Difference between logit and probit models

Question

Can anybody please tell me the difference between the logit and the probit model? I'm more interested here in knowing when to use logistic regression, and when to use probit. If there's any literature which define it using R software, that would be helpful as well.

Answer 1

The difference mainly in the link function.

In Logit: $\Pr(Y=1 \mid X) = [1 + e^{-X'\beta}]^{-1} $

In Probit: $\Pr(Y=1 \mid X) = \Phi(X'\beta)$ (Cumulative normal pdf)

In other way, logistic has slightly flatter tails. i.e probit curve approaches the axes more quickly than the curve.

Logit has better interpretation than probit. Logistic regression can be interpreted as modeling log odds. Usually people start the modeling with logit. You could use likelihood value to decide logit or probit.

Answer 2

+1 to both @vinux and @Elvis. Perhaps I can still contribute something here by providing a broader framework within which to think about these things and then using that to address the choice between logit and probit.

A standard linear model (e.g., a simple regression model) can be thought of as having two 'parts'. These are called the structural component and the random component. For example:
$$ Y=\beta_0+\beta_1X+\varepsilon \\ \text{where } \varepsilon\sim\mathcal{N}(0,\sigma^2) $$ The first two terms (i.e., $\beta_0+\beta_1X$) constitute the structural component, and the $\varepsilon$ (which indicates a normally distributed error term) is the random component. When the response variable is not normally distributed (e.g., if your response variable is binary) this approach may no longer be valid. The generalized linear model was developed to address such cases, and logit and probit models are special cases of GLiMs that are appropriate for binary variables (or multi-category response variables with some adaptations to the process). A GLiM has three parts, a structural component, a link function, and a response distribution. For example:
$$ g(\mu)=\beta_0+\beta_1X $$ Here $\beta_0+\beta_1X$ is again the structural component, $g()$ is the link function, and $\mu$ is a mean of a conditional response distribution at a given point in the covariate space. The way we think about the structural component here doesn't really differ from how we think about it with standard linear models; in fact, that's one of the great advantages of GLiMs. Because for many distributions the variance is a function of the mean, having fit a conditional mean (and given that you stipulated a response distribution), you have automatically accounted for the analog of the random component in a linear model (n.b., this can be more complicated in practice).

The link function is the key to GLiMs: since the distribution of the response variable is non-normal, it's what lets us connect the structural component to the response--it 'links' them (hence the name). It's also the key to your question, since the logit and probit are links (as @vinux explained), and understanding link functions will allow us to intelligently choose when to use which one. Although there can be many link functions that can be acceptable, often there is one that is special. Without wanting to get too far into the weeds (this can get very technical) the predicted mean, $\mu$, will not necessarily be mathematically the same as the response distribution's canonical location parameter; the link function that does equate them is the canonical link function. The advantage of this "is that a minimal sufficient statistic for $\beta$ exists" (German Rodriguez). The canonical link for binary response data (more specifically, the binomial distribution) is the logit. However, there are lots of functions that can map the structural component onto the interval $(0,1)$, and thus be acceptable; the probit is also popular, but there are yet other options that are sometimes used (such as the complementary log log, $\ln(-\ln(1-\mu))$, often called 'cloglog'). Thus, there are lots of possible link functions and the choice of link function can be very important. The choice should be made based on some combination of:

1. Knowledge of the response distribution,
2. Theoretical considerations, and
3. Empirical fit to the data.

Having covered a little of conceptual background needed to understand these ideas more clearly (forgive me), I will explain how these considerations can be used to guide your choice of link. (Let me note that I think @David's comment accurately captures why different links are chosen in practice.) To start with, if your response variable is the outcome of a Bernoulli trial (i.e., $0$ or $1$), your response distribution will be binomial, and what you are actually modeling is the probability of an observation being a $1$ (i.e., $\pi(Y=1)$). As a result, any function that maps the real number line, $(-\infty,+\infty)$, to the interval $(0,1)$ will work.

From the point of view of your substantive theory, if you are thinking of your covariates as directly connected to the probability of success, then you would typically choose logistic regression because it is the canonical link. However, consider the following example: You are asked to model high_Blood_Pressure as a function of some covariates. BP itself is normally distributed in the population (I don't actually know that, but it seems reasonable prima fascie), nonetheless, clinicians dichotomized it during the study (i.e., they only recorded 'high-BP' or 'normal'). In this case, probit would be preferable a-priori for theoretical reasons. This is what @Elvis meant by "your binary outcome depends on a hidden Gaussian variable". Another consideration is that both logit and probit are symmetrical, if you believe that the probability of success rises slowly from zero, but then tapers off more quickly as it approaches one, the cloglog is called for, etc.

Lastly, note that the empirical fit of the model to the data is unlikely to be of assistance in selecting a link, unless the shapes of the link functions in question differ substantially (of which, the logit and probit do not). For instance, consider the following simulation:

set.seed(1)
probLower = vector(length=1000)

for(i in 1:1000){

  x = rnorm(1000)
  y = rbinom(n=1000, size=1, prob=pnorm(x))

  logitModel  = glm(y~x, family=binomial(link="logit"))
  probitModel = glm(y~x, family=binomial(link="probit"))

  probLower[i] = deviance(probitModel)<deviance(logitModel)
}

sum(probLower)/1000
[1] 0.695

Even when we know the data were generated by a probit model, and we have 1000 data points, the probit model only yields a better fit 70% of the time, and even then, often by only a trivial amount. Consider the last iteration:

deviance(probitModel)
[1] 1025.759
deviance(logitModel)
[1] 1026.366
deviance(logitModel)-deviance(probitModel)
[1] 0.6076806

The reason for this is simply that the logit and probit link functions yield very similar outputs when given the same inputs.

The logit and probit functions are practically identical, except that the logit is slightly further from the bounds when they 'turn the corner', as @vinux stated. (Note that to get the logit and the probit to align optimally, the logit's $\beta_1$ must be $\approx 1.7$ times the corresponding slope value for the probit. In addition, I could have shifted the cloglog over slightly so that they would lay on top of each other more, but I left it to the side to keep the figure more readable.) Notice that the cloglog is asymmetrical whereas the others are not; it starts pulling away from 0 earlier, but more slowly, and approaches close to 1 and then turns sharply.

A couple more things can be said about link functions. First, considering the identity function ($g(\eta)=\eta$) as a link function allows us to understand the standard linear model as a special case of the generalized linear model (i.e., the response distribution is normal, and the link is the identity function). It's also important to recognize that whatever transformation the link instantiates is properly applied to the parameter governing the response distribution (i.e., $\mu$), not the actual response data. Finally, because in practice we never have the underlying parameter to transform, in discussions of these models, often what is considered to be the actual link is left implicit and the model is represented by the inverse of the link function applied to the structural component instead. That is:
$$ \mu=g^{-1}(\beta_0+\beta_1X) $$ For instance, logistic regression is usually represented: $$ \pi(Y)=\frac{\exp(\beta_0+\beta_1X)}{1+\exp(\beta_0+\beta_1X)} $$ instead of: $$ \ln\left(\frac{\pi(Y)}{1-\pi(Y)}\right)=\beta_0+\beta_1X $$

For a quick and clear, but solid, overview of the generalized linear model, see chapter 10 of Fitzmaurice, Laird, & Ware (2004), (on which I leaned for parts of this answer, although since this is my own adaptation of that--and other--material, any mistakes would be my own). For how to fit these models in R, check out the documentation for the function ?glm in the base package.

(One final note added later:) I occasionally hear people say that you shouldn't use the probit, because it can't be interpreted. This is not true, although the interpretation of the betas is less intuitive. With logistic regression, a one unit change in $X_1$ is associated with a $\beta_1$ change in the log odds of 'success' (alternatively, an $\exp(\beta_1)$-fold change in the odds), all else being equal. With a probit, this would be a change of $\beta_1 ~ z$'s. (Think of two observations in a dataset with $z$-scores of 1 and 2, for example.) To convert these into predicted probabilities, you can pass them through the normal CDF, or look them up on a $z$-table.

Answer 3

In addition to vinux’ answer, which already tells the most important:

• the coefficients $\beta$ in the logit regression have natural interpretations in terms of odds ratio;

• the probistic regression is the natural model when you think that your binary outcome depends of a hidden gaussian variable $Z = X' \beta + \epsilon\ $ [eq. 1] with $\epsilon \sim \mathcal N(0,1)$ in a deterministic manner: $Y = 1$ exactly when $Z > 0$.

• More generally, and more naturally, probistic regression is the more natural model if you think that the outcome is $1$ exactly when some $Z_0 = X' \beta_0 + \epsilon_0$ exceeds a threshold $c$, with $\epsilon \sim \mathcal N(0,\sigma^2)$. It is easy to see that this can be reduced to the aforementioned case: just rescale $Z_0$ as $Z = {1\over \sigma}(Z_0-c)$; it’s easy to check that equation [eq. 1] still holds (rescale the coefficients and translate the intercept). These models have been defended, for example, in medical contexts, where $Z_0$ would be an unobserved continuous variable, and $Y$ eg a disease which appears when $Z_0$ exceeds some "pathological threshold".

Both logit and probit models are only models. "All models are wrong, some are useful", as Box once said! Both models will allow you to detect the existence of an effect of $X$ on the outcome $Y$; except in some very special cases, none of them will be "really true", and their interpretation should be done with cautiousness.

Answer 4

An important point that has not been addressed in the previous (excellent) answers is the actual estimation step. Multinomial logit models have a PDF that is easy to integrate, leading to a closed-form expression of the choice probability. The density function of the normal distribution is not so easily integrated, so probit models typically require simulation. So while both models are abstractions of real world situations, logit is usually faster to use on larger problems (multiple alternatives or large datasets).

To see this more clearly, the probability of a particular outcome being selected is a function of the $x$ predictor variables and the $\varepsilon$ error terms (following Train)

$$ P = \int I[\varepsilon > -\beta'x] f(\varepsilon)d\varepsilon $$ Where $I$ is an indicator function, 1 if selected and zero otherwise. Evaluating this integral depends heavily on the assumption of $f(x)$. In a logit model, this is a logistic function, and a normal distribution in the probit model. For a logit model, this becomes

$$ P=\int_{\varepsilon=-\beta'x}^{\infty} f(\varepsilon)d\varepsilon\\ = 1- F(-\beta'x) = 1-\dfrac{1}{\exp(\beta'x)} $$

No such convenient form exists for probit models.

Answer 5

Regarding your statement

I'm more interested here in knowing when to use logistic regression, and when to use probit

There are already many answers here that bring up things to consider when choosing between the two but there is one important consideration that hasn't been stated yet: When your interest is in looking at within-cluster associations in binary data using mixed effects logistic or probit models, there is a theoretical grounding for preferring the probit model. This is, of course, assuming that there is no a priori reason for preferring the logistic model (e.g. if you're doing a simulation and know it to be the true model).

First, To see why this is true first note that both of these models can be viewed as thresholded continuous regression models. As an example consider the simple linear mixed effects model for the observation $i$ within cluster $j$:

$$ y^{\star}_{ij} = \mu + \eta_{j} + \varepsilon_{ij} $$

where $\eta_j \sim N(0,\sigma^2)$ is the cluster $j$ random effect and $\varepsilon_{ij}$ is the error term. Then both the logistic and probit regression models are equivalently formulated as being generated from this model and thresholding at 0:

$$ y_{ij} = \begin{cases} 1 & \text{if} \ \ \ y^{\star}_{ij}≥0\\ \\ 0 &\text{if} \ \ \ y^{\star}_{ij}<0 \end{cases} $$

If the $\varepsilon_{ij}$ term is normally distributed, you have a probit regression and if it is logistically distributed you have a logistic regression model. Since the scale is not identified, these residuals errors are specified as standard normal and standard logistic, respectively.

Pearson (1900) showed that that if multivariate normal data were generated and thresholded to be categorical, the correlations between the underlying variables were still statistically identified - these correlations are termed polychoric correlations and, specific to the binary case, they are termed tetrachoric correlations. This means that, in a probit model, the intraclass correlation coefficient of the underlying normally distributed variables:

$$ {\rm ICC} = \frac{ \hat{\sigma}^{2} }{\hat{\sigma}^{2} + 1 } $$

is identified which means that in the probit case you can fully characterize the joint distribution of the underlying latent variables.

In the logistic model the random effect variance in the logistic model is still identified but it does not fully characterize the dependence structure (and therefore the joint distribution), since it is a mixture between a normal and a logistic random variable that does not have the property that it is fully specified by its mean and covariance matrix. Noting this odd parametric assumption for the underlying latent variables makes interpretation of the random effects in the logistic model less clear to interpret in general.