📄 pa 765 logistic regression.mht
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Subject: PA 765: Logistic Regression
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<CENTER>
<H1>Logistic Regression</H1></CENTER>
<P><BR>
<H2>Overview</H2><I>Binomial (or binary) logistic regression</I> is a =
form of=20
regression which is used when the dependent is a dichotomy and the =
independents=20
are continuous variables, categorical variables, or both. <I>Multinomial =
logistic regression</I> exists to handle the case of dependents with =
more=20
classes. Logistic regression applies maximum likelihood estimation after =
transforming the dependent into a logit variable (the natural log of the =
odds of=20
the dependent occurring or not). In this way, logistic regression =
estimates the=20
probability of a certain event occurring. Note that logistic regression=20
calculates changes in the log odds of the dependent, not changes in the=20
dependent itself as OLS regression does.=20
<P>Logistic regression has many analogies to OLS regression: logit =
coefficients=20
correspond to b coefficients in the logistic regression equation, the=20
standardized logit coefficients correspond to beta weights, and a pseudo =
R<SUP>2</SUP> statistic is available to summarize the strength of the=20
relationship. Unlike OLS regression, however, logistic regression does =
not=20
assume linearity of relationship between the independent variables and =
the=20
dependent, does not require normally distributed variables, does not =
assume=20
homoscedasticity, and in general has less stringent requirements. The =
success of=20
the logistic regression can be assessed by looking at the classification =
table,=20
showing correct and incorrect classifications of the dichotomous, =
ordinal, or=20
polytomous dependent. Also, goodness-of-fit tests are available as =
indicators of=20
success as is the Wald statistic and other tests of the model's =
significance.=20
<P>In SPSS 10, binomial logistic regression is under Analyze - =
Regression -=20
Binary Logistic, and the multinomial version is under Analyze - =
Regression -=20
Multinomial Logistic. The GENLOG and LOGLINEAR procedures in SPSS can =
also fit=20
logit models when all variables are categorical.=20
<P>
<H2>Key Terms and Concepts</H2><A=20
href=3D"http://www2.chass.ncsu.edu/garson/pa765/logit.htm">Logit =
regression</A>=20
has numerically identical results to logistic regression, but some =
computer=20
programs offer both, often with different output options. Logistic =
regression=20
has become more popular among social scientists.=20
<UL><A name=3Ddesign></A>
<LI><B>Design variables</B> are nominal or ordinal independents =
entered as=20
dummy variables. SPSS will convert categorical variables to dummies=20
automatically by leaving out the last category. Researchers may prefer =
to=20
create dummy variables manually so as to control which category is =
omitted and=20
thus becomes the reference category. For more on the selection of =
dummy=20
variables, click <A=20
=
href=3D"http://www2.chass.ncsu.edu/garson/pa765/regress.htm#dummy">here</=
A>.=20
<P></P>
<LI><B>Covariates</B> are interval independents.=20
<P><A name=3Dodds></A></P>
<LI><B>Odds, odds ratios, second-order odds ratios, partial odds =
ratios, and=20
logits</B> are all important basic terms in logistic regression. They =
are=20
defined in the separate section on <A=20
href=3D"http://www2.chass.ncsu.edu/garson/pa765/logit.htm">log-linear=20
analysis.</A> Those new to the subject should click on <A=20
href=3D"http://www2.chass.ncsu.edu/garson/pa765/logit.htm">log-linear=20
analysis.</A> before proceeding with the remainder of this section so =
that=20
terms such as "logits" are understood.=20
<P><A name=3Dlcoeff></A></P>
<LI><B>Logit coefficients</B>, also called unstandardized <I>logistic=20
regression coefficients</I> or <I>effect coefficients</I>, correspond =
to the b=20
(unstandardized regression) coefficients in ordinary least squares =
(OLS)=20
regression, and are used in the logistic regression equation to =
estimate=20
(predict) the odds that the dependent equals 1 (binomial logistic =
regression)=20
or that the dependent equals its highest/last value (multinomial =
logistic=20
regression). For the dichotomous case, if the logit for a given =
independent=20
variable is b<SUB>1</SUB>, then a unit increase in the independent =
variable is=20
associated with b<SUB>1</SUB> unit increase in the log odds of the =
dependent=20
variable (the natural log of the probability that the dependent =3D 1 =
divided by=20
the probability that the dependent =3D 0). In multinomial logistic =
analysis,=20
where the dependent may have more than the usual 0-or-1 values, the =
comparison=20
is always with the last value rather than with the value of 1. Note =
that OLS=20
had an identity link function while logistic regression has a logit =
link=20
function (that is, logistic regression calculates changes in the log =
odds of=20
the dependent, not changes in the dependent itself as OLS regression =
does).=20
<P>The probability that dependent event y=3D1 is a function of the =
logit=20
coefficients. For instance, let y=3D0 or y=3D1 and let x<SUB>1</SUB>,=20
x<SUB>2</SUB>, and x<SUB>3</SUB> be continuous independent variables =
for the=20
logistic model y =3D b<SUB>0</SUB> + b<SUB>1</SUB>x<SUB>1</SUB> +=20
b<SUB>2</SUB>x<SUB>2</SUB> + b<SUB>3</SUB>x<SUB>3</SUB>. The estimate =
of=20
p(y=3D1) is the natural logarithm e to the power of a term which is =
the logistic=20
regression equation. (Note that SPSS will use all the logit =
coefficients in=20
calculating p, whether they are significant or not. To avoid this, =
re-run the=20
logistic model dropping non-significant independents.)=20
<P>In SPSS output, the logit coefficients are labeled B.=20
<P></P>
<LI><B>Interpreting the logit coefficient</B>=20
<UL>
<P>
<LI><B>Odds ratio</B>. The logit can be converted easily into a =
statement=20
about <I>odds ratio</I> of the dependent rather than log odds simply =
by=20
using the exponential function (raising the natural log to the =
b<SUB>1</SUB>=20
power). For instance, if the logit b<SUB>1</SUB> =3D 2.303, then its =
log odds=20
ratio (the exponential function, e<SUP>b</SUP>) is 10 and we may say =
that=20
when the independent variable increases one unit, the odds that the=20
dependent =3D 1 increase by a factor of 10, when other variables are =
controlled. Thus the odds ratio can be used to compare the relative=20
importance of the independent variables through statements along the =
lines,=20
"The effect of x<SUB>1</SUB> on the predicted odds that y=3D1 is =
equivalent to=20
an increase (or decrease) of (odds ratio) units of y." Similar =
statements=20
can be made for each of the independent variables. The ratio of odds =
ratios=20
of the independents is the ratio of relative importance of the =
independent=20
variables in terms of effect on the dependent variable. Note =
standardized=20
logit coefficients may also be used, as discussed below.=20
<P></P>
<LI><B>Percent increase in odds</B>. Another way of saying the same =
thing is=20
to transform the logit coefficient by 100(e<SUP>b</SUP> - 1), where =
b is the=20
logit coefficient, then express the result as a percentage. To take =
an=20
example given by Allison (1999: 188), for a group of male professors =
the=20
logit coefficient for "number of articles published" is .0737, where =
the=20
dependent is being promoted. Rounded to the nearest hundredth, this=20
transforms to .08. Therefore one may say, "each additional article =
yields an=20
increase in the odds of promotion of about 8%." </LI></UL>
<P><A name=3Dconfidence></A><A name=3Dase></A></P>
<LI><B>Confidence interval for the logistic regression =
coefficient</B>. The=20
confidence interval around the logistic regression coefficient is plus =
or=20
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