%%% Title of object:	Logistic Regression
%%% Canonical Name:	LogisticRegression
%%% Type:	Topic
%%% Created on:	2010-08-20 18:34:36
%%% Modified on:	2010-08-24 00:37:13
%%% Creator:	jhilbe
%%% Modifier:	jkimmel
%%% Author:	jhilbe
%%% 
%%% Classification:	msc:62J12
%%% Keywords:	logistic, binary, nonlinear, binomial, Bernoulli
%%% Synonyms:	Logistic Regression=logit regression
%%% Logistic Regression=logistic models
%%% Logistic Regression=binary logistic regression
%%% Logistic Regression=Bernoulli regression
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\begin{center}

{\large \bf LOGISTIC REGRESSION}\\[4mm]

{\large \it Joseph M. Hilbe}\\[2mm]

Arizona State University

\end{center}
\vspace{2cc}

Logistic regression is the most common method used to model binary
response data. When the response is binary, it typically takes the
form of 1/0, with 1 generally indicating a success and 0 a failure.
However, the actual values that 1 and 0 can take vary widely,
depending on the purpose of the study. For example, for a study of
the odds of failure in a school setting, 1 may have the value of
{\it fail}, and 0 of {\it not-fail}, or pass. The important point is
that 1 indicates the foremost subject of interest for which a binary
response study is designed. Modeling a binary response variable
using normal linear regression introduces substantial bias into the
parameter estimates. The standard linear model assumes that the
response and error terms are normally or Gaussian distributed, that
the variance, $\sigma^2,$ is constant across observations, and that
observations in the model are independent. When a binary variable is
modeled using this method, the first two of the above assumptions
are violated. Analogical to the normal regression model being based
on the Gaussian probability distribution function $(pdf),$ a binary
response model is derived from a Bernoulli distribution, which is a
subset of the binomial $pdf$ with the binomial denominator taking
the value of 1. The Bernoulli $pdf$ may be expressed as:
\begin{equation}\label{eq1}
f(y_i;\pi_i)=\pi_i^{\;y_i}(1-\pi_i)^{1-y_i}.
\end{equation}

Binary logistic regression derives from the canonical form of the
Bernoulli distribution. The Bernoulli $pdf$ is a member of the
exponential family of probability distributions, which has
properties allowing for a much easier estimation of its parameters
than traditional Newton-Raphson-based maximum likelihood estimation
$(MLE)$ methods.

In 1972 Nelder and Wedderbrun discovered that it
was possible to construct a single algorithm for estimating models
based on the exponential family of distributions. The algorithm was
termed {\bf Generalized linear models} $(GLM),$ and became a standard
method to estimate binary response models such as logistic, probit,
and complimentary-loglog regression, count response models such as
Poisson and negative binomial regression, and continuous response
models such as gamma and inverse Gaussian regression. The standard
normal model, or Gaussian regression, is also a generalized linear
model, and may be estimated under its algorithm. The form of the
exponential distribution appropriate for generalized linear models
may be expressed as
\begin{equation}\label{eq2}
f(y_i;\theta_i,\phi)=\exp\{(y_i\theta_i-b(\theta_i))/\alpha(\phi)+c(y_i;\phi)\},
\end{equation}
with $\theta$ representing the link function, $\alpha(\phi)$ the
scale parameter, $b(\theta)$ the cumulant, and $c(y;\phi)$ the
normalization term, which guarantees that the probability function
sums to 1. The link, a monotonically increasing function, linearizes
the relationship of the expected mean and explanatory predictors.
The scale, for binary and count models, is constrained to a value of
1, and the cumulant is used to calculate the model mean and variance
functions. The mean is given as the first derivative of the cumulant
with respect to $\theta,$ $b'(\theta);$ the variance is given as the
second derivative, $b''(\theta).$ Taken together, the above four terms
define a specific $GLM$ model.

We may structure the Bernoulli distribution (Eq 3) into exponential
family form (Eq 2) as:
\begin{equation}\label{eq3}
f(y_i;\pi_i)=\exp\{y_i\ln(\pi_i/(1-\pi_i))+\ln(1-\pi_i)\}.
\end{equation}

The link function is therefore $\ln(\pi/(1-\pi)),$ and cumulant
$-\ln(1-\pi)$ or $\ln(1/(1-\pi)).$ For the Bernoulli, $\pi$ is
defined as the probability of a success. The first derivative of the
cumulant is $\pi,$ the second derivative, $\pi(1-\pi).$ These two
values are, respectively, the mean and variance functions of the
Bernoulli $pdf.$ Recalling that the logistic model is the canonical
form of the distribution, meaning that it is the form that is
directly derived from the $pdf,$ the values expressed in Eq 3, and
the values we gave for the mean and variance, are the values for the
logistic model.

Estimation of statistical models using the $GLM$ algorithm, as well
as $MLE,$ are both based on the log-likelihood function. The
likelihood is simply a re-parameterization of the $pdf$ which seeks
to estimate $\pi,$ for example, rather than $y.$ The log-likelihood
is formed from the likelihood by taking the natural log of the
function, allowing summation across observations during the
estimation process rather than multiplication.

The traditional $GLM$ symbol for the mean, $\mu,$ is typically
substituted for $\pi,$ when $GLM$ is used to estimate a logistic
model. In that form, the log-likelihood function for the
binary-logistic model is given as:
\begin{equation}\label{eq4}
L(\mu_i;y_i)=\sum_{i=1}^n\{y_i\ln(\mu_i/(1-\mu_i))+\ln(1-\mu_i)\},
\end{equation}
or
\begin{equation}\label{eq5}
L(\mu_i;y_i)=\sum_{i=1}^n\{y_i\ln(\mu_i)+(1-y_i)\ln(1-\mu_i)\}.
\end{equation}

The Bernoulli-logistic log-likelihood function is essential to
logistic regression. When $GLM$ is used to estimate logistic models,
many software algorithms use the deviance rather than the
log-likelihood function as the basis of convergence. The deviance,
which can be used as a goodness-of-fit statistic, is defined as
twice the difference of the saturated log-likelihood and model
log-likelihood. For logistic model, the deviance is expressed as
\begin{equation}\label{eq6}
D=2\sum_{i=1}^n\{y_i\ln(y_i/\mu_i)+(1-y_i)\ln((1-y_i)/(1-\mu_i))\}.
\end{equation}
Whether estimated using maximum likelihood techniques or as $GLM,$
the value of $\mu$ for each observation in the model is calculated
on the basis of the linear predictor, $x'\beta.$ For the normal model,
the predicted fit, $\hat{y},$ is identical to $x'\beta,$ the right side
of Equation 7. However, for logistic models, the response is
expressed in terms of the link function,
$\ln(\mu_i/(1-\mu_i)).$ We have, therefore,
\begin{equation}\label{eq7}
x_i'\beta=\ln(\mu_i/(1-\mu_i))=\beta_0+\beta_1x_1+\beta_2x_2+\cdots+\beta_n
x_n.
\end{equation}

\noindent The value of $\mu_i,$ for each observation in the logistic model, is calculated
as
\begin{equation}\label{eq8}
\mu_i=1/(1+\exp(-x_i'\beta))=\exp(x_i'\beta)/(1+\exp(x_i'\beta)).
\end{equation}
The functions to the right of $\mu$ are commonly used ways of
expressing the logistic inverse link function, which converts the
linear predictor to the fitted value. For the logistic model, $\mu$ is
a probability.

When logistic regression is estimated using a Newton-Raphson type of
$MLE$ algorithm, the log-likelihood function as parameterized to
$x'\beta$ rather than $\mu.$ The estimated fit is then determined by
taking the first derivative of the log-likelihood function with
respect to $\beta,$ setting it to zero, and solving. The first
derivative of the log-likelihood function is commonly referred to as
the gradient, or score function. The second derivative of the
log-likelihood with respect to $\beta$ produces the Hessian matrix,
from which the standard errors of the predictor parameter estimates
are derived. The logistic gradient and hessian functions are given
as {\arraycolsep 0.5mm
\begin{eqnarray}
\frac{\partial L(\beta)}{\partial\beta}&=&\sum_{i=1}^n
(y_i-\mu_i)x_i\\
\frac{\partial^2
L(\beta)}{\partial\beta\partial\beta'}&=&-\sum_{i=1}^n\big\{
x_ix'_i\mu_i(1-\mu_i)\big\}
\end{eqnarray}}\\
One of the primary values of using the logistic regression model is
the ability to interpret the exponentiated parameter estimates as
odds ratios. Note that the link function is the log of the odds of
$\mu,\,\ln(\mu/(1-\mu)),$ where the odds are understood as the
success of $\mu$ over its failure, $1-\mu.$ The log-odds is commonly
referred to as the {\it logit} function. An example will help
clarify the relationship, as well as the interpretation of the odds
ratio.

We use data from the 1912 Titanic accident, comparing the odds of
survival for adult passengers to children. A tabulation of the data
is given as:

\begin{center}
\smallskip{\small
\begin{tabular}{r|rr|r}
& \multicolumn{2}{l}{Age (Child vs Adult)} &
\\ Survived & \phantom{child}child & adults & Total
\\ \hline
no & 52 & 765 & 817
\\ yes & 57 & 442 & 499
\\\hline
Total & 109 & 1,207 & 1,316
\end{tabular}}
\end{center}


The odds of survival for adult passengers is 442/765, or 0.578. The
odds of survival for children is 57/52, or 1.096. The ratio of the
odds of survival for adults to the odds of survival for children is
(442/765)/(57/52), or 0.52709552. This value is referred to as the
{\it odds ratio}, or ratio of the two component odds relationships. Using
a logistic regression procedure to estimate the odds ratio of age
produces the following results

\bigskip{ \small
\begin{tabular}{r|rrrrrr}
\hline
survived & Odds Ratio & Std. Err. & $z$ & $P>|z|$ & \multicolumn{2}{c}{[95\% Conf.
Interval]}
\\\hline
age & .5270955 & .1058718 & -3.19 & 0.001 & .3555642 & .7813771
\\\hline
\end{tabular}}

\bigskip

With $1=adult$ and $0=child,$ the estimated odds ratio may be interpreted as:

\medskip

{\it The odds of an adult surviving were about half the odds of a child surviving.}

\medskip

By inverting the estimated odds ratio above, we may conclude that
children had $[1/.527\sim1.9]$ some 90\% --- or nearly two times --
greater odds of surviving than did adults.

For continuous predictors, a one-unit increase in a predictor value
indicates the change in odds expressed by the displayed odds ratio.
For example, if age was recorded as a continuous predictor in the
Titanic data, and the odds ratio was calculated as 1.015, we would
interpret the relationship as:

\medskip

{\it The odds of surviving is one and a half percent greater for each increasing
year of age.}

\medskip

\noindent Non-exponentiated logistic regression parameter estimates are
interpreted as log-odds relationships, which carry little meaning in
ordinary discourse. Logistic models are typically interpreted in
terms of odds ratios, unless a researcher is interested in
estimating predicted probabilities for given patterns of model
covariates; i.e., in estimating $\mu.$

Logistic regression may also be used for grouped or proportional
data. For these models the response consists of a numerator,
indicating the number of successes (1s) for a specific covariate
pattern, and the denominator $(m),$ the number of observations having
the specific covariate pattern. The response $y/m$ is binomially
distributed as:
\begin{equation}\label{eq11}
f(y_i;\pi_i,m_i)=\bi{m_i}{y_i}\pi_i^{\;y_i}(1-\pi_i)^{m_i-y_i},
\end{equation}
with a corresponding log-likelihood function expressed as
\begin{equation}\label{eq12}
L(\mu_i;y_i,m_i)=\sum_{i=1}^n\left\{y_i\ln(\mu_i/(1-\mu_i))+m_i\ln(1-\mu_i)+\bi{m_i}{y_i}\right\}.
\end{equation}

Taking derivatives of the cumulant, $-m_i\ln(1-\mu_i),$ as we
did for the binary response model, produces a mean of
$\mu_i=m_i\pi_i$ and variance, $\mu_i(1-\mu_i/m_i).$

Consider the data below:

\begin{center}
{\small
\begin{tabular}{ccccc}
\hline
$y$ & cases & $x_1$ & $x_2$ & $x_3$
\\ \hline
1 & 3 & 1 & 0 & 1
\\ 1 & 1 & 1 & 1 & 1
\\ 2 & 2 & 0 & 0 & 1
\\ 0 & 1 & 0 & 1 & 1
\\ 2 & 2 & 1 & 0 & 0
\\ 0 & 1 & 0 & 1 & 0
\\\hline
\end{tabular}
}

\end{center}
\noindent $y$ indicates the number of times a specific pattern of
covariates is successful. {\it Cases} is the binomial denominator.
The first observation in the table informs us that there are three
cases having predictor values of $x_1=1, x_2=0,\mbox{ and }x_3=1.$ Of
those three cases, one has a value of $y$ equal to 1, the other two
have values of 0. All current commercial software applications estimate
this type of logistic model using $GLM$ methodology.

\bigskip

{\small\begin{tabular}{r|rcrrrr} \hline
$y$ & Odds Ratio & OIM Std. Err. & $z$ & $P>|z|$ & \multicolumn{2}{c}{[95\% Conf.
Interval]}
\\\hline
$x_1$ & 1.186947 & 1.769584 & 0.11 & 0.908 & .0638853 & 22.05271
\\ $x_2$ & .2024631 & .3241584 & -1.00 & 0.318 & .0087803 & 4.668551
\\ $x_3$ & .5770337 & .9126937 & -0.35 & 0.728 & .025993 & 12.8099
\\\hline
\end{tabular}}

\bigskip

The data in the above table may be restructured so that it is in
individual observation format, rather than grouped. The new table
would have ten observations, having the same logic as described.
Modeling would result in identical parameter estimates. It is not
uncommon to find an individual-based data set of, for example,
10,000 observations, being grouped into 10-15 rows or observations
as above described. Data in tables is nearly always expressed in
grouped format.

Logistic models are subject to a variety of fit tests. Some of the
more popular tests include the Hosmer-Lemeshow goodness-of-fit test,
$ROC$ analysis, various information criteria tests, link tests, and
residual analysis. The Hosmer-Lemeshow test, once well used, is now
only used with caution. The test is heavily influenced by the manner
in which tied data is classified. Comparing observed with expected
probabilities across levels, it is now preferred to construct tables
of risk having different numbers of levels. If there is consistency
in results across tables, then the statistic is more trustworthy.


Information criteria tests, e.g., Akaike information Criteria
$(AIC)$ and Baye\-sian Information Criteria $(BIC)$ are the most used
of this type of test. Information tests are comparative, with lower
values indicating the preferred model. Recent research indicates
that $AIC$ and $BIC$ both are biased when data is correlated to any
degree. Statisticians have attempted to develop enhancements of
these two tests, but have not been entirely successful. The best
advice is to use several different types of tests, aiming for
consistency of results.

Several types of residual analyses are typically recommended for
logistic models. The references below provide extensive discussion
of these methods, together with appropriate caveats. However, it
appears well established that m-asymptotic residual analyses is most
appropriate for logistic models having no continuous predictors.
m-asymptotics is based on grouping observations with the same
covariate pattern, in a similar manner to the grouped or binomial
logistic regression discussed earlier. The Hilbe (2009) and Hosmer
and Lemeshow (2000) references below provide guidance on how best to
construct and interpret this type of residual.

Logistic models have been expanded to include categorical responses,
e.g. proportional odds models and multinomial logistic regression.
They have also been enhanced to include the modeling of panel and
correlated data, e.g. generalized estimating equations, fixed and
random effects, and mixed effects logistic models.

Finally, exact logistic regression models have recently been
developed to allow the modeling of perfectly predicted data, as well
as small and unbalanced datasets. In these cases, logistic models
which are estimated using GLM or full maximum likelihood will not
converge. Exact models employ entirely different methods of
estimation, based on large numbers of permutations. Refer to {\it
exact methods} or {\it computer intensive methods} for an
elaboration of this technique.

\vspace{1.5cc}

\noindent{\bf References}
\newcounter{ref}
\begin{list}{\small [\,\arabic{ref}\,]}{\usecounter{ref} \leftmargin 4mm \itemsep -1mm}
{\small

\item Collett, D. (2003). {\it Modeling Binary Regression}. 2nd edition, London: Chapman \& Hall/CRC.

\item Cox, D. R., E. J. Snell, (1989). {\it Analysis of Binary Data}. 2nd edition,
London: Chapman \& Hall.

\item Hardin, J. W., J.M. Hilbe (2007). {\it Generalized Linear Models and
Extensions}. 2nd edition, College Station: Stata Press.

\item Hilbe, J. M. (2009) {\it Logistic Regression Models}. Boca Raton,
FL: Chapman \& Hall/CRC.

\item Hosmer, D., S. Lemeshow (2000). {\it Applied Logistic Regression}. 2nd edition, New York: Wiley.

\item Kleinbaum, D. G. (1994) {\it Logistic Regression; A Self-Teaching
Guide}. New York: Springer.

\item Long, J. S. (1997) {\it Regression Models for Categorical and Limited
Dependent Variables.} Thousand Oaks, CA: Sage.

\item McCullagh, P., J. Nelder (1989). {\it Generalized Linear Models}. 2nd edition, London: Chapman \& Hall.

}
\end{list}

\end{document}