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Simple Linear Regression

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Simple Linear Regression Often we want to understand the relationships among variables, e.g., SAT scores and college GPA car weight and gas mileage – PowerPoint PPT presentation

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Title: Simple Linear Regression


1
Simple Linear Regression
  • Often we want to understand the relationships
    among variables, e.g.,
  • SAT scores and college GPA
  • car weight and gas mileage
  • amount of a certain pollutant in wastewater and
    bacteria growth in local streams
  • number of takeoffs and landings and degree of
    metal fatigue in aircraft structures
  • Simplest relationship ?
  • Y ß0 ß1x

1
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Example
  • An electric power cooperative is concerned about
    the cost of power outages in the winter and the
    analyst has an idea that these costs are directly
    related to the average temperature during the
    outage period. A random sampling of power outages
    over a number of years was conducted and the cost
    per 100 homes (adjusted for inflation) was
    determined, with these results

Temp, F Cost/ Outage
45 3,639
42 4,111
44 3,928
37 4,252
33 5,020
45 3,838
35 4,293
38 4,244
39 4,227
40 4,111
30 5,335
2
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Estimating the regression coefficients
  • Method of Least Squares
  • Determine estimates for ß0 and ß1 so that the sum
    of the squares of the residuals is minimized,
    that is
  • Solution to the minimization gives

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For our example,
Sample Temp, x Cost, y xiyi xi2
1 45 3,639 163,755 2025
2 42 4,111 172,662 1764
3 44 3,928 172,832 1936
4 37 4,252 157,324 1369
5 33 5,020 165,660 1089
6 45 3,838 172,710 2025
7 35 4,293 150,255 1225
8 38 4,244 161,272 1444
9 39 4,227 164,853 1521
10 40 4,111 164,440 1600
11 30 5,335 160,050 900
sum 428 46998 1805813 16898
4
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What does this mean?
  • We can draw the regression line that describes
    the relationship between temperature and outage
    cost
  • We can also predict the cost of outages based on
    expected temperatures.

5
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Dangers of regression analysis
  • You can regress any variable on any other
    variable
  • e.g., hair loss and heart disease hours playing
    video games and number of arrests for violent
    behavior consecutive hours in class and
    retention of material etc.
  • Which of these relationships can you legitimately
    claim reflect a causal relationship between the
    predictor and the response?
  • The regression equation is a best fit for the
    data on which it is based, but may lose validity
    for predictor values outside the range of the
    data.
  • For example, our outage cost data implies that
    the cost per outage decreases as the temperature
    increases do you believe that temperatures in
    the 80s or 90s will result in low-cost outages?

7
How good is our prediction?
  • Estimating the variance
  • Lack of fit test,
  • Tests the hypotheses
  • H0 the model adequately fits the data
  • H1 the model does not fit the data
  • As with our goodness-of-fit tests, a high p-value
    indicates that the model is adequate.

7
(see next page)
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How good is our prediction?
  • Coefficient of determination, R2
  • a measure of the quality of fit, or the
    proportion of the variability explained by the
    fitted model.
  • Use with care increasing the number of
    variables will usually increase R2, but this
    doesnt necessarily make it a better model!

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9
Linear regression in Excel
  • Step 1 Graph the data
  • Does it look like a straight line is the best
    fit?

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Step 2 Perform the analysis
  • Choose Regression from the Data Analysis menu
    (under Tools). Input the Y-range (Cost, including
    the label) and X-range (Temp, including the
    label), then select
  • Labels if you included those in your data
    range.
  • Your desired location for the output.
  • Residuals and Normal Probability Plot, as
    desired.
  • Choose OK

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Step 3 Check assumptions
  • Look at residuals plot and normal probability
    plots.

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Step 4. Evaluate the results.
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Step 5. Specify and use the model.
  • Simple linear model
  • Use the model to
  • Make predictions
  • expected costs
  • budgeting
  • Recommend actions
  • identify and address sources of cost increase

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In Minitab
  • Step 1 Graph the data (for one or two predictor
    variables)!
  • Again, do you think a simple linear relationship
    is the best fit?
  • Step 2 Select Stat ? Regression ?Regression
  • Step 3 Choose Response (y) and Predictor
    (x).
  • Step 4 In Options, check the Lack of Fit
    box. (Fit Intercept box should be checked by
    default.) Click OK.
  • Step 6 In Graphs select the appropriate
    residual plots to create.
  • Step 5 Click OK.
  • Step 6 Evaluate the residual plots and results.

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Transformation to a straight line ..,
  • If simple linear regression is not appropriate
    because the underlying function is nonlinear,
    then we have two choices
  • fit a more complex model
  • transform the model to a straight-line model
  • Simplest transformation logarithmic
    transformation
  • Original model
  • Transformed model
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