Coefficient interpretations

• Keep in mind that words expected, average, and mean are interchangeable.

• The following are required parts of the interpretation

  • Units of Y

  • Units of X

  • Discussing intercept: Mean or average or expected before Y

  • Discussing coefficient for continuous covariate: Mean or average or expected before difference, increase, or decrease

    • OR: Mean or average or expected before Y
    • Only need before difference or Y!!

  • Confidence interval

  • If other covariates in the model

    • Discussing intercept: Must state that variables are equal to 0

      • or at their centered value if centered!

    • Discussing coefficient for covariate: Must state “adjusting for all other variables”, “Controlling for all other variables”, or “Holding all other variables constant”

      • If only one other variable in the model, then replace “all other variables” with the single variable name

• When the estimate of the population coefficient is…

  • Positive: using the word “increase” in the place of “difference” helps with understanding the relationship

  • Negative: using the word “decrease” in the place of “difference” helps with understanding the relationship

Population Model	Variable information	Coefficient estimate interpretations
\(Y=\beta_0+\beta_1X_1+\epsilon\)	\(X_1\): continuous covariate	\(\widehat\beta_0\): Mean \(Y\) when \(X_1\) is 0 Example: For someone who is 0 years old, the expected peak exercise heart rate is 214.233 beats per minute (95% CI: 204.918, 223.548) \(\widehat\beta_1\): Mean difference in \(Y\) per 1 unit increase in \(X_1\) Example: For every one year increase in age, the expected peak exercise heart rate decreases 0.834 bpm (95% CI: ….)
\(Y=\beta_0+\beta_1X^c_1+\epsilon\)	\(X^c_1\): continuous covariate that is centered around its mean or median	\(\widehat\beta_0\): Mean \(Y\) when \(X^c_1\) is at its mean or median \(\widehat\beta_1\): Mean difference in \(Y\) per 1 unit increase in \(X^c_1\)
\(Y=\beta_0+\beta_1X_1+\beta_2X_2+\epsilon\)	\(X_1\): continuous covariate \(X_2\): continuous covariate