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st: RE: standard error obtained from ttest and linear regression


From   "Nick Cox" <n.j.cox@durham.ac.uk>
To   <statalist@hsphsun2.harvard.edu>
Subject   st: RE: standard error obtained from ttest and linear regression
Date   Thu, 19 Feb 2004 17:07:06 -0000

The standard error for domestic cars reported by -ttest- 
depends only on the data for domestic cars. The standard 
error for the intercept of the regression depends on all 
the data. So you can't expect exactly the same value. 

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

Aijing Shang
 
> By chance, I found the standard error of mean values obtained 
> from ttest is
> different from linear regression. Please look at the 
> following example.
> The data is auto.dta in STATA.
> 
> . ttest price, by(foreign)
> 
> Two-sample t test with equal variances
> 
> --------------------------------------------------------------
> --------------
> --
>    Group |     Obs        Mean    Std. Err.   Std. Dev.   [95% Conf.
> Interval]
> ---------+----------------------------------------------------
> --------------
> --
> Domestic |      52    6072.423    429.4911    3097.104    5210.184
> 6934.662
>  Foreign |      22    6384.682    558.9942    2621.915     5222.19
> 7547.174
> ---------+----------------------------------------------------
> --------------
> --
> combined |      74    6165.257    342.8719    2949.496    5481.914
> 6848.6
> ---------+----------------------------------------------------
> --------------
> --
>     diff |           -312.2587    754.4488               -1816.225
> 1191.708
> --------------------------------------------------------------
> --------------
> --
> Degrees of freedom: 72
> 
>                 Ho: mean(Domestic) - mean(Foreign) = diff = 0
> 
>      Ha: diff < 0               Ha: diff != 0              
> Ha: diff > 0
>        t =  -0.4139                t =  -0.4139              
> t =  -0.4139
>    P < t =   0.3401          P > |t| =   0.6802          P > 
> t =   0.6599
> 
> 
> . reg price foreign
> 
>       Source |       SS       df       MS              Number of obs =
> 74
> -------------+------------------------------           F(  1,    72) =
> 0.17
>        Model |  1507382.66     1  1507382.66           Prob > F      =
> 0.6802
>     Residual |   633558013    72  8799416.85           R-squared     =
> 0.0024
> -------------+------------------------------           Adj R-squared
> = -0.0115
>        Total |   635065396    73  8699525.97           Root MSE      =
> 2966.4
> 
> --------------------------------------------------------------
> --------------
> --
>        price |      Coef.   Std. Err.      t    P>|t|     [95% Conf.
> Interval]
> -------------+------------------------------------------------
> --------------
> --
>      foreign |   312.2587   754.4488     0.41   0.680    -1191.708
> 1816.225
>        _cons |   6072.423    411.363    14.76   0.000     5252.386
> 6892.46
> --------------------------------------------------------------
> --------------
> --
> 
> ttest and linear regression provide exactly the same estimates of mean
> difference of price between domestic and foreign autos, which 
> is 312.2587 in
> this example. As well as its standard error, which is 
> 754.4488. However,
> look at _cons in the linear regression. In this example, the 
> coefficient of
> _cons should be the mean price of domestic auto, since 
> varible "foreign" is
> coded as 0 if the autos were from domestic, 1 if foreign. The 
> coefficient of
> _cons is exactly the same as the mean value in ttest, which 
> is 6072.423. But
> its standard error in ttest is 429.4911, whereas 411.363 in linear
> regression. Can anybody tell me how this can happen? Thank 
> you very much in
> advance.

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