This page shows an example of truncated regression analysis with footnotes explaining the output. A truncated regression model predicts an outcome variable restricted to a truncated sample of its distribution. For example, if we wish to predict the age of licensed motorists from driving habits, our outcome variable is truncated at 16 (the legal driving age in the U.S.). While the population of ages extends below 16, our sample of the population does not. It is important to note the difference between truncated and censored data. In the case of censored data, there are limitations to the measurement scale that prevent us from knowing the true value of the dependent variable despite having some measurement of it. Consider the speedometer in a car. The speedometer may measure speeds up to 120 miles per hour, but all speeds equal to or greater than 120 mph will be read as 120 mph. Thus, if the speedometer measures the speed to be 120 mph, the car could be traveling 120 mph or any greater speed–we have no way of knowing. Censored data suggest limits on the measurement scale of the outcome variable, while truncated data suggest limits on the outcome variable in the sample of interest.
In this example, we will look at study of students in a special GATE (gifted and talented education) program. We wish to model achievement (achiv) as a function of gender, language skills and math skills (female, langscore and mathscore in the dataset). A major concern is that students require a minimum achievement score of 41 to enter the special program. Thus, the sample is truncated at an achievement score of 40.
First, we can examine the data.
use https://stats.idre.ucla.edu/stat/stata/dae/truncreg2, clear
summarize
Variable | Obs Mean Std. Dev. Min Max
-------------+--------------------------------------------------------
id | 178 103.6236 57.08957 3 200
achiv | 178 54.23596 8.96323 41 76
female | 178 .5505618 .4988401 0 1
langscore | 178 5.401124 .8944896 3.1 6.7
mathscore | 178 5.302809 .9483515 3.1 7.4
histogram achiv, bin(7) freq
tabulate female
female | Freq. Percent Cum.
------------+-----------------------------------
male | 80 44.94 44.94
female | 98 55.06 100.00
------------+-----------------------------------
Total | 178 100.00
Now, we can generate a truncated regression model in Stata using the truncreg command. We first list the outcome variable, then the predictors and the lower and/or upper limit.
Our data are only left-truncated, so we will only indicate a lower limit, ll(40).
truncreg achiv female langscore mathscore, ll(40)
(note: 0 obs. truncated)
Fitting full model:
Iteration 0: log likelihood = -580.98553
Iteration 1: log likelihood = -574.83026
Iteration 2: log likelihood = -574.53094
Iteration 3: log likelihood = -574.53056
Iteration 4: log likelihood = -574.53056
Truncated regression
Limit: lower = 40 Number of obs = 178
upper = +inf Wald chi2(3) = 89.85
Log likelihood = -574.53056 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
achiv | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
female | -2.290933 1.490333 -1.54 0.124 -5.211932 .6300663
langscore | 5.064698 1.037769 4.88 0.000 3.030709 7.098688
mathscore | 5.004054 .9555717 5.24 0.000 3.131168 6.87694
_cons | -.2940047 6.204858 -0.05 0.962 -12.4553 11.86729
-------------+----------------------------------------------------------------
/sigma | 7.739053 .5476443 14.13 0.000 6.66569 8.812416
------------------------------------------------------------------------------
Truncated Regression Output
(note: 0 obs. truncated)a Fitting full modelb: Iteration 0: log likelihood = -580.98553 Iteration 1: log likelihood = -574.83026 Iteration 2: log likelihood = -574.53094 Iteration 3: log likelihood = -574.53056 Iteration 4: log likelihood = -574.53056 Truncated regression Limit: lowerc = 40 Number of obsf = 178 upperd = +inf Wald chi2(3)g = 89.85 Log likelihoode = -574.53056 Prob > chi2h = 0.0000 ------------------------------------------------------------------------------ achivi| Coef.j Std. Err.k zl P>|z|m [95% Conf. Interval]n -------------+---------------------------------------------------------------- female | -2.290933 1.490333 -1.54 0.124 -5.211932 .6300663 langscore | 5.064698 1.037769 4.88 0.000 3.030709 7.098688 mathscore | 5.004054 .9555717 5.24 0.000 3.131168 6.87694 _cons | -.2940047 6.204858 -0.05 0.962 -12.4553 11.86729 -------------+---------------------------------------------------------------- /sigmao| 7.739053 .5476443 14.13 0.000 6.66569 8.812416 ------------------------------------------------------------------------------
a. (note: 0 obs. truncated) – This indicates how many observations in the model had outcome variable values below the lower limit or above the upper limit indicated in the function call. In this example, it is the number of observations where achiv \(\leq\) 40. The minimum value of achiv listed in the data summary was 41, so there were zero observations truncated.
b. Fitting full model – This is the iteration history of the truncated regression model. It lists the log likelihoods at each iteration. Truncated regression uses maximum likelihood estimation, which is an iterative procedure. The first iteration (called Iteration 0) is the log likelihood of the “null” or “empty” model; that is, a model with no predictors. At the next iteration (called Iteration 1), the specified predictors are included in the model. In this example, the predictors are female, langscore and mathscore. At each iteration, the log likelihood increases because the goal is to maximize the log likelihood. When the difference between successive iterations is very small, the model is said to have “converged” and the iterating stops. For more information on this process for binary outcomes, see Regression Models for Categorical and Limited Dependent Variables by J. Scott Long (page 52-61).
c. lower – This indicates the lower limit truncation specified for the outcome variable. In this example, the lower limit is 40.
d. upper – This indicates the upper limit specified for the outcome variable. In this example, we did not specify an upper limit, so it is assumed to be infinity.
e. Log likelihood – This is the log likelihood of the fitted model. It is used in the Likelihood Ratio Chi-Square test of whether all predictors’ regression coefficients in the model are simultaneously zero.
f. Number of obs – This is the number of observations in the dataset where the outcome and predictor variables all have non-missing values.
g. Wald chi2(3) -This is the Wald Chi-Square statistic. It is used to test the hypothesis that at least one of the predictors’ regression coefficient is not equal to zero. The number in the parentheses indicates the degrees of freedom of the Chi-Square distribution used to test the Wald Chi-Square statistic and is defined by the number of predictors in the model (3).
h. Prob > chi2 – This is the probability of getting a Wald test statistic as extreme as, or more so, than the observed statistic under the null hypothesis; the null hypothesis is that all of the regression coefficients across both models are simultaneously equal to zero. In other words, this is the probability of obtaining this chi-square statistic (89.85) or one more extreme if there is in fact no effect of the predictor variables. This p-value is compared to a specified alpha level, our willingness to accept a type I error, which is typically set at 0.05 or 0.01. The small p-value from the test, <0.0001, would lead us to conclude that at least one of the regression coefficients in the model is not equal to zero. The parameter of the chi-square distribution used to test the null hypothesis is defined by the degrees of freedom in the prior line, chi2(3).
i. achiv – This is the outcome variable predicted by the model.
j. Coef. – These are the regression coefficients. They are interpreted in the same manner as OLS regression coefficients: for a one unit increase in the predictor variable, the expected value of the outcome variable changes by the regression coefficient, given the other predictor variables in the model are held constant.
female – The expected achievement score for a female student is 2.290933 units lower than the expected achievement score for a male student while holding all other variables in the model constant. In other words, if two students, one female and one male, had identical language and math scores, the predicted achievement score of the male would be 2.290933 units higher than the predicted achievement score of the female student.
langscore – This is the estimated regression estimate for a one unit increase in langscore, given the other variables are held constant in the model. If a student were to increase her langscore by one point, her predicted achievement score would increase by 5.064698 units, while holding the other variables in the model constant. Thus, the students with higher language scores will have higher predicted achievement scores than students with lower language scores, holding the other variables constant.
mathscore – This is the estimated regression estimate for a one unit increase in mathscore, given the other variables are held constant in the model. If a student were to increase her mathscore by one point, her predicted achievement score would increase by 5.004054 units, while holding the other variables in the model constant. Thus, the students with higher math scores will have higher predicted achievement scores than students with lower math scores, holding the other variables constant.
_cons – This is the regression estimate when all variables in the model are evaluated at zero. For a male student (the variable female evaluated at zero) with langscore and mathscore of zero, the predicted achievement score is -0.2940047. Note that evaluating langscore and mathscore at zero is out of the range of plausible test scores.
k. Std. Err. – These are the standard errors of the individual regression coefficients. They are used in both the calculation of the z test statistic, superscript l, and the confidence interval of the regression coefficient, superscript n.
l. z – The test statistic z is the ratio of the Coef. to the Std. Err. of the respective predictor. The z value follows a standard normal distribution which is used to test against a two-sided alternative hypothesis that the Coef. is not equal to zero.
m. P>|z| – This is the probability the z test statistic (or a more extreme test statistic) would be observed under the null hypothesis that a particular predictor’s regression coefficient is zero, given that the rest of the predictors are in the model. For a given alpha level, P>|z| determines whether or not the null hypothesis can be rejected. If P>|z| is less than alpha, then the null hypothesis can be rejected and the parameter estimate is considered statistically significant at that alpha level.
female – The z test statistic for the predictor female is (-2.290933/1.490333) = -1.54 with an associated p-value of 0.124. If we set our alpha level to 0.05, we would fail to reject the null hypothesis and conclude that the regression coefficient for female has not been found to be statistically different from zero given langscore and mathscore are in the model.
langscore – The z test statistic for the predictor langscore is (5.064698/1.037769 ) = 4.88 with an associated p-value of <0.001. If we set our alpha level to 0.05, we would reject the null hypothesis and conclude that the regression coefficient for langscore has been found to be statistically different from zero given female and mathscore are in the model.
mathscore – The z test statistic for the predictor mathscore is (5.004054/0.9555717) = 5.24 with an associated p-value of <0.001. If we set our alpha level to 0.05, we would reject the null hypothesis and conclude that the regression coefficient for mathscore has been found to be statistically different from zero given female and langscore are in the model.
_cons – The z test statistic for the intercept, _cons, is (-0.2940047/6.204858) = -0.05 with an associated p-value of 0.962. If we set our alpha level at 0.05, we would fail to reject the null hypothesis and conclude that _cons has not been found to be statistically different from zero given female, langscore and mathscore are in the model and evaluated at zero.
n. [95% Conf. Interval] – This is the Confidence Interval (CI) for an individual coefficient given that the other predictors are in the model. For a given predictor with a level of 95% confidence, we’d say that we are 95% confident that the “true” coefficient lies between the lower and upper limit of the interval. It is calculated as the Coef. (zα/2)*(Std.Err.), where zα/2 is a critical value on the standard normal distribution. The CI is equivalent to the z test statistic: if the CI includes zero, we’d fail to reject the null hypothesis that a particular regression coefficient is zero given the other predictors are in the model. An advantage of a CI is that it is illustrative; it provides a range where the “true” parameter may lie.
o. /sigma – This is the estimated standard error of the regression. In this example, the value, 7.739053, is comparable to the root mean squared error that would be obtained in an OLS regression. If we ran an OLS regression with the same outcome and predictors, our RMSE would be 6.8549. This is indicative of how much the outcome varies from the predicted value. /sigma approximates this quantity for truncated regression.