# why does the same variable have a different slope when incorporated into a linear model with multiple x variables

When I call for the summary of my linear model it shows X2 to have a negative slope. But when i call the same variable in a linear model of its own it has a positive slope. why is it not just the same slope for x2 in both models?

summary(lm(y~x1+x2+x3+x4))

Call:
lm(formula = y ~ x1 + x2 + x3 + x4)

Residuals:
Min       1Q   Median       3Q      Max
-0.31054 -0.07664 -0.00567  0.06249  0.35183

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  0.05794    0.02342   2.473  0.01387 *
x1           0.25579    0.04514   5.666 3.09e-08 ***
x2*         -0.02356*   0.04781  -0.493  0.62248
x3           0.45445    0.04878   9.317  < 2e-16
x4           0.13801    0.04620   2.987  0.00302

> lm(y~x2)

Call:
lm(formula = y ~ x2)

Coefficients:
(Intercept)           x2
0.2607       **0.2110**

Let some there be some data distributed according to a quadratic curve:

$$y \sim \mathcal{N}(\mu = a+bx+cx^2, \sigma^2 = 10^{-3})$$

For instance with $$x \sim \mathcal{U}(0,1)$$ and $$a=0.2$$, $$b=0$$ and $$c=1$$. Then a linear curve and a polynomial curve will have very different coefficients for the linear term.

set.seed(1)
x <- runif(100, 0,1)
y <- rnorm(100, mean = 0.2+0*x+1*x^2,
sd = 10^-1.5)
plot(x,y, ylim = c(0,1.5),
pch = 21, col = 1 , bg = 1, cex = 0.7)

mod1 <- lm(y~x)
mod2 <- lm(y~poly(x,2, raw =TRUE))

xs <- seq(0,10,0.01)
lines(xs,predict(mod1,newdata = list(x = xs)), lty = 2)
lines(xs,predict(mod2,newdata = list(x = xs)),lty =1)

legend(0,1.5,c("y = 0.009 + 1.023 x", "y = 0.193 + 0.016 x + 0.994 x^2"), lty = c(2,1))

### Correlation

The reason is that the variables/regressors $$x$$ and $$x^2$$ correlate.

The coefficient estimates computed with a linear regression are not a simple correlation (perpendicular projection onto each regressor seperately):

$$\hat{\beta} \neq \alpha = \mathbf{X^t} y$$

(this would give coefficients $$\alpha_1$$ and $$\alpha_2$$ in the image below, and these coordinates/coefficients/correlations do not change when you add or remove other regressors)

Using the correlation/projection $$\mathbf{X^t}y$$ is wrong, because if there is a correlation between the vectors in $$\mathbf{X}$$, then there will be an overlap between some vectors. This part that overlaps will be redundant and added too much. The predicted value $$\hat{y} = \alpha \mathbf{X}$$ would be too large.

For this reason there is a correction with a term $$(\mathbf{X^t}\mathbf{X})^{-1}$$ that accounts for the overlap/correlation between the regressors. This might be clear in the image below which stems from this question: Intuition behind $(X^TX)^{-1}$ in closed form of w in Linear Regression

### Intuitive view

So the regressors $$x$$ and $$x^2$$ both correlate with the data $$y$$ and they both will be able to express the variation in the dependent data. But when we use them together then we are not gonna add them according to their single independent effects (according to correlation with $$y$$) because that would be too much.

If we use both $$x$$ and $$x^2$$ in the regression then obviously the coefficient for the linear term $$x$$ should be very small since this is the same in the true relation.

However, when we are not using the quadratic term $$x^2$$ in the regression (or otherwise add a bias to the coefficient for the quadratic term), then the coefficient for $$x$$ which correlates somewhat with $$x^2$$ will partly take correct this (take over) and... the value of the estimate for the coefficient of the linear term will change.

• Sextus Empiricus, thank you for your detailed explanation and helpful reading suggestions.. – D.man May 6 at 6:19

You can think of the other variables regression with all the variables as something you are controlling for.

Let's say you are estimating weight gain, with the weight of the food eaten as a predictor. You see than you lose weight when you eat more, i.e. the slope is negative, when it should be the opposite by common sense. You realize that days when you eat more as measured by the weight of the food, you have eaten more vegetables (that have very little calories) since they don't make you full as fast as the more calorie-rich foods.

So then you add a predictor of the amount of calories in the food you eat. You'll see that the weight of the food most likely turns insignificant (as in your case the predictor x2) and the slope can change the direction, since it is actually the calories which make you gain weight, not the weight of the food.

This is closely related to the Simpson's Paradox, where missing variables would lead to the opposite conclusion instead of the correct one. In your case, it is most likely due to multicollinearity. In my example the weight of the food and the calories were negatively correlated. You can check for multicollinearity using the variance inflation factor (VIF) like this: car::vif(lm(y~x1+x2+x3+x4))

• kkz, thanks for your explination. I have run the vif and am trying to interpret the results.(below). the vif for x2 seems low, just 1.215179. Does this mean that there might be somethingelse going on in my data? > car::vif(lm(y~x1+x2+x3+x4)) x1 x2 x3 x4 1.083421 1.215179 1.264715 1.134739 – D.man May 3 at 2:46
• Yes, then it's something else. If your goal is to make accurate predictions instead of inference, then the model with highest accuracy on the test set should be used and you don't have to worry about the flip of the sign. If your goal is inference then you could still say that the sign changes because you control for the other variables. – kkz May 3 at 13:11
• Yes, my goal is to make predictions. I have concluded that the radical change indicates that X2 is a reciprocal suppression variable. Would you agree with this estimation? – D.man May 6 at 6:27