Simple Linear Regression

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Simple Linear Regression
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Simple Linear Regression

• Regression analysis is used to predict the value
  of one variable on the basis of other variables.
• The technique involves developing a mathematical
  equation that describes the relationship between
  the variable to be forecast, which is called the
  dependent variable, and variables that the
  statistician believes are related to the
  dependent variable.
• The dependent variable is denoted by "y", while
  the related variables are called independent
  variables and are denoted x1, x2, , xk (where
  "k" is the number of independent variables).
• Equations such as
• E mc2
• F ma

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Simple Linear Regression

• Are deterministic models because with exception
  of small errors, these equations allow us to
  determine the value of the independent variable
  (on the left side of the equation) from the value
  of the independent variables.
• These equations do not represent the random
  nature of real life. Equations that contain some
  measure of randomness are called probabilistic
  models.
• To build a probabilistic model, we start with a
  deterministic model that approximates the
  relationship we want to model. We then add a
  random term that measures the error of the
  deterministic component.

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Example Predict Cost of Building a House

• Suppose that the cost of building a new house is
  about 75 per square foot and that most lots sell
  for about 25,000. The approximate selling price
  would be
• y 25000 75x
• Where y selling price and x Size of the house
  in square feet
• Thus a house of 2000 square feet would be
  estimated to sell for
• y 25000 75(2000) 175,000

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Example Predict Cost of Building a House

• We know the price is not exactly 175,000, but
  between 150,000 - 200,000. To represent this
  situation properly, we should use the
  probabilistic model
• y 25000 75x ?
• where " ? " ( the Greek letter epsilon)
  represents the random term (Called the error
  variable).
• ? is the difference between the actual selling
  price and the estimated price based on the size
  of the house.
• Thus the random term accounts for all the
  variables, both measurable and immeasurable, that
  are not part of the model such as number of
  bedrooms and location.

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Example Predict Cost of Building a House

• The value of ? will vary from house to house
  depending on location, number of bedrooms, etc
• In this chapter we will only consider equation
  (models) that have one independent variable. This
  model we will use is called the first-order
  linear model or the simple linear regression
  model.

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The Model

• The first order linear model
• y dependent variable
• x independent variable
• b0 y-intercept
• b1 slope of the line
• error variable

b0 and b1 are unknown, therefore, are estimated
from the data.
y
Rise
b1 Rise/Run
Run
b0
x
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Estimating the Coefficients

• The estimates are determined by
• drawing a sample from the population of interest,
• calculating sample statistics.
• producing a straight line that cuts into the data.

y
w
The question is Which straight line fits best?
w
w
w
w
w w w w
w
w w
w w
w
x
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Estimating the Coefficients
The best line is the one that minimizes the sum
of squared vertical differences between the
points and the line.
Sum of squared differences
(2 - 1)2
(4 - 2)2
(1.5 - 3)2
(3.2 - 4)2 6.89
Let us compare two lines
(2,4)
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The second line is horizontal
w
(4,3.2)
w
3
2.5
2
w
(1,2)
The smaller the sum of squared differences the
better the fit of the line to the data.
(3,1.5)
w
3
4
2
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Estimating the Coefficients
To calculate the estimates of the
coefficients that minimize the differences
between the data points and the line, use the
formulas
The regression equation that estimates the
equation of the first order linear model is
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Example 17.1 Relationship between odometer
reading and a used cars selling price.

• A car dealer wants to find the relationship
  between the odometer reading and the selling
  price of used cars.
• A random sample of 100 cars is selected, and the
  data recorded.
• Find the regression line.

Independent variable x
Dependent variable y
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Example 17.1 Relationship between odometer
reading and a used cars selling price.

• Solving by handTo calculate b0 and b1 we need to
  calculate several statistics first

where n 100.
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Example 17.1 Relationship between odometer
reading and a used cars selling price.

• Using the computer (see file Xm17-01.xls)

Tools Data analysis Regression Shade the y
range and the x range OK
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Example 17.1 Relationship between odometer
reading and a used cars selling price.
6533
0
No data
This is the slope of the line. For each
additional mile on the odometer, the price
decreases by an average of 0.0312
The intercept is b0 6533.
Do not interpret the intercept as the Price of
cars that have not been driven
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Example 17.1 Relationship between odometer
reading and a used cars selling price.

• Using Minitab
• Type or import the data into 2 columns
• Click Stat, Regression, and Regression
• Type the name of the dependent variable (Response
  - Price or C2)
• Hit tab, and type the name of the independent
  variable (Predictors - Odometer or C1)
• Click O.K.
• Click Stat, Regression, and Fitted Line Plot
• Type the name of the dependent variable (Response
  - Y - Price or C2)
• Hit tab, and type the name of the independent
  variable (Predictors - X - Odometer or C1)
• Click O.K.

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Error Variable Required Conditions

• b 1 -.0312 which means that for each additional
  mile on the odometer, the price decreases by an
  average of .0312 or 3.12 cents.
• b 0 6533 since this is the y - intercept, we
  might think that a car with 0 miles would sell
  for 6533.
• In this case however, the intercept is probably
  meaningless. Because our sample did not include
  any cars with 0 miles on the odometer, we have no
  basis for interpreting b 0.
• As a general rule, we cannot determine the value
  of y for a value of x that is far outside the
  range of sample values of x. In this sample, x
  ranged from 19075 to 49223.

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Error Variable Required Conditions

• The error ? is a critical part of the regression
  model.
• Four requirements involving the distribution of ?
  must be satisfied.
• The probability distribution of ? is normal.
• The mean of e is zero E(?) 0.
• The standard deviation of ? is ?? for all values
  of x.
• The set of errors associated with different
  values of y are all independent.

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From the first three assumptions we have y is
normally distributed with mean E(y) b0 b1x,
and a constant standard deviation se
The standard deviation remains constant,
m3
m2
but the mean value changes with x
m1
x1
x2
x3
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Assessing the Model

• The least squares method will produce a
  regression line whether or not there is a linear
  relationship between x and y.
• Consequently, it is important to assess how well
  the linear model fits the data.
• Several methods are used to assess the model
• Testing and/or estimating the coefficients.
• Using descriptive measurements.

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Standard Error of Estimate

• The mean error is equal to zero.
• If se is small the errors tend to be close to
  zero (close to the mean error). Then, the model
  fits the data well.
• If se is large, some of the errors will be large,
  which implies that the model's fit is poor.
• We could use se to measure the suitability of
  using a linear model, but se is a population
  parameter which is usually unknown.
• We can estimate se from the data using the Sum of
  Squares Error (SSE).
• Therefore, we can, use se as a measure of the
  suitability of using a linear model.

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Standard Error of Estimate

• An unbiased estimator of se2 is given by se2

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Sum of Squares For Errors

• SSE the minimized sum of squares differences
  between the points and the regression line, also
  known as "sum of squares about the regression
  (line)", "sum of squares of the residuals", and
  "sum of squares errors (SSE)"
• It can serve as a measure of how well the line
  fits the data.
• This statistic plays a role in every statistical
  technique we employ to assess the model.

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Sum of Squares For Errors
Shortcut formula
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Example 17.2

• Calculate the standard error of estimate for
  example 17.1, and describe what does it tell you
  about the model fit?
• Solution

Calculated before
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Testing The Slope

• When no linear relationship exists between two
  variables, the regression line should be
  horizontal.

q
q
Linear relationship. Different inputs (x)
yield different outputs (y).
No linear relationship. Different inputs (x)
yield the same output (y).
The slope is not equal to zero
The slope is equal to zero
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Testing The Slope

• We can draw inference about b1 from b1 by testing
• H0 b1 0
• H1 b1 ? 0 (or 0)
• The test statistic is

where
If the error variable is normally distributed,
the statistic is Student t distribution with d.f.
n-2.
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Example 17.3

• SolutionSolving by hand
• To compute t we need the values of b1 and sb1.

Test Statistic
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Example 17.3

• The rejection region is

• Conclusion Reject the null hypothesis since
• 13.49

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Example 17.3

• Interpreting the Results
• The value of the test statistic is t -13.49
  with a p-value of 0.00. There is overwhelming
  evidence to infer that a linear relationship
  exists.
• What this means is that the odometer reading does
  affect the auction selling price of the cars.

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Coefficient of Determination

• When we want to measure the strength of the
  linear relationship, we use the coefficient of
  determination.

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Coefficient of Determination

• To understand the significance of this
  coefficient note

The regression model
Overall variability in y
The error
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Coefficient of Determination
Two data points (x1,y1) and (x2,y2) of a certain
sample are shown.
y2
y1
x1
x2
Total variation in y
Variation explained by the regression line)
Unexplained variation (error)
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Coefficient of Determination
Variation in y SSR SSE

• R2 measures the proportion of the variation in y
  that is explained by the variation in x.

• R2 takes on any value between zero and one.
• R2 1 Perfect match between the line and the
  data points.
• R2 0 There are no linear relationship between
  x and y.

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Example 17.4

• Find the coefficient of determination for example
  17.1 what does this statistic tell you about the
  model?
• Solution
• Solving by hand
• Using the computer
• From the regression output we have

65 of the variation in the auction selling price
is explained by the variation in odometer
reading. The rest (35) remains unexplained
by this model.
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Example 17.4 Interpreting the Results

• We found that R2 65. This statistic tells us
  that 65 of the variation in the auction selling
  price is explained by the variation in the
  odometer readings. The remaining 35 is
  unexplained.
• Unlike test statistics, R2 does not have a
  critical value that enables us to draw
  conclusions.
• We know that the higher the value of R2 the
  better the model fits the data.
• R2 provides us a measure of the strength of the
  linear relationship between the independent and
  dependent variables.

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Cause and Effect Relationships

• Do not assume that a linear relationship between
  variables means that there is a cause and effect
  relationship between those variables.
• We cannot infer a causal relationship from
  statistics alone. Causal relationships must be
  justified by reasonable theoretical
  relationships.
• Do not interpret the work "explained" in relation
  to R2 to mean "caused".
• In the Taurus example, we might conclude that
  decreasing the odometer reading would cause the
  price to rise. This conclusion may not be
  entirely true.
• It is theoretically possible that the price is
  determined by the overall condition of the car
  and that condition worsens when the car is driven
  longer.

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Using the Regression Equation

• Before using the regression model, we need to
  assess how well it fits the data.
• If we are satisfied with how well the model fits
  the data, we can use it to make predictions for
  y.
• Illustration
• Predict the selling price of a three-year-old
  Taurus with 40,000 miles on the odometer (Example
  17.1)

Thus, the dealer would predict that the car would
sell for 5285. This does not provide any
information about how closely the value will
match the true selling price. To do that we must
construct a confidence interval.
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Prediction interval and confidence interval

• Two intervals can be used to discover how closely
  the predicted value will match the true value of
  y.
• Prediction interval - for a particular value of
  y,
• Confidence interval - for the expected value of y.

The prediction interval is wider than the
confidence interval
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Example 17.6 Interval Estimates for the Car
Auction Price

• Provide an interval estimate for the bidding
  price on a Ford Taurus with 40,000 miles on the
  odometer.
• Solution
• The dealer would like to predict the price of a
  single car
• The prediction interval(95)

t.025,98
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Example 17.6 Interval Estimates for the Car
Auction Price

• The car dealer wants to bid on a lot of 250 Ford
  Tauruses, where each car has been driven for
  about 40,000 miles.
• Solution
• The dealer needs to estimate the mean price per
  car.
• The confidence interval (95)

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The Effect of the Given Value of X on the
Interval

• As xg moves away from x the interval becomes
  longer. That is, the shortest interval is found
  at x.

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Coefficient of Correlation

• The coefficient of correlation is used to measure
  the strength of association between two
  variables.
• The coefficient values range between -1 and 1.
• If r -1 (negative association) or r 1
  (positive association) every point falls on the
  regression line.
• If r 0 there is no linear pattern.
• The coefficient can be used to test for linear
  relationship between two variables.

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Coefficient of Correlation

• Use in cases where we are interested only if a
  linear relationship exists and not the form of
  the relationship

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Coefficient of Correlation

• Testing the coefficient of correlation
• When there are no linear relationship between two
  variables, r 0.
• The hypotheses are
• H0 r 0H1 r 0
• The test statistic is

The statistic is Student t distributed with d.f.
n - 2, provided the variables are bivariate
normally distributed.
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Example 17.7 Testing for Linear Relationship

• Test the coefficient of correlation to determine
  if linear relationship exists in the data of
  example 17.1.
• Solution
• We test H0 r 0 H1 r 0.
• Solving by hand
• The rejection region ist ta/2,n-2 t.025,98
  1.984 or so.

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Example 17.7 Testing for Linear Relationship

• The sample coefficient of correlation
• r cov(X,Y)/sxsy -1,356,256/(6597.6)(254.9)
  -.806

The value of the t statistic is
Conclusion There is sufficient evidence at a
5 to infer that there are linear relationship
between the two variables.
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Example 17.7 Relationship between odometer
reading and a used cars selling price.

• Using Minitab
• Type or import the data into 2 columns
• Click Stat, Basic Statistics, and Correlation
• Type the variable names (Price or C2, Odometer
  or C1)
• Click O.K.

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Spearman Rank Correlation Coefficient

• The Spearman rank test is used to test whether
  relationship exists between variables in cases
  where
• at least one variable is ranked, or
• both variables are quantitative but the normality
  requirement is not satisfied

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Spearman Rank Correlation Coefficient

• The hypotheses are
• H0 rs 0
• H1 rs 0
• The test statistic is
• a and b are the ranks of the data.
• For a large sample (n 30) rs is approximately
  normally distributed

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Example 17.8 Spearman Rank Correlation
Coefficient

• A production manager wants to examine the
  relationship between
• aptitude test score given prior to hiring, and
• performance rating three months after starting
  work.
• A random sample of 20 production workers was
  selected. The test scores as well as performance
  rating was recorded.

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Example 17.8 Spearman Rank Correlation
Coefficient

• The aptitude test results range from 0 100.
• The performance ratings are as follows
• 1 Employee has performed well below average
• 2 Employee has performed somewhat below average
• 3 Employee has performed at the average level
• 4 Employee has performed somewhat above average
• 5 Employee has performed well above average
• Can the firms manager infer at the 5
  significance level that aptitude test scores are
  correlated with performance rating?

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Example 17.8 Spearman Rank Correlation
Coefficient

• The problem objective is to analyze the
  relationship between two variables.
• The aptitude test score is quantitative, but the
  performance rating is ranked
• We will treat the aptitude test score as is it is
  ranked and calculate the Spearman rank
  correlation coefficient

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Example 17.8 Spearman Rank Correlation
Coefficient
Scores range from 0 to 100
Scores range from 1 to 5

• The hypotheses are
• H0 rs 0
• H1 rs 0

• The test statistic is rs, and the rejection
  region is rs rcritical (taken from the
  Spearman rank correlation table).

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Example 17.8 Spearman Rank Correlation
Coefficient
Ties are broken by averaging the ranks.

• Solving by hand
• Rank each variable separately.

Conclusion Do not reject the null
hypothesis. At 5 significance level there is
insufficient evidence to infer that the two
variable are related to one another.

• Calculate sa 5.92 sb 5.50 cov(a,b) 12.34
• Thus rs cov(a,b)/sasb .379.
• The critical value for a .05 and n 20 is .450.

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Example 17.8 Relationship between odometer
reading and a used cars selling price.

• Interpret the results
• There is not enough evidence to believe that the
  aptitude test scores and performance ratings are
  related.
• This suggests that the aptitude test should be
  improved to better measure the knowledge and
  skill required by a production line worker.
• If this proves impossible, the aptitude test
  should be discarded.

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Example 17.8 Relationship between odometer
reading and a used cars selling price.

• Using Minitab
• Type or import the data into 2 columns
• Click Manip, and Rank
• Type the name of the first variable (arbitrary
  choice, Aptitude or C1)
• Hit Tab and specify the column where the ranks
  are to be stored. Click O.K.
• Click Manip, and Rank
• Type the name of the second variable (arbitrary
  choice, C2)
• Hit Tab and specify the column where the ranks
  are to be stored. Click O.K.
• Click Stat, Basic statistics and Correlation
• Type the names of the ranked variables
• Click O.K.

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Regression Diagnostics - I

• The three conditions required for the validity of
  the regression analysis are
• the error variable is normally distributed.
• the error variance is constant for all values of
  x.
• The errors are independent of each other.
• How can we diagnose violations of these
  conditions?

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Residual Analysis

• Examining the residuals (or standardized
  residuals), we can identify violations of the
  required conditions
• Example 17.1 - continued
• Nonnormality.
• Use Excel to obtain the standardized residual
  histogram.
• Examine the histogram and look for a bell shaped
  diagram with mean close to zero.

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Residual Analysis
A Partial list of Standard residuals
Standardized residual i Residual i / Standard
deviation
We can also apply the Lilliefors test or the c2
test of normality.
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Heteroscedasticity

• When the requirement of a constant variance is
  violated we have heteroscedasticity.

Residual

















y








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Heteroscedasticity

• When the requirement of a constant variance is
  not violated we have homoscedasticity.

Residual

















y








The spread of the data points does not change
much.
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Heteroscedasticity

• When the requirement of a constant variance is
  not violated we have homoscedasticity.

Residual
















y









As far as the even spread, this is a much better
situation
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Nonindependence of Error Variables

• A time series is constituted if data were
  collected over time.
• Examining the residuals over time, no pattern
  should be observed if the errors are independent.
• When a pattern is detected, the errors are said
  to be autocorrelated.
• Autocorrelation can be detected by graphing the
  residuals against time.

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Nonindependence of Error Variables
Patterns in the appearance of the residuals over
time indicates that autocorrelation exists.
Residual
Residual














0
0

Time
Time













Note the runs of positive residuals, replaced by
runs of negative residuals
Note the oscillating behavior of the residuals
around zero.
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Outliers

• An outlier is an observation that is unusually
  small or large.
• Several possibilities need to be investigated
  when an outlier is observed
• There was an error in recording the value.
• The point does not belong in the sample.
• The observation is valid.
• Identify outliers from the scatter diagram.
• It is customary to suspect an observation is an
  outlier if its standard residual 2

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Outliers
An influential observation
An outlier


but, some outliers may be very influential














The outlier causes a shift in the regression line
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Procedure for regression diagnostics

• Develop a model that has a theoretical basis.
• Gather data for the two variables in the model.
• Draw the scatter diagram to determine whether a
  linear model appears to be appropriate.
• Check the required conditions for the errors.
• Assess the model fit.
• If the model fits the data, use the regression
  equation.