AP Statistics Curriculum 2007 GLM MultLin
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General Advance-Placement (AP) Statistics Curriculum - Multiple Linear Regression
In the previous sections we saw how to study the relations in bivariate designs. Now we extend that to any finite number of variables (multivariate case).
Multiple Linear Regression
We are interested in determining the linear regression, as a model, of the relationship between one dependent variable Y and many independent variables X_{i}, i = 1, ..., p. The multilinear regression model can be written as
- , where is the error term.
The coefficient β_{0} is the intercept ("constant" term) and β_{i}s are the respective parameters of the p independent variables. There are p+1 parameters to be estimated in the multilinear regression.
- Multilinear vs. non-linear regression: This multilinear regression method is "linear" because the relation of the response (the dependent variable Y) to the independent variables is assumed to be a linear function of the parameters β_{i}. Note that multilinear regression is a linear modeling technique not because is that the graph of Y = β_{0} + βx is a straight line nor because Y is a linear function of the X variables. But the "linear" terms refers to the fact that Y can be considered a linear function of the parameters ( β_{i}), even though it is not a linear function of X. Thus, any model like
is still one of linear regression, that is, linear in x and x^{2} respectively, even though the graph on x by itself is not a straight line.
Parameter Estimation in Multilinear Regression
A multilinear regression with p coefficients and the regression intercept β_{0} and n data points (sample size), with allows construction of the following vectors and matrix with associated standard errors:
or, in vector-matrix notation
Each data point can be given as , . For n = p, standard errors of the parameter estimates could not be calculated. For n less than p, parameters could not be calculated.
- Point Estimates: The estimated values of the parameters β_{i} are given as
- Residuals: The residuals, representing the difference between the observations and the model's predictions, are required to analyse the regression and are given by:
The standard deviation, for the model is determined from
The variance in the errors is Chi-square distributed:
- Interval Estimates: The 100(1 − α)% confidence interval for the parameter, β_{i}, is computed as follows:
- ,
where t follows the Student's t-distribution with n − p − 1 degrees of freedom and denotes the value located in the i^{th} row and column of the matrix.
The regression sum of squares (or sum of squared residuals) SSR (also commonly called RSS) is given by:
- ,
where and is an n by 1 unit vector (i.e. each element is 1). Note that the terms y^{T}u and u^{T}y are both equivalent to , and so the term is equivalent to .
The error (or explained) sum of squares (ESS) is given by:
The total sum of squares (TSS) is given by
Examples
In the simple linear regression case, we were able to compute by hand some (simple) examples). Such calculations are much more involved in the multilinear regression situations. Thus we demonstrate multilinear regression only using the SOCR Multiple Regression Analysis Appet.
Use the SOCR California Earthquake dataset to investigate whether Earthquake magnitude (dependent variable) can be predicted by knowing the longitude, latitude, distance and depth of the quake. Clearly, we do not expect these predictors to have a strong effect on the earthquake magnitude, so we expect the coefficient parameters not to be significantly distinct from zero (null hypothesis). SOCR Multilinear regression applet reports this model:
References
- SOCR Home page: http://www.socr.ucla.edu
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