Compute a linear equation between two or more variables

Are the two variables correlated (no equation)?

Parametric Approach: Pearson's r. Requires a linear relationship and that residuals follow a normal distribution.

Nonparametric Approach. Spearman's rho or Kendall's tau. Relationship may be linear or nonlinear. No normality requirement.

See Helsel & others (2020) Chapter 8.

Is my one explanatory (X) variable a significant predictor of Y?

Parametric Approach: Linear Regression. Requires a linear relationship and that residuals follow a normal distribution.

Bootstrap Approach: Computes significance test with a bootstrap. Requires a linear relationship, but not normality of residuals.

Nonparametric Approach. Theil-Sen nonparametric line. Requires a linear relationship, but not normality of residuals.

See Helsel & others (2020) Chapters 9 and 10.

Which of several possible explanatory variables are significant predictors of Y?

Parametric Approach: Multiple Regression. Requires a linear relationship on partial plots and that residuals follow a normal distribution.

Nonparametric Approach. A two-step process. Compute the residuals of Y from a LOESS surface smoothed versus one or more X variables. In step two, model and test the relationship between the residuals and the remaining primary X variable of interest using the Theil-Sen line. Assumes a linear relationship between residuals and the primary X variable, but does not require normal residuals.

See Helsel & others (2020) Chapters 11 and 12.

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Chapter references are to the 2020 Second Edition of Statistical Methods in Water Resources.