Scatterplots

Scatterplots of achievement and language test score by program type / overall programs.

Boxplots

In this dataset, “prog” is the type of program the student is in. It is a categorical (nominal) variable that takes on three values: academic (prog = 1), general (prog = 2), and vocational (prog = 3). Note that here a major concern is that students are required to have a minimum “achiv” (achievement) score of 40 to enter the special program. Thus, the sample is truncated at an achievement score of 40. Looking at the histogram, we can see the scale for achiv is truncated at 40.

Truncated Regression

The variable “langscore” (language score) is statistically significant. Interpreation: . A unit increase in language score leads to a 0.71 increase in predicted achievement.

One of the indicator variables for prog is also statistically significant (reference program: general).

. Compared to general programs, academic programs are about 4.07 higher.

model <- truncreg(achiv ~ langscore + prog, data = dat, point = 40, direction = "left")

summary(model)


Call:
truncreg(formula = achiv ~ langscore + prog, data = dat, point = 40, 
    direction = "left")

BFGS maximization method
57 iterations, 0h:0m:0s 
g'(-H)^-1g = 2.5E-05 
 


Coefficients :
             Estimate Std. Error t-value  Pr(>|t|)    
(Intercept)  11.29942    6.77173  1.6686   0.09519 .  
langscore     0.71267    0.11446  6.2264 4.773e-10 ***
progacademic  4.06267    2.05432  1.9776   0.04797 *  
progvocation -1.14422    2.66958 -0.4286   0.66821    
sigma         8.75368    0.66647 13.1343 < 2.2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Log-Likelihood: -591.31 on 5 Df

Testing Significance

We saw that compared to general programs, academic programs are about 4.07 higher in achievement. To determine if program itself is statistically significant, we can test models with and without it for the two degree-of-freedom test of this variable using a likelihood ratio test (LRT). The LRT is associated with a p-value of 0.0252, indicating that the overall effect of prog is statistically significant.

'log Lik.' 0.02516651 (df=3)