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Regression Modelling (What :question: (Equation line, if linear r'ship…

- - - - Independent/
        Explanatory
        
        X'(s)
      - Dependent/
        Target
        
        Y
  - - - :one: Does a set of predictor variables predict well an outcome variable
      - :two: Which variables are significant predictors, and the magnitude of impact on outcome variable
  - - - scatter plot
        
        check
        
        linear or non-linear pattern
        
        If non-linear, consider transformation
        
        deviations from patterns i.e. outliers
        
        remove outliers, if there are a non-statistical reasons :!!:
    - - fit least squares regression line
        
        check assumptions
        
        Residual plot
        
        linear pattern
        
        constant standard deviation assumption
        
        if good fit model
        
        write regression equation
        
        & identify
        
        standard errors of estimates
        
        estimate of variance/sd
        
        R squared
    - - test of significance
        
        significance of predictor variable
        
        t-test
        
        or Ftest
        
        confidence Interval for the estimate of regression coefficient
- - - - value of the intercept and slope
    - - test the accuracy of coefficient estimates
        
        the best estimate of variance
        
        represents how far data will fall from the lm predictions,
        on the scale of Y measurement
        
        can be used to compute Confidence Interval
        
        e.g. 𝜷’₁ ± 2.SE( 𝜷’₀) 95% CI
        
        approximately a 95% chance that the interval will contain the true value of 𝜷₁ (under a scenario where we got
        repeated samples like the present sample)
        
        can be used to perform hypothesis test i.e. t statistics
        
        reflects that the estimator varies under repeated sampling
  - - - probability of observing any value equal or > |t|
        
        related to t-statistics value
      - to make the decisions whether to retain or reject the null hypotheses
      - P-value < 5% is reasonable to indicate the relationship between Y and X exists not by chance and thus the H0 can be rejected
    - - Indicates how far the data points are from the regression line on average.
      - estimate standard deviation of the residuals, e
      - :+1: lower values signifies that the distances between the data points and the fitted values are smaller
      - http://statisticsbyjim.com/regression/standard-error-regression-vs-r-squared/
    - - proportion of variance in Y
        explained/predictable from x
      - [0,1]
      - For R-squared, you want the regression model to explain higher percentages of the variance. Higher R-squared values indicate that the data points are closer to the fitted values. While higher R-squared values are good, they don’t tell you how far the data points are from the regression line. (i.e. Residual Standard Error)
    - - t-statistics can only evaluate one term, while F stats can do > 1
      - test hypothesis
    - - t-statistics, to test null hypothesis
        
        for independent var, H0 : = 0
        
        if = 0, then Y is not associated with X
        
        if t value further away from 0,
        then greater evidence to reject H0
        
        can only evaluate one term, while F stats can do > 1
- - - - EXACT r'ship btween Y & X
      - e.g. r'ship between Celcius & Fahrenheit
    - - Not exact
      - a r'ship, in which trend exists between X & Y but with scatter
  - - - Linear
      - Non-linear
    - - Linear
      - Non-linear
      - More than 1 explanatory variable
- - - - refer to notes
      - least squares method, such that RSS, residual sum of squares is minimized
    - - proportion of variance in Y
        explained/predictable from x
      - squared of Correlaton coefficient
        for linear regression only
      - R2 ~near 0
        
        Don't use regression :forbidden:
      - R2 nearest to
        significantly > 0
        
        Use regression :check:
    - - Don't use values of x outside the range of values used to create regression equation
      - check range() of x sample
    - - Residual = Observed - Predicted Value
        
        properties
        
        Sum of e = 0
        
        mean of e = 0
      - e = Y - Y'
      - Residual plot
        
        a scatter plot
        
        for simple linear regression (1 x var)
        
        y-axis: e
        
        x-axis: X
        
        If the points are randomly dispersed around the horizontal axis i.e. 0
        
        Linear Regression is good :check:
        
        typically (DA notes)
        Residual vs. fit plot
        
        y-axis: e
        
        x-axis: fitted/predicted value of y
      - there is a residual, e value for each data point