Goal

In the last lab, we focused on visualizations as well as summary statistics for one numeric variable. Today, we are going to use visualizations and numeric measures to describe the relationship between two numeric variables. The primary visualization we will use is a scatter plot, and the primary numeric measure we will use is called correlation. We will also see how to create a least squares linear regression line. This is another tool we use to describe the relationship between two numeric variables.

The Data

We are going to continue to work on the data set on student stress and sleep from our last lab. Recall that the data were collected from \(n=1442\) university students who wore a Fit Bit 3 device that recorded information on student sleep, stress, and motion.

The data set can be found on Canvas and is also linked here: https://www.dropbox.com/scl/fi/vj2cr3xdjo4gces1273d1/StressStudy.csv?rlkey=napepcy0dpo56ysw4xpeskpkd&st=9a1o84vi&dl=0

As we did in Lab 2, we are going to use StatKey to do our analysis. However, we will use a different StatKey tool today:https://www.lock5stat.com/StatKey/descriptive_2_quant/descriptive_2_quant.html

Graphing the Relationship between two numeric variables

Suppose a counselor from a University Counseling Center ask you if the quality of sleep a student gets each night has a strong relationship with the Fit Bit recorded stress score. The client is specifically interested in having you use a linear regression model to describe the relationship between X = sleep score and Y = stress score. Remember that response variables are usually denoted by a capital letter Y, and explanatory variables are usually denoted by a capital letter X.

To complete this task, we need to

1. Determine if using linear regression is appropriate and
2. If so, build and interpret a linear regression line.

We need to start with (1), because it will not make sense to use linear regression if the relationship between X and Y is not the right shape!

In order to check the shape, we need to create a visualization that explores the relationship between sleep score and stress score.

Now that we know what type of plot we need, let’s make it! Once you upload your data, you should see this on your screen:

Click the column for sleep score first, and then the column for stress. The order you click tells the applet which variable is X and which is Y.

Creating the plot is an excellent first step to answering the client’s question, but once we have created the plot we then have to be able to interpret what this plot is telling us. Generally, when we are comparing two numeric variables using this type of plot, we are interested in four things:

1. Does the relationship seem linear?
2. Is the relationship positive, negative, or neither?
3. Are there any points that seem not to follow the trend of the rest of the data? In other words, are there any visually evident outliers?
4. Does the relationship between these variables seem to be strong or weak?

There is a reason why we are interested in these specific questions. The first question (Does the relationship seem linear?) tells us whether or not using a regression line makes sense for these two variables. All models have assumptions. A big one for regression lines is that the relationship between the two variables we are modeling should be able to be described using a line. In other words, the relationship should look linear. If it does not, we should not use a regression line.

The second question (Is the relationship positive, negative, or neither?) tells us what sort of slope to expect when we fit a regression line to the data. Do we expect a positive slope? A negative one? This intuition will allows us to check our line once we fit it. Do the results of the line make sense with what the plot is telling us?

The third question (Are there any points that seem not to follow the trend of the rest of the data?) helps with identifying outliers that can strongly impact the fit of a line.

The final question asks about the strength of the relationship between the two variables. This is important because a regression line like the client has asked us for uses X to describe the variation in Y. In other words, we are hoping to be able to say something like as X goes up by 1, Y generally goes up by 2.5. If the relationship between X and Y is not very strong, than the line may not be a good tool for describing the relationship.

It can be difficult to assess the strength of a linear relationship just by looking at a plot. Luckily, we have learned that a numerical way to quantify the strength of a relationship is with the correlation.

Based on all of this, regression seems reasonable! We do not need a strong fit to use regression, we just need a line to look like a reasonable choice.

Fitting the regression line

Once we have determined that regression is an appropriate choice, we can move into the actual process of estimating our line. The question then becomes…which line.

Making Predictions