Correlation and Multiple Regression

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Correlation and Multiple Regression

• Robert K. Toutkoushian
• Associate Professor
• Educational Leadership and Policy Studies
• Indiana University

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Objectives of Module

• Review statistical procedures such as correlation
  and multiple regression analysis
• Examine ways in which these procedures can be
  applied to institutional research
• Practice using SPSS to implement these procedures
• Discuss more involved procedures and applications

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My Approach

• Aim for a middle ground in terms of difficulty
  (higher UG/lower G level)
• Focus more on intuition behind procedures rather
  than proofs derivations
• Assume familiarity with descriptive stats and
  hypothesis testing
• STRONGLY encourage questions at any time!

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Covariance and Correlation

• Both measure the extent to which two variables
  move together. They differ only in units of
  measure
• Positive covariance/correlation Both variables
  tend to move in the same direction
• Negative covariance/correlation Both variables
  tend to move in the opposite direction

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Remember...

• When looking for correlations, you may have to
  first reorder one of the variables
• If two variables are related, then knowing the
  value of one variable may help with guesses as to
  the value of the other (e.g., retention and
  SAT/ACT scores)
• Correlation does not imply causation!

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Calculating the Covariance

• Calculate the means for X and Y (denoted x-bar
  and y-bar)
• Subtract the mean for X from each X value and
  repeat for Y
• Multiply the differences together for each
  observation, then sum and divide by degrees of
  freedom (n-1)

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Covariance -132,000/(4-1) -44,000
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Correlation Coefficient

• Properties of correlation coefficient
• A standardized measure of covariance that
  ranges between -1 and 1
• Positive 0 lt r ? 1
• Negative -1 ? r lt 0
• No correlation r 0
• Cov(x,y) and r will have the same sign
• Stronger relationship as r moves away from zero

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Calculating Correlation Coefficient

• Calculate cov(x,y) as before
• Calculate st. devs for X and Y
• Divide cov(x,y) by product of standard deviations

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Institutional Research Applications

• Correlations can be useful in IR when one
  variable of interest is unobservable, and a
  correlated variable is observable
• College performance (correlated with HS
  performance)
• Faculty experience (correlated with age, years
  since degree)
• Teaching quality (correlated with student
  evaluations)

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Limitations

• Weak correlations are less useful for making
  inferences
• Correlations vary across factors, so it is
  difficult to compare across factors (e.g., stock
  prices and faculty salaries)
• May be multiple factors affecting a single factor
  of interest
• Does not measure non-linear relationships

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Class Example 1

• Filename TUITION.SAV contains data on average
  public tuition rates, state appropriations, and
  median family income by state in 1994. In SPSS
• Calculate the means and standard deviations for
  these three variables.
• Calculate the covariances and correlations
  between state appropriations and (a) public
  tuition rates, (b) median family income.

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Linear Regression (OLS)

• Objective find the best linear (straight line)
  relationship between two or more variables.
• Ordinary Least Squares (OLS) is the technique
  most often used to choose the best line.
• This linear relationship is based on the
  covariance between two variables.
• Regression analysis requires the analyst to
  specify the direction of causation.

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Advantages of Linear Regression

• Can predict/forecast one variable (Y) based on
  values of another variable (X)
• Can perform hypothesis tests to determine if X
  affects Y
• Can control for differences in Y due to X
• Very flexible with regard to functional form,
  model specification, etc.

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Example Gender Equity in Salaries

• Your President asks you to examine faculty
  salaries at your institution and determine if
  there is a gender equity problem. Descriptive
  stats show that on average men earn more than
  women.
• How can you control for salary differences due to
  justifiable factors such as experience,
  productivity?
• How can you determine if the remaining pay
  difference is large enough to conclude that this
  is a problem?

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Ordinary Least Squares
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Three Formulations

• Slope ß in population, b in sample
• Error term (e or e) encompasses effects of all
  omitted factors
• Parameters in the population model is
  unobservable
• Sample line is what you estimate with OLS

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Assumptions in Linear Regression

• The error term has a mean of zero and constant
  variance
• The errors are unrelated to each other
• The errors are unrelated to the independent
  variable(s) in the model
• The error term is normally distributed (needed
  for hypothesis testing)

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Ordinary Least Squares
OLS specifies that the best line is the one
that minimizes the sum of squared errors (
minimize S ei2 )
Intercept (a)
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Notes on OLS

• The slope formula is the covariance between X and
  Y divided by the variance for X
• The slope and covariance will always have the
  same sign
• b gt 0 indicates a positive relationship
• b lt 0 indicates a negative relationship
• b 0 indicates no linear relationship

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Example An IR analyst is asked to help forecast
applications. She believes there is a
relationship between HS grads and resident
applications each year
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• Regression line Y -358.28 0.29X
• Interpretation For each additional HS grad,
  predicted applications will rise by 0.29.
• The intercept may not have much meaning.
• Can predict applications given projections of HS
  grads. If HS grads 36,000, then
• Y -358.28 0.29(36,000) 10,082

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Goodness of Fit

• Measures the strength of the relationship between
  X and Y
• R-squared (or coefficient of determination)
  proportion of total deviation in Y that is
  explained by X(s)
• R-squared is bounded between 0 and 1 (R2 1 if
  perfect fit, R2 0 if no fit)
• R-squared square of correlation coefficient
  (with only one X variable in the model)

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More on R-squared...

• When there is no covariance, the slope of the
  regression line is zero and R2 0.
• Adding variables to the regression model will
  almost always raise R2, but this does not mean
  that the resulting model is better
• Adjusted R2 attempts to correct for this, but no
  longer has the same interpretation
• R2 varies depending on the dependent variable.
  Do not use this to compare regression models with
  different Ys.

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Predicting Resident Applications
Note that HS grads account for 88.5 of the total
deviation in applications.
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Class Example 2

• Using TUITION.SAV, in SPSS
• Calculate a regression line showing how median
  income affects average tuition
• Calculate R2, TSS, RSS, ESS, and corr(x,y). SPSS
  syntax
• REGRESSION
• /MISSING LISTWISE
• /STATISTICS COEFF OUTS R ANOVA
• /CRITERIAPIN(.05) POUT(.10)
• /NOORIGIN
• /DEPENDENT tuition
• /METHODENTER income .

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Equation Tuition 313.119 0.0719Income
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Hypothesis Testing for ß

• In most situations in the social sciences, it is
  rarely known for sure if X affects Y
• A hypothesis test can be used to determine if the
  data provide sufficient evidence of a
  relationship
• For most variables, the sample slope b will not
  exactly equal zero. How far from zero must it be
  in order to safely conclude that ß ? 0??

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Steps in Hypothesis Testing

• Specify null (H0) and alternative (HA)
  hypotheses
• Identify test statistic and find critical
  value(s) based on degrees of freedom and
  significance level
• Calculate test statistic and compare to critical
  value(s)

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Common Hypotheses for ß

• ß 0 (X has no effect on Y)
• ß gt 0 (X has a positive effect on Y)
• ß lt 0 (X has a negative effect on Y)
• ß ? 0 (X has some effect on Y or - )
• Choose two hypotheses that are mutually exclusive
  and exhaustive.
• The null hypothesis (H0) should always contain
  some form of equal sign.

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Test Statistic for ß
If e N(0, s2), then b N(ß, Var(b))
Therefore the t-ratio
Will follow a Student t-distribution with n-k
degrees of freedom (k parameters to be
estimated)
The t-ratio is defined as the random variable
minus its mean (when H0 is true), divided by its
standard deviation.
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Notes on Hypothesis Testing

• The t-ratio simply counts the standard
  deviations the slope is from zero (distance)
• The greater the distance, the less likely you
  would have found the value of b if ß 0.
• For significance tests, since ß 0, the t-ratio
  is the slope divided by its standard deviation
  (or standard error)

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Example t-ratio of 2.40
This shows that there is only a 1.3 chance of
finding a t-ratio of 2.40 or greater if in fact ß
0. Therefore, if you found a t-value this
high, it is unlikely that ß 0.
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R2 0.025
TSS 1.9E10 ESS 1.9E10 se v(ESS/826) 4766
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Do undergraduate enrollments have a significant
effect on average costs per student? Null
Hypothesis ß 0, Alternative Hypothesis ß ?
0 For 826 df, 1 significance level, reject the
null when the calculated t-ratio exceeds 2.575 in
absolute value.
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P-value Probability of drawing a more extreme
sample value given that the null hypothesis is
true P-value Pr(b lt -0.175) Pr(t lt -4.577)
0.000
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Units of Measurement

• The significance levels of any variable will not
  be influenced by the units of measure used for X
  or Y
• The coefficient represents the units change in
  Y due to a one-unit change in X
• When the units of measure change, both the
  coefficients and standard errors change
  proportionately (t-ratio remains the same)

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Out-of-Sample Forecasts/Predictions

• The regression model can be used to derive
  predictions of Y given values of X(s)
• Point estimates are found by substituting X into
  the equation and solving for Y (I predict that
  the grad rate will be 70)
• Interval estimates are predictions that Y will
  fall within a certain interval (I am 95 certain
  that the grad rate will be between 68 and 72)
• Interval estimates are more conservative, and
  convey the uncertainty in predictions.

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Two Types of Intervals

• C.I. For expected value (mean) of Y
• For given X, what is the predicted average value
  of Y
• C.I. For a single value of Y
• For given X, what is the predicted single value
  of Y (more uncertainty, so wider interval)
• The two methods yield very similar intervals.
  Most IR applications use C.I.s for single value.
• Intervals can be obtained in SPSS using the
  save subcommand.

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Predict HS Grads in New Hampshire

• An IR analyst is charged with developing a model
  to help predict changes in HS grads in the state
  through 2006.
• File AIR1.SAV in SPSS has two vars HS grads in
  year t (HSGRAD), and 2nd grade enrollments in
  year t-10 (GRADE2).
• Find correlation between HSGRAD and GRADE2
• Estimate a regression model
• Form point and 95 CI estimates of high school
  grads for the next ten years.

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Under statistics gt correlate gt bivariate
Note that r 0.959, cov(x,y) 701,160 (n12)
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Under statistics gt regression gt linear
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In 2006, the model predicts there will be 14,919
high school grads
95 certain that in 2006 there will be between
14,185 and 15,652 high school grads
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Multiple Regression Analysis

• In most IR applications, the dependent variable
  may be influenced by multiple factors
• Grad rate f(avg. SAT, gender composition, avg.
  HS rank, students on campus,...)
• Faculty Salary f(education, experience,
  productivity, field,...)
• Education Costs f(enrollments, research
  intensity, student/faculty ratio,...)

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Assumptions in Multiple Regression

• Error term has a mean of zero and constant
  variance
• Error terms are unrelated to each other
• Error term is unrelated to independent vars
• Error term is normally distributed
• Independent variables are not collinear with each
  other (no multicollinearity)

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Ordinary Least Squares
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Least Squares Estimates

• The coefficients are referred to as partial
  effects because they show the effect of one
  variable on Y holding other vars constant.
• The OLS formula takes into account any
  relationships between the X variables. For this
  reason, the coefficients usually change when
  variables are dropped/added from model.

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Other Stats in Multiple Regression

• Hypothesis tests for significance of coefficients
  can be performed as before, except degrees of
  freedom change (n-k-1).
• Goodness of fit measures are calculated as
  before. R-squared now represents the deviation
  in Y explained by all Xs together. Thus, R2
  usually rises as Xs are added.
• Confidence intervals and point estimates can be
  calculated as before.

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Example Average Public Tuition Rates

• An IR analyst is asked to help explain why there
  are variations across states in their tuition
  rates at public institutions. She feels that
  factors such as state aid given to students and
  state appropriations help account for these
  differences.
• Open the file TUITION.SAV in SPSS.

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Question 1 How do state appropriations affect
average tuition?
State appropriations account for 13.5 of
differences in tuition.
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Question 2 How do state appropriations and aid
to students affect average tuition?
These two variables account for 40.4 of
differences in tuition.
A 1 increase in appropriations reduces tuition
by 22.6 cents, holding constant state aid per
student.
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Extensions of Regression Model

• So far, we have only considered linear models
  where Xs and Ys were continuous. We will now
  examine how to handle
• Categorical Xs
• Interactions among Xs
• Non-linear relationships between X and Y

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Categorical Variables

• There are many examples of independent variables
  that are not numerical (ex gender, race,
  institution attended, attitudes/beliefs)
• Likert scale variables (assign s to
  categorical responses) should not be used in
  regression models in their present form due to
  problems in interpreting changes in units.
• Slope units change in Y due to a one-unit
  change in X (but Likert s are artificial)

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Dummy Variables

• However, categorical Xs can be used if they are
  first recoded into dummy variables
• Dummy variable has only two values (0,1)
• Need to specify an assignment rule. Can be used
  for categorical, Likert, and continuous
  variables.
• The variable can now be used in regression
• It does not matter which group is assigned 1
• Coef represents the difference in intercepts for
  the two groups
• Must omit one of the dummy variables for a
  construct to avoid multicollinearity

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Examples of Assignment Rules

• Let X 1 if (0 otherwise)
• Teaches in Psychology Department
• Enrolled in public university
• Family income exceeds 100,000
• Student is very satisfied with the quality of
  instruction
• Student graduated from campus
• Student dropped out of college

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Note Both equations have the same slope for RANK.
Question Does living on campus matter?
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Variable Interactions

• It is possible that the joint occurrence of two
  Xs has an effect on Y separate from each Xs
  effect
• Academic performance of students with high SAT
  scores and HS ranks
• State appropriations for higher ed in states with
  low incomes and high tax rates
• The salary increase from promotions for men and
  women may be different

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Interactions (contd)

• In these examples, there is something special
  about the joint occurrence of two variables.
• To test these assertions, an interaction
  variable can be created and added to the
  regression model.
• Interaction variables are created by defining a
  third variable as the product of the two
  variables in question.

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The interaction variable is then added to the
regression model and treated as any other
variable
To find the effect of x1 on y, you need to
differentiate the equation with respect to x1
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Non-linear Functional Forms

• Regression analysis can also be used in
  situations where X has a non-linear relationship
  with Y
• Linear The change in Y due to a one-unit change
  in X is constant.
• Non-linear The change in Y due to a one-unit
  change in X can vary with the level of X.

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Graphs of Non-linear Functions
Y
Y
X
X
Exponential Y exp(X)
LogarithmicY ln(X)
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Graphs of Quadratic Functions
Y
Y
X
X
Maximize Y
Minimize Y
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Possible IR Examples

• Exponential Implies that as X increases, Y
  increases at a faster rate.
• Y salary, X years of experience
• Logarithmic Implies that as X increases, Y
  increases at a slower rate.
• Y college GPA, X hrs/week studying
• Y retention rate, X avg. student SAT score

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Possible IR Quadratic Examples

• Maximize Y There is some value of X at which Y
  is maximized.
• Y Tuition revenue, X tuition rate
• Y Student gains, X class size
• Minimize Y There is some value of X at which
  Y is minimized.
• Y costs/student, X enrollments

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Using Non-linear Functions

• Regression analysis requires a linear
  relationship between X and Y.
• When there is a non-linear relationship, you can
  transform one or more variables and then use the
  transformed variables in the regression model.
• As long as there is a linear relationship between
  the transformed variables, regression analysis is
  appropriate.

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Exponential Transformations

• The coefficient estimate for ß represents the
  approximate percentage change in Y due to a
  one-unit increase in x.
• The variable x always has the same directional
  effect on Y (positive or negative)
• The change in Y due to a change in x increases
  at an increasing rate

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Natural Log Function
The natural log function is the inverse of the
exponential function ln (exp (X)) X
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Logarithmic
This can also be used for a subset of Xs.
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Double-Log Function Elasticities
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Quadratic Transformations
If X is believed to have a quadratic effect on Y,
then create a new variable as the square of X and
add this to the regression model
The change in Y due to a one-unit change in X1
would be found by differentiating the equation
with respect to this variable
Hill-shaped if ß3 lt 0, U-shaped if ß3 gt 0, linear
if ß3 0
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More on Quadratic Functions

• The value of X that maximizes or minimizes Y can
  have important implications. This is found by
  solving for X in the first-derivative.
• Higher-order functions (ex., cubic) can also be
  used in regression. They can yield better
  representations of relationships, but are harder
  to explain and interpret.

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SPSS Exercise Faculty Salaries

• An IR analyst is asked to investigate if female
  faculty are paid less than comparable males. She
  draws a sample of 432 faculty and creates these
  variables
• Salary monthly base salary (in dollars)
• Rank 1 if Full, 2 if Associate, 3 if Assistant
• Gender if if male, 0 otherwise
• Prevexp days of experience before current job
• Npleave days of non-professional leave
• Potenexp days since highest degree
• Nine12 1 if nine-month appointment, 0 otherwise
• Cite85 Citations in 1985 to all publications

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Tasks

• Open the SPSS system file FACSAL.SAV
• Estimate a regression model showing how gender
  affects salary. How do these results compare to
  a two-sample t-test?
• How do your findings change when potential
  experience and citations are added?
• An economist argues that salaries rise
  exponentially with potential experience,
  citations, and gender. How can this be addressed?

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Answer to first task...
Note mean difference is 916, which has a
t-value of 6.227 and is significant.
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Answer to second task...
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Answer to third task...
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More Tasks...

• The VP for Finance argues that individuals with
  high experience levels often get smaller
  percentage salary increases than others. How
  could this be addressed (use same function as in
  previous example)?
• A female faculty member claims that women face
  discrimination in part because they are rewarded
  less for each citation they receive. How could
  you test this?

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Answer to fourth task...
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Answer to fifth task...
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Model Selection

• For most IR problems, there are many alternative
  models from which to choose. How should the
  best model be selected?
• Begin with published studies that look at the
  same (similar) Ys. What variables and
  functional forms do they use?
• Is there a theory that can be used to guide
• human capital theory gt salary models
• median voter theory gt state funding for HE
• Tintos model gt student retention

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More model selection comments

• Better to include too many factors than to omit
  important variables (omitted variable bias)
• Can estimate several competing model
  specifications and compare results. Be careful
  not to simply select model with the most
  appealing results!
• Keep in mind trade-off between simplicity and
  accuracy. A simple model is worth its weight in
  gold when explaining to decisionmakers!

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Faculty Salary Example

• Return to FACSAL.SAV and create a dummy variable
  for full professors
• Estimate a model explaining salary as a function
  of gender, then gender and full professor.
• Estimate a model explaining salary as a function
  of gender, full professor, and potential
  experience.

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Problems in Regression Analysis

• There are three main problems which may arise in
  multiple regression
• Autocorrelation
• Heteroscedasticity
• Multicollinearity
• We will briefly discuss what each means, how they
  can be detected, and what can be done about them
  when they occur.

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Autocorrelation

• This can occur in time-series data when the error
  in one period is related to the error in the
  next.
• Violates the assumption E(eiej) for i ? j
• Causes the computer to calculate incorrect
  standard errors, thereby affecting t-ratios.
  Usually, st.errs are too small, so t-ratios are
  too high (making X appear significant when it
  isnt.)
• Possible IR Examples Predicting applications, HS
  grads, state funding for HE.

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First-order autocorrelation
Error
et

0
-
Time
t4
t9
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Durbin-Watson test
Calculates a d-statistic that reflects the
correlation among subsequent error terms
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Correcting Autocorrelation

• If autocorrelation is detected, it can be
  corrected through transforming the data to yield
  correct standard errors (generalized least
  squares).
• Cochrane-Orcutt or Prais-Winston two
  commonly-used methods
• Standard autocorrelation option in SPSS does
  not do this. Use SPSS Trends or another program.
• Keep in mind that autocorrelation affects the
  standard errors and not coefficients.

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Heteroscedasticity

• May occur in cross-section data when the variance
  of the error term is related to one or more
  independent variable (si2 not constant).
• Affects standard errors, and hence t-ratios (but
  not coefficient estimates)
• Potential IR examples
• Effects of enrollments on average costs
• Effect of tax revenues on state appropriations
• Effect of program size on expenditures

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Graph of Heteroscedasticity
Dependent variable
Regression line
Independent variable
As X increases, the possible errors become larger.
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Testing for Heteroscedasticity

• Visual Plot residuals against the variable
  thought to be causing the problem.
• Park-Glesjer test Estimate model and save
  residuals. Regress the log of squared residuals
  against the log of variable thought to cause the
  problem.
• Other tests White (1978), Goldfeld-Quandt.
• SPSS will not do these by default (must do by
  hand or with other software).

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Correcting for Heteroscedasticity

• Weighted least squares Weight observations by
  the variable causing heteroscedasticity. However,
  you must know the form.
• For example, if si2 s2X1i, then weighting each
  observation by the square root of X1 will yield
  correct standard errors.
• An option that does not require knowing the form
  of heteroscedasticity is by White.

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Multicollinearity

• Multicollinearity arises when there is an
  extremely high correlation between two or more
  independent variables in the model.
• The coefficients are biased the stats program
  does not know how to assign proper weights
• Standard errors increase, making t-ratios small

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Multicollinearity (contd)

• Potential IR examples include (1) effect of
  current and previous experience on faculty
  salaries, (2) effect of SAT score and high school
  rank on academic performance, (3) effect of
  family income and wealth on student demand for
  higher education.
• A significant correlation between Xs does not
  necessarily lead to multicollinearity. Only when
  the correlation is very high does this occur.

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Testing/Correcting Multicollinearity

• There is no universally-accepted test for
  multicollinearity.
• Variance inflation factors (VIF) estimate how
  much the standard errors increase due to
  correlation with other Xs. No single cutoff
  point for VIFs.
• Signs of multicollinearity include
• Two similar variables have widely different
  effects on Y (e.g., only one is signif.)
• The standard errors are large
• To test, drop one of the variables from the model
  and compare results. If the coef and st. err.
  change considerably, this may be a problem.

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Correcting Multicollinearity

• There is also no uniformly-accepted solution to
  this problem. However, you can drop one of the
  problem variables from the model.
• Multicollinearity may not be an important issue
  if the collinearity occurs between unimportant
  variables.

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Makin Multicollinearity

• Return to the faculty salary data and create a
  new variable
• newpot potenexp / 365 (years of exper) and
  add this to the regression model
• Then, make slight changes to first two data
  points change 27.02 to 13 and change 19.01
  to 27.
• Estimate regression model again, using gender,
  potenexp, newpot

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Using gender and potenexp
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Using gender, potenexp and newpot
Variable POTENEXP drops out of the equation
because it is perfectly correlated with NEWPOT.
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Using gender, potenexp, newpot (after changes)
Gender is significant throughout all three models
Standard errors are about forty-three times
larger than before!
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Limited Dependent Variables

• Thus far, we have considered instances where Y
  was continuous and unbounded. However, there are
  many situations where this is violated
• Individual student data are often dichotomous
  (0,1) variables 1 if graduate, 1 if return, 1 if
  apply/enroll.
• Some data are discrete counts number of journal
  articles or citations, number of times a student
  changes his/her major

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Problems with OLS when Y is (0,1)

• Predictions can be gt 1 or lt 0
• Coefficients may be biased
• Heteroscedasticity is present (s2 P(1-P))
• Error term is not normally distributed (only two
  possible values), so hypothesis tests are invalid
• Of these problems, the last is the most severe.

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Maximum Likelihood Estimation

• In this instance, there are advantages to using a
  technique (MLE) in place of OLS.
• MLE Find the coefficients that maximize the
  likelihood of generating the observations on Y in
  the sample.
• Recall that OLS chooses the coefficients based on
  those that minimize the sum of squared errors.

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Logit and probit analysis

• When Y (0,1), the two most commonly-used
  functional forms in MLE are the cumulative
  logistic distribution (logit analysis or
  logistic regression) and the cumulative normal
  distribution (probit analysis).
• The two choices usually yield similar results
• Each avoids the four problems noted with OLS

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Logistic regression
For logistic regression, the following functional
form is used
Ln P/(1-P) a b1X1 b2X2
where P probability that Y1
All you have to do, however, is create the dummy
variable for Y and tell SPSS to use logistic
regression to estimate the model. SPSS will
create the log odds ratio for you.
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Interpreting results

• The coefficients from logistic regression are
  hard to interpret and explain.
• Focus on the signs of the coefficients
• If the sign is positive and significant, then as
  X increases, the probability that Y1 will also
  increase
• If the sign is negative and significant, then as
  X increases, the probability that Y1 will
  decrease
• If the coefficient is not significant, then X has
  no effect on the probability that Y1.

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Example Faculty Rank

• Return to the faculty dataset, and estimate a
  logistic regression model to explain whether a
  faculty member is a Full professor (under
  Regression / bivariate logistic)
• Xs include gender, potenexp, prevexp, and cite85
• Need to create a dummy variable for Full
  Professor first
• SPSS Probit module is different than used here.

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Wald Chi-square statistic (coefficient /
standard error) 2
Note that these Chi-square values are the square
of the standard t-ratios
Coefficients Effect of X on log odds
Standard errors
Odds Ratio
P-value
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Results from rank analysis

• Since the coefficient for GENDER is positive and
  significant, it means that men and more likely
  than women to hold the rank of Full professor
  after controlling for experience and citations.
• The positive and significant coef for CITE85
  means that a faculty member is more likely to be
  a Full Prof as citations rise

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Final Exam SATDATA.SAV
File contains data on 1,999 NH high school
seniors in 1996 who have taken the SAT

• ASSOC 1 if highest planned degree AA
• MA Masters
• PHD Doctorate
• MALE 1 if male
• FIRSTGEN 1 if 1st generation
• INCOME family income
• INCOME2 income squared
• PUBHS 1 if attend public high school

• SATCOMB Combined SAT score
• SATCOMB2 SAT squared
• ANYAP 1 if taken any AP course
• GRADEAVG high school GPA
• UNH 1 if sent SAT score to UNH
• KSC 1 if sent SAT score to KSC
• PSC 1 if sent SAT score to PSC

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Questions

• How does family income, student ability, and
  student intentions affect whether a student
  submits SAT scores to UNH, KSC, or PSC?
• Do SAT takers from poor families and/or first
  generation families do worse on the SAT than
  other students?