130 Chapter 4. Models for Binary Outcomes
Then, the discrete change for a change of δ in xk equals
ΔPr (y = 1 x)
Δxk
= Pr (y = 1 x, xk + δ) − Pr (y = 1 x, xk)
which can be interpreted as
For a change in variable xk from xk to xk + δ, the predicted probability of an event
changes by ΔPr(y=1x)
Δxk
, holding all other variables constant.
As shown in Figure 4.5, in general, the two measures of change are not equal. That is,
∂ Pr(y = 1 x)
∂xk =
ΔPr (y = 1 x)
Δxk
The measures differ because the marginal change is the instantaneous rate of change, while the
discrete change is the amount of change in the probability for a given finite change in one independent
variable. The two measures are similar, however, when the change occurs over a region of the
probability curve that is roughly linear.
The value of the discrete change depends on
1. The start level of the variable that is being changed. For example, do you want to examine the
effect of age beginning at 30? At 40? At 50?
2. The amount of change in that variable. Are you interested in the effect of a change of 1 year
in age? Of 5 years? Of 10 years?
3. The level of all other variables in the model. Do you want to hold all variables at their mean?
Or, do you want to examine the effect for women? Or, to compute changes separately for men
and women?
Accordingly, a decision must be made regarding each of these factors. See Chapter 3 for further
discussion.
For our example, let’s look at the discrete change with all variables held at their mean, which
is computed by default by prchange, where the help option is used to get detailed descriptions of
what the measures mean:
. prchange, help
logit: Changes in Predicted Probabilities for lfp
min->max 0->1 -+1/2 -+sd/2 MargEfct
k5 -0.6361 -0.3499 -0.3428 -0.1849 -0.3569
k618 -0.1278 -0.0156 -0.0158 -0.0208 -0.0158
age -0.4372 -0.0030 -0.0153 -0.1232 -0.0153
wc 0.1881 0.1881 0.1945 0.0884 0.1969
hc 0.0272 0.0272 0.0273 0.0133 0.0273
lwg 0.6624 0.1499 0.1465 0.0865 0.1475
inc -0.6415 -0.0068 -0.0084 -0.0975 -0.0084
NotInLF inLF
Pr(yx) 0.4222 0.5778
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4.6 Interpretation using predicted values 131
k5 k618 age wc hc lwg inc
x= .237716 1.35325 42.5378 .281541 .391766 1.09711 20.129
sd(x)= .523959 1.31987 8.07257 .450049 .488469 .587556 11.6348
Pr(yx): probability of observing each y for specified x values
AvgChg: average of absolute value of the change across categories
Min->Max: change in predicted probability as x changes from its minimum to
its maximum
0->1: change in predicted probability as x changes from 0 to 1
-+1/2: change in predicted probability as x changes from 1/2 unit below
base value to 1/2 unit above
-+sd/2: change in predicted probability as x changes from 1/2 standard
dev below base to 1/2 standard dev above
MargEfct: the partial derivative of the predicted probability/rate with
respect to a given independent variable
First consider the results of changes from the minimum to the maximum. There is little to be learned
by analyzing variables whose range of probabilities is small, such as hc, while age, k5, wc, lwg,
and inc have potentially important effects. For these we can examine the value of the probabilities
before and after the change by using the fromto option:
. prchange k5 age wc lwg inc, fromto
logit: Changes in Predicted Probabilities for lfp
from: to: dif: from: to: dif: from:
x=min x=max min->max x=0 x=1 0->1 x-1/2
k5 0.6596 0.0235 -0.6361 0.6596 0.3097 -0.3499 0.7398
age 0.7506 0.3134 -0.4372 0.9520 0.9491 -0.0030 0.5854
wc 0.5216 0.7097 0.1881 0.5216 0.7097 0.1881 0.4775
lwg 0.1691 0.8316 0.6624 0.4135 0.5634 0.1499 0.5028
inc 0.7326 0.0911 -0.6415 0.7325 0.7256 -0.0068 0.5820
to: dif: from: to: dif:
x+1/2 -+1/2 x-1/2sd x+1/2sd -+sd/2 MargEfct
k5 0.3971 -0.3428 0.6675 0.4826 -0.1849 -0.3569
age 0.5701 -0.0153 0.6382 0.5150 -0.1232 -0.0153
wc 0.6720 0.1945 0.5330 0.6214 0.0884 0.1969
lwg 0.6493 0.1465 0.5340 0.6204 0.0865 0.1475
inc 0.5736 -0.0084 0.6258 0.5283 -0.0975 -0.0084
NotInLF inLF
Pr(yx) 0.4222 0.5778
k5 k618 age wc hc lwg inc
x= .237716 1.35325 42.5378 .281541 .391766 1.09711 20.129
sd(x)= .523959 1.31987 8.07257 .450049 .488469 .587556 11.6348
We learn, for example, that varying age from its minimum of 30 to its maximum of 60 decreases
the predicted probability from .75 to .31, a decrease of .44. Changing family income (inc) from its
minimum to its maximum decreases the probability of a women being in the labor force from .73 to
.09. Interpreting other measures of change, the following interpretations can be made:
Using the unit change labeled -+1/2: For a woman who is average on all characteristics,
an additional young child decreases the probability of employment by .34.
Using the standard deviation change labeled -+1/2sd: A standard deviation change in
age centered around the mean will decrease the probability of working by .12, holding
other variables to their means.
Using a change from 0 to 1 labeled 0->1: If a woman attends college, her probability
of being in the labor force is .18 greater than a woman who does not attend college,
holding other variables at their mean.
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132 Chapter 4. Models for Binary Outcomes
What if you need to calculate discrete change for changes in the independent values that are not
the default for prchange (e.g., a change of 10 years in age rather than 1 year)? This can be done in
two ways:
Nonstandard discrete changes with prvalue command The command prvalue can be used to
calculate the change in the probability for a discrete change of any magnitude in an independent
variable. Say we want to calculate the effect of a ten-year increase in age for a 30-year old woman
who is average on all other characteristics:
. prvalue, x(age=30) save brief
Pr(y=inLFx): 0.7506 95% ci: (0.6771,0.8121)
Pr(y=NotInLFx): 0.2494 95% ci: (0.1879,0.3229)
. prvalue, x(age=40) dif brief
Current Saved Difference
Pr(y=inLFx): 0.6162 0.7506 -0.1345
Pr(y=NotInLFx): 0.3838 0.2494 0.1345
The save option preserves the results from the first call of prvalue. The second call adds the dif
option to compute the differences between the two sets of predictions. We find that an increase in
age from 30 to 40 years decreases a woman’s probability of being in the labor force by .13.
Nonstandard discrete changes with prchange Alternatively, we can use prchange with the
delta() and uncentered options. delta(#) specifies that the discrete change is to be computed
for a change of # units instead of a one-unit change. uncentered specifies that the change should
be computed starting at the base value (i.e., values set by the x() and rest() options), rather than
being centered around the base. In this case, we want an uncentered change of 10 units, starting at
age=30:
. prchange age, x(age=30) uncentered d(10) rest(mean) brief
min->max 0->1 +delta +sd MargEfct
age -0.4372 -0.0030 -0.1345 -0.1062 -0.0118
The result under the heading +delta is the same as what we just calculated using prvalue.
4.7 Interpretation using odds ratios with listcoef
Effects for the logit model, but not probit, can be interpreted in terms of changes in the odds. Recall
that for binary outcomes, we typically consider the odds of observing a positive outcome versus a
negative one:
Ω =
Pr(y = 1)
Pr(y = 0)
=
Pr(y = 1)
1 − Pr(y = 1)
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either electronically or in printed form, to others.
4.7 Interpretation using odds ratios with listcoef 133
Recall also that the log of the odds is called the logit and that the logit model is linear in the logit,
meaning that the log odds are a linear combination of the x’s and β’s. For example, consider a logit
model with three independent variables:
ln

Pr(y = 1 x)
1 − Pr(y = 1 x)
= lnΩ(x) = β0 + β1x1 + β2x2 + β3x3
We can interpret the coefficients as
For a unit change in xk, we expect the logit to change by βk, holding all other variables
constant.
This interpretation does not depend on the level of the other variables in the model. The problem
is that a change of βk in the log odds has little substantive meaning for most people (including the
authors of this book). Alternatively, by taking the exponential of both sides of this equation, we can
create a model that is multiplicative instead of linear, but in which the outcome is the more intuitive
measure, the odds:
Ω(x, x2) = eβ0eβ1x1eβ2x2eβ3x3
where we take particular note of the value of x2. If we let x2 change by 1,
Ω(x, x2 + 1) = eβ0eβ1x1eβ2(x2+1)eβ3x3
= eβ0eβ0eβ1x1eβ2x2eβ2eβ3x3
which leads to the odds ratio:
Ω(x, x2 + 1)
Ω(x, x2)
= eβ0eβ1x1eβ2x2eβ2eβ3x3
eβ0eβ1x1eβ2x2eβ3x3
= eβ2
Accordingly, we can interpret the exponential of the coefficient as
For a unit change in xk, the odds are expected to change by a factor of exp(βk), holding
all other variables constant.
For exp(βk) > 1, you could say that the odds are “exp(βk) times larger”. For exp(βk) < 1,
you could say that the odds are “exp(βk) times smaller”. We can evaluate the effect of a standard
deviation change in xk instead of a unit change:
For a standard deviation change in xk, the odds are expected to change by a factor of
exp(βk × sk ), holding all other variables constant.
The odds ratios for both a unit and a standard deviation change of the independent variables can be
obtained with listcoef:
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either electronically or in printed form, to others.
134 Chapter 4. Models for Binary Outcomes
. listcoef, help
logit (N=753): Factor Change in Odds
Odds of: inLF vs NotInLF
lfp b z P>z e^b e^bStdX SDofX
k5 -1.46291 -7.426 0.000 0.2316 0.4646 0.5240
k618 -0.06457 -0.950 0.342 0.9375 0.9183 1.3199
age -0.06287 -4.918 0.000 0.9391 0.6020 8.0726
wc 0.80727 3.510 0.000 2.2418 1.4381 0.4500
hc 0.11173 0.542 0.588 1.1182 1.0561 0.4885
lwg 0.60469 4.009 0.000 1.8307 1.4266 0.5876
inc -0.03445 -4.196 0.000 0.9661 0.6698 11.6348
b = raw coefficient
z = z-score for test of b=0
P>z = p-value for z-test
e^b = exp(b) = factor change in odds for unit increase in X
e^bStdX = exp(b*SD of X) = change in odds for SD increase in X
SDofX = standard deviation of X
Examples of interpretations are
For each additional young child, the odds of being employed decrease by a factor of
.23, holding all other variables constant.
For a standard deviation increase in the log of the wife’s expected wages, the odds of
being employed are 1.43 times greater, holding all other variables constant.
Being ten years older decreases the odds by a factor of .53 (=e[−.063]×10), holding all
other variables constant.
Other ways of computing odds ratios Odds ratios can also be computed with the or option for
logit. This approach does not, however, report the odds ratios for a standard deviation
change in the independent variables.
Multiplicative coefficients
When interpreting the odds ratios, remember that they are multiplicative. This means that positive
effects are greater than one and negative effects are between zero and one. Magnitudes of positive
and negative effects should be compared by taking the inverse of the negative effect (or vice versa).
For example, a positive factor change of 2 has the same magnitude as a negative factor change of
.5=1/2. Thus, a coefficient of .1=1/10 indicates a stronger effect than a coefficient of 2. Another
consequence of the multiplicative scale is that to determine the effect on the odds of the event not
occurring, you simply take the inverse of the effect on the odds of the event occurring. listcoef
will automatically calculate this for you if you specify the reverse option:
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either electronically or in printed form, to others.
4.7 Interpretation using odds ratios with listcoef 135
. listcoef, reverse
logit (N=753): Factor Change in Odds
Odds of: NotInLF vs inLF
lfp b z P>z e^b e^bStdX SDofX
k5 -1.46291 -7.426 0.000 4.3185 2.1522 0.5240
k618 -0.06457 -0.950 0.342 1.0667 1.0890 1.3199
age -0.06287 -4.918 0.000 1.0649 1.6612 8.0726
wc 0.80727 3.510 0.000 0.4461 0.6954 0.4500
hc 0.11173 0.542 0.588 0.8943 0.9469 0.4885
lwg 0.60469 4.009 0.000 0.5462 0.7010 0.5876
inc -0.03445 -4.196 0.000 1.0350 1.4930 11.6348
Note that the header indicates that these are now the factor changes in the odds of NotInLF versus
inLF, whereas before we computed the factor change in the odds of inLF versus NotInLF. We can
interpret the result for k5 as follows:
For each additional child, the odds of not being employed are increased by a factor of
4.3 (= 1/.23), holding other variables constant.
Effect of the base probability
The interpretation of the odds ratio assumes that the other variables have been held constant, but it
does not require that they be held at any specific values. While the odds ratio seems to resolve the
problem of nonlinearity, it is essential to keep the following in mind: A constant factor change in
the odds does not correspond to a constant change or constant factor change in the probability. For
example, if the odds are 1/100, the corresponding probability is .01.9 If the odds double to 2/100,
the probability increases only by approximately .01. Depending on one’s substantive purposes, this
small change may be trivial or quite important (such as when one identifies a risk factor that makes
it twice as likely that a subject will contract a fatal disease). Meanwhile, if the odds are 1/1 and
double to 2/1, the probability increases by .167. Accordingly, the meaning of a given factor change
in the odds depends on the predicted probability, which in turn depends on the levels of all variables
in the model.
Percent change in the odds
Instead of a multiplicative or factor change in the outcome, some people prefer the percent change,
100 [ exp (βk × δ) − 1]
which is listed by listcoef with the percent option.
9The formula for computing probabilities from odds is p = Ω
1+Ω.
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either electronically or in printed form, to others.
136 Chapter 4. Models for Binary Outcomes
. listcoef, percent
logit (N=753): Percentage Change in Odds
Odds of: inLF vs NotInLF
lfp b z P>z % %StdX SDofX
k5 -1.46291 -7.426 0.000 -76.8 -53.5 0.5240
k618 -0.06457 -0.950 0.342 -6.3 -8.2 1.3199
age -0.06287 -4.918 0.000 -6.1 -39.8 8.0726
wc 0.80727 3.510 0.000 124.2 43.8 0.4500
hc 0.11173 0.542 0.588 11.8 5.6 0.4885
lwg 0.60469 4.009 0.000 83.1 42.7 0.5876
inc -0.03445 -4.196 0.000 -3.4 -33.0 11.6348
With this option, the interpretations would be
For each additional young child, the odds of being employed decrease by 77%, holding
all other variables constant.
A standard deviation increase in the log of the wife’s expected wages increases the odds
of being employed by 83%, holding all other variables constant.
Percentage and factor change provide the same information; which you use for the binary model is
a matter of preference. While we both tend to prefer percentage change, methods for the graphical
interpretation of the multinomial logit model (Chapter 6) only work with factor change coefficients.
4.8 Other commands for binary outcomes
Logit and probit models are the most commonly used models for binary outcomes and are the only
ones that we consider in this book, but other models exist that can be estimated in Stata. Among
them, cloglog assumes a complementary log-log distribution for the errors instead of a logistic or
normal distribution. scobit estimates a logit model that relaxes the assumption that the marginal
change in the probability is greatest when Pr(y = 1) = .5. hetprob allows the assumed variance
of the errors in the probit model to vary as a function of the independent variables. blogit and
bprobit estimate logit and probit models on grouped (“blocked”) data. Further details on all of
these models can be found in the appropriate entries in the Stata manuals.
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either electronically or in printed form, to others.