Lecture 21
Regression

ABD 3e Chapter 17

Chris Merkord

Learning Objectives

• Identify the explanatory variable and response variable in a regression study
• Recognize when a scatterplot suggests an approximately linear relationship between two numerical variables
• Explain how the least squares regression line summarizes the relationship between variables
• Interpret the slope, intercept, and predicted values of a regression equation in context
• Distinguish confidence bands for the mean response from prediction intervals for individual outcomes
• Interpret the slope test, \(R^2\), and other evidence beyond the p-value
• Use residual plots to assess linearity, constant variance, and unusual observations

What is regression?

• Regression is used to predict values of one numerical variable from values of another numerical variable.

• A regression line summarizes the relationship between two variables in a scatter plot.

• It can be used to estimate the expected value of a response variable from an explanatory variable.

• In this chapter, we focus on linear regression, where the relationship is described by a straight line.

Regression example

• Genetic diversity in human populations is related to distance from East Africa.

• A fitted line can be used to predict expected genetic diversity from migration distance.

• The downward slope suggests diversity decreases as distance increases.

Figure 1: Fossil crania from Herto, Ethiopia, dated to approximately 160,000–154,000 years ago, representing early Homo sapiens from East Africa. Included here as contextual evidence for human origins relevant to geographic-distance examples in regression. Source: White et al. (2003), Nature.

Figure 2: Genetic diversity declines with increasing geographic distance from East Africa across sampled human populations. Points represent populations grouped by world region; line shows the fitted linear regression. Source: Prugnolle et al. (2005), Current Biology.

Regression vs correlation

• Both methods describe relationships between two numerical variables.

• Correlation measures strength and direction of association.

• Regression fits an equation to predict one variable from another.

• Regression also measures how much the response changes when the explanatory variable changes.

Two variables in regression

• Response variable (\(Y\)): the outcome we want to predict.

• Explanatory variable (\(X\)): the variable used to explain or predict \(Y\).

• In scatter plots for regression:

  • \(X\) is placed on the horizontal axis.
  • \(Y\) is placed on the vertical axis.

Figure 3: Axes used in regression scatterplots, with the explanatory variable on the horizontal axis and the response variable on the vertical axis.

Study designs for regression

• Regression can be used with observational data.

• Example: randomly sample individuals and measure both \(X\) and \(Y\).

• Regression can also be used in experiments.

• Example: choose treatment levels of \(X\), then measure response \(Y\).

Linear regression

• The most common type of regression is linear regression.

• It fits a straight line through data to predict \(Y\) from \(X\).

• A key assumption is that the true relationship is approximately linear.

• If the relationship is strongly curved, a straight line may be inappropriate.

Figure 4: Example scatterplot showing a positive linear relationship between an explanatory variable and a response variable, with a fitted regression line.

Example 17.1: The lion’s nose

• Managers want to estimate the ages of male lions.

• Older males may be removed with less disruption than younger males.

• Black pigmentation on the nose increases with age.

• We use proportion black on the nose to predict age.

Figure 5: Photo of a male African Lion (Panthera leo). Image: Clément Bardot (CC BY-SA 4.0)

Lion data

• Data from 32 male lions of known age.

• \(X\) = proportion black on the nose.

• \(Y\) = age (years).

• A scatter plot is the first step.

Figure 6: Age of 32 male lions plotted against the proportion of black pigmentation on the nose. Data from Whitlock & Schluter 3e.

The method of least squares

• Many lines can be drawn through a scatter plot.

• We need a rule for choosing the best line.

• Least squares chooses the line with the smallest total squared vertical deviations from the points.

• These vertical deviations are called residuals.

Comparing possible lines

• Poorly chosen lines have large deviations from the data.

• Better lines have smaller deviations.

• The least squares line minimizes the sum of squared deviations.

Figure 7: Lion age data shown with three candidate regression lines that produce large, smaller, and smallest vertical deviations from the observed points.

Why square the deviations?

• Points above the line have positive residuals.

• Points below the line have negative residuals.

• If we simply added deviations, positives and negatives could cancel.

• Squaring avoids cancellation and gives more weight to large errors.

Formula for the line

• A regression line is written as:

\[ Y = a + bX \]

• \(a\) is the intercept.

• \(b\) is the slope.

• Together, they determine the location and tilt of the line.

The intercept and slope

• Intercept : the predicted value of \(Y\) when \(X = 0\)

  • Where the line crosses the \(y\)-axis
  • Units are same as response variable

• Slope : the change in \(Y\) for a one-unit increase in \(X\).

  • Positive slope: larger \(X\) predicts larger \(Y\).
  • Negative slope: larger \(X\) predicts smaller \(Y\).
  • Zero slope: no linear trend.
  • The rate of change in \(Y\) per unit of \(X\).

Figure 8: Example scatterplots showing positive, negative, and zero slopes, and a comparison of higher and lower intercepts.

Calculating the slope

• For a sample, the least squares slope is:

\[ b = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2} \]

• The numerator measures whether \(X\) and \(Y\) tend to increase or decrease together.
• The denominator measures how much the \(X\) values vary.
• Their ratio determines the slope of the fitted regression line.

Calculating the intercept

• Once the slope is known, the intercept can be calculated as:

\[ a = \bar{Y} - b\bar{X} \]

• The fitted regression line always passes through the point:

\[ (\bar{X}, \bar{Y}) \]

• This means the line passes through the sample means of \(X\) and \(Y\).

Lion regression equation

• For the lion data, the fitted line is approximately:

\[ \hat{Y} = 0.88 + 10.65X \]

• Equivalent interpretation:

\[ \widehat{\text{Age}} = 0.88 + 10.65(\text{proportion black}) \]

Figure 9: Lion age plotted against the proportion of black pigmentation on the nose, with the fitted linear regression line and equation.

Interpreting the lion slope

• Each 1.0 increase in proportion black predicts about 10.65 more years of age.

• More practically, a 0.10 increase predicts about:

  \[ 10.65 \times 0.10 = 1.065 \]

  years older on average.

As shown in Figure 9, age increases with pigmentation.

Lion fitted line

• The line summarizes the average trend.

• Individual lions vary around the line.

• Not every point falls exactly on the prediction line.

Figure 9

Populations and samples

• The fitted line from data is a sample estimate.

• We use it to estimate the true regression line in the population.

• Population form:

\[ Y = \alpha + \beta X \]

• Sample estimates:

  • \(a\) estimates \(\alpha\)
  • \(b\) estimates \(\beta\)

Predicted values

• A predicted value lies on the regression line.

• We use:

  \[ \hat{Y} \]

  for a prediction.

• \(\hat{Y}\) estimates the mean response for all individuals with a given \(X\) value.

Lion prediction example

• Predict age when proportion black is \(X = 0.50\):

\[ \hat{Y} = 0.88 + 10.65(0.50) = 6.2 \]

• Lions with \(X = 0.50\) are predicted to average about 6.2 years old.

Use caution with prediction

• Predictions most reliable within the observed range of \(X\) values

• Predicting far outside the data range is called extrapolation

• May misleading because relationship may change beyond sampled data

Figure 10: Extrapolation can produce misleading predictions when a linear regression line is extended beyond the observed data range. Image: Kostia, via Stack Exchange

Figure 11: Very serious examples of extrapolation gone awry. Images: xkcd (CC BY-NC 2.5)

Confidence in predictions

• A regression line gives the best predicted mean value of \(Y\) for each value of \(X\).

• Predictions are most reliable within the observed range of the explanatory variable.

• Every prediction includes uncertainty because the fitted line was estimated from sample data.

• Two common goals:

  • predict the mean response at a given \(X\)
  • predict a single future observation at that \(X\)

Mean response vs individual response

• These two questions are different:

  • What is the average value of \(Y\) when \(X= x\)?

  • What value of \(Y\) might one individual case have when \(X= x\)?

• Both use the same regression line.

• The prediction for an individual is less precise because individuals vary around the line.

Confidence bands vs prediction intervals

Confidence band : interval for the mean response

• narrower
• uncertainty in the fitted line
• Used when studying the overall trend.

Prediction interval : interval for a single observation

• wider
• includes uncertainty in the line plus individual variation
• Used when predicting one case.

Figure 12: Confidence bands and prediction intervals for lion age predicted from the proportion of black pigmentation on the nose.

Uncertainty increases near the edges

• Confidence bands are narrowest near \(\bar{X}\), the center of the observed \(X\) values.
• Confidence bands widen toward the smallest and largest observed values of \(X\).
• There is less information about the mean response at the extremes of the data.
• Prediction intervals are wider overall because they must also include individual variation.
• Prediction intervals may widen somewhat near the edges, but this is often less noticeable.

Testing whether the slope is zero

• A common hypothesis test in regression asks whether the population slope equals zero.

\[ H_0:\beta = 0 \]

\[ H_A:\beta \ne 0 \]

• If \(\beta = 0\), there is no linear relationship between \(X\) and the mean of \(Y\)

• If the p-value is small, we reject \(H_0\)

  (small = less than significance level \(\alpha\) )

• Usually called the t-test for the slope of the regression line

• Generated automatically in R when you use summary(lm_object)

Interpreting the slope test

• A significant result (\(p<\alpha\)) suggests evidence that \(X\) is linearly associated with \(Y\) and may help predict \(Y\).

• It does not establish causation.

• Statistical significance does not necessarily mean a strong relationship or a large slope.

• A small slope can be statistically significant if the sample size is large.

Figure 13: Two example datasets showing that statistical significance can occur with either a strong relationship in a small sample or a weak relationship in a very large sample.

Beyond the p-value

• A \(p\)-value alone does not describe the size, precision, or practical importance of a relationship.

• Interpret the slope test together with other evidence from the model and the data.

• Always examine:

  • slope estimate (effect size)

  • confidence interval (precision)

  • scatterplot (pattern, outliers, linearity)

  • practical importance (real-world relevance)

Using \(R^2\) to measure fit

• \(R^2\) measures the proportion of variation in \(Y\) explained by the regression model

\[ 0 \le R^2 \le 1 \]

• Larger values mean points lie closer to the fitted line

• Smaller values mean more unexplained scatter

• Example: \(R^2 = 0.22\) means 22% of variation is explained

Do not confuse these regression statistics

• Correlation coefficient (\(r\))

  • measures strength and direction of a linear relationship

  • ranges from \(-1\) to \(1\)

• Coefficient of determination (\(R^2\))

  • measures the proportion of variation in \(Y\) explained by the model

  • ranges from \(0\) to \(1\)

• P-value for the slope

  • tests whether the population slope differs from zero

  • does not measure effect size or model fit

• These statistics are related, but they answer different questions.

Assumptions of linear regression

• Relationship between \(X\) and mean \(Y\) is approximately linear
• Residuals have similar spread across values of \(X\)
• Observations are independent
• Residuals are approximately normal (mainly important for tests and intervals)
• Use graphs to evaluate assumptions

Figure 14: Illustration of the assumptions of linear regression: at each value of X, the response variable is normally distributed around the true regression line with equal variance.

Residual plots help assess regression model assumptions

• A residual is the difference between the observed value and the predicted value:

\[ e_i = y_i - \hat{y}_i \]

• Positive residuals: points above the regression line

• Negative residuals: points below the line

• Plot residuals against \(X\) or against fitted values (\(\hat{y}\))

• Help assess linearity, constant variance, and unusual observations

Figure 15: Example residual plots used to assess model assumptions. In the left panel, residuals are centered around zero with a similar vertical spread across values of X, consistent with constant variance. In the right panel, the spread of residuals decreases as X increases, forming a funnel shape that suggests unequal variance (heteroscedasticity).

Detecting nonlinearity

• If the scatterplot bends or levels off, a straight-line model may be inappropriate.

• Residual plots can reveal curved patterns more clearly.

• A curved residual pattern suggests the line is not fitting well.

• Random scatter around zero is more consistent with a good linear fit.

• Do not force a straight line onto clearly curved data.

Figure 16: Residual plot showing a curved pattern caused by fitting a straight-line regression to a nonlinear relationship. Residuals tend to be positive at low and high values of X and negative in the middle, indicating that a linear model is inappropriate.

Transformations

• Transformations can improve linearity or stabilize variance.

Common examples:

• \(\log(Y)\)

• \(\log(X)\)

• \(\log(X)\) and \(\log(Y)\)

• \(\sqrt{Y}\) for counts

• After transformation, interpret results on the transformed scale unless back-transformed.

Regression toward the mean

• Extremely high or low observations often appear closer to average when measured again

• This occurs when repeated measurements are imperfectly correlated and include random variation

• Example:

  • Very high cholesterol at a first visit often tends to be lower later, even without treatment

• Improvement after an extreme starting value does not necessarily mean the treatment worked

• A control group helps separate real treatment effects from regression toward the mean

Summary

• Regression predicts one numerical variable from another

• Linear regression fits the least squares line:

\[ \hat{Y} = a + bX \]

• Slope describes rate of change

• Intercept gives predicted \(Y\) when \(X=0\)

• Predictions are strongest within the data range

• Confidence bands estimate the mean response; prediction intervals estimate individual outcomes

• Slope test: \(H_0: \beta = 0\)

• \(R^2\) : proportion of variation in \(Y\) explained by the model

• Residual plots help assess linearity, constant variance, and unusual observations

• Curved patterns or unequal spread may require transformations or a different model

• Extreme values often move closer to average on remeasurement (regression toward the mean)