Step 2: Direction of Each Marginal Effect

The interpretation of the direction of each coefficient is a bit different than in a simple linear regression. In a simple linear regression, which has one $x$ variable, the coefficient is interpreted the same as a slope in a linear equation. In a multiple regression, the coefficient $\beta_i$ reported for each variable $x_i$ represents the marginal effect of that variable on the outcome, $y$. The marginal effect shows how much the outcome, $y$, changes as one $x$ variable changes and all other $x$ variables are held constant (i.e. they do not change). These effects are depicted in the image below.


	% shift to Trump, 2012-2016

County median income ($1,000s)	-0.013
	(0.007)

County college experience	-0.344^*
	(0.012)

Constant	20.103^*
	(0.512)


Observations	3,111
Adjusted R²	0.371

Note:	* p<0.05

Interpreting marginal effects

The regression surface, $y \sim x_1 + x_2$, is shown in blue, with grey confidence intervals. The marginal effects are shown in red and orange lines. Hover over and spin the graphic to examine the effects described below.

The marginal effect of $x_1$ (county median income) is shown with the red line. It depicts the predicted change in $y$ (shift to Trump) when $x_2$ (county college experience) is held constant. That is, county college experience is held constant at 51.24% in the red line, while county median income values rage from $20k to $135k.
The marginal effect of $x_2$ (county college experience) with the orange line. It depicts the predicted change in $y$ (shift to Trump) when $x_1$ (county median income) is held constant.
In this visualization, the marginal effect of $x_1$ is shown holding $x_2$ at its mean, and vice versa. Because the marginal effect is the same for any value $x_2$ is set at, this is an arbitrary decision with no meaningful impact on the interpretation of the marginal effects.

County median income

The coefficient for county income, -0.013, is negative. However, it is not statistically significant at the p-value noted in the regression results, $p < 0.05$. Therefore we are not confident that this effect is negative. The coefficient could be positive or effectively zero

County college experience

There is a negative relationship between county level education and shifting to Trump. Because it has stars, we think with 90% confidence* that this negative sign is not due to chance alone.
Therefore, we believe that when county level education increases and county level median income remains constant, there is a decreased shift in the county’s Trump vote.
This value of the coefficient, -0.344, is called the estimated effect size of county median income on shifting to Trump between 2012 and 2016.
We cannot be confident in the exact value of this coefficient. We can only be confident that the direction is negative.
In technical terms, we have rejected the null hypothesis that the coefficient is zero, and accept the alternate hypothesis that the coefficient is negative.

* Recall that when $p<0.05$, we are 95% confident that the coefficient is non-zero, and 90% confident in the sign (positive/negative) of the coefficient.