Statistics%20and%20Quantitative%20Analysis%20U4320

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Statistics and Quantitative Analysis U4320

• Segment 8
• Prof. Sharyn OHalloran

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I. Introduction

• A. Overview
• 1. Ways to describe, summarize and display data.
• 2.Summary statements
• Mean
• Standard deviation
• Variance
• 3. Distributions
• Central Limit Theorem

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I. Introduction (cont.)

• A. Overview
• 4. Test hypotheses
• 5. Differences of Means
• B. What's to come?
• 1. Analyze the relationship between two or more
  variables with a specific technique called
  regression analysis.

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I. Introduction (cont.)

• A. Overview
• B. What's to come?
• 2. This tools allows us to predict the impact of
  one variable on another.
• For example, what is the expected impact of a
  SIPA degree on income?

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II. Causal Models

• Causal models explain how changes in one variable
  affect changes in another variable.
• Incinerator -------------------------gt Bad Public
  Health
• Regression analysis gives us a way to analyze
  precisely the cause-and-effect relationships
  between variables.
• Directional
• Magnitude

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II. Causal Models (cont.)

• A. Variables
• Let us start off with a few basic definitions.
• 1. Dependent Variable
• The dependent variable is the factor that we want
  to explain.
• 2. Independent Variables
• Independent variable is the factor that we
  believe causes or influences the dependent
  variable.
• Independent variable-------gt Dependent Variable
• Cause ------------------gt Effect

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II. Causal Models (cont.)

• A. Variables
• B. Voting Example
• Let us say that we have a vote in the House of
  Representatives on health. And we want to know
  if party affiliation influenced individual
  members' voting decisions?
• 1. The raw data looks like this

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II. Causal Models (cont.)

• A. Variables
• B. Voting Example
• 2. Percentages look like this
• 3. Does party affect voting behavior?
• Given that the legislator is a Democrat, what is
  the chance of voting for the health care
  proposal?

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II. Causal Models (cont.)

• A. Variables
• B. Voting Example
• 3. Does party affect voting behavior? (cont.)
• What is the Probability of being a democrat?
• What is the Probability of being a Democrat and
  voting yes?

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II. Causal Models (cont.)

• A. Variables
• B. Voting Example
• 4. Casual Model
• This is the simplest way to state a causal model
• A-------------gt B
• Party ---------gt Vote
• 5. Interpretation
• The interpretation is that if party influences
  vote, then as we move from Republicans to
  Democrats we should see a move from a No vote to
  a YES vote.

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II. Causal Models (cont.)

• A. Variables
• B. Voting Example
• C. Summary
• 1. Regression analysis helps us to explain the
  impact of one variable on another.
• We will be able to answer such questions as what
  is the relative importance of race in explaining
  one's income?
• Or perhaps the influence of economic conditions
  on the levels of trade barriers?

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II. Causal Models (cont.)

• A. Variables
• B. Voting Example
• C. Summary
• 2. Univariate Model
• For now, we will focus on the univariate case, or
  the causal relation between two variables.
• We will then relax this assumption and look at
  the relation of multiple variables in a couple of
  weeks.

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III. Fitted Line

• Although regression analysis can be very
  complicated, the heart of it is actually very
  simple.
• It centers on the notion of fitting a line
  through the data.

• 1. Example
• Suppose we have a study of how wheat yield
  depends on fertilizer. And we observe this
  relation

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III. Fitted Line (cont.)

• 1. Example (cont.)
• The observed relation between Fertilizer and
  Yield then can be plotted as follows

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III. Fitted Line (cont.)

• 1. Example
• 2. What line best approximates the relation
  between these observations?
• a) Highest and Lowest Value

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III. Fitted Line (cont.)

• 1. Example
• 2. What line best approximates the relation
  between these observations? (cont.)
• b) Median Value

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III. Fitted Line (cont.)

• 1. Example
• 2. What line best approximates the relation
  between these observations?
• 3. Predicted Values
• a) Example 1
• The line that is fitted to the data gives the
  predicted value of Y for any give level of X.

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III. Fitted Line (cont.)

• 1. Example
• 2. What line best approximates the relation
  between these observations?
• 3. Predicted Values (cont.)
• a) Example 1

• If X is 400 and all we know was the fitted line
  then we would expect the yield to be around 65.

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III. Fitted Line (cont.)

• 1. Example
• 2. What line best approximates the relation
  between these observations?
• 3. Predicted Values (cont.)
• b) Example 2
• Many times we have a lot of data and fitting the
  line becomes rather difficult.

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III. Fitted Line (cont.)

• 1. Example
• 2. What line best approximates the relation
  between these observations?
• 3. Predicted Values (cont.)
• b) Example 2

• For example, if our plotted data looked like this

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IV. OLS Ordinary Least Squares

• We want a methodology that allows us to be able
  to draw a line that best fits the data.
• A. The Least Square Criteria
• What we want to do is to fit a line whose
  equation is of the form
• This is just the algebraic representation of a
  line.

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IV. OLS Ordinary Least Squares (cont.)

• A. The Least Square Criteria (cont.)
• 1. Intercept
• a represents the intercept of the line. That is,
  the point at which the line crosses the Y axis.
• 2. Slope of the line
• b represents the slope of the line.

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IV. OLS Ordinary Least Squares (cont.)

• A. The Least Square Criteria (cont.)
• 1. Intercept
• 2. Slope of the line
• Remember the slope is just the change in Y
  divided by the change in X. Rise/Run
• 3. Minimizing the Sum or Squares
• a) Problem
• How do we select a and b so that we minimize the
  pattern of vertical Y deviations (predicted
  errors)?
• We what to minimize the deviation

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IV. OLS Ordinary Least Squares (cont.)

• A. The Least Square Criteria (cont.)
• 1. Intercept
• 2. Slope of the line
• 3. Minimizing the Sum or Squares
• b) There are several ways in which we can do
  this.
• 1. First, we could minimize the sum of d.

• We could find the line that will give us the
  lowest sum of all the d's.
• The problem of course is that some d's would be
  positive and others would be negative and when we
  add them all up they would end up canceling each
  other.
• In effect, we would be picking a line so that the
  d's add up to zero.

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IV. OLS Ordinary Least Squares (cont.)

• A. The Least Square Criteria (cont.)
• 1. Intercept
• 2. Slope of the line
• 3. Minimizing the Sum or Squares
• b) There are several ways in which we can do
  this.
• 2. Absolute Values
• 3. Sum of Squared Deviations

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IV. OLS Ordinary Least Squares (cont.)

• A. The Least Square Criteria
• B. OLS Formulas
• 1. Fitted Line
• The line that we what to fit to the data is
• This is simply what we call the OLS line.
• Remember we are concerned with how to calculate
  the slope of the line b and the intercept of the
  line

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IV. OLS Ordinary Least Squares (cont.)

• A. The Least Square Criteria
• B. OLS Formulas
• 1. Fitted Line
• 2. OLS Slope
• The OLS slope can becalculated from the formula

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IV. OLS Ordinary Least Squares (cont.)

• A. The Least Square Criteria
• B. OLS Formulas
• 1. Fitted Line
• 2. OLS Slope
• In the book they use the abbreviations

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IV. OLS Ordinary Least Squares (cont.)

• A. The Least Square Criteria
• B. OLS Formulas
• 1. Fitted Line
• 2. OLS Slope
• 3. Intercept
• Now that we have the slope b it is easy to
  calculate a
• Note when b0 then the intercept is just the
  mean of the dependent variable.

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IV. OLS Ordinary Least Squares (cont.)

• A. The Least Square Criteria
• B. OLS Formulas
• C. Example 1 Fertilizer and Yield

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IV. OLS Ordinary Least Squares (cont.)

• A. The Least Square Criteria
• B. OLS Formulas
• C. Example 1 Fertilizer and Yield
• So to calculate the slope we solve
• We can then use the slope b to calculate the
  intercept

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IV. OLS Ordinary Least Squares (cont.)

• A. The Least Square Criteria
• B. OLS Formulas
• C. Example 1 Fertilizer and Yield
• Remember
• Plugging these estimated values into our fitted
  line equation, we get

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IV. OLS Ordinary Least Squares (cont.)

• A. The Least Square Criteria
• B. OLS Formulas
• C. Example 1 Fertilizer and Yield
• What is the predicted bushels produced with 400
  lbs of fertilizer?
• What if we add 700 lbs of fertilizer what would
  be the expected yield?

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IV. OLS Ordinary Least Squares (cont.)

• A. The Least Square Criteria
• B. OLS Formulas
• C. Example 1 Fertilizer and Yield
• D. Interpretation of b and a
• 1. Slope b
• Change in Y that accompanies a unit change X.
• The slope tells us that when there is a one unit
  change in the independent variable what is the
  predicted effect on the dependent variable?

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IV. OLS Ordinary Least Squares (cont.)

• A. The Least Square Criteria
• B. OLS Formulas
• C. Example 1 Fertilizer and Yield
• D. Interpretation of b and a
• 1. Slope b
• The slope then tells us two things
• i) The directional effect of the independent
  variable on the dependent variable.
• There was a positive relation between fertilizer
  and yield.

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IV. OLS Ordinary Least Squares (cont.)

• A. The Least Square Criteria
• B. OLS Formulas
• C. Example 1 Fertilizer and Yield
• D. Interpretation of b and a
• 1. Slope b
• The slope then tells us two things
• ii) It also tells you the magnitude of the effect
  on the dependent variable.
• For each additional pound of fertilizer we expect
  an increased yield of .059 bushels.

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IV. OLS Ordinary Least Squares (cont.)

• A. The Least Square Criteria
• B. OLS Formulas
• C. Example 1 Fertilizer and Yield
• D. Interpretation of b and a
• 2. The Intercept
• The intercept tells us what we would expect if
  there is no fertilizer added, we expect a yield
  of 36.4 bushels.
• So independent of the fertilizer you can expect
  36.4 bushels.
• Alternatively, if fertilizer has no effect on
  yield, we would simply expect 36.4 bushels. The
  yield we expected with no fertilizer.

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IV. OLS Ordinary Least Squares (cont.)

• A. The Least Square Criteria
• B. OLS Formulas
• C. Example 1 Fertilizer and Yield
• D. Interpretation of b and a
• E. Example II Radio Active Exposure
• 1. Casual Model
• We want to know if exposure to radio active
  waste is linked to cancer?
• Radio Active Waste --------------gt Cancer

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IV. OLS Ordinary Least Squares (cont.)

• A. The Least Square Criteria
• B. OLS Formulas
• C. Example 1 Fertilizer and Yield
• D. Interpretation of b and a
• E. Example II Radio Active Exposure
• 2. Data

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IV. OLS Ordinary Least Squares (cont.)

• A. The Least Square Criteria
• B. OLS Formulas
• C. Example 1 Fertilizer and Yield
• D. Interpretation of b and a
• E. Example II Radio Active Exposure
• 3. Graph

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IV. OLS Ordinary Least Squares (cont.)

• A. The Least Square Criteria
• B. OLS Formulas
• C. Example 1 Fertilizer and Yield
• D. Interpretation of b and a
• E. Example II Radio Active Exposure
• 4. Calculate the regression line for predicting Y
  from X
• i) Slope
• How do we interpret the slope coefficient?
• For each unit of radioactive exposure, the cancer
  mortality rate rises by 9.03 deaths per 10,000
  individuals.

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IV. OLS Ordinary Least Squares (cont.)

• A. The Least Square Criteria
• B. OLS Formulas
• C. Example 1 Fertilizer and Yield
• D. Interpretation of b and a
• E. Example II Radio Active Exposure
• ii) Calculate the intercept
• Plugging these estimated values into our fitted
  line equation, we get

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IV. OLS Ordinary Least Squares (cont.)

• A. The Least Square Criteria
• B. OLS Formulas
• C. Example 1 Fertilizer and Yield
• D. Interpretation of b and a
• E. Example II Radio Active Exposure
• 5. Predictions
• Let's calculate the mortality rate if X were 5.0.
• How about if X were 0?

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IV. OLS Ordinary Least Squares (cont.)

• A. The Least Square Criteria
• B. OLS Formulas
• C. Example 1 Fertilizer and Yield
• D. Interpretation of b and a
• E. Example II Radio Active Exposure
• How can we interpret this result?
• Even with no radioactive exposure, the
  mortality rate would be 118.5.

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III. Advantages of OLS

• A. Easy
• 1. The least square method gives relative easy or
  at least computable formulas for calculating a
  and b.

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III. Advantages of OLS (cont.)

• A. Easy
• B. OLS is similar to many concepts we have
  already used.
• 1. We are minimizing the sum of the squared
  deviations. In effect, this is very similar to
  how we find the variance.
• 2. Also, we saw above that when b0,
• The interpretation of this is that the best
  prediction we can make of Y is just the sample
  mean .
• This is the case when the two variables are
  independent.

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III. Advantages of OLS (cont.)

• A. Easy
• B. OLS is similar to many concepts we have
  already used.
• C. Extension of the Sample Mean
• Since OLS is just an extension of the sample
  mean, it has many of the same properties like
  efficient and unbiased.
• D. Weighted Least Squares
• We might want to weigh some observations more
  heavily than others.

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V. Homework Example

• In the homework assignment, you are asked to
  select two interval/ratio level variables and
  calculate the fitted line that minimizes the sum
  of the squared deviations (the regression line).
• A. Choose 2 Variables
• What effect does the number of years of education
  have on the frequency that one reads the
  newspaper?
• The independent variable is Education
• And the dependent variable is Newspaper reading.

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V. Homework Example(cont.)

• A. Choose 2 Variables
• B. Coding the Variables
• First, I made a new variable called PAPER.
• Recode all the missing data values to a single
  value.
• Remove missing values from the data set.
• Then do the same for education

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V. Homework Example(cont.)

• A. Choose 2 Variables
• B. Coding the Variables
• C. Getting the number of valid observations
• Next, see how many valid observations are left by
  using the Summarize command under the Data
  menu.

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V. Homework Example(cont.)

• A. Choose 2 Variables
• B. Coding the Variables
• C. Getting the number of valid observations
• D. Sampling five observations
• 1. So we randomly sample 5 from 1019.
• 2. As before, use the Select command under the
  Data menu to get 5 random observations.
• 3. Then go to the Statistics menu and use the
  Summarize gt List command to get the entries
  for the variables of interest.

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V. Homework Example(cont.)

• A. Choose 2 Variables
• B. Coding the Variables
• C. Getting the number of valid observations
• D. Sampling five observations
• E. Calculate the OLS Line
• Finally, you will have to compute the fitted line
  for these data.

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V. Homework Example(cont.)

• A. Choose 2 Variables
• B. Coding the Variables
• C. Getting the number of valid observations
• D. Sampling five observations
• E. Calculate the OLS Line
• 1. Calculate b
• 2 . Calculate the intercept
• 3 . Calculate the OLS line

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V. Homework Example(cont.)

• A. Choose 2 Variables
• B. Coding the Variables
• C. Getting the number of valid observations
• D. Sampling five observations
• E. Calculate the OLS Line
• 4. Plot

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V. Homework Example(cont.)

• A. Choose 2 Variables
• B. Coding the Variables
• C. Getting the number of valid observations
• D. Sampling five observations
• E. Calculate the OLS Line
• 5. Interpretation
• A person with no education would read 3.3
  newspapers a day.

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V. Homework Example(cont.)

• A. Choose 2 Variables
• B. Coding the Variables
• C. Getting the number of valid observations
• D. Sampling five observations
• E. Calculate the OLS Line
• 5. Interpretation (cont.)
• Our results further tell us that each additional
  year of education reduces the number of
  newspapers a person reads by 0.14.
• So for every year of education you read 14 less.

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V. Homework Example(cont.)

• A. Choose 2 Variables
• B. Coding the Variables
• C. Getting the number of valid observations
• D. Sampling five observations
• E. Calculate the OLS Line
• 5. Interpretation (cont.)
• This example suggests some of the problems with
  drawing inferences about the underlying
  population from small samples.