Linear Statistical Models: Regression

Regression with Measurement Error


As you will most likely recall, one of the assumptions of regression is that the predictor variables are measured without error.

Measurement error in the response variable does not bias the regression coefficient. It does however increase the the standard error of the coefficient, thereby weakening tests of statistical significance.

Measurement error in the predictor variables, on the other hand, has an impact of the estimates of the regression coefficients. With a single predictor, measurement error leads to underestimation of the coefficient. The degree of attenuation is a function of the reliability of the predictor.

where rtt is the reliabity coefficient for the predictor, b is the observed regression coefficient computed on the sample, and β is the "true" value of the paramenter.

Let's look at a regression using the hsbdemo dataset.

use http://www.philender.com/courses/data/hsbdemo, clear

regress write read female

  Source |       SS       df       MS                  Number of obs =     200
---------+------------------------------               F(  2,   197) =   77.21
   Model |  7856.32118     2  3928.16059               Prob > F      =  0.0000
Residual |  10022.5538   197  50.8759077               R-squared     =  0.4394
---------+------------------------------               Adj R-squared =  0.4337
   Total |   17878.875   199   89.843593               Root MSE      =  7.1327

------------------------------------------------------------------------------
   write |      Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
    read |   .5658869   .0493849     11.459   0.000        .468496    .6632778
  female |   5.486894   1.014261      5.410   0.000        3.48669    7.487098
   _cons |   20.22837   2.713756      7.454   0.000       14.87663    25.58011
------------------------------------------------------------------------------

The predictor read is a standardized test score. Every test has measurement error. I don't know the exact reliability of read, but I would say that using .9 for reliability would not be far off. We will now estimate the same regression model with the Stata eivreg command, which stands for errors-in-variables regression.

eivreg write read female, r(read .9)

               assumed                          errors-in-variables regression
variable     reliability
------------------------                               Number of obs =     200
    read       0.9000                                  F(  2,   197) =   83.41
       *       1.0000                                  Prob > F      =  0.0000
                                                       R-squared     =  0.4811
                                                       Root MSE      = 6.86268

------------------------------------------------------------------------------
   write |      Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
    read |   .6289607   .0528111     11.910   0.000        .524813    .7331085
  female |   5.555659   .9761838      5.691   0.000       3.630548     7.48077
   _cons |   16.89655   2.880972      5.865   0.000       11.21504    22.57805

Note that the F-ratio and the R2 increased along with the regression coefficient for read. Additionally, there was an increase in the standard error for read.

Now, let's try a model with multiple predictors, read math and socst. First, we will run a standard multiple regression.

regress write read math socst female

  Source |       SS       df       MS                  Number of obs =     200
---------+------------------------------               F(  4,   195) =   64.37
   Model |  10173.7036     4  2543.42591               Prob > F      =  0.0000
Residual |  7705.17137   195  39.5136993               R-squared     =  0.5690
---------+------------------------------               Adj R-squared =  0.5602
   Total |   17878.875   199   89.843593               Root MSE      =   6.286

------------------------------------------------------------------------------
   write |      Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
    read |   .2065341   .0640006      3.227   0.001       .0803118    .3327563
    math |   .3322639   .0651838      5.097   0.000       .2037082    .4608195
   socst |   .2413236   .0547259      4.410   0.000        .133393    .3492542
  female |   5.006263   .8993625      5.566   0.000       3.232537     6.77999
   _cons |   9.120717   2.808367      3.248   0.001       3.582045    14.65939
------------------------------------------------------------------------------

Now, let's try to account for the measurement error by using the following reliabilities: read - .9, math - .9, socst - .8.

eivreg write read math socst female, r(read .9 math .9 socst .8)

               assumed                          errors-in-variables regression
variable     reliability
------------------------                               Number of obs =     200
    read       0.9000                                  F(  4,   195) =   70.17
    math       0.9000                                  Prob > F      =  0.0000
   socst       0.8000                                  R-squared     =  0.6047
       *       1.0000                                  Root MSE      = 6.02062

------------------------------------------------------------------------------
   write |      Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
    read |   .1506668   .0936571      1.609   0.109      -.0340441    .3353776
    math |    .350551   .0850704      4.121   0.000       .1827747    .5183273
   socst |   .3327103   .0876869      3.794   0.000        .159774    .5056467
  female |   4.852501   .8730646      5.558   0.000        3.13064    6.574363
   _cons |    6.37062   2.868021      2.221   0.027       .7142973    12.02694
------------------------------------------------------------------------------

Note that the overall F and R2 went up, but that the coefficient for read is no longer significant.


Linear Statistical Models Course

Phil Ender, 22dec00