Regression
        Introduction
        •   We will introduce multiple regression, in particular we will:
             •   Learn when we can use multiple regression
             •   Learn how multiple regression extends simple regression
             •   Learn how to use multiple regression in real applications
        •   This presentation is intended for students in initial stages of 
            Statistics. No previous knowledge is required. It is advised to first 
            read the presentation on simple linear regression.
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                            Regression
       • Regression is used to study the relationship 
        between one dependent variable and two or more 
        independent variables.
       • Just as in single regression, we need the dependent 
        variable to be numerical. The independent variables 
        can be numerical or categorical. 
       •However, if all the independent variables are 
        categorical, it is best to use ANOVA.
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                            Motivation
        •  Single regression allows us to study the relationship 
         between two variables only.
        •  However, in reality, we do not believe that only a single 
         variable explains all the variation of the dependent variable.
        •  For example, in the scenario of IQ and income, we do not 
         expect IQ only to explain income, but we expect that there 
         are also other variables, such as level of education, to explain 
         income.
        •  Hence, to make the model more realistic, it makes sense to 
         include multiple independent variables in the regression.
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                         Examples
    The following are situations where we can use 
    multiple regression:
    • Testing if IQ and level of education affect income 
      (IQ and level of education are the IV and income is 
      the DV).
    • Testing if hours of work and level of stress affect 
      hours of sleep (DV is hours of sleep, and the hours 
      of work and level of stress are the IV).
    •Testing if the number of cigarettes smoked and 
      amount of salt in the diet affect blood pressure 
      (number of cigarettes smoked and salt are the IV 
      and blood pressure is the DV).
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                  Displaying the data
     As opposed to the simple linear regression case, we 
     do not have a way to plot all the variables at the 
     same time. 
     Hence, the scatterplot can be performed only for 
     each continuous independent variable 
     independently.
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