SOC6078 SOC6078

Advanced Statistics:
4. Models for Categorical
Dependent Variables II
Robert Andersen
Department of Sociology
University of Toronto
Thursday, 24 May 2012
Extending the Logit and Probit Models
We can extend the logit model to dependent
variables with more than two categories
Specific models correspond to specific types of
relations between the categories of the variable
Some examples:
proportional- odds models fit to ordered data p p
binomial logit models fit to grouped data
multinomial logit and multinomial probit
models fit to a variable with several categories models fit to a variable with several categories
that have no order among them
All of these models are part of the more general
family of binomial models that fit under the larger family of binomial models that fit under the larger
umbrella of generalized linear models
2
Multinomial Logit Models
The multinomial logit model is a direct extension of the
binary logit model to a dependent variable with several
unordered categories (i.e., the model does not assume
any order) any order)
Note: Although the model assumes no order in the
categories, it is still sometimes sensible to fit a
multinomial logit model to an ordered variable, especially multinomial logit model to an ordered variable, especially
if the data fail to meet the assumptions for ordered
models, in particular the proportional odds assumption
A multinomial logit model simultaneously fits m-1 logit
models, contrasting each m-1 category of the dependent
variable with a reference category, m
When fit to a dichotomous variable, the multinomial logit
model is identical to the binary logit model model is identical to the binary logit model
Following directly from the binary logit case, the model is fit
using maximum likelihood
3
Multinomial Logit Models
The model is given by the equations:
As this equation shows, there is one set of parameters,

0j
,
1j
, ...
kj
for each category of the dependent variable
accept the reference category p g y
Although technically we could simply fit a series of
separate binary logit models to find the coefficients, these
models would not give us a single overall measure of the
4
g g
deviance
The regression coefficients from multinomial logit models The regression coefficients from multinomial logit models
reflect the log odds of membership in category j
versus the reference category, m
Although the coefficients refer to a baseline, it is possible
to calculate the log odds of being in any pair of categories
j and j
0
j j
5
Multinomial Logit Models in R
The gl mfunction in R does not handle multinomial logit The gl mfunction in R does not handle multinomial logit
models. Instead, the mul t i nomfunction in the nnet library
must be used
The following example is from the Womens Labour-Force The following example is from the Women s Labour-Force
data in Fox (1997)
The goal is to predict propensity to work outside the
home (full-time or part-time) versus not working home (full-time or part-time) versus not working
according to husbands income and whether children
are at home
6
Multinomial Logit Models in R (2)
The odds of
working full-
time versus time versus
not at all for
women with
children at
home are .077
times the odds
of working for
h h those without
children
7
Multinomial Logit Models in R (3)
Effect Displays p y
.
0
Chi l dren Absent
.
0
Chi l dren Present
0
.
8
1
.
not working
part-time
full-time
0
.
8
1
.
4
0
.
6
e
d

P
r
o
b
a
b
i
l
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t
y
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0
.
6
e
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P
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a
b
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t
y
0
.
2
0
.
4
F
i
t
t
e
0
.
2
0
.
4
F
i
t
t
e
0 10 20 30 40
0
.
0
0 10 20 30 40
0
.
0
8
Husband's Income Husband's Income
R-script for Multinomial Logit Effect Display
9
Multinomial Logit Models
Choice of Reference Category g y
In terms of model fit, choice of the reference category
is arbitrary
Although the coefficients will differ the model fit is Although the coefficients will differ, the model fit is
the same
As shown earlier, it is also possible to determine odds
ratios for the contrasts between any two categories of y g
the dependent variable (though this requires some
calculations or refitting the model with a different
reference category)
Still th t thi t id Still, there are two things to consider:
1. To which category do you want to compare? It is
sensible to make this category the reference
category category.
2. Less change of a convergence problem if the largest
category is the reference.
10
Independence of Irrelevant
Alternatives (IIA) Assumption (1) ( ) p ( )
Relevant with respect to choice models
Assumes that the odds of choosing one option ( say A)
over another ( B) are the same regardless of the over another ( B) are the same regardless of the
existence of other options
In short, the errors associated with predicting each
category are uncorrelated g y
Ex 1: The odds of voting Conservative versus Liberal
should be the same regardless of other parties
i.e., the odds should not change if another party enters
the race
If strategic voting, IIA assumption is not met
Ex 2: If given the choice between Pepsi and Milk but
thi l th h i diff t h th t th nothing else, the choices are different enough that they are
likely independent
If choices are Coke, Pepsi and Milk, the IIA assumption
will be violatedCoke and Pepsi are similar enough that
11
will be violatedCoke and Pepsi are similar enough that
the odds ratio for Coke/Milk will change drastically if
Pepsi is also given as a choice
Independence of Irrelevant
Alternatives (IIA) Assumption (2) ( ) p ( )
The multinomial probit model overcomes the IIA
assumption, by allowing the response errors to correlate
As said earlier however the probit model is more As said earlier, however, the probit model is more
complicated and thus can have problems converging.
As a result, most statistical software programs do not
yet have built-in functions for multinomial probit y p
models.
It can, however, now be done in R using the MNP
package. See:
Imai, Kosuke and David van Dyk (2005). MNP: R
Package for Fitting the Multinomial Probit Model,
Journal of Statistical Software, 14(3):1-32.
In any event the IIA assumption is really not of concern In any event, the IIA assumption is really not of concern
as long as all possible options have been included in the
model
In such cases, the multinomial logit model will suffice
12
In such cases, the multinomial logit model will suffice
Ordered Logit Models
Often we have ordinal measures for which we cannot
necessarily assume that the categories are equally
spacedthis means, of course, that OLS is spaced this means, of course, that OLS is
inappropriate
Likert questionnaire items for opinions, social class
(depending on the conceptualization), level of
d i ( h h ) ll l education (rather than years) etc. are all examples
of dependent variables for which OLS may not be
appropriate
If we can we would also like to preserve the ordered If we can, we would also like to preserve the ordered
character of the data, so a multinomial logit model is
not our first choice
Ordered logit models or ordered probit models g p
can often (but not always) provide a better alternative
The simplest and most commonly used of these
models is the proportional- odds model
13
Ordered Logit Models:
Proportional-odds Model p
Assume a latent continuous variable (e.g., attitudes
toward abortion) is a linear function of the Xs plus a
random error
A survey is unable to tap the concept in great detail, so we A survey is unable to tap the concept in great detail, so we
rely on a 5-point scale. In other words, the latent
continuous variable has been divided into 5 categories
The range of is then divided by m-1=4 boundaries into The range of is then divided by m 1 4 boundaries into
m=5 regions, o
1
<o
2
<<o
m-1
14
Ordered Logit Models:
Proportional-odds Model (2) p ( )
The cumulative probability distribution of Y is easily
f d found:
Where j is the category of interest
15
j g y
Ordered Logit Models:
Proportional-odds Model (3) p ( )
The ordered logit model is then:
or, equivalently or, equivalently
The logits are then cumulative logits that contrast
categories above category j with category j and below categories above category j with category j and below
This means, then, that the regression slopes moving
from one category to the next are equal
I h d fi d l ill i l i
16
In other words, fitted values will give us cumulative
probabilities
Ordered Logit Models:
Proportional-odds Model (4) Proportional odds Model (4)
The Parallel
Slopes
A ti Assumption
(also known as
the proportional
odds odds
assumption)
requires that the
separate
equations for equations for
each category
differ only in their
intercepts p
In other words,
the slopes are
the same when
going from each
17
going from each
category to the
next
Ordered Logit Models:
Proportional-odds Model (5) p ( )
The ordered logit model fits only one coefficient | for
each X, but a separate intercept for each category
Th fi t t ( l t) i t f The first category (or last) is set as a reference
category to which all the intercepts relate
I nterpretation I nterpretation
We interpret the coefficients in exactly the same way
as for the binary logit model except rather than
referring to a single baseline category, we contrast referring to a single baseline category, we contrast
each category and those below it with all the
categories above it
In other words, the logit tells us the log of the odds In other words, the logit tells us the log of the odds
that Y is in one category and below versus above
that category (or vice versa, depending on the
software)
18
)
Ordered Logit Models:
A Caution
Often the categories for an ordinal variable are chosen
arbitrarily
A i t t b t ti it i f th d d l it An important substantive criteria of the ordered logit
model is that the results remain consistent regardless of
how the dependent variable is cut into categories
If t i dd d t ld d l b j i d If a new category is added to an old model by joined
adjacent categories, the substantive conclusions of
the model should remain the samei.e., the
regression coefficients should be the same regardless regression coefficients should be the same regardless
of how many categories the variable is divided into
If this condition cant be met, we cannot have
confidence in the results and might instead consider a confidence in the results, and might instead consider a
multinomial logit model
19
Proportional Odds models in R
Example: Womens Participation (1) p p ( )
Proportional odds models in R are fit using the pol r
function in the MASS package
We specify logit or probit using the method argument: We specify logit or probit using the method argument:
Logit: pol r ( f or mul a, dat a, met hod = l ogi st i c)
Probit: pol r ( f or mul a, dat a, met hod = pr obi t )
Continuing with the Womens Labor-Force Data in Fox Continuing with the Women s Labor Force Data in Fox
(1997) , we examine the propensity to work outside the
home (measured as full-time, part-time, not working)
but this time treating the DV as an ordered factor
20
Proportional Odds models in R
Example: Womens Participation (2) p p ( )
The odds of moving up one category in participation The odds of moving up one category in participation
for women with children present are e
-1.972
=.139 times
what they are for women who do not have children at
home
21
An effect display can shed more light on the model
R-script for Effect Display
22
Effect Display for the
Ordered Logit Model g
1
.
0
Chi l dren Absent
1
.
0
Chi l dren Present
0
.
8
1
y
not working
part-time
full-time
0
.
8
1
y
.
4
0
.
6
e
d

P
r
o
b
a
b
i
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4
0
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F
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F
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0 10 20 30 40
0
.
0
0 10 20 30 40
0
.
0
23
Husband's Income Husband's Income
Effect Display for the
Multinomial Logit Model g
1
.
0
Chi l dren Absent
not orking
1
.
0
Chi l dren Present
0
.
8
1
y
not working
part-time
full-time
0
.
8
1
y
.
4
0
.
6
e
d

P
r
o
b
a
b
i
l
i
t
y
.
4
0
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6
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P
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0
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2
0
F
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0 10 20 30 40
0
.
0
0 10 20 30 40
0
.
0
24
Husband's Income Husband's Income
Testing the
Proportional Odds Assumption p p
The fitted probabilities for the ordered logit and multinomial
logit models differ somewhat, especially for the part-timers
It is of interest then to assess further which model fits the It is of interest, then, to assess further which model fits the
data best. We can do two things:
1. Compare the Akaike information criterion (AIC) for
the two models
AIC=-2*log-likelihood+ 2*p, where p is the number
of parameters in the model
2. The ordered model is nested within the multinomial
model, allowing the use of an Analysis of Deviance to
compare the fit of the two models. If the multinomial
logit models fits better, the proportional odds
assumption has not been met assumption has not been met
There is no function for this test in Rnor does the
anova function apply to proportional odds and
multinomial logit modelsbut it is simple to create a
25
g p
function that compares the fit of the two models
Testing the
Proportional Odds Assumption (2) Proportional Odds Assumption (2)
R function
26
Testing the
Proportional Odds Assumption (3) p p ( )
The statistically significant difference between the two
models indi ates that the p opo tional odds ass mption is models indicates that the proportional odds assumption is
not met
The AIC is also smaller for the multinomial model,
indicating it fits best
27
indicating it fits best
In this case, then, I choose the multinomial logit model
Ordered Probit Models
These models parallel the proportional odds logit models These models parallel the proportional odds logit models
That is, all the same assumptions and tests apply
As with logit models, fitted probabilities can be
calculated to help with interpretation calculated to help with interpretation
As mentioned earlier, the pol r function will fit these
models, but an alternative function, vgl m, is included in
the VGAMpackage (from Thomas Yee) the VGAMpackage (from Thomas Yee)
Continuing with the Womens labour force example, the
ordered probit model is fit simply:
28
Ordered Probit Models (2)
29
Generalized Linear Models (GLM)
Linear model is usually not suitable for non-continuous
dependent variables
1. Non- normal errors poses problem for efficiency p p y
2. Heteroskedasticity can cause both efficiency
problems and bias in the standard errors
3. Nonlinearity: Assumption of linearity, E(c
i
)=0, is y p y (
i
)
usually not met
4. Nonsensical predictions can occur
These failures lead us to the Generalized Linear
d l h h l d d h Model, which generalizes and adapts the assumptions
of the linear model to different types of outcomes other
than continuous variables
A large set of families of models can be A large set of families of models can be
accommodated under this framework
30
Comparing the GLM to the Linear Model
The linear model finds the conditional mean of Y given
the Xs. It assumes then,
Th GLM ll l hi i h h f ll i h The GLM parallels this with the following, much more
flexible assumptions:
31
Generalized Linear Model (GLM):
Three components
1. As in the general linear model, the influence of the
explanatory variables is still linear:
p
where q is called the linear predictor
2. The relationship between q and the modelled mean is
generalized from
i
=q
i
to:
where g is the link function which links the response to

i
q
i
where g is the link function, which links the response to
the linear predictor through a transformation
3. The assumption that the errors c are normally distributed
is generalized to the assumption that c (and thus the
32
is generalized to the assumption that c (and thus the
residuals) has a specified exponential family distribution
Commonly GLMs
The main requirement is that g() can take any value
(positive or negative) so that linear dependence on the
explanatory variables makes sense
The link must also be monotonic and differentiable
The table below gives some important examples
GLMs are fitted using maximum likelihood and Iterative
Weighted Least Squares (see McCullagh and Nelder
33
Weighted Least Squares (see McCullagh and Nelder,
1999: Chapter 2)
GLMs: Interpretation (1)
Interpretation depends on the link for the model
1 Linear model 1. Linear model
A unit increase in X increases the mean () of Y by |
1
A unit increase in X increases the mean () of Y by |
1
2 L it li d l ( l it d l) 2. Logit- linear model ( logit model)
A unit increase in X changes the logit by | and multiples
the odds by exp(|
1
)
the effects are multiplicative on the odds
34
the effects are multiplicative on the odds
Used for binary dependent variables
GLMs: Interpretation (2)
3. Poisson Model
A unit increase in X increases log by |
1
i.e., multiples by exp(|
1
); the effect of explanatory
i bl th i lti li ti variables on the mean is multiplicative
Used for count data
4. Gamma Model
A unit increase in X increases log by |
1
i.e., multiples by exp(|
1
); the effect of explanatory
variables on the mean is multiplicative
35
variables on the mean is multiplicative
Used for continuous data with a gamma distribution
GLMs: Interpretation (3)
Interpretation of the individual coefficients for a multiple
regression is the same as the single predictor case, but in
all cases follows the OLS interpretation in that it is
holding other variables constant
Interpretation of the coefficients from GLMs requires care,
however
Aside from the OLS model, these models are
nonlinear so the coefficients alone are difficult to
interpret
This is especially the case with interaction terms
Usually complementary measuressuch as fitted
values and graphsare needed for a good g p g
understanding of the results
36
Transformations versus GLM link (1)
Using OLS with a transformed dependent variable is
often an acceptable alternative to the GLM
Transformation of Y and using a link function with the
same transformation are NOT the same, however
The OLS model assumes normally distributed errors
when using the transformed variable
If we cannot make this assumption the GLM is a better
37
If we cannot make this assumption, the GLM is a better
alternative
Transformations versus GLM link (2)
Here is an example comparing an OLS regression with a log
transformation of Y with a Gamma model with a log link
0
0
1
5
0
0
n
k
1
0
0
0
0
o
d
e
l

w
i
t
h

l
o
g

l
i
n
5
0
0
0
G
a
m
m
a

m
o
0 5000 10000 15000
0
38
0 5000 10000 15000
OLS with log transformation
Model Fit and Deviance for GLMs
Evaluation of GLMs relies on the deviance
Recall that the deviance of a model can be equated to
the residual sum of squares from OLS regression
Each family of models requires a different calculation for
the deviance, however:
W th d i f t t l d l We can compare the deviance of a constant only model
to the full model, thus meaning we can also find an R
2
analogue
39
Significance Tests (1)
Hypothesis tests and confidence intervals follow the same
general procedures as for all MLEs:
Tests for individual slopes, H
0
: |
j
=|
j
(0)
are based on
j j
the Wald statistic:
Tests that several slopes are 0, H
0
: |
j
= |
q
=0 are Tests that several slopes are 0, H
0
: |
j
|
q
0 are
carried out using the now familiar analysis of deviance,
which is based on the generalized likelihood-ratio test.
The test is distributed as _
2
with degrees of freedom The test is distributed as _ with degrees of freedom
equal to the number of parameters removed from the
model
40
R
2
Analogue
An R
2
analogue can be obtained from the log likelihood
If the fitted model perfectly predicts Y-values (i.e., P
i
=1
when y
i
=1, and P
i
=0 when y
i
=0) then the log
e
L=0. In
other words, the maximized likelihood is L=1.
If the model predictions are less than perfect, log
e
L<0 and
the maximized likelihood is 0<L<1
The degree to which our model improves predictions over
the constant only can be assessed by comparing the
deviance under the two models
The R
2
analogue is then easily calculated:
Where G
1
2
is the deviance for the fitted model and G
0
2
is
the deviance for the intercept only model
41
p y
GLMs in R
GLMs can be fit easily with the l function (it will not fit GLMs can be fit easily with the gl mfunction (it will not fit
multinomial or ordered models, howevermul t i nomin
the nnet library fits multinomial models; pol r in MASS fits
ordered models) ordered models)
Most of the functions available for the l mfunction are also
available for the gl mfunction (e.g., r esi dual s( ) ,
coef f i ci ent s( ) anova( ) Anova( ) etc ) coef f i ci ent s( ) , anova( ) , Anova( ) etc.)
It is important that the correct family is specified
The default is f ami l y=gaussi an which means that a
regular OLS model will be fit regular OLS model will be fit
For families with more than one possible link function,
it may also be necessary to specify the link
42
Poisson Regression Models
It is common to have response variables that are counts
For example, number of associations respondents
belong to, number of corporate interlocks etc. belong to, number of corporate interlocks etc.
Count variables typically have a Poisson distribution
(i.e., the highest frequency of counts is at 0, and the
frequency rapidly decreases through the range of the eque cy ap d y dec eases oug e a ge o e
variable)
By implication, the residuals from a linear model will
not be normally distributed, and OLS is no longer the y , g
appropriate model
Poisson regression models assume a Poisson
distribution for the random component p
Other possible models for count data include:
Quasi- Poisson-
Negative binomial
43
Negative binomial
Zero- inflated Poisson ( ZI P)
Poisson Regression Models (2)
The link function for a Poisson model is the log link:
This looks similar to an OLS with a logged dependent
variable but differs because it also assumes the Poisson variable but differs because it also assumes the Poisson
distribution for the residuals and IWLS is used for
estimation
I n other words we treat the variable as counts I n other words, we treat the variable as counts
rather than as continuous
For Poisson models, a unit increase in X has a
multiplicative impact of e
B
on multiplicative impact of e
B
on .
The mean of Y at x+1 is equal to the mean of Y at x
multiplied by e
B
If B=0, the multiplicative effect is 1:
44
If B 0, the multiplicative effect is 1:
e
B
=e
0
=1
Fitting a Poisson Regression Model:
Example: Voluntary organizations p y g
The data are a subset from the 1981-91 World Values
Survey (n=4803). The variables are as follows:
ASSOC: Number of voluntary associations to which
the respondent belongs (ranges from 0-15)
SEX, AGE S , G
SES: unskilled, skilled, middle upper
COUNTRY: Great Britain, Canada, USA
The goal is to assess differences in membership by SES The goal is to assess differences in membership by SES
and COUNTRY and whether SES effects are different
across countries (i.e., an interaction between SES and
COUNTRY) COUNTRY)
We begin by inspecting the distribution of the dependent
variable, ASSOC
45
Distribution of ASSOC
We see here that
the distribution of
the number of
Hi stogram of ASSOC
3
0
0
0
the number of
associations to
which respondents
belong has many
0
0
0
2
5
0
0
belong has many
0s and a positive
skew
A Poisson
F
r
e
q
u
e
n
c
y
1
5
0
0
2
0
A Poisson
Regression
Model is thus
appropriate
F
5
0
0
1
0
0
0
pp p
0
5
46
ASSOC
0 5 10 15
Poisson Regression Model
No interaction model (1) ( )
I begin by fitting a model excluding the interaction
(notice that I specify f ami l y=poi sson, and thus the log
link is specified by default): link is specified by default):
47
Poisson Regression Model
No interaction model (2) ( )
For effects without interactions, we can determine the
multiplicative effect on the fitted value for each 1 unit
i i X b l l ti (B) increase in X by calculating exp(B)
Below we see that holding everything else constant, on
average, Americans belong to approximately 20% more
voluntary organizations than Canadians (the reference voluntary organizations than Canadians (the reference
category)
This simple interpretation is lost, however, in the presence
of interactions
48
Poisson Regression Model (2)
49
Assessing the Interaction
As usual, the first step is to determine whether the
interaction is statistically significant using an analysis of
deviance
The Anova function in the car package works here (as it The Anova function in the car package works here (as it
does for all models of gl mclass)
Assuming the effects are significant, it is helpful to
calculate fitted values for the different groups calculate fitted values for the different groups
The al l Ef f ect s function in the car package gives fitted
values using the same principles as for the linear and
logit models (it does so for all gl mobjects)
For all Xs not of interest, the means are substituted
into the fitted equation
Fitted values are then found by varying the Xs of
i t t interest
Because we use the log link to map the linear
predictor on to Y, the exponent of the fitted value is
the effect on the Y-scale (al l Ef f ect s does this
50
the effect on the Y scale (al l Ef f ect s does this
automatically)
Assessing the Interaction (2)
The interaction is statistically significant, so I proceed to
find the fitted values
51
Assessing the Interaction (3)
The class
differences are
largest in the US,
and smallest in
Canada
Holding all other
variables to their
means, upper SES
Americans belong
t 2 85 l t to 2.85 voluntary
organizations on
average, while
unskilled unskilled
Americans belong
to only .87
organizations on
52
organizations on
average
Some other models for count data
An assumption of the Poisson model is that the variance
is equal to the mean
If the variance is larger than the meani.e., there is
overdispersion the Poisson model is not appropriate overdispersionthe Poisson model is not appropriate
In such cases, three alternative models are the quasi-
Poisson model, the negative binomial model, and the zero-
inflated Poisson model (ZIP) inflated Poisson model (ZIP)
All three models explicitly model the overdispersion,
though they can give slightly different results
The quasi- Poisson model models the variance as a q
linear function of the mean
The negative binomial model models the variance as
a quadratic function of the mean
The zero- inflated Poisson explicitly takes into
consider a distribution characterized by an abundance
of zeros.
A practical way to assess which fits best is to compare the
53
A practical way to assess which fits best is to compare the
AIC or BIC values
Summary and Conclusions
The linear model fails to accommodate categorical
dependent variables, but a transformation of t allows us
to fit more sensible models
The most commonly used is the logit model The most commonly used is the logit model
Binary logit models (dichotomous variable)
Proportional-odds logit models (ordered variable)
Multinomial logit models (several non ordered Multinomial logit models (several non-ordered
categories)
Interpretation of logit models:
e
b
tells us the odds ratio for an event occurring as X e
b
tells us the odds ratio for an event occurring as X
goes up by onethis interpretation is convenient in a
simple model without interaction terms
Models with interaction terms are intrinsically Models with interaction terms are intrinsically
complicated but their results can be simplified with
fitted probabilities
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Summary and Conclusions (2)
The probit model gives nearly identical results to those
of the logit model when using binary data
Interpretation of probit models:
The coefficient B
1
from a probit model is interpreted as
the average increase in Z (the underlying latent
variable) for a one unit increase in X
1
, holding all other
Xs constant
Simply put, fitted probits are standardized values of Z
As with logit models, interpretation of the probit g , p p
models can often be improved by calculating fitted
probabilities
If the I I A assumption is violated, multinomial probit p , p
models perform better than multinomial logit models
Since multinomial logit models are typically easier to
estimate, however, they are usually preferred
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, , y y p
Summary and Conclusions (3)
Generalized linear models are a generalization of the
linear model to outcomes other than quantitative
variables
A different link function depending on the distribution of
the dependent variable is used to map the response to a
linear predictor
Logit models and probit models are two frequently
used sets of models
Binary logit models are for dichotomous variables
Proportional-odds logit models are for ordered data
Multinomial logit models are for a dependent
variable with many categories that do not have an y g
systematic order
Poisson models typically work well for count data
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Next Week:
Wednesday: Detecting nonnormality,
heteroskedasticity and collinearity
Th d D t ti tli Thursday: Detecting outliers
Assignment #2 due on Wednesday