Chapter 7: Linear Regression Flashcards
Define ‘Linear model’.
An equation of the form
y-hat = bo + b1 x
where the x-variable is being used as an explanatory variable to help predict the response variable y. To interpret a linear model, we need to know the variables (along with their W’s) and their units.
Define ‘Model’.
An equation or formula that simplifies and represents reality.
Define ‘Predicted (fitted) value’.
The valye of y-hat found for a given x-value in the data. A predicted value is found by substituting the x-value in the regression equations. The predicted values are the values on the fitted line; the points (x, y-hat) all lie exactly on the fitted line.
The predicted values are found from the linear model that we fit:
y-hat = bo + b1 x.
Define ‘Residuals’.
The differences between the observed values of the response variable y and the corresponding values predicted by the regression model - or, more generally, values predicted by any model (y-hat).
Residual = Obs. y-value - Pred. y-value = y - y-hat.
Define ‘Regression line (Line of best fit)’.
The particular linear equation
y-hat = bo + b1 x
that satisfies the least squares criterion is called the least squares regression line. Casually, we often just call it the regression line, or the line of best fit.
Define ‘Least squares’.
The least squares criterion specifies the unique line that minimizes the variance of the residuals or, equivalently, the sum of the squared residuals.
Define ‘Slope’.
The slope, b1, gives a value in “y-units per x-units.” Changes of one unit in x are associated with changes in b1 units in predicted values of y.
The slope can be found by
b1 = r (sy / sx)
Define ‘Intercept’.
The intercept, bo, gives a starting value in y-units. It’s the y-hat-value when x is 0. You can find the intercept from bo = y-bar - b1 x-bar.
Define ‘Regression to the mean’.
Because the correlation is always less than 1.0 in magnitude, each predicted y-hat tends to be fewer standard deviations from its mean that its corresponding x was from its mean.
Define ‘Standard deviation of the residuals (se)’.
The standard deviation of the residuals is found by se = sqrt (∑e^2 / (n-2) ).
When the residuals are roughly Normally distributed (check their histogram), their sizes can be well described by using this standard deviation and the 68-95-99.7 Rule.
Define ‘Coefficient of determination R^2’.
The square of the correlation between y and x.
- R^2 gives the fraction of the variability of y accounted for by the least squares linear regression on x.
- R^2 is an overall measure of how successful the regression is in linearly relating y to x.
Understand the correlation coefficient as the number of ___ ___ by which one vairbale is expected to change for a one ___ ___ change in the other. (r is always less than 1 in magnitude, recall sign)
Standard deviation. Sections that discussed it.
Always check the ___ to check for violations of assumptions and conditions and to identify any outliers.
Residuals.
- Any bends (Straight Enough Condition)?
- Outliers?
- Change in spread?
What are the conditions for regression?
- Quantitative Variables Condition
- Straight Enough
- Does the Plot Thicken?
- Outlier Condition
