1. Multiple Regression Assumptions & Diagnostics

2. Regression: Outliers
Note: Even if regression assumptions are met, slope estimates can have problems
Example: Outliers -- cases with extreme values that differ greatly from the rest of your sample
More formally: “influential cases”
Outliers can result from:
Errors in coding or data entry
Highly unusual cases
Or, sometimes they reflect important “real” variation
Even a few outliers can dramatically change estimates of the slope, especially if N is small.

3. Regression: Outliers Outlier Example:
Extreme case that pulls regression line up 4 2 -2 -4
Regression line with extreme case removed from sample

4. Regression: Outliers Strategy for identifying outliers:
1. Look at scatterplots or regression partial plots for extreme values
Easiest. A minimum for final projects
2. Ask SPSS to compute outlier diagnostic statistics
Examples: “Leverage”, Cook’s D, DFBETA, residuals, standardized residuals.

5. Regression: Outliers SPSS Outlier strategy: Go to Regression – Save
Choose “influence” and “distance” statistics such as Cook’s Distance, DFFIT, standardized residual
Result: SPSS will create new variables with values of Cook’s D, DFFIT for each case
High values signal potential outliers
Note: This is less useful if you have a VERY large dataset, because you have to look at each case value.

7. Scatterplots Example: Study time and student achievement.
X variable: Average # hours spent studying per day
Y variable: Score on reading test
Y axis
X axis 30 20 10
Case X Y 1
2.6 28 2
1.4 13 3
.65 17 4
4.1 31 5
.25 8 6
1.9 16 7
3.5

8. Outliers
Results with outlier:

9. Outlier Diagnostics Residuals: The numerical value of the error
Error = distance that points falls from the line
Cases with unusually large error may be outliers
Note: residuals have many other uses!
Standardized residuals
Z-score of residuals… converts to a neutral unit
Often, standardized residuals larger than 3 are considered worthy of scrutiny
But, it isn’t the best outlier diagnostic.

10. Outlier Diagnostics
Cook’s D: Identifies cases that are strongly influencing the regression line
SPSS calculates a value for each case
Go to “Save” menu, click on Cook’s D
How large of a Cook’s D is a problem?
Rule of thumb: Values greater than: 4 / (n – k – 1)
Example: N=7, K = 1: Cut-off = 4/5 = .80
Cases with higher values should be examined.

11. Outlier Diagnostics Example: Outlier/Influential Case Statistics Hours
Score
Resid
Std Resid
Cook’s D
2.60 28
9.32
1.01
.124
1.40 13
-1.97
-.215
.006
.65 17
4.33
.473
.070
4.10 31
7.70
.841
.640
.25 8
-3.43
-.374
.082
1.90 16
-.515
-.056
.0003
3.50 6
-15.4
-1.68
.941

12. Outliers
Results with outlier removed:

13. Regression: Outliers
Question: What should you do if you find outliers? Drop outlier cases from the analysis? Or leave them in?
Obviously, you should drop cases that are incorrectly coded or erroneous
But, generally speaking, you should be cautious about throwing out cases
If you throw out enough cases, you can produce any result that you want! So, be judicious when destroying data.

14. Regression: Outliers
Circumstances where it can be good to drop outlier cases:
1. Coding errors
2. Single extreme outliers that radically change results
Your results should reflect the dataset, not one case!
3. If there is a theoretical reason to drop cases
Example: In analysis of economic activity, communist countries may be outliers
If the study is about “capitalism”, they should be dropped.

15. Regression: Outliers Circumstances when it is good to keep outliers
1. If they form meaningful cluster
Often suggests an important subgroup in your data
Example: Asian-Americans in a dataset on education
In such a case, consider adding a dummy variable for them
Unless, of course, research design is not interested in that sub-group… then drop them!
2. If there are many
Maybe they reflect a “real” pattern in your data.

16. Regression: Outliers
When in doubt: Present results both with and without outliers
Or present one set of results, but mention how results differ depending on how outliers were handled
For final projects: Check for outliers!
At least with scatterplots
In the text: Mention if there were outliers, how you handled them, and the effect it had on results.

17. Multicollinearity Another common regression problem: Multicollinearity
Definition: collinear = highly correlated
Multicollinearity = inclusion of highly correlated independent variables in a single regression model
Recall: High correlation of X variables causes problems for estimation of slopes (b’s)
Recall: variable denominators approach zero, coefficients may wrong/too large.

18. Multicollinearity Multicollinearity symptoms:
Unusually large standard errors and betas
Compared to if both collinear variables aren’t included
Betas often exceed 1.0
Two variables have the same large effect when included separately… but…
When put together the effects of both variables shrink
Or, one remains positive and the other flips sign
Note: Not all “sign flips” are due to multicollinearity!

19. Multicollinearity What does multicollinearity do to models?
Note: It does not violate regression assumptions
But, it can mess things up anyway
1. Multicollinearity can inflate standard error estimates
Large standard errors = small t-values = no rejected null hypotheses
Note: Only collinear variables are effected. The rest of the model results are OK.

20. Multicollinearity What does multicollinearity do?
2. It leads to instability of coefficient estimates
Variable coefficients may fluctuate wildly when a collinear variable is added
These fluctuations may not be “real”, but may just reflect amplification of “noise” and “error”
One variable may only be slightly better at predicting Y… but SPSS will give it a MUCH higher coefficient
Note: These only affect variables that are highly correlated. The rest of the model is OK.

21. Multicollinearity Diagnosing multicollinearity:
1. Look at correlations of all independent vars
Correlation of .7 is a concern, .8> is often a problem
But, sometimes problems aren’t always bivariate… and don’t show up in bivariate correlations
Ex: If you forget to omit a dummy variable
2. Watch out for the “symptoms”
3. Compute diagnostic statistics
Tolerances, VIF (Variance Inflation Factor).

22. Multicollinearity Multicollinearity diagnostic statistics:
“Tolerance”: Easily computed in SPSS
Low values indicate possible multicollinearity
Start to pay attention at .4; Below .2 is very likely to be a problem
Tolerance is computed for each independent variable by regressing it on other independent variables.

23. Multicollinearity If you have 3 independent variables: X1, X2, X3…
Tolerance is based on doing a regression: X1 is dependent; X2 and X3 are independent.
Tolerance for X1 is simply 1 minus regression R-square.
If a variable (X1) is highly correlated with all the others (X2, X3) then they will do a good job of predicting it in a regression
Result: Regression r-square will be high… 1 minus r-square will be low… indicating a problem.

24. Multicollinearity
Variance Inflation Factor (VIF) is the reciprocal of tolerance: 1/tolerance
High VIF indicates multicollinearity
Gives an indication of how much the Standard Error of a variable grows due to presence of other variables.

25. Multicollinearity Solutions to multcollinearity
It can be difficult if a fully specified model requires several collinear variables
1. Drop unnecessary variables
2. If two collinear variables are really measuring the same thing, drop one or make an index
Example: Attitudes toward recycling; attitude toward pollution. Perhaps they reflect “environmental views”
3. Advanced techniques: e.g., Ridge regression
Uses a more efficient estimator (but not BLUE – may introduce bias).

26. What is Model Specification?
Model Specification is two sets of choices:
The set of variables that we include in a model
The functional form of the relationships we specify
These are central theoretical choices
Can’t get the right answer if we ask the wrong question

27. What is the “Right” Model?
In truth, all our models are misspecified to some extent.
Our theories are always a simplification of reality, and all our measures are imperfect
Our task is to seek models that are reasonably well specified – keeps our errors relatively modest

28. Omitting Variables and Model Specification
These criteria give us our conceptual standards for determining when a variable must be included
A model must be included in a regression equation IF:
The variable is correlated with other X’s
AND
The variable is also a cause of Y

29. The Meaning of b in Multiple Regression
Each element of the vector b is a slope coefficient for one of the X’s
Same as in bivariate context except that b1 is the expected change in Y for a 1 unit increase in X1, while holding X2…Xn constant
Thus b1 represents the direct effect of X1 on Y, controlling for X2…Xn

30. Illustrating Omitted Variable Bias
Imagine a true model where X1 has a small effect on Y and is correlated with X2 that has a large effect on Y.
Specifiying both variables can distinguish these effects X1 Y X2

31. Illustrating Omitted Variable Bias
But when we run simple model excluding X2, we attribute all causal influence to X1
Coefficient is too big and variance of coefficient is too small X1 Y

32. Omitted Variable Bias: Causes
This problem is explicitly theoretical rather than “statistical.”
No statistical test can reveal a specification error or omitted variable bias
Scholars can form hypotheses about other X’s that may be a source of omitted variable bias

33. Including Irrelevant Variables
In this case:
If b2=0, our estimate of b1 is not affected
If X1’X2=0, our estimate of b1 is not affected
But including X2 if it is not relevant does unnecessarily inflate the σb2

34. Including Irrelevant Variables: Consequences
σb2 increases for two reasons:
Addition of parameter for X2 reduces the degrees of freedom
Part of estimator for σu2
If b2=0 but X1’X2 is not, then including X2 unnecessarily reduces independent variation of X1
Thus parsimony remains a virtue

35. Rules for Model Specification
Model specification is fundamentally a theoretical exercise. We build models to reflect our theories
Theorizing process cannot be replaced with statistical tests
Avoid mechanistic rules for specification such as stepwise regression

36. The Evils of Stepwise Regression
Stepwise regression is a method of model specification that chooses variables on:
Significance of their t-scores
Their contribution to R2
Variables will be selected in or out depending on the order they are introduced into the model

37. Multiple Regression Analysis: Further Issues
y = b0 + b1x1 + b2x bkxk + u

38. Functional Form
OLS can be used for relationships that are not strictly linear in x and y by using nonlinear functions of x and y – will still be linear in the parameters
Can take the natural log of x, y or both
Can use quadratic forms of x
Can use interactions of x variable

39. Interpretation of Log Models
If the model is ln(y) = b0 + b1ln(x) + u
b1 is the elasticity of y with respect to x
If the model is ln(y) = b0 + b1x + u
b1 is approximately the percentage change in y given a 1 unit change in x
If the model is y = b0 + b1ln(x) + u
b1 is approximately the change in y for a 100 percent change in x

40. Why use log models?
Log models are invariant to the scale of the variables since measuring percent changes
They give a direct estimate of elasticity
For models with y > 0, the conditional distribution is often heteroskedastic or skewed, while ln(y) is much less so
The distribution of ln(y) is more narrow, limiting the effect of outliers

41. Adjusted R-Squared
Recall that the R2 will always increase as more variables are added to the model
The adjusted R2 takes into account the number of variables in a model, and may decrease

42. Adjusted R-Squared (cont)
It’s easy to see that the adjusted R2 is just (1 – R2)(n – 1) / (n – k – 1), but most packages will give you both R2 and adj-R2
You can compare the fit of 2 models (with the same y) by comparing the adj-R2
You cannot use the adj-R2 to compare models with different y’s (e.g. y vs. ln(y))

43. The Use and Interpretation of the Constant Term
General Rule
Do Not Suppress the Constant Term
( Even if theory specifically calls for it )
Do Not Rely on estimates of the Constant Term

44. Multiple Regression Assumptions
1. a. Linearity: The relationship between dependent and independent variables is linear
Just like bivariate regression
Points don’t all have to fall exactly on the line; but error (disturbance) must be random
Check scatterplots of X’s and error (residual)
Watch out for non-linear trends: error is systematically negative (or positive) for certain ranges of X
There are strategies to cope with non-linearity, such as including X and X-squared to model curved relationship.
C-D production function

45. Multiple Regression Assumptions
1. b. And, the model is properly specified:
No extra variables are included in the model, and no important variables are omitted. This is HARD!
Correct model specification is critical
If an important variable is left out of the model, results are biased (“omitted variable bias”)
Example: If we model job prestige as a function of family wealth, but do not include education
Coefficient estimate for wealth would be biased
Use theory and previous research to decide what critical variables must be included in your model.
For final paper, it is OK if model isn’t perfect.

46. Multiple Regression Assumptions
2. All variables are measured without error
Unfortunately, error is common in measures
Survey questions can be biased
People give erroneous responses (or lie)
Aggregate statistics (e.g., GDP) can be inaccurate
This assumption is often violated to some extent
We do the best we can:
Design surveys well, use best available data
And, there are advanced methods for dealing with measurement error.

47. Multiple Regression Assumptions
3. The error term (ei) has certain properties
Recall: error is a cases deviation from the regression line
Not the same as measurement error!
After you run a regression, SPSS can tell you the error value for any or all cases (called the “residual”)
3. a. Error is conditionally normal
For bivariate, we looked to see if Y was conditionally normal… Here, we look to see if error is normal
Examine “residuals” (ei) for normality at different values of X variables.

48. Regression Assumptions
Normality:
Examine residuals at different values of X. Make histograms and check for normality.
Good
Not very good

49. Multiple Regression Assumptions
3. b. The error term (ei) has a mean of 0
This affects the estimate of the constant. (Not a huge problem)
3. c. The error term (ei) is homoskedastic (has constant variance)
Note: This affects standard error estimates, hypothesis tests
Look at residuals, to see if they spread out with changing values of X
Or plot standardized residuals vs. standardized predicted values.

50. Regression Assumptions
Homoskedasticity: Equal Error Variance
Examine error at different values of X. Is it roughly equal?
Here, things look pretty good.

51. Regression Assumptions
Heteroskedasticity: Unequal Error Variance
At higher values of X, error variance increases a lot.
This looks pretty bad.

52. Multiple Regression Assumptions
3. d. Predictors (Xis) are uncorrelated with error
This most often happens when we leave out an important variable that is correlated with another Xi
Example: Predicting job prestige with family wealth, but not including education
Omission of education will affect error term. Those with lots of education will have large positive errors.
Since wealth is correlated with education, it will be correlated with that error!
Result: coefficient for family wealth will be biased.

53. Multiple Regression Assumptions
4. In systems of equations, error terms of equations are uncorrelated
This is not a concern for us in this class
Worry about that later!

54. Multiple Regression Assumptions
5. Sample is independent, errors are random
Technically, part of 3.c.
Not only should errors not increase with X (heteroskedasticity), there should be no pattern at all!
Things that cause patterns in error (autocorrelation):
Measuring data over long periods of time (e.g., every year). Error from nearby years may be correlated.
Called: “Serial correlation”.

55. Multiple Regression Assumptions
More things that cause patterns in error (autocorrelation):
Measuring data in families. All members are similar, will have correlated error
Measuring data in geographic space.
Example: data on 50 US states. States in a similar region have correlated error
Called “spatial autocorrelation”
There are variations of regression models to address each kind of correlated error.

56. Multiple Regression Estimation
Calculating b’s involves solving a set of equations to minimize squared error
Analogous to bivariate, but math is more complex
The optimal estimator has minimum variance and is referred to as “BLUE”:
Best Linear, Unbiased Estimate
The BLUE Multiple Regression has more assumptions than bivariate.