### slides - Chrissnijders

```Advanced Models and Methods
in Behavioral Research
• Chris Snijders
• [email protected]
ToDo
(if not done yet):
Enroll in 0a611
• 3 ects
• http://www.chrissnijders.com/ammbr
(=studyguide)
• literature: Field book + separate course material
• laptop exam (+ assignments)
Advanced Methods and Models in Behavioral Research –
The methods package
• MMBR (6 ects)
– Blumberg: questions, reliability, validity, research design
– Field: SPSS: factor analysis, multiple regression, ANcOVA,
sample size etc
• AMMBR (3 ects)
- Field (1 chapter): logistic regression
- literature through website:
conjoint analysis  multi-level regression
Advanced Methods and Models in Behavioral Research –
Models and methods: topics
• t-test, Cronbach's alpha, etc
• multiple regression, analysis of (co)variance and
factor analysis
• logistic regression
• conjoint analysis / repeated measures
– Stata next to SPSS
– “Finding new questions”
– Some data collection
In the background:
“now you should be able to deal with data on your own”
Advanced Methods and Models in Behavioral Research –
Methods in brief (1)
• Logistic regression: target Y, predictors Xi.
Y is a binary variable (0/1).
-
Why not just multiple regression?
Interpretation is more difficult
goodness of fit is non-standard
...
(and it is a chapter in Field)
Advanced Methods and Models in Behavioral Research –
Methods in brief (2)
• Conjoint analysis
Underlying assumption: for
each user, the "utility" of an
offer can be written as
-10 Euro p/m
- 2 years fixed
- free phone
- ...
How attractive is this
offer to you?
U(x1,x2, ... , xn) = c0 + c1 x1 + ... + cn xn
Advanced Methods and Models in Behavioral Research –
Conjoint analysis as an “in between method”
Between
Which phone do you like and why?
What would your favorite phone be?
And:
Let’s keep track of what people buy.
We have:
Advanced Methods and Models in Behavioral Research –
Local Master Thesis example:
Fiber to the home
Speed:
Price:
Installation:
really fast
sort of high
free!
are in!
(Roel Schuring)
How attractive is this to you?
Advanced Methods and Models in Behavioral Research –
Coming up with new ideas (3)
“More research is necessary”
But on what?
YOU: come up with sensible new
ideas, given previous research
Advanced Methods and Models in Behavioral Research –
Stata next to SPSS
•
It’s just better
•
Multi-level regression
is much easier than in
SPSS
•
It’s good to be
exposed to more than
just a single statistics
(faster,
better written, more
possibilities, better
programmable …)
should not be based on
“where to click” arguments)
•
More stable
•
BTW Supports OSX as
well… (anybody?)
Advanced Methods and Models in Behavioral Research –
• Output less “polished”
• It takes some extra work
to get you started
• The Logistic Regression
chapter in the Field book
for the larger part)
• (and it’s not campus
software, but subfaculty
software)
• Installation …
Advanced Methods and Models in Behavioral Research –
Logistic Regression Analysis
That is: your Y variable is 0/1:
Now what?
The main points
1.
Why do we have to know and sometimes use logistic
regression?
2.
What is the underlying model? What is maximum
likelihood estimation?
3.
Logistics of logistic regression analysis
1.
2.
3.
4.
4.
Estimate coefficients
Assess model fit
Interpret coefficients
Check residuals
An example (with some output)
Advanced Methods and Models in Behavioral Research
Suppose we have 100 observations with information
about an individuals age and wether or not this indivual
had some kind of a heart disease (CHD)
ID
age
CHD
1
2
3
4
…
98
99
100
20
23
24
25
0
0
0
1
64
65
69
0
1
1
A graphic representation of the data
CHD
Age
Let’s just try regression analysis
pr(CHD|age) = -.54 +.022*Age
... linear regression is not a suitable model for probabilities
pr(CHD|age) = -.54 +.0218107*Age
In this graph for 8 age groups, I plotted the probability of
having a heart disease (proportion)
A nonlinear model is probably better here
Something like this
This is the logistic regression model
Pr( Y | X ) 
1
1 e
 ( b 0  b1 X 1   1 )
Predicted probabilities are always between 0 and 1
Pr( Y | X ) 
1
1 e
 ( b 0  b1 X 1   1 )
similar to classic regression
analysis
Side note: this is similar to MMBR …
Suppose Y is a percentage (so between 0 and 1).
Then consider
…which will ensure that the estimated Y will vary between 0 and 1
and after some rearranging this is the same as
Advanced Methods and Models in Behavioral Research –
… (continued)
And one “solution” might be:
- Change all Y values that are 0 to 0.001
- Change all Y values that are 1 to 0.999
Now run regression on log(Y/(1-Y)) …
… but that really is sort of higgledy-piggledy …
Advanced Methods and Models in Behavioral Research –
Logistics of logistic regression
1.
2.
3.
4.
How do we estimate the coefficients?
How do we assess model fit?
How do we interpret coefficients?
How do we check regression assumptions?
Kinds of estimation in regression
• Ordinary Least Squares (we fit a line through a cloud
of dots)
• Maximum likelihood (we find the parameters that are
the most likely, given our data)
We never bothered to consider maximum likelihood in standard
multiple regression, because you can show that they lead to
exactly the same estimator (in MR, that is, normally they
differ).
Actually, maximum likelihood has superior statistical
properties (efficiency, consistency, invariance, …)
Advanced Methods and Models in Behavioral Research –
Maximum likelihood estimation
• Method of maximum likelihood yields values
for the unknown parameters that maximize
the probability of obtaining the observed set
of data
Pr( Y | X ) 
1
1 e
 ( b 0  b1 X 1   1 )
Unknown parameters
Maximum likelihood estimation
• First we have to construct the “likelihood
function” (probability of obtaining the
observed set of data).
Likelihood = pr(obs1)*pr(obs2)*pr(obs3)…*pr(obsn)
Assuming that observations are independent
Log-likelihood
• For technical reasons the likelihood is
transformed in the log-likelihood (then you
just maximize the sum of the logged
probabilities)
LL= ln[pr(obs1)]+ln[pr(obs2)]+ln[pr(obs3)]…+ln[pr(obsn)]
Some subtleties
• In OLS, we did not need stochastic assumptions to
be able to calculate a best-fitting line (only for the
estimates of the confidence intervals we need that).
With maximum likelihood estimation we need this
from the start
(and let us not be bothered at this point by how
the confidence intervals are calculated in
maximum likelihood)
Advanced Methods and Models in Behavioral Research –
And this is what it looks like …
Advanced Methods and Models in Behavioral Research –
Note: optimizing log-likelihoods is difficult
• It’s iterative (“searching the landscape”)
 it might not converge
 it might converge to the wrong answer
Advanced Methods and Models in Behavioral Research –
Nasty implication:
extreme cases should be left out
(some handwaving here)
Advanced Methods and Models in Behavioral Research –
Example (with some SPSS output)
Advanced Methods and Models in Behavioral Research –
Estimation of coefficients: SPSS Results
Pr( Y | X ) 
1
1 e
 (  5 . 3  . 11 X 1 )
Variables in the Equation
B
Step 1a
age
Constant
S.E.
Wald
df
Sig.
Exp(B)
,111
,024
21,254
1
,000
1,117
-5,309
1,134
21,935
1
,000
,005
a. Variable(s) entered on step 1: age.
Pr( Y | X ) 
1
1 e
 (  5 . 3  . 11 X 1 )
This function fits best: other values of b0 and b1 give worse results
(that is, other values have a smaller likelihood value)
Pr( Y | X ) 
1
1 e
 (  5 . 3  . 11 X 1 )
Illustration 1: suppose we chose .05X instead of .11X
Pr( Y | X ) 
1
1 e
 (  5 . 3  . 05 X 1 )
Illustration 2: suppose we chose .40X instead of .11X
Pr( Y | X ) 
1
1 e
 (  5 . 3  . 40 X 1 )
Logistics of logistic regression
• Estimate the coefficients (and their conf.int.)
• Assess model fit
– Between model comparisons
– Pseudo R2 (similar to multiple regression)
– Predictive accuracy
• Interpret coefficients
• Check regression assumptions
Model fit:
comparisons between models
The log-likelihood ratio test statistic can
be used to test the fit of a model
  2[ LL ( New )  LL ( baseline )]
2
The test statistic has a
chi-square distribution
full model
reduced model
NOTE This is sort of similar to the variance decomposition
tables you see in MR!
42
Advanced Methods and Models in Behavioral Research
Between model comparisons:
the likelihood ratio test
  2[ LL ( New )  LL ( baseline )]
2
full model
P (Y ) 
1
1 e
 ( b 0  b1 X 1 )
reduced model
P (Y ) 
1
1 e
 ( b0 )
The model including only an intercept
Is often called the empty model. SPSS uses this
model as a default.
Between model comparison: SPSS output
  2 LL ( New )  2 LL ( baseline )]
2
Omnibus Tests of Model Coefficients
Chi-square
Step 1
df
Sig.
Step
29,310
1
,000
Block
29,310
1
,000
Model
29,310
1
,000
Model Summary
Step
1
-2 Log likelihood
107,353a
Cox & Snell R
Nagelkerke R
Square
Square
,254
,341
a. Estimation terminated at iteration number 5 because
parameter estimates changed by less than ,001.

This is the test statistic,
and it’s associated
significance
Overall model fit
pseudo R2
log-likelihood of the model
that you want to test
R
2
LOGIT

 2 LL ( Model )
 2 LL ( Empty )
Just like in multiple
regression, pseudo
R2 ranges 0.0 to 1.0
– Cox and Snell
• cannot theoretically
reach 1
– Nagelkerke
log-likelihood of model
before any predictors were
entered
can reach 1
NOTE: R2 in logistic regression tends to be (even) smaller than in multiple regression
46
Overall model fit: Classification table
Classification Table
a
Predicted
chd
Percentage
Observed
Step 1
chd
0
1
Correct
0
45
12
78,9
1
14
29
67,4
Overall Percentage
74,0
a. The cut value is ,500
We predict 74% correctly
47
Overall model fit: Classification table
Classification Table
a
Predicted
chd
Percentage
Observed
Step 1
chd
0
1
Correct
0
45
12
78,9
1
14
29
67,4
Overall Percentage
74,0
a. The cut value is ,500
14 cases had a CHD while according to our model
this shouldnt have happened
48
Overall model fit: Classification table
Classification Table
a
Predicted
chd
Percentage
Observed
Step 1
chd
0
1
Correct
0
45
12
78,9
1
14
29
67,4
Overall Percentage
74,0
a. The cut value is ,500
12 cases didn’t have a CHD while according to our model
this should have happened
49
Logistics of logistic regression
• Estimate the coefficients
• Assess model fit
• Interpret coefficients
– Direction
– Significance
– Magnitude
• Check regression assumptions
The Odds Ratio
p (Y ) 
1
1 e
 ( b 0  b1 X 11  ...  b n X n )

e
( b 0  b1 X 11  ...  b n X n )
1 e
( b 0  b1 X 11  ...  b n X n )
And after some rearranging we can get
51
Magnitude of association: Percentage change in odds
Odds
i
 prob event
 
 1  prob event




Probability
Odds
25%
0.33
50%
1
75%
3
Interpreting coefficients: direction
• original b reflects changes in logit: b>0 implies positive relationship
logit  ln
p( y)
1  p( y)
 b0  b1 x1  b 2 x 2  ...  b n x n
• exponentiated b reflects the “changes in odds”: exp(b) > 1 implies a
positive relationship
53
3. Interpreting coefficients: magnitude
• The slope coefficient (b) is interpreted as the rate of change in
the "log odds" as X changes … not very useful.
logit  ln
p( y)
1  p( y)
 b0  b1 x1  b 2 x 2  ...  b n x n
• exp(b) is the effect of the independent variable on the odds,
more useful for calculating the size of an effect
Odds 
54
p( y)
1  p( y)
e
b0
e
b1 x1
e
b2 x 2
 ...  e
bn x n
Another way to get an idea of the size of effects:
Calculating predicted probabilities
Pr( Y | X ) 
1
1 e
 (  5 . 3  . 11 X 1 )
For somebody of 20 years old, the predicted probability is .04
For somebody of 70 years old, the predicted probability is .91
But this gets more complicated
when you have more than a single X-variable
Pr(Y | X) =
1
1+ e
-(-5.3+.11X1+1*X2 )
(see blackboard)
Conclusion: if you consider the effect of a variable on
the predicted probability, the size of the effect of X1
depends on the value of X2! (yuck!)
Advanced Methods and Models in Behavioral Research –
Testing significance of coefficients
•
In linear regression
analysis this statistic is
used to test
significance
b
•
In logistic regression
something similar
exists
SE b
•
however, when b is
large, standard error
tends to become
inflated, hence
underestimation (Type
II errors are more
likely)
estimate
Wald 
t-distribution
standard error of estimate
Note: This is not the Wald Statistic SPSS presents!!!
Interpreting coefficients: significance
• SPSS presents
Wald 
b
2
SE
2
b
• While Andy Field thinks SPSS presents this (at least in the 2nd
version of the book):
Wald 
b
SE b
Advanced Methods and Models in Behavioral Research –
Logistics of logistic regression
•
•
•
•
Estimate the coefficients
Assess model fit
Interpret coefficients
Check regression assumptions
Checking assumptions
0. Independent data points
Problem: likelihood function is wrong otherwise + confidence intervals too small
1. Influential data points & Residuals
– Follow Samanthas tips in Field; we will get back to this later
2. No multi-collinearity (Stata: “collin”)
3. All relevant variables included
4. Hosmer & Lemeshow
(Stata: “estat gof”)
– Divides sample in subgroups
– Checks whether there are differences between observed and predicted between
subgroups
– Test should not be significant, if so: indication of lack of fit
1. Residual statistics: Field’s rules of thumb
1. Examining residuals in logistic regression
Isolate points for which the model fits poorly
Isolate influential data points
2. No multi-collinearity
• Problem = same as in regression, the net effect of
two (or more) collinear variables will be zero (see
MMBR)
• In regression: Stata-command is “vif”:
reg y x
vif
// Stata’s regression command
// the variance-inflation-factors
• In logistic regression: Stata-command is “collin”
logit y x
collin
// Stata’s logit regr. Command
// the variance-inflation-factors
Advanced Methods and Models in Behavioral Research –
NOTE: “collin” is not standard Stata
help ...
(if you know and have the command)
net search …
(otherwise)
findit …
(otherwise)
Advanced Methods and Models in Behavioral Research –
3. All relevant variables included:
Model specification
• Note that this refers to the inclusion of given
variables (not the inclusion of totally other variables)
(compare Stata’s “ovtest” in multiple regression)
Many specification tests consider whether including yhat and (y-hat)^2 would improve your model. If yes 
Advanced Methods and Models in Behavioral Research –
4. Hosmer & Lemeshow
Test divides sample in subgroups, checks whether
difference between observed and predicted is about
equal in these groups
Test should not be significant (indicating no difference)
Advanced Methods and Models in Behavioral Research –
Logistic regression
•
•
•
•
•
•
Y = 0/1
Multiple regression (or ANcOVA) is not right
You consider either the odds or the log(odds)
It is estimated through “maximum likelihood”
Interpretation is a bit more complicated than normal
Assumption testing is a bit more concrete than in
multiple regression (also because now we can do
this with Stata)
Advanced Methods and Models in Behavioral Research –
8 groups – run a logistic regression in Stata
• Create groups, choose a data set
• Create a do-file that reads in the data, and runs a
logistic regression (along the lines of the commands
in the example file, BUT WITH MORE COMMENTS