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RE: st: significance of mean and median


From   "Nick Cox" <n.j.cox@durham.ac.uk>
To   <statalist@hsphsun2.harvard.edu>
Subject   RE: st: significance of mean and median
Date   Tue, 25 Nov 2008 15:28:29 -0000

I take Ronan's point, but there is a difference here. -regress- and
-ttest- are working with the standard Gaussian or normal machinery that
statistically minded people expect to be used for means. -qreg- as a
test for medians does not correspond to any of the customary
non-parametric tests that might be expected to be used for medians. I am
not saying that's a problem; it is more by way of a signal. 

Nick
n.j.cox@durham.ac.uk 

Ronan Conroy

On 24 Nov 2008, at 11:55, Bastian Steingros wrote:

> After creating the descriptive statistics of my sample I want to  
> show that the mean of a certain variable is statistically significant.
> For example the mean of var1 is 0,45. Now I want to show that the  
> positive sign is significant.

A regression without any predictor variable is a constant-only model,  
and the test for significance of the constant is a test that the  
constant is zero.

. sysuse auto
(1978 Automobile Data)

. regress mpg

       Source |       SS       df       MS              Number of obs  
=      74
-------------+------------------------------           F(  0,    73)  
=    0.00
        Model |           0     0           .           Prob > F       
=       .
     Residual |  2443.45946    73  33.4720474           R-squared      
=  0.0000
-------------+------------------------------           Adj R-squared  
=  0.0000
        Total |  2443.45946    73  33.4720474           Root MSE       
=  5.7855

------------------------------------------------------------------------
------
          mpg |      Coef.   Std. Err.      t    P>|t|     [95% Conf.  
Interval]
------------- 
+----------------------------------------------------------------
        _cons |    21.2973   .6725511    31.67   0.000      19.9569     
22.63769
------------------------------------------------------------------------
------

The coefficient for the constant is the mean of the data (since the  
mean is the statistic that minimises prediction error, when we define  
prediction error as the squared difference between observed and  
expected).

Similarly, quantile regression will test that the median is zero

. qreg mpg
Iteration  1:  WLS sum of weighted deviations =  330.13202

Iteration  1: sum of abs. weighted deviations =        330
Iteration  2: sum of abs. weighted deviations =        328

Median regression                                    Number of obs  
=        74
   Raw sum of deviations      328 (about 20)
   Min sum of deviations      328                     Pseudo R2      
=    0.0000

------------------------------------------------------------------------
------
          mpg |      Coef.   Std. Err.      t    P>|t|     [95% Conf.  
Interval]
------------- 
+----------------------------------------------------------------
        _cons |         20   .7799751    25.64   0.000     18.44551     
21.55449
------------------------------------------------------------------------
------

the median being the statistic which minimises error, when we define  
error as the absolute difference between the observed and expected.

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