Week 12: Linear Probability Models, Logistic and

Probit

Marcelo Coca Perraillon

University of Colorado
Anschutz Medical Campus

Health Services Research Methods I

HSMP 7607
2019

These slides are part of a forthcoming book to be published by Cambridge

University Press. For more information, go to perraillon.com/PLH. c This
material is copyrighted. Please see the entire copyright notice on the book’s
website.
Updated notes are here: https://clas.ucdenver.edu/marcelo-perraillon/
teaching/health-services-research-methods-i-hsmp-7607 1
Outline

Modeling 1/0 outcomes

The “wrong” but super useful model: Linear Probability Model
Deriving logistic regression
Probit regression as an alternative

2
Binary outcomes

Binary outcomes are everywhere: whether a person died or not, broke

a hip, has hypertension or diabetes, etc
We typically want to understand what is the probability of the binary
outcome given explanatory variables
It’s exactly the same type of models we have seen during the
semester, the difference is that we have been modeling the conditional
expectation given covariates: E [Y |X ] = β0 + β1 X1 + · · · + βp Xp
Now, we want to model the probability given covariates:
P(Y = 1|X ) = f (β0 + β1 X1 + · · · + βp Xp )
Note the function f() in there

3
Linear Probability Models

We could actually use our vanilla linear model to do so

If Y is an indicator or dummy variable, then E [Y |X ] is the proportion
of 1s given X , which we interpret as the probability of Y given X
The parameters are changes/effects/differences in the probability of Y
by a unit change in X or for a small change in X
If an indicator variable, then change from 0 to 1
For example, if we model diedi = β0 + β1 agei + i , we could interpret
β1 as the change in the probability of death for an additional year of
age

4
Linear Probability Models
The problem is that we know that this model is not entirely
correct. Recall that in the linear model we assume
Y ∼ N(β0 + β1 X1 + · · · + βp Xp , σ 2 ) or equivalently, i ∼ N(0, σ 2 )
That is, Y distributes normal conditional on Xs or the error
distributes normal with mean 0
Obviously, a 1/0 variable can’t distribute normal, and i can’t be
normally distributed either
We also know that we needed the normality assumption for
inference, not to get best betas
The big picture: Using the linear model for a 1/0 outcomes is
mostly wrong in the sense that the SEs are not right
Yet, the effects of covariates on the probability of the outcome are
more often than not fine
So LPM is the wrong but super useful model because changes can
be interpreted in the probability scale
5
Example
Data from the National Health and Nutrition Examination Survey
(NHANES)
We model the probability of hypertension given age
reg htn age

Source | SS df MS Number of obs = 3,529

-------------+---------------------------------- F(1, 3527) = 432.78
Model | 59.3248737 1 59.3248737 Prob > F = 0.0000
Residual | 483.471953 3,527 .137077389 R-squared = 0.1093
-------------+---------------------------------- Adj R-squared = 0.1090
Total | 542.796826 3,528 .153853976 Root MSE = .37024

------------------------------------------------------------------------------
htn | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
age | .0086562 .0004161 20.80 0.000 .0078404 .009472
_cons | -.1914229 .0193583 -9.89 0.000 -.2293775 -.1534682
------------------------------------------------------------------------------

An additional year of life increases the probability of hypertension by

0.8 percent. See, neat
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Example
Plot predicted values against observed values (note the jittering)
reg htn age
predict htnhat_lpm
scatter htn age, jitter(5) msize(tiny) || line htnhat_lpm age, sort ///
legend(off) color(red) saving(hlpm.gph, replace)
graph export hlpm.png, replace
di _b[_cons] + _b[age]*20

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Linear Probability Models

You can see the first problem with the LPMs

The relationship between age (or any other variable) cannot be
linear. Probabilities need to be constrained to be between 0 and 1
In this example, the probability of hypertension for a 20 y/o is
-.0182996
Is this a big problem in this example? No, because on average the
probability of hypertension is 0.19, which is not close to zero
Lowess is neat for seeing 0/1 variables conditional on a variable

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Lowess
Lowess can show you how the relationship between the indicator
variable and the explanatory variable looks like
scatter htn age, jitter(5) msize(vsmall) || line lhtn age, sort ///
legend(off) color(red) saving(hlowess.gph, replace) ///
title("Lowess")

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LPN, more issues

The variance of a 1/0 (binary) depends on the values of X so there is

always heteroskedasticity: var (y |x) = p(x)[1 − p(x)]
We know that we need this assumption for correct SEs and F tests.
So we can correct SEs in LPMs using the robust option (Huber-White
SEs; aka sandwich estimator)
Still, we do know that SEs are not totally correct because they
do not distribute normal either, even is we somehow correct for
heteroskedasticity
But at the very least, use the robust option by default

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So why do we use LPMs?
Not long ago, maybe 10 or 15 years ago (it’s always 10 to 15 to 20
years ago), you couldn’t use other alternatives with large datasets
(logistic, probit)
It would take too long to run the models or they wouldn’t run;
researchers would take a sample and run logit or probit as a sensitivity
analysis
The practice still lingers in HSR and health economics
The main reason to keep using LPM as a first step in modeling, it’s
because the coefficients are easy to interpret
In my experience, if the average of the outcome is not close to 0 or 1,
not much difference between LPM or logit/probit (but SEs can
change, although not by a lot)
But not a lot of good reasons to present LPM results in papers
anymore, except maybe in difference-in-difference models
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One more time

Other than interpretation of coefficients or a first pass to modeling,

there are NO GOOD REASONS TO USE THE LPM model
Some researches (ok, economists, mostly) truly love the LPN because
the parameters are easy to interpret and often the effects are close
enough
Yet, in some cases, the effects could be off, too
But it’s the wrong model. Use a probit or logit, period

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Logistic or logit model
Logistic models can be derived in several ways, which makes learning
confusing since you can read different versions
In the MLE lecture we derived the model assuming that the outcome
1/0 distributes Bernoulli and that observations were iid . We will
extend that example today
An (almost) identical way is to assume that the outcome comes
from a Binomial distribution since the Binomial is the sum of iid
Bernoulli random variables
A third way is to assume that there is a latent and continuous
variable that distributes logistic (yes, there is also a logistic pdf), or
probit, but we only get to observe a 1 or 0 when the latent variable
crosses a threshold
You get to the same model but the latent interpretation has a
bunch of applications ins economics (for example, random utility
models) and psychometrics (the latent variable is “ability” but you
only observed if a person answers a question correctly, a 1/0)
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Recall the Logit MLE from the MLE lecture

Remember that if we have an iid Bernoulli rv we can write the joint

probability of observing the data as:
L(p) = ni=1 p yi (1 − p)1−yi
Q

As we saw before, taking the log makes the analytical and

computational calculations easier:
lnL(p) = ni=1 yi ln(p) + ni=1 (1 − yi )ln(1 − p)
P P

The log likelihood simplifies to:

lnL(p) = nȳ ln(p) + (n − nȳ )ln(1 − p)
If we take the derivative with respect to p and set it equal to zero we
find the MLE estimators. The SEs can be calculated using the second
derivatives (see the MLE lecture)

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Adding covariates

We can of course make p a function of covariates but the function

can’t be something like p = β0 + β1 X1 + · · · + βp Xp since the p must
be bounded between 0 and 1
So we use a transformation. One of those transformations is the
logistic response function:
ex 1
π= 1+e x = 1+e −x
, where x is any real number
π(x) is then restricted to be between 0 and 1
Confusion alert: Note that there is a logistic response function and
also the logistic distribution. Here we are using the response function.
In the latent derivation, the logistic distribution is used
See more here:
https://en.wikipedia.org/wiki/Logistic_distribution

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Logistic response function
If we constrain the response to be between 0 and 1, it can’t be linear
with respect to X
twoway function y=exp(x) / (1+ exp(x)), range(-10 10) saving(l1.gph, replace)
twoway function y=exp(-x) / (1+ exp(-x)), range(-10 10) saving(l2.gph, replace)
graph combine l1.gph l2.gph, xsize(20) ysize(10)
graph export lboth.png, replace

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Logistic or logit model

Notice a couple of things.The effect of x on π is not linear; the effect

depends on the value of x
But we can make the function linear using the so-called logit
transformation
π
ln( 1−π )=x
I made you go the other way in one homework. If you solve for π you
get to the logistic response function
More general, the model is:
p
ln( 1−p ) = β0 + β1 X1 + · · · + βp Xp , which transformed is
e β0 +β1 X1 +···+βp Xp
p= 1+e β0 +β1 X1 +···+βp Xp
1
Can also be written as: p = 1+e −(β0 +β1 X1 +···+βp Xp )

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Logistic MLE

When extending the log-likelihood function to make p a function of

covariates we use the logistic response function to constraint p
And that’s what creates a bunch of interpretation problems. The
estimated betas are changes in the log-odds scale
From now on, always, always write logistic models like this:
pi
ln( 1−p i
) = β0 + β1 X1i + · · · + βp Xpi
That’s what Stata (SAS or R) estimate. The betas are changes in
the log-odds scale
We will see next class that we could write effects as odds ratios but
I’ll tell you that odds ratios are confusing

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Yet another way

What about if we assume there is a latent (unobserved) variable y ∗

that is continuous. Think about it as a measure of illness
If y ∗ crosses a threshold, then the person dies. We only observe if the
person died but we can’t observe the latent variable
We can write this problem as
P(y = 1|X ) = P(β0 + β1 X1 + · · · + βp Xp + u > 0) = P(−u <
β0 + β1 X1 + · · · + βp Xp ) = F (β0 + β1 X1 + · · · + βp Xp )
F() is the cdf of -u. If we assume logistic distribution, we get logistic
regression, if we assume cumulative normal, we get a probit model
See Cameron and Trivedi Chapter 14, section 14.3.1
Both models are similar because the logistic distribution and the
normal and almost the same

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Compare logistic distribution and normal

Stata doesn’t have a logistic distribution but you can simulate any
probability distribution using the uniform distribution and plugging in
into the inverse of the pdf you want
clear
set seed 123456
set obs 5000
gen u = uniform()

* Simulate logistic distribution

gen l = -ln((1 - u)/u)
sum l

* Simulated normal with same parameters

gen n = rnormal(r(mean), r(sd))

* Plot
kdensity l, bw(0.3) gen(xl dl)
kdensity n, bw(0.3) gen(xn dn)
line dl xl, sort color(red) || line dn xn, sort ///
title("Logistic (red) vs normal distribution") ytitle("Density") ///
xtitle("x") legend(off)

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Standard logistic vs standard normal

Does it make much of a difference if we use one vs the other?

clear
set seed 123456
set obs 5000
gen u = uniform()

* Simulate logistic distribution

gen l = -ln((1 - u)/u)
sum l

* Simulated normal with same parameters

gen n = rnormal(r(mean), r(sd))

* Plot
kdensity l, bw(0.3) gen(xl dl)
kdensity n, bw(0.3) gen(xn dn)
line dl xl, sort color(red) || line dn xn, sort ///
title("Logistic (red) vs normal distribution") ytitle("Density") ///
xtitle("x") legend(off)
graph export logvsnorm.png, replace

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Logistic vs normal
Assuming either one as the latent distributions makes little difference

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Big picture

Not a big difference in the probability scale between probit and logit
If you are an economist you run probit models; for the rest of the
world, there is the logistic model
IMPORTANT: There is a big difference in terms of interpreting a
regression output because the coefficients are estimated in different
scales
In the logistic model the effect of a covariate can be made linear in
terms of the odds-ratio; you can’t do the same in probit models
We will see examples of both

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Digression: How to simulate a logistic model

set seed 12345678

set obs 10000
gen x1 = rchi2(1)+2

* Make the probability a function of x1

gen pr1 = exp(-1+0.1*x1) /(1+exp(-1+0.1*x1))

* Generate a one-trial binomial (Bernoulli)

gen y = rbinomial(1, pr1)
list y in 1/10
+---+
| y |
|---|
1. | 0 |
2. | 1 |
3. | 1 |
4. | 0 |
5. | 1 |
|---|
6. | 0 |
7. | 0 |
8. | 0 |
9. | 1 |
10. | 0 |
+---+

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Simulation

Note that the parameters match the ones I used in the gen step in
previous slide
logit y x1

Iteration 0: log likelihood = -6351.6567

Iteration 1: log likelihood = -6328.5462
Iteration 2: log likelihood = -6328.4883
Iteration 3: log likelihood = -6328.4883

Logistic regression Number of obs = 10,000

LR chi2(1) = 46.34
Prob > chi2 = 0.0000
Log likelihood = -6328.4883 Pseudo R2 = 0.0036

------------------------------------------------------------------------------
y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x1 | .0998315 .0145838 6.85 0.000 .0712478 .1284153
_cons | -1.004065 .0493122 -20.36 0.000 -1.100715 -.9074147
------------------------------------------------------------------------------

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Big picture

In the next weeks, we will (again) focus on parameter interpretation

You already have most of the tools to do modeling but we need to
adapt them to these types of models
The tricky part about logistic or probit models or other types of
models is to move from the world in which relationships are linear to
the world in which they are not
There is no R 2 but there are other ways to check the predictive ability
and fit of models. Mostly, we do it using the estimated log-likelihood

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Comparing LPM, logit, and probit models

For now, let’s compare models

* LPM
qui reg htn age
est sto lpm
predict hatlpm

* Logit
qui logit htn age
est sto logit
predict hatlog

* Probit
qui probit htn age
est sto prob
predict hatprob

line hatlpm age, sort color(blue) || line hatlog age, sort || ///
line hatprob age, sort legend(off) saving(probs.gph, replace)
graph export prob.png, replace

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Predicted probabilities
Note that probabilities are not linear with probit and logit even
though we wrote models in the same way
Note that there is practically no difference between the logit and
probit models in this example

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But wait, look closely... What about effects?
In the linear model the effect of X (age in this case) is always the
same regardless of age. It’s the slope for small changes and when we
use marginal effects
See graph again. Around 45 the effect will be identical (similar
slopes). But not around 25 or 65

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Effect at different points

Comparing logit and LPM for now (not probit)

For the linear model, the effect is, well, linear, so always the same:
0.0086562
qui reg htn age
margins, dydx(age) at(age=(25 45 65))
Conditional marginal effects Number of obs = 3,529
Model VCE : OLS
Expression : Linear prediction, predict()
dy/dx w.r.t. : age
1._at : age = 25
2._at : age = 45
3._at : age = 65
------------------------------------------------------------------------------
| Delta-method
| dy/dx Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
age |
_at |
1 | .0086562 .0004161 20.80 0.000 .0078404 .009472
2 | .0086562 .0004161 20.80 0.000 .0078404 .009472
3 | .0086562 .0004161 20.80 0.000 .0078404 .009472
------------------------------------------------------------------------------

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Effect at different points

But for logit the effect will be different. Note that the difference is
quite large
qui logit htn age
margins, dydx(age) at(age=(25 45 65)) vsquish
Conditional marginal effects Number of obs = 3,529
Model VCE : OIM
Expression : Pr(htn), predict()
dy/dx w.r.t. : age
1._at : age = 25
2._at : age = 45
3._at : age = 65
------------------------------------------------------------------------------
| Delta-method
| dy/dx Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
age |
_at |
1 | .0030922 .0001522 20.32 0.000 .0027939 .0033904
2 | .0084346 .0004166 20.25 0.000 .007618 .0092511
3 | .0149821 .0009574 15.65 0.000 .0131057 .0168585
------------------------------------------------------------------------------

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But, does this matter?

Well, yes and no

If you are using the model to understand how age affects the
probability of hypertension, yes, of course it matters
If you want to understand, the effect of age on p(htn) for most
people (on average) then it doesn’t matter because the average age
on the sample is 45 and at 45 the effect is the same
Yet another reason to use the LPM as a first pass, not as your final
tool
At the very least you must understand these issues
I’m with Will Manning on this issue. The LPM is the wrong model
(but useful)

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Coefficients in the estimation scale?

The coefficients are very different because they are measured in

different scales
. est table lpm logit prob, p

-----------------------------------------------------
Variable | lpm logit prob
-------------+---------------------------------------
_ |
age | .00865616
| 0.0000
_cons | -.19142286
| 0.0000
-------------+---------------------------------------
htn |
age | .0623793 .0352797
| 0.0000 0.0000
_cons | -4.4563995 -2.5542014
| 0.0000 0.0000
-----------------------------------------------------
legend: b/p

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Similar in probability scale
We will learn how to make them comparable using the probability
scale, which is what we really care about. The margins command is
computing an average effect across values of age
. * LPM
qui reg htn age
margins, dydx(age)
------------------------------------------------------------------------------
| Delta-method
| dy/dx Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
age | .0086562 .0004161 20.80 0.000 .0078404 .009472
------------------------------------------------------------------------------

* Logit
qui logit htn age
margins, dydx(age)
------------+----------------------------------------------------------------
age | .0084869 .0004066 20.87 0.000 .00769 .0092839
------------------------------------------------------------------------------

* Probit
qui probit htn age
margins, dydx(age)
-------------+----------------------------------------------------------------
age | .0084109 .0003917 21.47 0.000 .0076432 .0091787
------------------------------------------------------------------------------

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Summary

There are many ways to arrive to the logistic model

The big picture is that we are trying to model a probability, which
must be bounded between 0 and 1
If it’s bounded, then the effect of any covariate on the probability
cannot be linear
The tricky part is to learn how to interpret parameters. The short
story is that we estimate (assuming one predictor)
p e β0 +β1 x
log ( 1−p ) = β0 + β1 x but we care about p = 1+e β0 +β1 x

To make life easier, logistic models are often interpreted using odds
ratios but odds ratios can be misleading
In the probit model, we interpret parameters as shifts in the
cumulative normal, even less intuitive

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