Bivatiate Plots

Note that there are two GPA responses for each observation, lgpa for the lower end of the interval and ugpa for the upper end.

Details

Means of the variables by each type of program.

dat$type: vocational
   lgpa    ugpa   write  rating 
 1.7500  2.4375 71.8750 52.5000 
------------------------------------------------------------ 
dat$type: general
  lgpa   ugpa  write rating 
  2.78   3.24 148.00  56.80 
------------------------------------------------------------ 
dat$type: academic
      lgpa       ugpa      write     rating 
  3.016667   3.416667 113.333333  61.500000

Interval Regression

In this particular case, all observations had interval censoring.

Interpretation of coefficients (reference program: vocational):

. The variable “write” is statistically significant. A one unit increase in writing score leads to a 0.005 increase in predicted GPA.

. One of the indicator variables for type of program, “academic”, is also statistically significant. Compared to vocational programs, the predicted achievement for academic programs is about 0.71 higher.


Call:
survreg(formula = Y ~ write + rating + type, data = dat, dist = "gaussian")
                Value Std. Error     z       p
(Intercept)   1.10386    0.44529  2.48  0.0132
write         0.00528    0.00169  3.12  0.0018
rating        0.01331    0.00912  1.46  0.1443
typegeneral   0.37485    0.19275  1.94  0.0518
typeacademic  0.70975    0.16684  4.25 2.1e-05
Log(scale)   -1.23726    0.15964 -7.75 9.2e-15

Scale= 0.29 

Gaussian distribution
Loglik(model)= -33.1   Loglik(intercept only)= -51.7
    Chisq= 37.24 on 4 degrees of freedom, p= 1.6e-07 
Number of Newton-Raphson Iterations: 5 
n= 30

Testing Significance

We saw that compared to vocational programs, academic programs are about 0.71 higher in GPA score.

To determine if program type itself is statistically significant, we can either test models with and without it for the two degree-of-freedom test of this variable using a likelihood ratio test (LRT), or get a test of the overall effect of type by examining an analysis of deviance table, which reports the sequential deviances (-2*LL) adding one term at a time.

Analysis of Deviance Table

 distribution with  link

Response: Y

Scale estimated

Terms added sequentially (first to last)
       Df Deviance Resid. Df   -2*LL  Pr(>Chi)    
NULL                      28 103.495              
write   1  16.6891        27  86.805 4.403e-05 ***
rating  1   6.0972        26  80.708  0.013540 *  
type    2  14.4505        24  66.258  0.000728 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The LRT is associated with a p-value of 0.000728, indicating that the overall effect of program type is statistically significant.