 0.2024, Adjusted R-squared: 0.1942
## F-statistic: 24.86 on 1 and 98 DF, p-value: 2.661e-06
summary(lm(y ~ x2))
##
## Call:
## lm(formula = y ~ x2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.62687 -0.75156 -0.03598 0.72383 2.44890
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.3899 0.1949 12.26 < 2e-16 ***
## x2 2.8996 0.6330 4.58 1.37e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.072 on 98 degrees of freedom
## Multiple R-squared: 0.1763, Adjusted R-squared: 0.1679
## F-statistic: 20.98 on 1 and 98 DF, p-value: 1.366e-05
x1 <- c(x1, 0.1)
x2 <- c(x2, 0.8)
y <- c(y, 6)
x1 <- c(x1, 0.1)
x2 <- c(x2, 0.8)
y <- c(y, 6)
summary(lm(y ~ x1 + x2))
##
## Call:
## lm(formula = y ~ x1 + x2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.69309 -0.68184 -0.04583 0.75224 2.29389
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.2665 0.2303 9.840 2.45e-16 ***
## x1 0.1671 0.5246 0.318 0.751
## x2 3.1371 0.7703 4.073 9.37e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.079 on 99 degrees of freedom
## Multiple R-squared: 0.246, Adjusted R-squared: 0.2308
## F-statistic: 16.15 on 2 and 99 DF, p-value: 8.501e-07
summary(lm(y ~ x1))
##
## Call:
## lm(formula = y ~ x1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.8848 -0.6542 -0.0769 0.6137 3.4510
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.3921 0.2454 9.747 3.55e-16 ***
## x1 1.5691 0.4255 3.687 0.000369 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.16 on 100 degrees of freedom
## Multiple R-squared: 0.1197, Adjusted R-squared: 0.1109
## F-statistic: 13.6 on 1 and 100 DF, p-value: 0.0003686
summary(lm(y ~ x2))
##
## Call:
## lm(formula = y ~ x2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.66396 -0.67794 -0.06181 0.75541 2.32512
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.3085 0.1879 12.28 < 2e-16 ***
## x2 3.2981 0.5786 5.70 1.21e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.074 on 100 degrees of freedom
## Multiple R-squared: 0.2452, Adjusted R-squared: 0.2377
## F-statistic: 32.49 on 1 and 100 DF, p-value: 1.214e-07
par(mfrow = c(2, 2))
plot(lm(y ~ x1 + x2), cex = 0.2)
par(mfrow = c(2, 2))
plot(lm(y ~ x1), cex = 0.2)
par(mfrow = c(2, 2))
plot(lm(y ~ x2), cex = 0.2)
plot(x1, x2)
points(0.1, 0.8, col = "red", pch = 19)
