Introduction to Linear Models

CHECK-IN : Post Mini-Exam Survey!

The Linear Model : Prediction is a Line

RECAP : The Mean as a Model (with Error)

DISCUSS. Measurement Error. The Cantril Ladder Score asks people to rank themselves on a ladder in terms of their well-being. What do y’all think of this measure? What benefits / problems are there of this approach? What questions do you have?

RECAP : The Mean as Prediction?

\(\huge y_i = \hat{Y} + \epsilon_i\)

\(\Large y_i\) = the DV = the individual’s actual score we are trying to predict.

  • on the graph: each individual dot

\(\Large \hat{Y}\) = our prediction (the mean).

  • on the graph: the solid red line

\(\Large \epsilon\) = residual error = difference between predicted and actual values of y

  • on the graph: the distance between each dot and the line.

RECAP : The Mean as Prediction With Error

\(\huge \epsilon_i = y_i - \hat{Y}\)

Error = actual score - prediction (the mean)

residual <- d$SWLS_24 - mean(d$SWLS_24, na.rm = T)
SST <- sum(residual^2, na.rm = T)
SST
[1] 233.827
  1. This number is critical - quantified error in our prediction.
  2. This number makes no sense (total unsquared error??!?!)
  • sd gives some context (average error)
  • use number as a starting place to see if we can make better predictions

KEY IDEA : The mean is an okay starting place for our predictions, but we can try to do better!

the mean the linear model ™ ©

DISCUSS : Making Predictions

  • THINK ABOUT A LINEAR MODEL : how do you think the variables (below) would help (or not help) us predict the happiness of a country in 2024? Why / why not???

  • THINK ABOUT SOURCES OF DATA. What other variables do you think would be important to include in this dataset?

TIME TO PLAY : WHERE’S THE LINE

What’s Going On : It’s a Line

\(\Huge y_i = a + b_1 * X_i + \epsilon_i\)

   (Intercept) LifeExpectancy 
         -3.14           0.12

\(\Large y_i\) = the DV = each actual score on the DV.

  • on the graph:** each dot on the y-axis

\(\Large a\) = the intercept = starting place for our prediction (“the predicted value of y when all x values are zero”.)

  • on the graph:** the value of the line at X = 0

\(\Large X_i\) = the IV = the actual score on the IV.

  • on the graph: the value of each dot on the x-axis

\(\Large b_1\) = the slope = an adjustment to our prediction of y based changes in x

  • on the graph: how much the line increases in y value when x-values increase by 1 unit.

\(\Large \epsilon_i\) = residual error = the distance between actual y and predicted y

  • on the graph: the distance between each individual data point and the line.

The Linear Model : Error in Our Predictions

Which model has dots that are closer to the line??? (Mean of Satisfaction or Life Expectancy)

The Linear Model : Quantifying Error

SWLS24 <- mod$model$SWLS_24 # using data that also have life expectancy
residual <- SWLS24 - mean(SWLS24) # error
SST <- sum(residual^2, na.rm = T) # sum of squared errors
SST # total sum of squared errors
[1] 195.867
SSM <- sum(mod$residuals^2) # R saves the residuals for you
SSM # hooray!
[1] 89.63457

Some math helps us evaluate.

SST - SSM # the total difference
[1] 106.2324
(SST - SSM)/SST # the relative difference
[1] 0.5423702
summary(mod)$r.squared # R^2
[1] 0.5423702

BREAK TIME : MEET BACK AT 3:30

PART 2 : DEFINING CONTINUOUS MEASURES

Milestone #2 (Measures)

The Likert Scale

Example : Satisfaction With Life Scale

  • scale : The variable that you want to measure as a continuous variable.

  • items : The specific question(s) in the scale. Each item measures some aspect of the variable the researcher is interested in.

    • positively keyed items : An item that measures the high end of the scale, where answering “yes” to the question means you are high on this variable.

    • negatively keyed items : An item that measures the low end of the scale, where answering “yes” to the question means you are low on the variable.

  • response scale : How people answer the scale items.

Some Student Examples : Professor Office Hours