Multiple Regression Analysis
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1 Multiple Regression Analysis Basic Concept: Extend the simple regression model to include additional explanatory variables: Y = β 0 + β1x1 + β2x βp-1xp + ε p = (number of independent variables + 1) = number of β s to be estimated Make the same assumptions as in simple linear regression regarding the error terms: independent, normally distributed, constant variance σ at all levels of the X s. Use statistical software to calculate least squares estimates. Example data set: Prod % Price Index Cost where the independent variable "Prod %" represents the production level, as a percent of rated capacity, that a production line is run at; the independent variable "Price Index" represents a materials/labor price index; and the dependent variable "Cost" represents the resulting average manufacturing cost per unit. 2 ε Multiple Regression 1 st Handout 1
2 Regression Analysis The regression equation is Cost = Prod & Price Index Predictor Coef StDev T P VIF Constant Prod & Price In S = R-Sq = 91.4% R-Sq(adj) = 90.4% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Source DF Seq SS Prod & Price In Obs Prod & Cost Fit StDev Fit Residual St Resid R R denotes an observation with a large standardized residual Multiple Regression 1 st Handout 2
3 Residuals Versus the Fitted Values (response is Cost) 3 2 Deleted Residual Fitted Value 6 Multiple Regression 1 st Handout 3
4 Other goodies: Correlation Matrix: Correlations (Pearson) Cost Prod & Prod & Price In Scatterplot Matrix: Cost Prod & Price Index Multiple Regression 1 st Handout 4
5 Diagnostics in Multiple Regression The assumptions of multiple regression are basically the same as in simple regression: the model is properly specified and the random error terms are independent and normally distributed with constant variance σ 2. Simple regression diagnostics (plot of residuals vs. fitted values, etc.) are also appropriate for multiple regression. In addition, you may wish to consider plotting the residuals versus each of the independent variables: patterns here may represent curvilinearity or heteroskedasticty with respect to a specific independent variable. A problem unique to multiple regression is that of multicollinearity, which occurs when linear relationships exist among the independent variables in the model. If MC is present in the model: (1) The "usual" interpretations of the regression coefficients are not valid, since it doesn't make sense to increase the value of one independent variable without changing the values of the others. (2) The standard deviations of the regression coefficients (the b's) are inflated, which leads to t-values that are artificially low. This can lead to an apparent contradiction, namely a model that's quite significant (as evidenced by the F-statistic) but which appears to have no significant individual independent variables. (3) Although problems (1) and (2) may be present with respect to the "right hand side" of the model, there is generally no negative impact on the model's usefulness to forecast or predict the dependent variable (Y). If that's your primary concern, multicollinearity is not a problem. Variance inflation factors are a common tool for identifying multicollinearity You can also detect simple MC problems (involving only a pair of independent variables) by generating a correlation matrix involving all of the independent variables. A good rule of thumb is that multicollinearity is present and likely to be a problem if any independent variable has a VIF which exceeds 10 (corresponding, for pairwise comparison, to a correlation of.95). Variables having such high VIFs are highly correlated with other independent variables in the model, either individually or in combination. A naive approach to dealing with MC is to remove the offending variable, however this may lead to loss of explanatory power and to "omitted variable bias." Multiple Regression 1 st Handout 5
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