Statistical Models for the Analysis of Single-Case Intervention Data Introduction to: Regression Models (models for a single participant) Multilevel Models (models for multiple participants)

Why consider statistical models? Can provide effect size estimates and confidence intervals for those estimates. e.g., We are 95% confident the immediate shift in level for Jenny was an increase of between 11 and 14 minutes of time spent reading, or between 1.5 and 2.0 standard deviations

Regression Imagine a scatter plot showing the relationship between motivation and achievement.

Regression allows us to summarize the relationship between the variables.

Often when we think of regression we think of each data point coming from a different individual, but all the observations could come from the same individual.

What is the rate of change for Jody? Rise Run

For single-case studies we expect a discontinuity Baseline Intervention

What is the shift in level for Jody? Effect =

What is the immediate shift in level and the shift in slope for Jody? 𝑌= 𝛽 0 + 𝛽 1 Phase + 𝛽 2 Time + 𝛽 3 Phase*Time + e 𝛽 3 = ∆ 𝑠𝑙𝑜𝑝𝑒 𝛽 1 𝛽 2 = slopeA 𝛽 0 -5 -4 -3 -2 -1 0 1 2 3 4 5

Issues to Keep in Mind You may have to choose at what point in time you should calculate the effect size

You may want to standardize the effect size If so, what should be use?

There needs to be a match between the trajectory specified in the model and what is seen in the data Effect = b1? This seems incorrect

What is seen may require specification of a complex growth trajectory Do you think specification tends to be easier when there are more or less observations in a phase?

Correct model specification requires more than just correctly specifying the growth trajectory Should you assume the errors (ei): are independent? have common variance? are normally distributed? If so, could use OLS, but if not GLS or Bayesian estimation