- Logit vs probit: both model the probability of a yes/no outcome with an S-shaped curve. Logit uses the logistic curve, probit the normal curve. That is the only difference.
- Their coefficients are on different scales: logit coefficients are usually about 1.6 to 1.8 times the probit ones. In our 2,000 loan applications the ratio is 1.67 to 1.73.
- Neither coefficient is a change in probability. Report average marginal effects (AMEs): here a 10-point higher credit score raises the approval probability by about 2.2 percentage points in both models.
- Logit gives you odds ratios (self-employed applicants have 0.53 times the odds of approval); probit fits naturally into normal-error models such as selection and bivariate models.
- In practice the two give almost identical predictions (largest gap here: 1.7 percentage points). Choose by convention and convenience, not by fit.
Logit vs probit is one of the first choices you make when the dependent variable is binary: approved or rejected, employed or not, voted or stayed at home. This guide answers the questions students search for most: what the difference between logit and probit actually is, why the coefficients look so different, how to interpret them, what marginal effects and odds ratios are, and which model to use. Every number comes from one worked example, the loan-approval data from Econometrics Made Simple, Chapter 14, with code in Python, R and Stata so you can reproduce it.
The case: who gets a loan?
The file loans.csv holds 2,000 applications for personal loans. The outcome approved is 1 for the 72.15% of applications that were approved and 0 otherwise. The explanatory variables are the applicant’s credit score (450 to 850), annual income and the loan amount requested (both in $000, used in logs because they are skewed), the debt ratio (existing debt payments as a share of income) and a dummy for being self-employed. Like most of the book’s datasets, the loan data are simulated from a known model, which lets us check that the methods recover the truth.
The questions a bank, a regulator or an examiner asks are about probabilities. How much does a better credit score raise the chance of approval? How much lower is the chance for a self-employed applicant with the same finances? A binary model answers them, and the first decision is which one.
Why not just run OLS?
You can. OLS on a 0/1 outcome is the linear probability model (LPM), and its slopes are changes in probability, which is convenient. Its weakness is the straight line. Fitted to the loan data, the LPM predicts a probability below 0 for one applicant and above 1 for 153 others: 154 impossible “probabilities” out of 2,000. It also forces the effect of a credit score to be the same for an applicant who is almost certain to be rejected as for one who is almost certain to be approved, which is not how approval decisions work.[1]
Logit and probit fix this by passing the linear index \( \beta_0 + \beta_1x_1 + \cdots + \beta_kx_k \) through a function that always lies between 0 and 1. The function rises steeply in the middle and flattens towards 0 and 1, so a better credit score matters most for borderline applicants.
Logit vs probit: the only difference is the S-curve
Write \( z \) \( = x\beta \) \( = \beta_0 + \beta_1x_1 + \cdots + \beta_kx_k \). Both models say \( P(y = 1 \mid x) \) \( = G(z) \) for an increasing function \( G \) with \( G(-\infty) = 0 \) and \( G(\infty) = 1 \). They differ only in their choice of \( G \).
Here \( P \) is short for \( P(y=1 \mid x) \), and \( \Phi(z) \) \( = \int_{-\infty}^{z} \phi(t)\,dt \) is the area under the standard normal density \( \phi(t) \) \( = e^{-t^2/2}/\sqrt{2\pi} \) to the left of \( z \).
A neat way to see both models is the latent-variable story. Suppose the bank computes an unobserved score \( y^* = x\beta + e \) and approves whenever \( y^* > 0 \). If the error \( e \) has a logistic distribution you get logit; if it is standard normal you get probit. The logistic distribution has slightly fatter tails and a variance of \( \pi^2/3 \approx 3.29 \) instead of 1, which is why the two sets of coefficients come out on different scales.[2] Both models began in bio-assay, the study of how doses affect survival: Bliss introduced the probit in 1934[10] and Berkson argued for the logistic alternative in 1944.[9]
Chapter 14 of Econometrics Made Simple builds the LPM, logit and probit from scratch, including maximum likelihood, marginal effects and LR tests.Get the textbook · $15
Running logit and probit in Python, R and Stata
The code below fits both models to the same specification. The variable names match the data file; lincome and lamount are the logs of income and loan amount.
import numpy as np, pandas as pd
import statsmodels.formula.api as smf
loans = pd.read_csv("loans.csv")
loans["lincome"] = np.log(loans["income"])
loans["lamount"] = np.log(loans["loan_amount"])
f = "approved ~ credit_score + lincome + debt_ratio + self_employed + lamount"
logit = smf.logit(f, data=loans).fit()
probit = smf.probit(f, data=loans).fit()
print(logit.summary(), probit.summary())
print(logit.get_margeff(at="overall", dummy=True).summary()) # AMEslogit <- glm(f, family = binomial(link = "logit"), data = loans)
probit <- glm(f, family = binomial(link = "probit"), data = loans)
# AME of a continuous regressor: mean of g(xb) times its coefficient
mean(dlogis(predict(logit, type = "link"))) * coef(logit)["credit_score"]
mean(dnorm(predict(probit, type = "link"))) * coef(probit)["credit_score"]generate lincome = ln(income)
generate lamount = ln(loan_amount)
logit approved credit_score lincome debt_ratio i.self_employed lamount
margins, dydx(*) // average marginal effects
logit approved credit_score lincome debt_ratio i.self_employed lamount, or // odds ratios
probit approved credit_score lincome debt_ratio i.self_employed lamount
margins, dydx(*)In R, the margins and marginaleffects packages automate the AME calculation; the two lines above show what they compute. In Stata, writing i.self_employed tells margins to treat the dummy as a 0-to-1 switch rather than a tiny change.
Here are the estimates. Python, R and Stata give the same numbers.
| Variable | Logit coef. (z) | Probit coef. (z) | Logit ÷ probit |
|---|---|---|---|
| Intercept | −9.2401 (−11.28) | −5.3340 (−11.47) | 1.73 |
| credit_score | 0.0138 (15.03) | 0.0080 (15.62) | 1.72 |
| lincome | 0.9611 (4.31) | 0.5593 (4.30) | 1.72 |
| debt_ratio | −4.0723 (−10.23) | −2.3866 (−10.42) | 1.71 |
| self_employed | −0.6440 (−4.00) | −0.3834 (−3.99) | 1.68 |
| lamount | −0.2993 (−1.63) | −0.1793 (−1.68) | 1.67 |
| Log-likelihood | −974.11 | −974.20 | |
| McFadden pseudo R² | 0.1766 | 0.1765 |
Look at the pattern. The signs agree, the \( z \) statistics are almost identical, and the log-likelihoods differ by less than 0.1. What differs is the scale of the coefficients: every logit coefficient is about 1.7 times its probit partner.
Why logit coefficients are about 1.6 to 1.8 times probit coefficients
The coefficients only matter through \( G(x\beta) \). Because the logistic error has a standard deviation of \( \pi/\sqrt{3} \approx 1.81 \) and the probit error has a standard deviation of 1, logit has to stretch its index to produce the same probabilities. Dividing by 1.81 would match the spreads of the two curves, but it overcorrects in the middle, where most observations lie.
A better guide is the slope of each curve at its centre. The logistic density at 0 is \( 0.25 \); the normal density at 0 is \( 0.3989 \). Their ratio, \( 0.3989/0.25 \approx 1.6 \), is Amemiya’s well-known rule of thumb: logit coefficients are roughly 1.6 times probit coefficients.[3] When many observations are away from the centre, as here (72% are approved, so the typical applicant is up the curve), the ratio drifts towards 1.8. In the loan data the average logistic density is 0.1599 and the average normal density is 0.2743; their ratio, 1.72, is exactly the ratio of the coefficients. That is no coincidence: it is what makes the marginal effects in the next sections agree.
How to interpret logit and probit coefficients
A coefficient on its own tells you three things reliably: its sign (does the variable raise or lower the probability?), its significance (the \( z \) test) and its size relative to other coefficients in the same model. In both models the credit score and the debt ratio are strongly significant, and the loan amount is not significant at 5% (\( p \approx 0.10 \)).
What a coefficient is not is a change in probability. In logit, \( \beta_j \) is the change in the log-odds, \( \ln[p/(1-p)] \), for a one-unit rise in \( x_j \). In probit, \( \beta_j \) is the change in the index \( z \), measured in standard deviations of the latent error. Neither is something a bank manager can use directly.
Odds ratios: the logit bonus
Logit has one interpretation that probit lacks. Because \( \ln[p/(1-p)] \) \( = x\beta \), exponentiating a coefficient gives an odds ratio: the factor by which the odds of approval are multiplied when \( x_j \) rises by one unit, whatever the values of the other variables.
Turn the logit coefficients on self-employment, credit score and debt ratio into odds ratios and interpret them.
Odds ratios are constant, which makes them easy to report, but odds are not probabilities. Halving the odds of approval reduces the probability by a lot for a borderline applicant and by very little for someone who is almost certain to be approved.
Marginal effects: the number you should report
A marginal effect is the change in the probability of \( y = 1 \) for a small change in \( x_j \). Differentiating \( P = G(x\beta) \) gives
The effect depends on \( x \): it is largest where \( g \) is largest, in the middle of the curve. So you have to choose where to evaluate it. There are two standard choices.
| Measure | How it is computed | When to use it |
|---|---|---|
| Average marginal effect (AME) | Compute \( g(x_i\beta)\beta_j \) for every observation and average | The default in modern work: the average effect in your sample |
| Marginal effect at the mean (MEM) | Compute \( g(\bar{x}\beta)\beta_j \) at the sample means | Quick, but “the average applicant” may not exist (12% self-employed is not a person) |
| Discrete change (dummies) | \( P(y=1 \mid d=1, x) \) \( – P(y=1 \mid d=0, x) \), averaged over the sample | Always, for 0/1 regressors such as self_employed |
Find the AME of the credit score in the logit model by hand, and compare it with probit and the LPM.
The full set of average marginal effects tells the same story for every variable:
| Variable | LPM slope | Logit AME | Probit AME | Logit MEM |
|---|---|---|---|---|
| credit_score | 0.00222 | 0.00221 | 0.00220 | 0.00244 |
| lincome | 0.1550 | 0.1537 | 0.1534 | 0.1701 |
| debt_ratio | −0.6775 | −0.6511 | −0.6546 | −0.7206 |
| self_employed (0→1) | −0.1185 | −0.1110 | −0.1126 | −0.1282 |
| lamount | −0.0526 | −0.0479 | −0.0492 | −0.0530 |
Being self-employed lowers the approval probability by about 11 percentage points on average; a 0.10 rise in the debt ratio lowers it by about 6.5 points. The logit and probit AMEs agree to the second decimal place. The MEMs are about 10% larger, because the “average applicant” sits closer to the steep part of the curve than the typical real applicant does.[4]
Predicted probabilities for one applicant
Marginal effects are averages. To see the models at work, follow one applicant through both.
An employee applies with a credit score of 680, income of $50,000, a debt ratio of 0.30 and a loan request of $100,000. Find the approval probability in both models, then for the same person if self-employed.
The index numbers (1.31 against 0.77) look nothing alike. The probabilities they produce (0.787 against 0.781) are almost the same.
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Figure 1 shows the whole picture. The logit and probit curves are almost on top of each other across the range of credit scores, and the largest gap between their fitted probabilities for any of the 2,000 applicants is 0.017. The LPM line, by contrast, overshoots 1 above a score of about 800 and is too high at the bottom.
Which should you choose: logit or probit?
For a standard binary outcome with a reasonable share of both 0s and 1s, it rarely matters. In the loan data:
- log-likelihood −974.11 (logit) against −974.20 (probit); AIC 1960.2 against 1960.4;
- the fitted probabilities have a correlation of 0.9995;
- at a 0.5 cut-off, 76.55% of applications are classified correctly by logit and 76.50% by probit.
Those differences are far too small to choose on. Choose on what you need from the model.
| Choose logit when… | Choose probit when… |
|---|---|
| you want odds ratios (common in medicine, epidemiology and much of finance) | the model has other normal errors: Heckman selection, bivariate probit, IV probit, tobit-style models |
| the sample was drawn on the outcome (case-control); logit slopes remain consistent | you work with random-effects panel models, where the normal is standard |
| you plan to extend to several unordered choices (multinomial or conditional logit)[5] | your field’s convention is probit (much of labour economics and political science) |
The choice matters more when the outcome is rare (say under 5% of ones) or when you predict far into the tails, because the logistic tails are fatter than the normal tails. Even then, the honest approach is to estimate both and report whether your marginal effects change. If they do not, say so in a footnote and move on.
And the LPM? With AMEs this close (0.00222 against 0.00221), many applied economists report the LPM with robust standard errors as the main result and logit or probit as a check. Angrist and Pischke make this case.[6] Use logit or probit when you need sensible predicted probabilities, especially near 0 or 1.
Measuring fit: pseudo R² and percent correctly predicted
There is no ordinary \( R^2 \) in a maximum-likelihood model. The usual substitute is McFadden’s pseudo \( R^2 \), \( 1 – \ln L_{\text{model}} / \ln L_{\text{null}} \), where the null model has only an intercept.[7] Here it is \( 1 – (-974.11)/(-1183.06) \) \( = 0.1766 \). Values between 0.2 and 0.4 are often described as an excellent fit; it is not on the same scale as an OLS \( R^2 \), so do not compare the two.
The percent correctly predicted classifies an applicant as approved when \( \hat{p} > 0.5 \). Logit gets 76.55% right, which sounds impressive until you notice that predicting “approved” for everybody would be right 72.15% of the time. Always compare with that naive benchmark.
The joint test that all five slopes are zero is a likelihood-ratio test: \( LR \) \( = 2(-974.11 + 1183.06) \) \( = 417.9 \), against a \( \chi^2_5 \) 5% critical value of 11.07. The regressors clearly matter.
Practice questions
Use the estimates in the tables above. Give probabilities to 3 decimal places.
EasyQ1. The logit coefficient on self_employed is −0.6440. Compute the odds ratio and interpret it in one sentence. Show worked answer
EasyQ2. The credit-score coefficient is 0.0138 in logit and 0.0080 in probit. Does this mean the logit model finds a stronger effect of credit scores? Show worked answer
MediumQ3. For one applicant the logit index is \( x\beta = 1.3094 \). (a) Find the approval probability. (b) Find the marginal effect of the credit score (coefficient 0.013808) for this applicant. (c) Use it to predict the effect of 50 extra points, and explain why the exact change, which is 0.093, is smaller. Show worked answer
(b) \( p(1-p) \) \( = 0.7874 \times 0.2126 \) \( = 0.1674 \); \( 0.1674 \times 0.013808 \) \( = 0.00231 \) per point.
(c) Linear approximation: \( 50 \times 0.00231 \) \( = 0.116 \). The exact probability at the higher score is \( \Lambda(1.3094 + 50 \times 0.013808) \) \( = \Lambda(2.0) \) \( = 0.881 \), a rise of 0.093. The curve flattens as the probability approaches 1, so the slope at the starting point overstates a large change.
MediumQ4. In the probit model an employee has index 0.7741 and the self_employed coefficient is −0.3834. Find the change in approval probability if this person were self-employed. Show worked answer
ExamQ5. A logit model of loan approval gives a debt-ratio coefficient of −4.0723 and an AME of −0.6511; the LPM slope is −0.6775. (a) Interpret the AME for a rise in the debt ratio from 0.25 to 0.35. [2] (b) Find the odds ratio for the same rise. [2] (c) Explain why the AME and the LPM slope are similar while the logit coefficient is six times larger. [3] (d) The log-likelihood is −974.11, and −1183.06 with only an intercept. Test whether the five slopes are jointly zero at the 5% level. [3] Show worked answer
(b) \( e^{0.10 \times (-4.0723)} \) \( = e^{-0.4072} \) \( = 0.666 \): the odds of approval fall by about a third. [2]
(c) The logit coefficient measures the change in log-odds, not probability [1]. The AME multiplies it by the average of \( p(1-p) \), here 0.1599, which converts it to probability units [1]. Both the AME and the LPM slope are average effects on the probability, so they are close; the coefficient itself is on a different scale [1].
(d) \( LR \) \( = 2[-974.11 – (-1183.06)] \) \( = 417.9 \) [1]. Under \( H_0 \) it is \( \chi^2_5 \), 5% critical value 11.07 [1]. 417.9 > 11.07, so reject: the regressors jointly affect approval [1].
ExamQ6. A student reports that her logit model “predicts 76.55% of approvals correctly, so it fits well”. The sample approval rate is 72.15%. Evaluate the claim. [3] Show worked answer
Frequently asked questions
What is the difference between logit and probit?
Only the link function. Logit puts the linear index through the logistic CDF; probit puts it through the standard normal CDF. Equivalently, logit assumes a logistic latent error and probit a normal one. The predictions and marginal effects are usually almost the same.
Why are logit coefficients bigger than probit coefficients?
Because the logistic distribution is more spread out (standard deviation about 1.81 against 1). Matching the slopes of the two curves near their centre gives a factor of about 1.6; in samples where many probabilities are far from 0.5 the ratio is closer to 1.8. In the loan data it is 1.67 to 1.73.
Which is better, logit or probit?
Neither is better in general. Compare their log-likelihoods or AIC if you like, but differences are usually tiny (here 0.09 in log-likelihood). Pick logit if you want odds ratios or multinomial extensions, probit if the model sits in a normal-error framework, and report average marginal effects either way.
How do you interpret a probit coefficient?
Its sign and significance tell you the direction and whether the effect is real. Its size is the change in the normal index \( z \), so convert it to probability with \( \phi(x\beta)\beta_j \), averaged over the sample (the AME), or compute predicted probabilities at chosen values.
Is a marginal effect the same as an odds ratio?
No. An odds ratio, \( e^{\beta_j} \), is a constant multiplicative change in the odds. A marginal effect is an additive change in the probability, and it varies across observations. Odds ratios only exist for logit; marginal effects exist for both models.
Can I use OLS (the linear probability model) instead?
Often, yes, for average effects: the LPM slopes here are within a few per cent of the logit AMEs. Use robust standard errors, because the LPM error is heteroskedastic by construction, and switch to logit or probit when you need probabilities that stay between 0 and 1.
Is McFadden’s pseudo R² comparable to an OLS R²?
No. It is based on log-likelihoods, not on explained variance, and it tends to be much lower than an OLS \( R^2 \) for a model that fits equally well. Use it to compare binary models for the same outcome, not to compare with OLS.
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References
- Wooldridge, J. M. (2020). Introductory Econometrics: A Modern Approach, 7th edn, Chapter 7 (the linear probability model) and Chapter 17 (logit and probit). Boston: Cengage Learning.
- Stock, J. H. and Watson, M. W. (2020). Introduction to Econometrics, 4th edn, Chapter 11 “Regression with a Binary Dependent Variable”. Harlow: Pearson.
- Amemiya, T. (1981). Qualitative response models: a survey. Journal of Economic Literature, 19(4), 1483–1536.
- Cameron, A. C. and Trivedi, P. K. (2005). Microeconometrics: Methods and Applications, Chapter 14 “Binary Outcome Models”. Cambridge: Cambridge University Press.
- McFadden, D. (1974). Conditional logit analysis of qualitative choice behavior. In P. Zarembka (ed.), Frontiers in Econometrics, 105–142. New York: Academic Press.
- Angrist, J. D. and Pischke, J.-S. (2009). Mostly Harmless Econometrics: An Empiricist’s Companion, Section 3.4.2 on limited dependent variables and marginal effects. Princeton: Princeton University Press.
- Long, J. S. and Freese, J. (2014). Regression Models for Categorical Dependent Variables Using Stata, 3rd edn. College Station, TX: Stata Press.
- Ai, C. and Norton, E. C. (2003). Interaction terms in logit and probit models. Economics Letters, 80(1), 123–129.
- Berkson, J. (1944). Application of the logistic function to bio-assay. Journal of the American Statistical Association, 39(227), 357–365.
- Bliss, C. I. (1934). The method of probits. Science, 79(2037), 38–39.
