least squares regression

Examples of least squares regression in the following topics:

• Least-Squares Regression

  • The criteria for determining the least squares regression line is that the sum of the squared errors is made as small as possible.
  • The process of fitting the best- fit line is called linear regression.
  • The criteria for the best fit line is that the sum of squared errors (SSE) is made as small as possible.
  • Therefore, this best fit line is called the least squares regression line.
  • Ordinary Least Squares (OLS) regression (or simply "regression") is a useful tool for examining the relationship between two or more interval/ratio variables assuming there is a linear relationship between said variables.

• Introduction to fitting a line by least squares regression

  • In this section, we use least squares regression as a more rigorous approach.

• A Graph of Averages

  • A graph of averages and the least-square regression line are both good ways to summarize the data in a scatterplot.
  • The most common method of doing this is called the "least-squares" method.
  • The least-squares regression line is of the form $\hat{y} = a+bx$, with slope $b = \frac{rs_y}{s_x}$ ($r$ is the correlation coefficient, $s_y$ and $s_x$ are the standard deviations of $y$ and $x$).
  • The points on a graph of averages do not usually line up in a straight line, making it different from the least-squares regression line.
  • The graph of averages plots a typical $y$ value in each interval: some of the points fall above the least-squares regression line, and some of the points fall below that line.

• Conditions for the least squares line

  • If there is a nonlinear trend (e.g. left panel of Figure 7.13), an advanced regression method from another book or later course should be applied.
  • The variability of points around the least squares line remains roughly constant.
  • Be cautious about applying regression to data collected sequentially in what is called a time series.
  • Should we have concerns about applying least squares regression to the Elmhurst data in Figure 7.12?
  • Least squares regression can be applied to these data.

• The Equation of a Line

  • In statistics, linear regression can be used to fit a predictive model to an observed data set of $y$ and $x$ values.
  • In statistics, simple linear regression is the least squares estimator of a linear regression model with a single explanatory variable.
  • Simple linear regression fits a straight line through the set of $n$ points in such a way that makes the sum of squared residuals of the model (that is, vertical distances between the points of the data set and the fitted line) as small as possible.
  • Linear regression was the first type of regression analysis to be studied rigorously, and to be used extensively in practical applications.
  • If the goal is prediction, or forecasting, linear regression can be used to fit a predictive model to an observed data set of $y$ and $X$ values.

• Estimating the Target Parameter: Point Estimation

  • Another popular estimation approach is the linear least squares method.
  • Linear least squares is an approach fitting a statistical model to data in cases where the desired value provided by the model for any data point is expressed linearly in terms of the unknown parameters of the model (as in regression).
  • In statistics, linear least squares problems correspond to a statistical model called linear regression which arises as a particular form of regression analysis.
  • One basic form of such a model is an ordinary least squares model.
  • Contrast why MLE and linear least squares are popular methods for estimating parameters

• Polynomial Regression

  • For this reason, polynomial regression is considered to be a special case of multiple linear regression.
  • Polynomial regression models are usually fit using the method of least-squares.
  • The least-squares method minimizes the variance of the unbiased estimators of the coefficients, under the conditions of the Gauss–Markov theorem.
  • The least-squares method was published in 1805 by Legendre and in 1809 by Gauss.
  • Although polynomial regression is technically a special case of multiple linear regression, the interpretation of a fitted polynomial regression model requires a somewhat different perspective.

• Model Assumptions

  • The following are the major assumptions made by standard linear regression models with standard estimation techniques (e.g. ordinary least squares):
  • This is to say there will be a systematic change in the absolute or squared residuals when plotted against the predicting outcome.
  • In effect, residuals appear clustered and spread apart on their predicted plots for larger and smaller values for points along the linear regression line, and the mean squared error for the model will be wrong.
  • (Actual statistical independence is a stronger condition than mere lack of correlation and is often not needed, although it can be exploited if it is known to hold. ) Some methods (e.g. generalized least squares) are capable of handling correlated errors, although they typically require significantly more data unless some sort of regularization is used to bias the model towards assuming uncorrelated errors.
  • For standard least squares estimation methods, the design matrix $X$ must have full column rank $p$; otherwise, we have a condition known as multicollinearity in the predictor variables.

• Checking the Model and Assumptions

  • These assumptions are similar to those of standard linear regression models.
  • That is, there will be a systematic change in the absolute or squared residuals when plotted against the predicting outcome.
  • Error will not be evenly distributed across the regression line.
  • In effect, residuals appear clustered and spread apart on their predicted plots for larger and smaller values for points along the linear regression line; the mean squared error for the model will be incorrect.
  • Most experts recommend that there should be at least 10 to 20 times as many observations (cases, respondents) as there are independent variables, otherwise the estimates of the regression line are probably unstable and unlikely to replicate if the study is repeated.

• Exercises

  • What is the equation for a regression line?
  • What criterion is used for deciding which regression line fits best?
  • In a regression analysis, the sum of squares for the predicted scores is 100 and the sum of squares error is 200, what is R2?
  • The sum of squares total is 1000.
  • What is the sum of squares of the predicted values?