Transcript Assumptions in regression analysis

ASSUMPTION CHECKING
• In regression analysis with Stata
• In multi-level analysis with Stata
(not much extra)
• In logistic regression analysis with
Stata
NOTE: THIS WILL BE EASIER IN
STATA THAN IT WAS IN SPSS
Assumption checking
in “normal” multiple regression
with Stata
Assumptions in regression
•No multi-collinearity
analysis
•All relevant predictor variables
included
•Homoscedasticity: all residuals are
from a distribution with the same variance
•Linearity: the “true” model should be
linear.
•Independent errors: having information
about the value of a residual should not
give you information about the value of
other residuals
•Errors are distributed normally
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FIRST THE ONE THAT LEADS TO
NOTHING NEW IN STATA
(NOTE: SLIDE TAKEN LITERALLY FROM MMBR)
Independent errors: having information about
the value of a residual should not give you
information about the value of other residuals
Detect: ask yourself whether it is likely that
knowledge about one residual would tell you
something about the value of another residual.
Typical cases:
-repeated measures
-clustered observations
(people within firms /
pupils within schools)
Consequences: as for heteroscedasticity
Usually, your confidence intervals are estimated
too small (think about why that is!).
Cure: use multi-level analyses
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In Stata:
Example:
the Stata “auto.dta” data set
sysuse auto
corr
vif
(correlation)
(variance inflation factors)
ovtest (omitted variable test)
hettest (heterogeneity test)
predict e, resid
swilk (test for normality)
Finding the commands
• “help regress”
•  “regress postestimation”
and you will find most of them
(and more) there
Multi-collinearity
A strong correlation between two or more of
your predictor variables
You don’t want it, because:
1. It is more difficult to get higher R’s
2. The importance of predictors can be difficult to
establish (b-hats tend to go to zero)
3. The estimates for b-hats are unstable under slightly
different regression attempts (“bouncing beta’s”)
Detect:
1. Look at correlation matrix of predictor variables
2. calculate VIF-factors while running regression
Cure:
Delete variables so that multi-collinearity
disappears, for instance by combining them
into a single variable
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Stata: calculating the correlation matrix
(“corr”) and VIF statistics (“vif”)
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Misspecification tests
(replaces: all relevant predictor
variables included)
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Homoscedasticity: all residuals
are from a distribution with the
same variance
Consequences: Heteroscedasticiy does not necessarily
lead to biases in your estimated coefficients (b-hat),
but it does lead to biases in the estimate of the width
of the confidence interval, and the estimation
procedure itself is not efficient.
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Testing for heteroscedasticity in Stata
• Your residuals should have the same variance for
all values of Y  hettest
• Your residuals should have the same variance for
all values of X  hettest, rhs
Errors distributed normally
Errors are distributed normally
(just the errors, not the variables themselves!)
Detect: look at the residual plots, test for
normality
Consequences: rule of thumb: if n>600, no
problem. Otherwise confidence intervals are
wrong.
Cure: try to fit a better model, or use more
difficult ways of modeling instead (ask an
expert).
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Errors distributed normally
First calculate the errors:
predict e, resid
Then test for normality
swilk e
Assumption checking
in multi-level multiple regression
with Stata
In multi-level
• Test all that you would test for
multiple regression – poor man’s
test: do this using multiple
regression! (e.g. “hettest”)
Add:
• xttest0 (see last week)
Add (extra):
Test visually whether the
normality assumption holds, but
do this for the random 
Note: extra material
(= not on the exam, bonus points if
you know how to use it)
tab school, gen(sch_)
reg y sch2 – sch28
gen coefs = .
for num 2/28: replace coefs =_b[schX] if _n==X
swilk coefs
Assumption checking
in multi-level multiple regression
with Stata
Assumptions
• Y is 0/1
• Ratio of cases to variables should
be “reasonable”
• No cases where you have
complete separation (Stata will
remove these cases
automatically)
• Linearity in the logit (comparable
to “the true model should be
linear” in multiple regression)
• Independence of errors (as in
multiple regression)
Further things to do:
• Check goodness of fit and
prediction for different groups (as
done in the do-file you have)
• Check the correlation matrix for
strong correlations between
predictors (corr)
• Check for outliers using regress
and diag (but don’t tell anyone I
suggested this)