BASIC REGRESSION CONCEPTS

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• BASIC REGRESSION CONCEPTS

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Basic Idea Behind Regression

• To to determine how a particular variable (called
  the dependent variable -- y) is influenced by
  one or more other variables (called independent
  variables -- x1, x2, etc.)

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5-step Regression Approach

• Hypothesize a form of the model
• Hypothesize whether a linear relation, quadratic
  relation, etc. exists between y and the xs
• Determine the best estimates for the values of
  the parameters
• Make and check any necessary assumptions
• Evaluate the model
• Determine how good the model is
• If the model is good -- use it for prediction and
  estimation

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Confidence Interval Review

• Suppose sales at Dollar Only Stores for 10
  randomly selected weeks were
• Week Sales
• 1 101,000
• 2 92,000
• 3 110,000
• 4 120,000
• 5 90,000
• 6 82,000
• 7 93,000
• 8 75,000
• 9 91,000
• 10 105,000

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Confidence Interval for Average Weekly Sales

• Assuming that weekly sales are normally
  distributed, a 95 confidence interval for
  average weekly sales is

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Using Excel
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Regression Concepts

• But couldnt the sales (y) have been affected by
  one or more factors?
• Advertising dollars (x1)
• Average number of salesmen (x2)
• Hours of operation (x3)
• The weather (good or not good) -- (x4)
• In this case we may want to regress y on one or
  more of these variables

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The Basic Linear Regression Relation

• We might hypothesize that sales are linearly
  dependent on all four of the previous variables,
  i.e.
• y ?0 ?1x1 ?2x2 ?3x3 ?4x4 ?
• ltRegression gt Error
• The ?s are (unknown) constants
• We shall estimate them by b0, b1, b2, b3 and b4
• ? is a random variable for the variability
  (error) when x1, x2, x3, and x4 take on a
  specific set of values
• ? has a distribution, a mean, and a standard
  deviation

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Simple Linear Regression

• Simple linear regression is when we regress
• y (Sales) on only one variable x (Ad )
• y ?0 ?1x ?
• Here, ?0 the true value of the y-intercept
• ?1 the true slope of the line
• ? a random variable of the error

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INPUT DATA

• The Dollar Only Stores advertising for the
  corresponding 10 sample weeks is
• Week(i) Ad (xi) Sales (yi)
• 1 1200 101,000
• 2 800 92,000
• 3 1000 110,000
• 4 1300 120,000
• 5 700 90,000
• 6 800 82,000
• 7 1000 93,000
• 8 600 75,000
• 9 900 91,000
• 10 1100 105,000

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Step 1 -- Hypothesizing the form of the model

• If we are regressing on only one variable -- use
  a scatterplot to determine an appropriate model
• Does it look like the data is relatively linear?
• y ?0 ?1x ?
• Does it look curved?
• Perhaps y ?0 ?1x ?2x2 ?
• etc.
• LETS SEE!

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Scatterplot
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Step 1

• It looks like a straight line fits through the
  points fairly well.
• Thus, we hypothesize
• y ?0 ?1x ?
• We now must get the best estimates for ?0 and ?1
  -- This is step 2!

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Review

• Regression seeks to explain how a dependent
  variable (y) is affected by independent variables
  (x1, x2, x3, etc.)
• Regression is a multi-step procedure.
• The first step is to hypothesize a form of the
  model.
• If there is only one variable, plot y vs. x to
  assist in forming the hypothesis.