Left and Right Truncation in Regression

Why Truncation Direction Matters

In a truncated sample, some units are not observed at all because their outcomes fail a known inclusion rule. This differs from recording every unit but withholding or limiting some outcome values. In regression, the likelihood must account for the probability that an outcome would have met the rule, and that probability can vary with the covariates.

The direction of truncation determines which tail of the outcome distribution remains in the sample. If only outcomes above a lower threshold are observed, units with lower predicted outcomes may be less likely to appear. If only outcomes below an upper threshold are observed, units with higher predicted outcomes may be less likely to appear. The regression likelihood therefore includes a different normalizing probability in each case.

Definition: Left truncation at \(c_i\) means that unit \(i\) is observed only when its outcome exceeds the known threshold \(c_i\). Right truncation at \(c_i\) means that unit \(i\) is observed only when its outcome is below \(c_i\). For a continuous outcome, whether the rule uses a strict or non-strict inequality does not change the probabilities.

Regression Model and Inclusion Probabilities

Let \(Y_i^*\) denote the outcome that would be observed without truncation, and let \(x_i\) be a vector of observed covariates. Assume the normal linear regression model

Y_i^* = x_i'\beta + u_i,\qquad u_i\mid x_i \sim N(0,\sigma^2),

where \(\beta\) is the coefficient vector, \(\sigma>0\), and the errors are independent across units conditional on the covariates. Write \(\mu_i=x_i'\beta\). The threshold \(c_i\) is known; it may be the same for every unit or may vary in a known way across units.

For left truncation, define the inclusion event \(S_i=1\) when \(Y_i^*>c_i\). Standardizing the outcome gives

P(S_i=1\mid x_i) =P(Y_i^*>c_i\mid x_i) =1-\Phi\!\left(\frac{c_i-\mu_i}{\sigma}\right) =\Phi\!\left(\frac{\mu_i-c_i}{\sigma}\right).

For right truncation, \(S_i=1\) when \(Y_i^*<c_i\), so

P(S_i=1\mid x_i) =P(Y_i^*<c_i\mid x_i) =\Phi\!\left(\frac{c_i-\mu_i}{\sigma}\right).

Here \(\Phi\) is the standard normal cumulative distribution function. In either direction, the inclusion probability depends on the distance between the threshold and the conditional mean, measured in standard deviations. Consequently, two units facing the same threshold can have different inclusion probabilities when their covariates—and therefore their conditional means—differ.

Worked Example: A Left-Truncated Observation

Suppose \(Y_i^*\mid x_i\sim N(0,2^2)\), and a unit is observed only if \(Y_i^*>2\). For an observed value \(y_i=3\), calculate its density conditional on inclusion.

The inclusion probability is the upper-tail probability at standardized threshold \(2/2=1\):

P(Y_i^*>2\mid x_i)=1-\Phi(1)\approx 0.158655.

The untruncated normal density at \(y_i=3\) is

f(3\mid x_i) =\frac{1}{2}\phi\!\left(\frac{3-0}{2}\right) =\frac{\phi(1.5)}{2} \approx \frac{0.129518}{2} =0.064759.

Conditional on inclusion, the density is the untruncated density divided by the inclusion probability:

f(3\mid x_i,Y_i^*>2) =\frac{\phi(1.5)/2}{1-\Phi(1)} \approx \frac{0.064759}{0.158655} \approx 0.4082.

This is a density, not the probability of observing exactly \(3\); it has units of inverse outcome units. The conditional density is larger than the untruncated density at this point because it is part of a distribution normalized over outcomes above 2.

Conditional Likelihoods for the Two Directions

When only selected units are observed, the outcome density for an included unit is the original density restricted to the permitted region and divided by the probability of entering that region. Thus, for a left-truncated observation \(y_i>c_i\),

f_L(y_i\mid x_i,S_i=1) = \frac{ \frac{1}{\sigma}\phi\!\left(\frac{y_i-\mu_i}{\sigma}\right) }{ \Phi\!\left(\frac{\mu_i-c_i}{\sigma}\right) }.

For a right-truncated observation \(y_i<c_i\), the corresponding density is

f_R(y_i\mid x_i,S_i=1) = \frac{ \frac{1}{\sigma}\phi\!\left(\frac{y_i-\mu_i}{\sigma}\right) }{ \Phi\!\left(\frac{c_i-\mu_i}{\sigma}\right) }.

These formulas assume the sample consists of units satisfying the stated rule, the thresholds and covariates are observed for included units, and there is no additional unmodeled selection mechanism. They give the likelihood conditional on the observed covariates and on inclusion. The likelihood does not model the distribution of \(x_i\).

For \(n\) independent included observations, the conditional likelihood is the product of the appropriate density contributions. In the left-truncated case,

L_L(\beta,\sigma) = \prod_{i=1}^{n} \frac{ \frac{1}{\sigma}\phi\!\left(\frac{y_i-x_i'\beta}{\sigma}\right) }{ \Phi\!\left(\frac{x_i'\beta-c_i}{\sigma}\right) }, \qquad y_i>c_i.

In the right-truncated case,

L_R(\beta,\sigma) = \prod_{i=1}^{n} \frac{ \frac{1}{\sigma}\phi\!\left(\frac{y_i-x_i'\beta}{\sigma}\right) }{ \Phi\!\left(\frac{c_i-x_i'\beta}{\sigma}\right) }, \qquad y_i<c_i.

The denominator is not a constant adjustment: it contains \(\beta\) and \(\sigma\), so it changes as those parameters change. Omitting it would treat the selected observations as if they had been sampled from the untruncated regression distribution.

Theorem: Under the stated normal regression model and a known deterministic truncation rule, the density of an observed outcome conditional on its covariates and inclusion is its untruncated density divided by its inclusion probability. The denominator is the upper-tail probability for left truncation and the lower-tail probability for right truncation.

Why the Conditional Density Has This Form

For either rule, let \(A_i\) be the permitted outcome region: \(A_i=(c_i,\infty)\) under left truncation or \(A_i=(-\infty,c_i)\) under right truncation. For a value \(y_i\) inside that region, the definition of a conditional density gives

f(y_i\mid x_i,Y_i^*\in A_i) = \frac{f(y_i,Y_i^*\in A_i\mid x_i)} {P(Y_i^*\in A_i\mid x_i)}.

Because \(y_i\in A_i\), the event that the outcome equals \(y_i\) is already within the permitted region. Its joint density with that event is therefore \(f(y_i\mid x_i)\). Substitution yields

f(y_i\mid x_i,Y_i^*\in A_i) = \frac{f(y_i\mid x_i)} {P(Y_i^*\in A_i\mid x_i)}.

The result is a valid density on the permitted region: integrating its numerator over that region gives exactly the denominator, so the conditional density integrates to one. Applying the normal model to the numerator and the appropriate tail probability to the denominator produces the two likelihood contributions above.

Worked Example: A Two-Observation Left-Truncated Likelihood

Consider a hypothetical regression with \(\mu_i=\beta_0+\beta_1x_i\), candidate values \(\beta_0=0\), \(\beta_1=1\), and \(\sigma=1\). Both observations are included only when \(Y_i^*>0\). The observed covariates and outcomes are \((x_1,y_1)=(0,0.5)\) and \((x_2,y_2)=(1,1.5)\). Calculate the conditional likelihood at these candidate parameters.

For the first observation, \(\mu_1=0\), and its inclusion probability is \(P(Y_1^*>0\mid x_1)=\Phi(0)=0.5\). Its untruncated density at \(0.5\) is \(\phi(0.5)\approx0.352065\), so its conditional density contribution is

\frac{\phi(0.5)}{\Phi(0)} = \frac{0.352065}{0.5} \approx 0.7041.

For the second observation, \(\mu_2=1\), so its inclusion probability is \(P(Y_2^*>0\mid x_2)=\Phi(1)\approx0.841345\). Its residual is also \(0.5\), giving

\frac{\phi(1.5-1)}{\Phi(1)} = \frac{0.352065}{0.841345} \approx 0.4185.

By conditional independence, the likelihood at the candidate parameters is

L_L \approx 0.7041(0.4185) \approx 0.2947.

The two denominators differ even though both observations face the same threshold: their conditional means differ. This value is the likelihood at the specified parameters, not an estimate of the parameters; estimation would compare likelihood values across candidate parameter values.

Worked Example: Right Truncation and Covariate-Dependent Inclusion

Suppose a hypothetical model has \(\mu_i=1+0.5x_i\), \(\sigma=1\), and a known upper threshold \(c_i=2\). Units are observed only when \(Y_i^*<2\). Find the inclusion probabilities for \(x=0\) and \(x=2\), and calculate conditional densities for the observed values \(y=1.5\) and \(y=1\), respectively.

At \(x=0\), the mean is \(\mu=1\). The inclusion probability is

P(Y^*<2\mid x=0) =\Phi\!\left(\frac{2-1}{1}\right) =\Phi(1) \approx 0.841345.

The untruncated density at \(y=1.5\) is \(\phi(0.5)\approx0.352065\), so the conditional density is \(0.352065/0.841345\approx0.4185\).

At \(x=2\), the mean is \(\mu=2\). The inclusion probability is

P(Y^*<2\mid x=2) =\Phi\!\left(\frac{2-2}{1}\right) =\Phi(0) =0.5.

For the observed value \(y=1\), the untruncated density is \(\phi((1-2)/1)=\phi(-1)\approx0.241971\). Its conditional density is \(0.241971/0.5\approx0.4839\). In this example, the larger covariate value raises the conditional mean toward the upper cutoff, reducing the probability of inclusion. This conclusion follows from the specified model and rule; it should not be generalized to other coefficient signs or selection rules.

Common Pitfall and Practical Interpretation

A frequent error is to use the ordinary normal regression density for the observed outcomes and argue that the threshold is known, so it needs no likelihood adjustment. The threshold itself may indeed be known, but the probability of passing it is not generally fixed across observations or parameter values. For example, under left truncation the inclusion probability is \(\Phi((x_i'\beta-c_i)/\sigma)\), which changes with the covariates and with the parameters being estimated. The appropriate correction is to divide each untruncated density by that probability.

Another error is to use the same tail probability in both directions. For lower-tail exclusion with only \(y_i>c_i\) retained, use \(1-\Phi((c_i-\mu_i)/\sigma)\). For upper-tail exclusion with only \(y_i<c_i\) retained, use \(\Phi((c_i-\mu_i)/\sigma)\). Writing the selection event first is a reliable way to identify the correct tail.

Truncation can make the observed outcome distribution differ systematically from the population outcome distribution, even when the normal regression model for \(Y_i^*\mid x_i\) is correct. The conditional likelihood accounts for this selection in estimating the regression parameters. It does not, by itself, justify causal interpretation of coefficients or resolve selection processes beyond the stated threshold rule.

Key takeaway: State the inclusion rule before writing the likelihood. For each included observation, use its normal regression density divided by the probability—conditional on its covariates—that it satisfies that rule. Lower-tail truncation retains \(y_i>c_i\); upper-tail truncation retains \(y_i<c_i\).

Check Your Understanding

Use the selection event and the conditional normal model to answer these questions.

Under left truncation at \(c_i\), write the inclusion probability in terms of \(\mu_i=x_i'\beta\), \(\sigma\), and \(\Phi\).
Under right truncation at \(c_i\), which tail probability belongs in the denominator of the conditional density?
Why can two observations facing the same threshold have different inclusion probabilities?
In a left-truncated regression, what is wrong with maximizing only the product of the untruncated normal densities for the observed outcomes?
Suppose \(\mu=1\), \(\sigma=2\), and only outcomes below \(c=3\) are observed. Express the inclusion probability using \(\Phi\), and evaluate the standardized argument.