I have a dependent variable that is continuous and I have two independent variables: one continuous and one categorical (with 2 categories)
The interaction between the independent variables is significant. Which statistical analysis should I use (in R) to proceed with the analysis and document the interaction?
(Should I simply analyze each of the two categories separately using simple linear regression?)
In the scenario you describe least squares regression will allow you to tell a very straightforward story:
First of all, imagine that you have no dichotomous independent variable. So:
(1) $y_{i} = \beta_{0} + \beta_{1}x_{1i} + \varepsilon_{i}$
Your regression describes the relationship between your dependent variable $y$ and your continuous independent variable $x_{1}$ as a straight line, with intercept $\beta_{0}$ and slope $\beta_{1}$. Cool? Cool.
Now add both the dichotomous independent variable $x_{2}$ and the interaction between $x_{1}$ and $x_{2}$ to the model:
(2) $y_{i} = \beta_{0} + \beta_{1}x_{1i} + \beta_{2}x_{2i} + \beta_{3}x_{1i}x_{2i} + \varepsilon_{i}$
So now what is your model telling you? Well, (assuming $x_{2}$ is coded 0/1) when $x_{2} = 0$, then the model reduces to equation (1) because $\beta_{2} \times 0 = 0$ and $\beta_{3} \times x_{1} \times 0 = 0$. So that is easy-peasy puddin' pie.
What about when $x_{2} =1$? Well now the $y$-intercept is $\beta_{0} + \beta_{2}$ (Right? Because $\beta_{2} \times 1 = \beta_{2}$).
And the slope of the line relating $y$ to $x_{1}$ is now $\beta_{1} + \beta_{3}$ (Right? Because $\beta_{1}\times x_{1} + \beta_{3} \times x_{1} \times 1 = \beta_{1}\times x_{1} + \beta_{3} \times x_{1} = (\beta_{1} + \beta_{3})\times x_{1}$).
So when $x_{2}=1$ you simply have a second regression line relating $y$ to $x_{1}$, with a different intercept (if $\beta_{2} \ne 0$) and a different slope (if $\beta_{3} \ne 0$ which will be true if you tested a significant interaction term in, say, ANOVA).
How do you communicate this? A single graph with two regression lines overlaying your data (possibly with different colored/shaped/sized markers when $x_{2}=1$), and a label indicating which line corresponds to $x_{2}=0$ and $x_{2}=1$. Also providing your audience with the values of the $\beta$s and their standard errors and/or confidence intervals is good (like, in a table of multiple regression results).
Cool? Cool.
Finally, while all the above tells you about trend relationships between $y$ and $x_{1}$ given $x_{2}$, least squares regression also tells you about strength of association. If you had a single independent variable, you'd probably want to use something like $R^{2}$ to describe this strength of association, but when you add variables $R^{2}$ doesn't quite mean what it did before. So you might use generalized $R^{2}$, or Pseudo-$R^{2}$ or some such.
@Alexis seems to cover the equations pretty well. Here's some example code in r:
set.seed(8);d8a=data.frame(x=rnorm(99),z=rbinom(99,1,.5)) #Data sim'd to fit the scenario
d8a$y=(d8a$x+rnorm(99,0,3))*(2*d8a$z-1) #Guarantees an interaction
summary(lm(y~scale(x)*factor(z),d8a)) #Fits a GLM with OLS – this is the part you need
$$\rm Output$$
Residuals:
Min 1Q Median 3Q Max
-6.1575 -2.1416 -0.2051 1.8558 6.5765
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.17374 0.43557 -0.399 0.690867
scale(x) -1.11354 0.45224 -2.462 0.015608 *
factor(z)1 0.01546 0.58976 0.026 0.979144
scale(x):factor(z)1 2.24831 0.59689 3.767 0.000287 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.922 on 95 degrees of freedom
Multiple R-squared: 0.1328, Adjusted R-squared: 0.1054
F-statistic: 4.85 on 3 and 95 DF, p-value: 0.003492
$$\rm Plot$$
require(ggplot2);ggplot(d8a,aes(x,y,color=factor(z)))+stat_smooth(method=lm)+geom_point()
