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. Copy this link and paste it into a new browswer window: 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. Copy this link and paste it into a new browswer window: https://www.lock5stat.com/StatKey/descriptive_2_quant/descriptive_2_quant.html

Today we are interested in sleep score and stress score.

Sleep Score

• Poor Sleep (Under 60): Indicates significant disruptions, very short duration, or a highly elevated sleeping heart rate.
• Fair Sleep (60–79): The most common starting point for new users. Sleep may be adequate in length but lacking in deep or REM stages.
• Good Sleep (80–89): A strong indicator of healthy, restorative sleep. Most users aim to consistently land in this tier.
• Excellent Sleep (90–100): A highly optimized night of sleep. Achieving a score in the 90s requires a near-perfect alignment of duration, sleep stages, and cardiovascular recovery.

Stress Score

• Score Above 70: Low signs of physical stress in the body, meaning you are more able to handle stress, new exercise, etc., during the day.
• Score between 50 - 69: Moderate signs of physical stress in the body. You tend to be less able to handle physical stressors (including emotional stress and heavy thinking!) during the day.
• Score below 50: High signs of physical stress in the body. This means your body is under physical strain already and is likely less able to handle physical stressors (including emotional stress and heavy thinking!) during the day.

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 (sleep score) has a strong relationship with the Fit Bit recorded ability to handle stress (stress score). In other words, the higher the stress score, the better the body’s ability to handle stress! Think about stress score as a way to measure stress tolerance.

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 (tolerance). 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.

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.

Building 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.

The regression line is the line that minimizes the distance between data points and the line. In other words, we try to make the line describe the relationship between X and Y as well a possible.

To do this, you need (1) a slope and (2) an intercept. The line is then:

\[\hat{y} = intercept + slope \times x\]

For example, if the intercept were 2 (it’s not!!) and the slope were .5 (it’s not!!), the line would be

\[\hat{y} = 2 + .5x\] But how do we get the intercept and the slope for the regression line?? We are going to choose the intercept and slope that make the residuals as small as possible. Remember, the residuals measure the difference between the y coordinate of our points and the y value we predict using the line.

In the plot below, we can see residuals visualized as the green vertical lines.

Wow there are a lot of lines! Specifically, there is one line per data point. What we want to do is find the intercept and slope that makes the overall sum of these residuals squared as small as possible. Why squared? Well, just like with the standard deviation, we have to get rid of those negatives!

So, we need to find a slope and an intercept that make the sum of squared residuals as small as possible. The values that do this are:

• Slope: \(R\frac{s_y}{s_x}\)
• Intercept: \(\bar{y} - Slope (\bar{x})\)

where

• \(R\) = the correlation between \(X\) and \(Y\)
• \(s_y\) = the standard deviation of \(Y\)
• \(s_x\) = the standard deviation of \(X\)
• \(\bar{y}\) = the mean of \(Y\)
• \(\bar{x}\) = the mean of \(X\)

Any other slope or intercept you choose will result in a higher sum of squared residuals, meaning the line is further from the actual data. Want to know why? Ask me!!

Using the Line

Once we have the line, it helps to think about it to make sure all the components make sense. :::{.question} #### Question 9

Think back to your plot. Does it make sense that the slope coefficient is positive? Why or why not? :::

Once we have checked that the slope seems reasonable, we can use it to describe the relationship between sleep and stress!! We usually do this by interpreting the slope.

Interpretation is great, but we also need to reflect on whether or not the change the slope makes is practically important, meaning a large enough change to actually matter. For example, what if for every 1 hour you study, we predict your exam score increases by .0001%. Is that a large enough change to actually matter to you? Since that means you would need to study over 1000 hours to improve by 1%, no!

Sometimes we can tell if something is practically important for ourselves, and sometimes we need to bring in an expert. This is why statistics and data science is often a team sport!

Making Predictions

In addition to describing relationships, we also use regression lines for making predictions. Basically, we do this by taking a value for X, putting it in the equation for the regression line, and then solving for \(\hat{y}\).

When we do prediction, we have to be careful to avoid extrapolation, which means making predictions for an x that is far away from the x values in our data set.

Putting it together