Topics in Multiple Regression

Topics in Multiple Regression

In this set of notes we present different extensions to the regular multiple regression model formulated by y=β0+β1x1+β2x2+…+βk+ε. First we discuss the dummy variable that deals with categorical (qualitative) data and then we’ll add the interaction term; finally we’ll discuss non linear regression models

Regression with Categorical Data

The dummy variable is a mathematical tool that makes it possible to include non-numerical information in the regression model. This makes the model more useful in decision making setting, where data may or may not belong to a certain category.

Value | Size(1000ft2) | Fireplace |

84.4 | 2 | Yes |

77.4 | 1.71 | No |

75.7 | 1.45 | No |

85.9 | 1.76 | Yes |

79.1 | 1.93 | No |

70.4 | 1.2 | Yes |

75.8 | 1.55 | Yes |

85.9 | 1.93 | Yes |

78.5 | 1.59 | Yes |

79.2 | 1.5 | Yes |

86.7 | 1.9 | Yes |

79.3 | 1.39 | Yes |

74.5 | 1.54 | No |

83.8 | 1.89 | Yes |

76.8 | 1.59 | No |

Example 1

One would like to know the effects on assessed house value of the house size, and whether or not there is a fireplace in the house. Data from a sample of 15 houses was recorded and is provided below:

The independent variable Size is quantitative (ft2), but the variable Fireplace is qualitative (Yes, No). We define a new variable “FirePlace” and let it have the value 1 when there is a fireplace and 0 when there is none. The data set becomes:

Value | Size(1000ft2) | Fireplace |

84.4 | 2 | 1 |

77.4 | 1.71 | 0 |

75.7 | 1.45 | 0 |

85.9 | 1.76 | 1 |

79.1 | 1.93 | 0 |

70.4 | 1.2 | 1 |

75.8 | 1.55 | 1 |

85.9 | 1.93 | 1 |

78.5 | 1.59 | 1 |

79.2 | 1.5 | 1 |

86.7 | 1.9 | 1 |

79.3 | 1.39 | 1 |

74.5 | 1.54 | 0 |

83.8 | 1.89 | 1 |

76.8 | 1.59 | 0 |

Now we can formulate a multiple regression model of the form:

y=β0+β1Size+β2FirePlace+ε

Price

0

0 + 2

The house value for a house with a fireplace is described by the equation

Value=β0+β1Size+β21+ε which reduces to...