Transcript Some thoughts on statistical modelling in Stata

‘Interpreting results from
statistical modelling – a
seminar for social scientists’
Dr Vernon Gayle and Dr Paul Lambert
(Stirling University)
Tuesday 29th April 2008
‘Interpreting results from statistical
modelling – a seminar for social
scientists’
• Our experience has shown that the results of statistical
models can easily be misrepresented
• In this seminar we demonstrate that the correct
interpretation of results from statistical models often
requires more detailed knowledge than is commonly
appreciated
• We illustrate some approaches to best practice in this
area
• This seminar is primarily aimed at quantitative social
researchers working with micro-social survey data
Principles of model construction
and interpretation
yi=bo+ b1 X1+….+bkXk+ui
Today we are interested in b
– “What does b tell us?”
– “Where’s the action?”
– Going beyond “significance and sign”
Statistical Models
The idea of generalized linear models (glm)
brings together of wealth of disparate topics –
thinking of these models under a general
umbrella term aids interpretation
Now I would say that generalized linear and
mixed models (glmm) are the natural extension
Statistical Modelling Process
Model formulation
[make assumptions]
Model fitting
[quantify systematic
relationships & random
variation]
(Model criticism)
[review assumptions]
Model interpretation
[assess results]
Davies and Dale, 1994 p.5
Building Models
REMEMBER – Real data is much more messy,
badly behaved (in real life people do odd stuff),
models are harder to interpret etc. than the data
used in books and at workshops
Building Models
• Many of you are experienced data analysts
(otherwise see our handout)
• Always be guided by substantive theory
(the economists are good at this – but a bit rigid)
• Consider the “functional form” of the variables
(especially the outcome)
• Start with “main effects” – more complicated
models later
How Long are Three Pieces of String?
Some Common Models
•
•
•
•
•
•
Continuous Y
Binary Y
Categorical Y
Ordered Cat. Y
Count Y
Repeated Binary Y
Linear Regression
Logit; Probit
Multinomial Logit
Cont. Ratio; Cum Logit
Poisson
Panel Probit (Logit)
I must not use Stepwise Regression
I must not use Stepwise Regression
I must not use Stepwise Regression
I must not use Stepwise Regression
I must not use Stepwise Regression
I must not use Stepwise Regression
I must not use Stepwise Regression
I must not use Stepwise Regression
I must not use Stepwise Regression
I must not use Stepwise Regression
I must not use Stepwise Regression
I must not use Stepwise Regression
I must not use Stepwise Regression
I must not use Stepwise Regression
I must not use Stepwise Regression
I must not use Stepwise Regression
I must not use Stepwise Regression
I must not use Stepwise Regression
I must not use Stepwise Regression
I must not use Stepwise Regression
I must not use Stepwise Regression
I must not use Stepwise Regression
A very very simple example
• A fictitious data set based on a short steep Scottish
hill race (record time 31 minutes; 5 miles and 1,200
feet of ascent)
•
•
•
•
•
A group of 73 male runners
Times 32 mins - 60 minutes mean 42.7; s.d. 8.32
Heights 60 - 70 inches (5 ft to 6 ft)
Weights 140 lbs - 161 lbs (10 st to 11 st 7 lb)
Everyone finishes (i.e. no censored cases)
A (vanilla) Regression
Simple Stata Output…
regress time height weight
--------------------------------------------------------------------------------------------------------time |
Coef.
Std. Err.
t
P>|t| [95% Conf. Interval]
-------------+-----------------------------------------------------------------------------------------height | 1.010251 .0813485 12.42 0.000 .8480067 1.172495
weight | .7369447 .0370876 19.87 0.000 .6629759 .8109135
_cons | -131.5619 6.834839 -19.25 0.000 -145.1936 -117.9303
---------------------------------------------------------------------------------------------------------
A (vanilla) Regression
------------------------------------------------------------------------------------------------time |
Coef.
Std. Err.
t P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------------------------height | 1.010251 .0813485 12.42 0.000 .8480067 1.172495
weight | .7369447 .0370876 19.87 0.000 .6629759 .8109135
_cons | -131.5619 6.834839 -19.25 0.000 -145.1936 -117.9303
--------------------------------------------------------------------------------------------------
On average (ceteris paribus) a one unit change in weight
(lbs) leads to an increase of .74 minutes on the runner’s
time
A (vanilla) Regression
-----------------------------------------------------------------------------time |
Coef.
Std. Err.
t P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------height | 1.010251 .0813485 12.42 0.000 .8480067 1.172495
weight | .7369447 .0370876 19.87 0.000 .6629759 .8109135
_cons | -131.5619 6.834839 -19.25 0.000 -145.1936 -117.9303
------------------------------------------------------------------------------
On average (ceteris paribus) a one unit change in height
(inches) leads to an increase of 1 minute on the runner’s
time (remember this is a fell race being too tall does not
necessarily help you)
A (vanilla) Regression
-----------------------------------------------------------------------------time |
Coef.
Std. Err.
t P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------height | 1.010251 .0813485 12.42 0.000 .8480067 1.172495
weight | .7369447 .0370876 19.87 0.000 .6629759 .8109135
_cons | -131.5619 6.834839 -19.25 0.000 -145.1936 -117.9303
------------------------------------------------------------------------------
This is the intercept b0 –
In this model it is the time (on average) that a person who
was 0 inches and 0 pounds would take?
A (vanilla) Regression
-----------------------------------------------------------------------------time |
Coef.
Std. Err.
t
P>|t|
-------------+---------------------------------------------------------------height0 | 1.010251 .0813485 12.42 0.000
weight0 | .7369447 .0370876 19.87 0.000
_cons | 32.22542 .5126303
62.86 0.000
A better parameterized model - height
centred at 60 inches; weight centred at 140 lb
This is the intercept b0 –
In this model it is the time (on average) that a runner who is
60 inches and 140 pounds would take
A (vanilla) Regression
regress time height0 weight0, beta
time |
Coef.
Std. Err.
t
P>|t|
Beta
-------------+---------------------------------------------------------------height0 | 1.010251 .0813485 12.42 0.000
.4659946
weight0 | .7369447 .0370876 19.87 0.000
.7456028
_cons
| 32.22542 .5126303
62.86 0.000
Standardized beta coefficients are reported instead of confidence intervals
The beta coefficients are the regression coefficients obtained by first
standardizing all variables to have a mean of 0 and a standard deviation of 1
Beta coefficients can be useful when comparing the effects of variables
measured on different scales (i.e. in different units such as inches and pounds)
X Variable Measurement – e.g. Age
• Linear units (e.g. months)
• Resolution of measurement (are years better?)
• Is a squared term appropriate (e.g. age may not
be linear in employment models)
• A banded variable (age bands – allow the
direction of the effect to change; between 20-29
women’s employment behaviour might be
different to 30-39)
Binary Outcomes
• Logit model is popular in sociology, social
geography, social policy, education etc
• Probit model is more widely used in
economics
Example
Drew, D., Gray, J. and Sime, N. (1992)
Against the odds: The Education and Labour
Market Experiences of Black Young People
The Deviance is
sometimes called G2
-2 * Log Likelihood
It has a chi-squared
distribution with
associated degrees of
freedom
The degrees of freedom
for the explanatory
variable
The estimate.
Also known as the
‘coefficient’
‘log odds’
‘parameter estimate’
Beta (b)
Measured on the log scale
This the standard error of the
estimate
Measured on the log scale
This is the odds ratio.
It is the exponential (i.e.
the anti-log) of the
estimate.
Comparison of Odds
Greater than 1 “higher odds”
Less than 1 “lower odds”
Naïve Odds
• In this model (after controlling for other
factors)
White pupils have an odds of 1.0
Afro Caribbean pupils have an odds of 3.2
• Reporting this in isolation is a naïve
presentation of the effect because it ignores
other factors in the model
A Comparison
Pupil with
Pupil with
4+ higher passes
White
Professional parents
Male
Graduate parents
Two parent family
0 higher passes
Afro-Caribbean
Manual parents
Male
Non-Graduate parents
One parent family
Odds are multiplicative
4+ Higher Grades
Ethnic Origin
Social Class
Gender
Parental Education
No. of Parents
1.0
1.0
1.0
1.0
1.0
1.0
1.0
3.2
0.5
1.0
0.6
0.9
Odds
1.0
0.86
Naïve Odds
• Drew, D., Gray, J. and Sime, N. (1992) warn of this
danger….
• …Naïvely presenting isolated odds ratios is still
widespread (e.g. Connolly 2006 Brit. Ed. Res.
Journal 32(1),pp.3-21)
• We should avoid reporting isolated odds ratios
where possible!
Logit scale
• Generally, people find it hard to directly interpret results
on the logit scale – i.e. b
Log Odds, Odds, Probability
• Log odds converted to odds = exp(log odds)
• Probability = odds/(1+odds)
• Odds = probability / (1-probability)
Log Odds, Odds, Probability
Odds
ln odds
p
99.00
4.60
0.99
19.00
2.94
0.95
9.00
2.20
0.9
4.00
1.39
0.8
2.33
0.85
0.7
1.50
0.41
0.6
1.00
0.00
0.5
0.67
-0.41
0.4
0.43
-0.85
0.3
0.25
-1.39
0.2
0.11
-2.20
0.1
0.05
-2.94
0.05
0.01
-4.60
0.01
Odds are asymmetric – beware!
A Simple Stata Example
Youth Cohort Study (1990)
n= c.14,000 16-17 year olds
y=1; pupil has 5+ GCSE passes (grade A*-C)
X vars;
– Gender; Parents in Service Class (NS-SEC)
Stata output logit
Logistic regression
Log likelihood = -9260.22
Number of obs =
14022
LR chi2(2)
= 807.67
Prob > chi2 = 0.0000
Pseudo R2
= 0.0418
-----------------------------------------------------------------------------------------------------t0fiveac |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
------------- +---------------------------------------------------------------------------------------boys | -.1495507 .0349946 -4.27 0.000 -.218139 -.0809625
service class | 1.398813 .0526951 26.55 0.000 1.295532 1.502093
_cons | -.309116 .0247608 -12.48 0.000 -.3576462 -.2605857
Stata output logit
Logistic regression
Log likelihood = -9260.22
Number of obs =
14022
LR chi2(2)
= 807.67
Prob > chi2 = 0.0000
Pseudo R2
= 0.0418
-----------------------------------------------------------------------------------------------------t0fiveac |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
------------- +---------------------------------------------------------------------------------------boys | -.1495507 .0349946 -4.27 0.000 -.218139 -.0809625
service class | 1.398813 .0526951 26.55 0.000 1.295532 1.502093
_cons | -.309116 .0247608 -12.48 0.000 -.3576462 -.2605857
Estimates are log odds - Sign = direction; Size = strength
Stata output logit
Logistic regression
Log likelihood = -9260.22
Number of obs =
14022
LR chi2(2)
= 807.67
Prob > chi2 = 0.0000
Pseudo R2
= 0.0418
-----------------------------------------------------------------------------------------------------t0fiveac |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
------------- +---------------------------------------------------------------------------------------boys | -.1495507 .0349946 -4.27 0.000 -.218139 -.0809625
service class | 1.398813 .0526951 26.55 0.000 1.295532 1.502093
_cons | -.309116 .0247608 -12.48 0.000 -.3576462 -.2605857
Standard errors also measured on the logit scale
Small standard errors indicate better precision of the coefficient (estimate;
beta)
Stata output logit
Logistic regression
Log likelihood = -9260.22
Number of obs =
14022
LR chi2(2)
= 807.67
Prob > chi2 = 0.0000
Pseudo R2
= 0.0418
-----------------------------------------------------------------------------------------------------t0fiveac |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
------------- +---------------------------------------------------------------------------------------boys | -.1495507 .0349946 -4.27 0.000 -.218139 -.0809625
service class | 1.398813 .0526951 26.55 0.000 1.295532 1.502093
_cons | -.309116 .0247608 -12.48 0.000 -.3576462 -.2605857
 = b / s.e. beta; Wald c2 = (b / s.e. beta)2 @ 1 d.f.
A very crude test of significance is if b is twice its standard error
Stata output logit
Logistic regression
Log likelihood = -9260.22
Number of obs =
14022
LR chi2(2)
= 807.67
Prob > chi2 = 0.0000
Pseudo R2
= 0.0418
-----------------------------------------------------------------------------------------------------t0fiveac |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
------------- +---------------------------------------------------------------------------------------boys | -.1495507 .0349946 -4.27 0.000 -.218139 -.0809625
service class | 1.398813 .0526951 26.55 0.000 1.295532 1.502093
_cons | -.309116 .0247608 -12.48 0.000 -.3576462 -.2605857
Formal significance test (p values)
Stata output logit
Logistic regression
Log likelihood = -9260.22
Number of obs =
14022
LR chi2(2)
= 807.67
Prob > chi2 = 0.0000
Pseudo R2
= 0.0418
-----------------------------------------------------------------------------------------------------t0fiveac |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
------------- +---------------------------------------------------------------------------------------boys | -.1495507 .0349946 -4.27 0.000 -.218139 -.0809625
service class | 1.398813 .0526951 26.55 0.000 1.295532 1.502093
_cons | -.309116 .0247608 -12.48 0.000 -.3576462 -.2605857
Confidence interval of b (on the logit scale)
b ± (1.96 * standard error) e.g. -.15 – (1.96 * .035)
Remember if the confidence interval does not include zero b is significant
A Thought on Goodness of Fit
• Standard linear models: R2 is an easy, consistent
measure of goodness of fit
• Nested models: change in deviance (G2) follows a chisquare distribution (with associated d.f)
• Non-nested non-linear models: change in deviance
cannot be compared AND there is no direct equivalent
of R2 (e.g. logit model from two different surveys)
• Various ‘pseudo’ R2 measures – none take on full 0 –
1 range and technically should not be used to
compare non-nested models (but may be adequate in
many practical situations)
A Thought on Goodness of Fit
• Handy file in Stata spost9_ado produces a
number of ‘pseudo’ R2 measures
• Discussion at
http://www.ats.ucla.edu/stat/mult_pkg/faq/general/Psuedo_RSquareds.htm
• Some analysts use Bayesian Information
Criteria (BIC) type measures – these evaluate
(and favour) parsimony – possibly a good idea
for comparing across models
Probit / Logit
• Convert probit / logit
(Probit b * 1.6) or (Logit b /1.6)
(Amemiya 1981)
Logit or probit – some say
logit for a purely discrete Y (e.g. pregnancy)
probit appeals to an underlying continuous distribution
Some people make silly claims (e.g. the case of
unemployment in Germany)
Logit / Probit
Logit b
Logit
s.e.
Logit Z
Probit b
Probit
s.e.
Probit Z Conversion
boys
-0.15
0.03
-4.27
-0.09
0.02
-4.27
-0.15
service class
1.40
0.05
26.55
0.86
0.03
27.42
1.38
_cons
-0.31
0.02
-12.48
-0.19
0.02
-12.58
-0.31
Generally, substantive inference is the same
and models will have similar log likelihoods,
Pseudo R2 etc.
Probit
• b expressed on the standard cumulative
normal scale (F)
• Unlike logit a calculator might not have the
appropriate function
• Use software or Excel [=NORMSDIST() ]
Probit
•
Probability 5+GCSE (A*- C) passes
girl; non-service class family
F(-.19) = .42
boy; non-service class family
F (-.19 -.09) = .39
Gender effect .03
Probit
•
Stata has dprobit
Here the coefficient is dF/dx
i.e. the effect of discrete change of a dummy
variable from 0 to 1
•
•
Continuous X vars interpreted at their mean
Analysts often demonstrate specific values /
combinations
Categorical Data
(Multinomial Logit)
Categorical Y
Example YCS 1990
What the pupil was doing in Oct after Yr 11
0 Education
1 Unemployment
2 Training
3 Employment
Multinomial Logit
Multinomial logit model = pairs of logits
1 Education / 0 Unemployment
1 Education / 0 Training
1 Education / 0 Employment
Base category of y is y=1 for these pairs of
models
Betas are readily interpreted as in logit
Multinomial Logit
Multinomial logistic regression
Log likelihood = -12653.444
Number of obs
LR chi2(3)
Prob > chi2
Pseudo R2
=
=
=
=
13925
80.25
0.0000
0.0032
-----------------------------------------------------------------------------t1dooct4 |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------1. unemplo~t |
girls | -.0840972
.0977346
-0.86
0.390
-.2756536
.1074591
_cons | -3.041328
.0708045
-42.95
0.000
-3.180102
-2.902553
-------------+---------------------------------------------------------------2. training |
girls |
-.245671
.0526523
-4.67
0.000
-.3488675
-.1424744
_cons | -1.604877
.0369626
-43.42
0.000
-1.677322
-1.532431
-------------+---------------------------------------------------------------3. employm~t |
girls | -.3961514
.0477778
-8.29
0.000
-.4897941
-.3025087
_cons | -1.291088
.0325547
-39.66
0.000
-1.354894
-1.227282
-----------------------------------------------------------------------------(t1dooct4==0. education is the base outcome)
Multinomial logistic regression
Log likelihood = -12653.444
Number of obs
LR chi2(3)
Prob > chi2
Pseudo R2
=
=
=
=
13925
80.25
0.0000
0.0032
-----------------------------------------------------------------------------t1dooct4 |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------1. unemplo~t |
girls | -.0840972
.0977346
-0.86
0.390
-.2756536
.1074591
_cons | -3.041328
.0708045
-42.95
0.000
-3.180102
-2.902553
-------------+---------------------------------------------------------------(t1dooct4==0. education is the base outcome)
Logit Unemployment / Education
Logistic regression
Log likelihood = -1803.3779
Number of obs
LR chi2(1)
Prob > chi2
Pseudo R2
=
=
=
=
10051
0.74
0.3899
0.0002
-----------------------------------------------------------------------------t1dooct4 |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------girls | -.0840972
.0977345
-0.86
0.390
-.2756533
.1074589
_cons | -3.041328
.0708044
-42.95
0.000
-3.180102
-2.902554
Multinomial Logit
Multinomial logit model is NOT an ordinal model
1 Education / 0 Unemployment
1 Education / 0 Training
1 Education / 0 Employment
Says nothing about
Unemployment / Training, Unemployment /
Employment or Training / Employment
Data with Ordinal Outcomes
• A large amount of data analysed within
sociological studies consists of categorical
outcome variables that can plausibly be
considered as having a substantively interesting
order (for example levels of attainment of
educational qualifications)
• Standard log-linear models do not take
ordinality into account
Data with Ordinal Outcomes
• Two different models
Continuation Ratio Model
Proportional Odds Model
Both models have ‘logit’ style b interpretations
Reversing Category Codes –
Proportional Odds Model
Categories
Categories
Cut pt A
0
1
2
3
Cut pt A
3
2 1
0
Cut pt B
0
1
2
3
Cut pt B
3
2 1
0
Cut pt C
0
1
2
3
Cut pt C
3
2 1
0
Results reversed (signs) substantive meaning not
changed – This can work well with attitude scales!
Reversing Category Codes –
Continuation Ratio Model
Categories
Cut pt A
Cut pt B
Cut pt C
0
Categories
1
2
3
Cut pt A
1
2
3
2
3
3
2 1
0
Cut pt B
2 1
0
Cut pt C
1
0
Results and substantive meaning are changed – Not
Palindromically Invariant
• The b that refer to the cut points (or
partitions) in these two ordinal models
have slightly different interpretations
Some thoughts on these ordinal models
• Proportional Odds
– Palindromic invariance (e.g. attitudinal scores)
– Motivated by an appeal to the existence of an underlying
continuous and perhaps unobservable random variable –
proportional odds
• Continuation ratio model
–
–
–
–
Natural base line (hierarchy in data)
Single direction of movement
Categories of Y really are discrete
Y categories denotes a shift or change from one state to
another) not a coarse groupings of some finer scale
Poisson Regression
Poisson regression is used to fit models to the number
of occurrences (counts) of an event
– Especially relevant if outcome has few values, or is a rate
– Although, in some circumstances counts can reasonably be
modelled as continuous outcomes – e.g. a wide range of
different counts, and lack of clustering around 0
Examples of the poisson distribution
• Soldiers kicked to death by horses (Bortkewitsch 1898)
• Patterns of buzz bomb launch against London WWII (Clarke 1946)
• Telephone wrong numbers (Thorndike 1926)
Poisson Regression
Example Coronary heart disease among male
British doctors (Doll & Hill 1966)
y (count) deaths / person years
X variables age bands; smokers
b has logit style interpretation – when exp(b) is
reported this is often termed an incident rate ratio
poisson deaths smokes agecat2-agecat5, exposure(pyears) irr
Some More Complex Models
Panel Analysis
Fixed or Random effects estimators
• Fierce debate
– F.E. b will be consistent
– R.E. standard errors will be efficient but b may not
be consistent
– F.E. models can’t estimate time constant X vars
– R.E. assumes no correlation between observed X
variables and unobserved characteristics
Panel Analysis
Fixed or Random effects
– Economists tend towards F.E.
(attractive property of consistent b )
– With continuous Y – little problem, fit both F.E. and
R.E. models and then Hausman test b f.e. / b r.e.
(don’t be surprised if it points towards F.E. model)
– Some estimators (xtprobit) don’t have F.E.
equivalents (xtlogit F.E. is not equivalent to R.E.)
An example
Married women’s employment (SCELI Data)
y is the woman working yes=1; no=0
x woman has child aged under 1 year
I have contrived this illustration….
Probit
b
Child
under 1
Probit
s.e.
b
s.e.
-1.95
0.56
-1.95
0.40
Constant
0.67
0.14
0.67
0.10
Log
likelihood
-54.70
-109.39
n
101.00
202.00
0.13
0.13
-
-
Pseudo
R2
Clusters
Consistent b - smaller standard errors (double the sample size) but Stata
thinks that there are 202 individuals and not 101 people surveyed in two
waves!
Probit
b
Child
under 1
Probit
s.e.
b
Probit
s.e.
b
Robust
-1.95
0.56
-1.95
0.40
-1.95
0.56
Constant
0.67
0.14
0.67
0.10
0.67
0.14
Log
likelihood
-54.70
-109.39
-109.39
n
101.00
202.00
202.00
0.13
0.13
0.13
-
-
Pseudo
R2
Clusters
101.00
Consistent b - standard errors are now corrected – Stata
knows that there are 101 individuals (i.e. repeated
measures)
Probit
b
Child
under 1
Probit
s.e.
b
Probit
s.e.
b
R.E. Probit
Robust
b
s.e.
-1.95
0.56
-1.95
0.40
-1.95
0.56
-19.41
1.22
Constant
0.67
0.14
0.67
0.10
0.67
0.14
6.39
0.28
Log
likelihood
-54.70
-109.39
-109.39
-49.57
n
101.00
202.00
202.00
202.00
0.13
0.13
0.13
-
-
Pseudo
R2
Clusters
101.00
101.00
Beware b and standard errors are no longer measured on the same scale
Stata knows that there are 101 individuals (i.e. repeated measures)
b in Binary Panel Models
The b in a probit random effects model is scaled
differently– Mark Stewart suggests
b r.e. * (1-rho) compared with b pooled probit
rho (is analogous to an icc) – proportion of the
total variance contributed by the person level
variance
Panel logit models also have this issue!
b in Binary Panel Models
• Conceptually two types of b in a binary random
effects model
• X is time changing - b is the ‘effect’ for a
woman of changing her value of X
• X is fixed in time - b is analogous to the effect
for two women (e.g. Chinese / Indian) with the
same value of the random effect (e.g. ui=0) –
For fixed in time X Fiona Steele suggests simulating
to get more appropriate value of b
Population Ave Model / Marginal Models
• Time constant X variables are usually analytically important
• Is a model that accounts for clustering between individuals
adequate? Simple pop. average model: logit y x1, cluster(id)
• Population average models are becoming more popular
(Pickles – preference in USA in public health)
• Marginal Modelling GEE approaches are developing rapidly
(e.g. estimating a policy or ‘social group’ difference)
• When do we need ‘subject’ specific random/fixed effects?
When ‘frailty’ or unobserved heterogeneity are important
Conclusions
• The results of statistical models can easily be
misrepresented
• The correct interpretation of results from statistical models
often requires more detailed knowledge than is commonly
appreciated
• Social science analysts should pay more attention to
developing the appropriate model context
– Knowing about a wider range of glm’s / glmm’s is important
– Thinking about the exact interpretation of b will help