This tutorial shows how to fit a simple regression model (that is, a linear regression with a single independent variable) using SAS. The details of the underlying calculations can be found in our simple regression tutorial. The data used in this post come from the *More Tweets, More Votes: Social Media as a Quantitative Indicator of Political Behavior* study from DiGrazia J, McKelvey K, Bollen J, Rojas F (2013), which investigated the relationship between social media mentions of candidates in the 2010 and 2012 US House elections with actual vote results. The replication data in SAS format can be downloaded from our github repo.

In this example, we will assess the relationship between the percentage of social media posts that mention a Congressional candidate and how well the candidates did in the next election. The variables of interest are:

`vote_share`

(*dependent variable*): The percent of votes for a Republican candidate`mshare`

(*independent variable*): The percent of social media posts for a Republican candidate

Both variables are measured as percentages ranging from zero to 100.

## Data Visualization

It is always a good idea to begin any statistical modeling with a graphical assessment of the data. This allows you to quickly examine the distributions of the variables and check for possible outliers. The following code returns a histogram for the `vote_share`

variable, our outcome of interest.

```
proc sgplot data = twitter_data;
histogram vote_share;
density vote_share / type=kernel;
title "Vote Share Distribution";
xaxis label="Vote Share";
xaxis min = 0 max = 100;
run;
```

We start off with the `proc sgplot`

command, which tells SAS to use the `sgplot`

procedure. We set the data as `twitter_data`

. Next, we use the `histogram`

call to create the histogram of `vote_share`

; the default for the y-axis is percent. The `density`

option provides a density line, and the `type=kernel`

specifies that it should be a kernel density, which is a smoothed version of the histogram. We provide a chart title, and an xaxis label so that the figure displays informative labels rather than the variable name. The variable’s values (x-axis) fall within the range we expect. There is some negative skew in the distribution.

We can do the same thing for our independent variable.

```
proc sgplot data = twitter_data;
histogram mshare;
density mshare / type=kernel;
title "Tweet Share Distribution";
xaxis label="Tweet Share";
xaxis min = 0 max = 100;
run;
```

We again see that the values fall into the range we expect. Note that there are also spikes at zero and 100. These indicate races where a single candidate received either all of the share of Tweets or none of the share of Tweets.

It is also helpful to look at the bivariate association between the two variables. This allows us to see whether there is visual evidence of a relationship, which will help us assess whether the regression results we ultimately get make sense given what we see in the data. The syntax to use is the following.

```
proc sgplot data = twitter_data;
reg x = mshare y = vote_share;
title "Scatterplot with Linear Fit";
xaxis label = "Vote Share";
yaxis label = "Tweet Share";
run;
```

Here we use the `reg`

command to look at a scatterplot of our observations with the best linear fit (i.e. the regression line) to better see the positive relationship. There is a clear, positive association between these variables.

## Running the Regression

The following syntax runs the regression.

```
proc reg data=twitter_data;
title "Linear Regression Model";
model vote_share = mshare;
run;
```

This returns quite a few tables and figures.

The first table tells us that there were 406 observations in the data, and all 406 were used in the analysis.

The next table provides us with an ANOVA table that gives 1) the sum of squares for the model, often called the regression sum of squares, 2) the error, or residual, sum of squares, and 3) the total sum of squares. Dividing the `Sum of Squares`

column by the `DF`

(degrees of freedom) column returns the mean squares in the `Mean Square`

column. These values go into calculating the \(F\)-statistic, which is 141.17. The \(F\)-statistic tests the null hypothesis that the independent variable does not help explain any variance in the outcome. We clearly reject the null hypothesis with \(p < 0.001\), as seen by `Pr > F <.0001`

.

Looking at the next table, the `R-squared`

value tells us that the independent variable explains 25.89% of the variation in the outcome. The adjusted \(R^2\) provides a slightly more conservative estimate of the percentage of variance explained, 25.71%. The `Root MSE`

is the square root of the residual MS from the previous table, \(\sqrt{213.549258} = 14.613\). This value gives a summary of how much the observed values vary around the predicted values, with better models having lower RMSEs. It is also used in the formula for the standard error of the coefficient estimates, shown in the next table. The table also displays the dependent variable’s mean and the coefficient of variable (the root MSE divided by the mean and multiplied by 100).

The final table tells us the results of the regression model. For each increase of one on the `mshare`

variable, the vote share increases by 0.269. The standard error tells us how much sample-to-sample variability we should expect. Dividing the coefficient by the standard error gives us the \(t\)-statistic used to calculate the \(p\)-value. Here we see that the `mshare`

coefficient estimate is easily significant, \(p < 0.001\).

The estimate for the intercept in the simple regression equation is also presented here. This is the vote share we expect when Tweet share equals zero. Here we see that the predicted value is 37.04, which coincides with what we saw above in the scatterplot. The estimated constant value is significantly different from zero, \(p < 0.001\), though this test is of less interest to us compared to assessing the significance of the regression line slope.

SAS also creates fit diagnostics figures, but we will not cover those here. For more information on assessing model fit, see our tutorial here.

## Fun Facts about Simple Regression

In a simple regression only (that is, when there is just a single independent variable), the \(R^2\) is exactly equal to the Pearson correlation between the two variables. To see this, run:

```
proc corr data=twitter_data;
var vote_share mshare;
run;
```

The correlation between Tweet share and vote share is 0.5089. If we square this, we get

\[ 0.5089^2 = 0.2589, \]

which is the same as the \(R^2\) value from the regression.

Also, in simple regression only, the model \(F\)-test is the same as the test for the single independent variable. A \(t\)-statistic with \(k\) degrees of freedom is equal to an \(F\)-statistic with 1 and \(k\) degrees of freedom. When there are no other predictors in the model, the square root of \(F\) will equal the \(t\) for our coefficient,

\[ \sqrt{141.17} = 11.88. \]

For more detailed information on where these numbers come from, consult our simple regression tutorial.