Alfred DeMaris - Regression with Social Data, Modeling Continuous and Limited Response Variables
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from the covariates,
ln
ᎏ
1 ⫺
P
it
P
it
ᎏ
⫽⫺6.383 ⫹ 1.541t ⫺.151t
2
⫹ .004t
3
⫹ .000t
4
.
Give the equation for the effect of years in rank, apart from covariates, as a
function of gender, using a dummy, F, for being female, and a set of
coefficients for the interaction of gender with time. That is, the equation
should take the form
ln
ᎏ
1 ⫺
P
it
P
it
ᎏ
⫽ α ⫹ δF ⫹ at ⫹ bt
2
⫹ ct
3
⫹ dt
4
⫹ g
1
Ft ⫹ g
2
Ft
2
⫹ g
3
Ft
3
⫹ g
4
Ft
4
.
12.15 The Long et al. (1993) study employed the following covariates (among oth-
ers): (a) years between receiving the PhD and entering the current rank; (b)
prestige of the PhD-granting department; (c) whether the doctorate was in
a medical area; (d) prestige of the current employing institution; (e) whether
the current job was in the PhD-granting institution; and (f) the square root
of the number of articles published since entering the current rank. Which
of these are time-invariant, and which are time-varying, covariates? Why?
12.16 Using the cohabiting transitions dataset, estimate the multistate model in
Table 12.1 via the two-step approach discussed in the text, despite this
approach being theoretically questionable for this problem. Be sure to
include both DURATION and the log of SURVIVAL in the logit step of the
model. Show estimates for both equations, with significant effects starred,
and provide a general interpretation of the results.
12.17 Using the cohabiting transitions dataset, estimate the competing risks
model of Table 12.1, allowing nonproportional effects of HEHIT, SHEHIT,
and TERROR2. Show estimates for both equations, with significant effects
starred, and provide a general interpretation of the results.
12.18 Using the unemployment dataset, estimate a Cox multiepisode model for
the hazard of reemployment using the predictors PRVJBS, JOBDUR,
FEMALES, NEWEDUC, LOGINC, and CANADA. Show estimates for the
equation, with significant effects starred, and provide a general interpreta-
tion of the results.
12.19 Using the unemployment dataset, estimate the model of Exercise 12.18,
allowing a nonproportional effect for JOBDUR. Show estimates for the
equation, with significant effects starred. Test for a significant nonpropor-
tional effect using both LRχ
2
and Wald χ
2
tests. Interpret the nature of the
nonproportional effect by showing how the effect of JOBDUR varies with
survival time.
EXERCISES 445
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12.20 Reestimate the model of Exercise 12.19 using the discrete-time approach.
Employ five time intervals, as follows: DURUNEM ⬍ 1, 1 ⱕ DURUNEM ⬍ 2,
2 ⱕ DURUNEM ⬍ 4, 4 ⱕ DURUNEM ⬍ 6, and DURUNEM ⱖ 6. Then use
the complementary log-log model, along with the log of time interval, to
approximate the Weibull model. Show estimates for the equation, with
significant effects starred, and provide a general interpretation of the results,
particularly as they differ from those found in Exercise 12.19.
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APPENDIX A
Mathematics Tutorials
APPENDIX OVERVIEW
In the event that your mathematics skills are rusty, the following sections contain
brief tutorials in basic algebra, summation notation, covariance algebra, derivatives,
and matrix algebra. Although you certainly can read and digest all of this material
before proceeding with the rest of the book, most readers will probably not want to
do that. Therefore, these tutorials are each designed to be discrete sections that can
be reviewed separately, according to the reader’s need to refresh his or her skills.
I. BASIC ALGEBRA
Algebra involves operations using arbitrary numbers, or variables, as well as spe-
cific numbers, or constants. In the following sections, letters such as a, b, c, x, w, y,
or z denote any arbitrary real numbers (in this book we will only be concerned with
real numbers), whereas 3 or π denote constants. (The symbol π denotes an irrational
number representing the ratio of the area of a circle to the square of its radius, and
equaling approximately 3.14159.) Further, I follow the mathematical notational con-
vention that exponents only apply to the expression immediately to the left of the
exponent unless parentheses group elements for exponentiation. Thus, if x 3, then
2x
2
2(3
2
) 18. But (2x)
2
[2(3)]
2
36.
The following algebraic rules apply to any real numbers.
A. Commutative Property of Numbers
(1) a b b a.
Example:3 7 7 3 10.
447
Regression with Social Data: Modeling Continuous and Limited Response Variables,
By Alfred DeMaris
ISBN 0-471-22337-9 Copyright © 2004 John Wiley & Sons, Inc.
bapp01.qxd 30.8.04 11:45 Page 447
(2) ab ba.
Example: 6(9) 9(6) 54.
B. Associative Property of Numbers
(1) a (b c) (a b) c.
Example:7 (11 19) (7 11) 19 37.
(2) a(bc) (ab)c.
Example: 2[4(8)] [2(4)]8 64.
C. Distributive Property of Numbers
(1) a(b c) ab ac.
Example:3(4 5) 3(4) 3(5) 12 15 27.
(2) (a b)c ac bc.
Example:(6 2)4 6(4) 2(4) 24 8 32.
D. Repeated Application of Distributive Property
Since a(b c) ab ac, if we let a m n, then a(b c) (m n)(b c)
(m n)b (m n)c mb nb mc nc.
Note: Subtraction and division are neither commutative nor associative:
a b b a
a (b c) (a b) c
b
a
a
b
a
c
/b
b
a
/c
E. Identities
(1) 0 is the identity element for addition and subtraction. That is,
a 0 0 a a.
a 0 a (however, 0 a a).
(2) 1 is the identity element for multiplication and division. That is,
1(a) (a)1 a.
a /1 a (however, 1/a does not equal a).
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F. Inverses
For every real number a, a is the additive inverse of a, such that
a (a) 0.
For every real number a,1/a is the multiplicative inverse, or reciprocal, of a, such
that
(1/a)a a(1/a) (a/a) 1.
G. Other Properties of Addition, Subtraction, Multiplication, and Division
If a b, then for any number c:
(1) a c b c,
(2) a c b c,
(3) ac bc,
(4) a/c b/c provided that c is not 0.
H. Factorials
For any integer, N, the symbol N! (pronounced “en factorial”) is defined by
N ! N(N 1)(N 2)(N 3)
...
(2)(1). For example, 5 ! 5(4)(3)(2)(1) 120.
I. Properties Associated with Zero
(1) 0 (a) a(0) 0; therefore, it is also true that 0(0) 0.
(2) a/0 is an undefined operation; therefore, 0/0 is also undefined.
(3) Any number (except 0) raised to the 0 power equals 1.
(4) 0
0
is undefined (however, 0
1
0).
(5) 0 ! 1.
(6) If a/b 0, then a 0.
(7) If ab 0, then either a 0 or b 0 or both a and b equal zero.
J. Rules for Negative Signs
(1) (a) a.
(2) (a)b ab a(b).
(3) (a)(b) ab.
(4) (1)a a.
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K. Rules for Fractions
(1)
a
b
b
a
a
b
.
Example:
2
3
3
2
2
3
.
(2)
a
b
b
a
a
b
a
b
.
Example:
3
7
7
3
3
7
3
7
.
(3)
a
b
d
c
b
a
d
c
.
Example:
2
3
冢
5
7
冣
(
(
2
3
)
)
(
(
5
7
)
)
1
2
0
1
.
(4)
a
b
d
d
a
b
.
Example:
3
4
π
π
3
4
.
(5)
a
d
b
d
a
d
b
.
Example:
1
3
7
1
9
7
1
1
2
7
.
(6)
a
b
d
c
ad
b
d
bc
. Why? Since
a
b
a
b
d
d
and
d
c
b
b
d
c
,
then
a
b
d
c
a
b
d
d
b
b
d
c
ad
b
d
bc
.
Example:
x
y
z
x
y
w
2
xzy
y
2
xw
.
Elaboration: By rule I.E(2), multiplying xz/y by 1 leaves the term unchanged.
However, if we write 1 as y/y, we have, by rule I.K(3), (xz/y)(y/y) xyz/y
2
. Now both
terms in the example have the same denominator, hence the numerators can be
summed, as in rule I.K(5), and the result follows.
(7)
c
a
/
/
d
b
b
a
c
d
a
b
d
c
.
Example:
x
2
yz
π
/
/
3
x
π
x
6
2
π
y
2
z
.
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(8)
a
b
d
c
if and only if ad bc.
Example:
2
3
x
3
4
implies that 8x 9 or x
9
8
.
L. Laws of Exponents
Let x and y be real numbers, and suppose that a and b are positive real numbers.
Then
(1) b
x
b
y
b
xy
.
Example: (27
4/18
)(27
2/18
) 27
6/18
27
1/3
3.
(2)
b
b
y
x
b
xy
.
Example:
x
x
3
5
x
35
x
2
x
1
2
.
(3) (b
x
)
y
b
xy
.
Example:(3
π
)
2
3
2 π
995.04.
(4) (ab)
x
a
x
b
x
.
Example:(9x)
2
81x
2
.
(5)
冢
a
b
冣
x
a
b
x
x
.
Example:
冢
3
xy
z
冣
2
x
9
2
z
y
2
2
.
(6) b
x
b
1
x
.
Example:15
4
1
1
5
4
.
M. Working with Rational Exponents
Frequently, we encounter expressions such as 兹a
苶
or 兹
3
a
苶
or (兹
3
a
苶
)
2
. Each expres-
sion involves a rational exponent or an exponent that is the ratio of integers. In
particular:
兹a
苶
a
1/2
.
兹
3
a
苶
a
1/3
.
(兹
3
a
苶
)
2
a
2/3
.
In general, if p and q are integers, a
p/q
is defined to be (兹
q
a
苶
)
p
, or, equivalently,
兹
q
a
苶
p
苶
. The laws of exponents enumerated above apply also to rational exponents.
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Some examples are
(32)
4/3
(兹
5
苶
3
苶
2
苶
)
4
(2)
4
16
16
5/2
16
1
5/2
(兹
1
1
苶
6
苶
)
5
4
1
5
10
1
24
(16)
2/5
(1
1
6)
2/5
(兹
5
苶
1
1
苶
6
苶
)
2
(1.7
1
411)
2
3.0
1
314
.
Rational exponents are frequently easier to work with than numbers expressed using
root signs. For example, suppose that we wish to evaluate [兹
5
苶
(兹2
苶
苶
)
3
苶
][兹
5
苶
(兹2
苶
苶
)
1
苶
7
苶
].
Using rational exponents, we write
冤
兹
5
苶
(兹2
苶
苶
)
3
苶
冥
as [(2
1/2
)
3
]
1/5
(2
3/2
)
1/5
2
3/10
,
and we write
冤
兹
5
苶
(兹2
苶
苶
)
1
苶
7
苶
冥
as [(2
1/2
)
17
]
1/5
(2
17/2
)
1/5
2
17/10
.
Thus,
冤
兹
5
苶
(兹2
苶
苶
)
3
苶
冥冤
兹
5
苶
(兹2
苶
苶
)
1
苶
7
苶
冥
(2
3/10
)(2
17/10
)2
3/1017/10
2
20/10
2
2
4.
N. Logarithms
Definition. The logarithm to the base q of a number x, symbolized “log
q
x,” is the
power to which one would have to raise the number q in order to get x. For example:
log
2
8 3 because 2
3
8.
log
2
16 4 because 2
4
16.
log
4
1
1
6
2 because 4
2
1
1
6
.
log
4
2
1
2
because 4
1/2
2.
log
64
4
1
3
because 64
1/3
4.
Rules for Logarithms
(1) log
q
1 0.
(2) log
q
ac log
q
a log
q
c.
(3) log
q
a
c
log
q
a log
q
c for c 0.
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(4) log
q
a
r
r log
q
a.
(5) log
q
1
c
log
q
c for c 0.
The natural logarithm, denoted “ln” (or “log”) is very important in statistics. It is the
logarithm to the base e. The number e (after the German mathematician, Euler) is an
irrational number defined as follows: e lim
x→∞
(1 1/x)
x
. The expression to the right
of the equals sign here means (in English) “the value approached by (i.e., the limiting
value, or limit, of) the expression (1 1/x)
x
, as x is allowed to increase without bound.”
This value can be estimated by plugging a very large value of x into the expression
(1 1/x)
x
. For example, if x 999999999, we have (1 1/999999999)
999999999
2.71828182438, or 2.718, to three decimal places. The natural log of x, written ln x or
log x, is therefore the power to which one must raise e to get x. It is defined only for
x 0.
Examples:
ln 1 0 because e
0
1.
ln 10 2.3026 because e
2.3026
10.
ln
1
2
.693 because e
.693
1
2
.
ln 500 6.215 because e
6.215
500.
Antilog Function. The function e
x
or exp(x) is called the exponential function. This
is also known as the antilogarithm, or antilog, of x, since it “undoes” the log func-
tion. Another way of stating this is that the exponential function is the inverse of the
log function, and vice versa. That is, e
ln x
x and ln e
x
x.
For example, ln e
2
is the power to which one must raise e to get e
2
, and this is
obviously 2, hence ln e
2
2. On the other hand, if one has the log of a number, and
one raises e to that power, one recovers the number again. Thus, 6.215 is the log of
500, so e
6.215
, or e
log 500
, gives us 500 again.
O. Absolute Value
The absolute value of a number a, denoted 冟a冟, is defined as follows:
(1) 冟a冟 a if a 0,
(2) 冟a冟a if a 0.
Examples: 冟3冟 3 (since 3 0); and 冟3冟 3 [since 3 0, its absolute value is
(3), or 3].
(3) For any real number, a, 兹a
苶
2
苶
冟a冟.
Examples: 兹3
苶
2
苶
冟3冟 3; 兹(
苶
3
苶
)
2
苶
冟(3)冟 3.
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If a and b are real numbers and n is an integer:
(4) 冟ab冟 冟 a冟冟b冟.
Example: 冟3(4)冟 冟 12冟 12 冟 3冟冟4冟.
(5)
冨
b
a
冨
冟
冟
a
b
冟
冟
.
Example:
冨
5
2
冨
冟
冟5
2
冟
冟
2
5
.
(6) 冟a
n
冟 冟 a冟
n
.
Example: 冟(3)
3
冟 冟 3冟
3
3
3
27.
Note: In general, 冟a b冟 冟 a冟 冟 b冟. For example, 冟3 4冟 1 冟 3冟 冟 4冟 7.
P. Functions
Functions and functional notation are very important in mathematics and statistics. In
the models in this book, for example, an outcome variable is typically modeled as a
function of one or more explanatory variables. Let’s define this notion more precisely.
Definition. A function is a rule of correspondence between the elements of two sets,
set x and set y. The elements are the respective values of x and y. It is usually posited
that y is a function of x, denoted y f(x), meaning that the values of y depend on the
values of x, often via some mathematical formula involving x. The function itself is
the rule that assigns a unique element from the set y to each element in the set x.
The simplest function is a constant: y f(x) 5 [or f(x) 5; or y 5]. This says
that the set y consists of one element, 5, which corresponds to each and every ele-
ment in the set x. Another example is f(x) 4x 7. In this case, 4x 7 is the rule
that assigns y to a given x value. When x 0, y 7. When x 1, y 3, and so
on. The notation “f(x)” is arbitrary. We could just as well use g(x), h(x), θ(x), or π(x).
As another example: g(x) x
2
3x 1.
It is important to understand that in the notation “f(x),” x is just a generic label or
a dummy variable. That is, it represents whatever is the subject of the operations
called for in the function. For example, if we let f(x) x
2
10, this notation essen-
tially says: “The outcome—f(x), or y—of this function is arrived at by squaring the
input—x—and then adding 10 to that result.” Hence, f(x 3) (x 3)
2
10,
f(ln x) (ln x)
2
10, f(e
x
) (e
x
)
2
10, and so on.
To speak of y as being a function of x also implies dependence of y on x. That is,
the value of y is generated by applying the rule (or formula) to different values of x.
It is in this sense that we will use the term function in this book. In social research,
y is the outcome variable, whose value is generated (in a causal sense) by values of
one or more explanatory variables.
Functions can be expressed in many different ways, with many different variables:
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