Week 5

• What is the line of best fit defined by?

  y = ax + b
• What are the fitted values?

  The prediction for y based on the observed value of x
• What are the residuals?

  The vertical difference from the regression line to the recorded data point
• When can the outcome be misleading?

  When data doesn't have a linear relationship or contains anomalies
• When is a line of best fit not useful?

  Typically, the more complex a graph is, the less useful a line of best fit is.
• What are the 5 steps in linear regression

  1. Plot the data
  2. Consider assumptions
  3. Fit the regression
  4. Diagnostic plots
  5. Plot the results with uncertainties
• Step 1 in more detail

  When considering regression studies, it's important to consider third variables that may impact bout explanatory and response variables.
• Step 2 in more detail

  Linearity of expected value, constant variance, independence, normally distributed residuals
• What is the correlation coefficient?

  R, it is between -1 and 1

  1 means a perfect correlation with a positive gradient and -1 is a negative gradient
• If y = 2x, what is the correlation between x and y?

  1
• if y = -0.1, what is the correlation between x and y?

  -1
• if R=0 what does that mean?

  This means that there is no correlation and no linear relationship between x and y
• what does R^2 mean?

  This is the fraction of variance explained. If R^2 = 0.8, then 80% of the variance in y is explained by variance in x
• When is one way analysis of variance (ANOVA) useful?

  When two or more levels in categorical explanatory variables.

  When there are two categories, ANOVA contains an f-test analogous to the two sample t-test (it is not identical).
• What are the assumptions we make in ANOVA analysis?

  The validity of our data depends on the assumptions we make about our data

  • normally distributed residuals
  • independent errors (if errors are correlated)
  • random sampling - closely related to independent errors
  • homogeneity if variance (i.e residuals could all be sampled from same normal distribution)