Statistics and Quantitative Analysis U4320

1
Statistics and Quantitative Analysis U4320

• Lecture 11 Path Diagrams
• Prof. Sharyn OHalloran

2
Key Points

• Slope Coefficient as a Multiplication Factor
• Path Diagram and Causal Models
• Direct and Indirect Effects

3
Regression Coefficients as Multiplication Factors

• I. Regression Coefficients as Multiplication
  Factors
• A. Simple Regression
• 1. Basic Equation
• Remember our basic one variable regression
  equation is
• b is the slope of the regression line. It
  represents the change in Y corresponding to a
  unit change in X.

4
Regression Coefficients as Multiplication Factors
(cont.)

• 2. Multiplication Factor
• We can also think of b as a multiplication
  factor.
• 3. Example
• Take the first fertilizer equation
• Say we add 5 more pounds of fertilizer. Then the
  change in yield according to this equation will
  be

5
Regression Coefficients as Multiplication Factors
(cont.)

• B. Multiple Regression "Other Things Being
  Equal"
• Now consider the multiple regression equation
• We can still think of the slopes as
  multiplication factors.
• But now they are multiplication factors if we
  change only one variable and keep all others
  constant.

6
Regression Coefficients as Multiplication Factors
(cont.)

• Say we change X1 to (X1 DX1)
• Then we can write
• If X1 changes while all others remain constant,
  then change in Y b1(change in X1)

7
Regression Coefficients as Multiplication Factors
(cont.)

• C. Examples
• Let's try an example.
• Say we have the following single and multiple
  regression equations

8
Regression Coefficients as Multiplication Factors
(cont.)

• 1. What will be the change in yield if a farmer
  adds another 100 pounds of fertilizer?
• Answer Only the fertilizer will change, not the
  rain. So use the multiple regression equation
• DY b1 DX1
• DY 100 (.038)
• DY 3.8 bushels

9
Regression Coefficients as Multiplication Factors
(cont.)

• 2. What will be the change in yield if a farmer
  irrigates his fields with 3 inches of water?
• Answer Only the amount of water will change,
  not the fertilizer. So use the multiple
  regression equation
• DY b2 DX2
• DY 3 (.83)
• DY 2.5 bushels

10
Regression Coefficients as Multiplication Factors
(cont.)

• 3. Say the farmer adds both 100 pounds of
  fertilizer and 3 inches of irrigation. Now what
  will the difference in yield be?
• Answer The change in yield will reflect the
  changes in both independent variables
• DY b1 DX1 b2 DX2
• DY 0.38 (100) (0.83) (3)
• DY 3.8 2.5
• DY 6.3 bushels

11
Regression Coefficients as Multiplication Factors
(cont.)

• 4. Now say that we know the rainfall has
  increased 3 inches and we know that fertilizer is
  not necessarily held constant.
• Now what would your best guess be as to the
  difference in yield?

12
Regression Coefficients as Multiplication Factors
(cont.)

• Answer Since fertilizer is not held constant,
  we should use the single regression equation
• DY b DX
• DY 3 (1.5)
• DY 4.5 bushels.
• What we want to do is develop a technique that
  allows us to disaggregate the effects caused
  directly by the increase in rainfall and
  indirectly by other factors.

13
Path Analysis

• II. Path Analysis
• A. Fiji Women
• Say we have data on 4700 women from Fiji.
• 1. Basic Model
• We know for each woman
• Age
• Years of education, and
• Number children

14
Path Analysis (cont.)

• a. Path Diagram
• We might think that a woman's age and education
  correlate with how many children she has.
• We can write a causal model that looks like this

15
Path Analysis (cont.)

• b. Estimates
• When we estimate these relationships, we get the
  results
• CHILDREN 3.4 .059 AGE - .16 EDUC
• We can represent these results as follows

16
Path Analysis (cont.)

• 2. Additional Effects within the Model Direct
  and Indirect
• Now, let's say we think there might also be a
  relationship between a woman's age and education.
  
• a. Estimated Equation
• If we estimate this regression, we get the
  result
• EDUC 7.6 - .032 AGE.
• Older women have less education than younger
  women.

17
Path Analysis (cont.)

• b. Path Diagram
• We now add this new information into the causal
  model

18
Path Analysis (cont.)

• Question
• 1. What is the change in the expected number of
  children due to 1 extra year, holding education
  constant?
• 2. What is the change in the years of education
  from this same 1 extra year of age?

19
Path Analysis (cont.)

• 3. Direct and Indirect Effects
• Question
• What's the change in number of children from one
  extra year of age, letting education change too?
• The change in age has two effects a direct and
  an indirect effect.

20
Path Analysis (cont.)

• a) Direct Effect (Multiple regression
  coefficient)
• The direct effect is captured in the coefficient
  leading from AGE to CHILDREN.
• This is the multiple regression coefficient, and
  it represents the expected extra number of
  children from one extra year, holding education
  constant

21
Path Analysis (cont.)

• b) Indirect Effect
• We know that an extra year corresponds with -.032
  years of school.
• Each extra year of school corresponds with -.16
  extra children.
• We get the indirect effect by multiplying along
  the arrows leading from AGE to CHILDREN through
  EDUC
• (-.032) (-.16) .005.

22
Path Analysis (cont.)

• c) Total Effect
• So the total effect of AGE on CHILDREN letting
  EDUC vary too is the sum of the direct and
  indirect effects.
• That is,
• .059 .005 .064.
• Question
• What do you think would have happened if we ran a
  simple regression of CHILDREN on AGE? What would
  the coefficient have been?

23
Path Analysis (cont.)

• Summary
• A path diagram gives us some insight as to the
  relationship between simple and multiple
  regression.
• Multiple regression gives us the partial effects
  of the independent variables on the dependent
  variable holding all else constant.
• Simple regression gives us the total effect,
  which is the sum of the direct and the indirect
  effects.

24
Path Analysis (cont.)

• B. Brady, Cooper and Hurley
• 1. Defining Unity
• Party unity scores are calculated as
• ( voting in the majority - voting in the
  minority)
• 2. Building the Causal Model
• Two components to party unity internal and
  external factors.
• So we can write a causal model like this

25
Path Analysis (cont.)

• External factors define how homogeneous is the
  constituent base of the party.
• Internal factors have to do with the strength of
  party leadership.

26
Path Analysis (cont.)

• However, it is also thought that external factors
  influence internal factors.
• That is, when legislators from a party are united
  on the issues, they are more likely to give their
  leaders power to get things done.
• Thus we add another line to our model

27
Path Analysis (cont.)

• 3. Results
• When this model was estimated, the results were
• PARTY STRENGTH .61 INTERNAL .58 EXTERNAL
• INTERNAL .66 EXTERNAL.
• 4. Question
• What is the effect of External factors on Party
  Unity?
• Direct Effect 0.58
• Indirect Effect (.66)(.61) 0.40
• Total Effect .58 .40 .98

28
Path Analysis (cont.)

• C. Commie Model from Shapiro
• What determines people's attitudes towards
  whether communists should be allowed to teach
  college?
• 1. How to Build a Causal Model
• First of all, what constitutes a valid causal
  model?
• For now, the answer is no cycles.
• That is, you shouldn't be able to start at a
  point and follow arrows and end up back at the
  same point.

29
Path Analysis (cont.)

• How to build a causal model? (cont.)
• Second, once you have a causal model, how do you
  know which regressions to run?
• For each variable, see what arrows are going into
  it. Then run a regression with those variables
  as the independent variables.

30
Path Analysis (cont.)

• 2. Variables and the Causal Model
• Our hypothesis is that attitudes towards teaching
  depend on attitudes towards communism in general,
  party ID, education, and age.
• The full causal model can be written like this

31
Path Analysis (cont.)

• Variables
• Attitudes towards teaching are determined by all
  the other variables.
• Attitude towards the communist system depends on
  party ID, education, and age.
• Party ID depends on education and age.
• Finally, education depends on age.

32
Path Analysis (cont.)

• 3. Defining the Variables
• First we make our own copies of all the
  variables.
• 1. TeachCom is a dichotomous variable, coded 1 if
  the respondent thought it was OK for communists
  to teach college.
• 2. Smarts is years of education.
• 3. PartyOn is the respondent's party ID. 0
  stands for strongly Democrat, up to 6 for
  strongly Republican.

33
Path Analysis (cont.)

• Variables (cont.)
• 4. ComPhile is how you think about communism as a
  system of government. Higher values mean that
  it's a good system.
• 5. Finally, Years is your age.

34
Path Analysis (cont.)

• 4. Estimating the Model
• a) Regression commands
• How we specify our causal model determines what
  regression we run.
• For instance, TeachCom has arrows going into it
  from all other variables, so we run the
  regression with all the variables.
• Then we take ComPhile, and regress it on Years,
  Smarts and PartyOn.
• And so on down the line.

35
Path Analysis (cont.)

• b) Descriptive Statistics
• We then report our descriptive statistics that
  we'll use
• "Means" gives the mean of each variable.
• "Stddev" gives their standard deviation.
• N gives the number of valid observations.
• "Corr" gives the correlations between variables.
• "Sig" tells us the significance of each
  correlation.

36
Path Analysis (cont.)

• 5. Results
• a) Means Table
• Look at the means table.

37
Path Analysis (cont.)

• b) Correlation Matrix
• Next is the correlation matrix.
• Smarts is negatively correlated with years. That
  means that older people tend to have had fewer
  years of schooling.
• PartyOn is negatively correlated with years, so
  older people tend to be Republican.

38
Path Analysis (cont.)

• b) Correlation Matrix (cont.)
• Comphile and Teachcom is also negatively related
  to years.
• Older people tend to have more negative attitudes
  toward the communist system and be against
  communists teaching college.
• One-tailed p-values are reported beneath the
  correlation coefficient.

39
Path Analysis (cont.)

• c) Regression Results

40
Path Analysis (cont.)

• d) Question
• What is the effect of Years on Teachcom?
• 1. Direct Effect -.003 -0.003
• 2. Indirect Effect via Comphile
  (-.006)(.16) -.00096
• 3. Indirect Effect via Partyon
• Partyon alone (-.0047)(-.015) .0000705
• Partyon and Comphile
• (-.0047)(-.029)(.016) .0000218

41
Path Analysis (cont.)

• 4. Indirect via Smarts
• Smarts alone (-.044)(.035) -.00154
• Smarts Partyon (-.044)(.053)(-.015)
  .000035
• Smarts Comphile (-.044)(.032)(.16)
  -.000023
• Smarts, Partyon Comphile
• (-.044)(.053)(-.029)(.16) .000011
• Total Indirect Effects -.00259
• Total Effects Direct Indirect
• Total Effects -.003 - .0026 -.0056

42
Homework

• III. Homework
• A. Recap
• Your homework assignment is to write a Path
  Diagram.
• B. Issues in the Article
• There is a dispute between American and European
  researchers on the effectiveness of AZT.
  Americans say that it works, and Europeans say
  that there's not enough evidence.

43
Homework

• 1. The U.S. View
• The U.S. allowed AZT to be distributed to
  HIV-positive individuals on the basis of a study
  completed in 1989.
• Usually the FDA requires that to release a drug
  the experimenters show
• DRUG --------------gt HEALTH
• Instead of direct link, researchers showed an
  indirect link.
• AZT --------------gt MARKERS ---------------gt
  HEALTH
• If both of these correlations are positive, then
  so should be the total effect from AZT to health.

44
Homework

• 2. The European View
• The European researchers said that although it's
  true that AZT raised the level of CD-4 markers,
  these markers didn't indicate any long-term
  improvement in health.
• So they say that the model looks like this
• AZT --------------gt MARKERS ---------------gt
  HEALTH
• If there's no link between CD-4 and health, then
  the overall link between AZT and health is also 0
  on the basis of the information presented so far.
  

45
Homework

• 3. How To Resolve the Dispute
• What kind of evidence would they need to resolve
  this dispute?
• First, they could do studies to show that AZT has
  a direct effect on health. These studies take
  longer, but their conclusions are more reliable
  since they show a direct link.
• Or, they could find another marker. That is,
  another intermediate substance that AZT affects
  and that affects health.