[R] lm - significance disappears

[Prof Brian Ripley]

On Mon, 8 Mar 2004, Stuart Leask wrote:

> Hi there.
> I am trying to model the effect of "Age at onset of a condition" (AGEONSET)
> upon IQ (ie. do they have more problems if they get ill younger?). I also
> want to see if there is an interaction with the TYPE of test (reading &
> maths).
> 
> I am struggling to understand why the effect of AGEONSET, so very
> significant in the non-interaction models, ceases to be so in the
> interacting model? Am I using lm() wrongly here?

First, the meaning of AGEONSET changes.  Assuming TYPE is a 2-level factor 
and AGEONSET is continuous, this is not an interaction per se but a model 
with one line for both types vs separate lines for both types.  

>From now on I need to assume standard (that is treatment) contrasts.
In your final model, AGEONSET is the slope for the line in the first
group: maybe that group is the smaller, as the standard error is 3x
larger?  AGEONSET:TYPE is the difference in slopes: that is determined 
well but is small.

It helps to look at both the size of the coefficient and its se, as well 
as their ratio.

> 
> 
> Stuart
> 
> ****************************
> 
> # Regress IQ on AGEONSET, AGEONSET + TYPE, & AGEONSET*TYPE.
> 
> > fit.f<-lm(IQ~AGEONSET)
> > summary(fit.f)
> 
> Call:
> lm(formula = IQ ~ AGEONSET)
> 
> Residuals:
>      Min       1Q   Median       3Q      Max
> -3.07434 -0.78350  0.04031  0.71869  2.82266
> 
> Coefficients:
>              Estimate Std. Error t value Pr(>|t|)
> (Intercept) -0.975267   0.077362 -12.607  < 2e-16 ***
> AGEONSET     0.037158   0.006876   5.404 9.57e-08 ***
> ---
> Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
> 
> Residual standard error: 1.109 on 574 degrees of freedom
> Multiple R-Squared: 0.04841,    Adjusted R-squared: 0.04675
> F-statistic:  29.2 on 1 and 574 DF,  p-value: 9.571e-08
> 
> 
> > fit.f<-lm(IQ~AGEONSET+TYPE)
> > summary(fit.f)
> 
> Call:
> lm(formula = IQ ~ AGEONSET + TYPE)
> 
> Residuals:
>      Min       1Q   Median       3Q      Max
> -3.03301 -0.80625  0.03662  0.72066  2.86379
> 
> Coefficients:
>              Estimate Std. Error t value Pr(>|t|)
> (Intercept) -0.852151   0.158755  -5.368 1.16e-07 ***
> AGEONSET     0.037147   0.006877   5.401 9.71e-08 ***
> TYPE        -0.082107   0.092447  -0.888    0.375
> ---
> Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
> 
> Residual standard error: 1.109 on 573 degrees of freedom
> Multiple R-Squared: 0.04972,    Adjusted R-squared: 0.0464
> F-statistic: 14.99 on 2 and 573 DF,  p-value: 4.513e-07
> 
> 
> > fit.f<-lm(IQ~AGEONSET*TYPE)
> > summary(fit.f)
> 
> Call:
> lm(formula = IQ ~ AGEONSET * TYPE)
> 
> Residuals:
>      Min       1Q   Median       3Q      Max
> -3.04133 -0.80869  0.03557  0.72420  2.86796
> 
> Coefficients:
>                Estimate Std. Error t value Pr(>|t|)
> (Intercept)   -0.833399   0.244866  -3.403 0.000712 ***
> AGEONSET       0.035070   0.021758   1.612 0.107555
> TYPE          -0.094608   0.154887  -0.611 0.541562
> AGEONSET:TYPE  0.001386   0.013767   0.101 0.919868
> ---
> Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
> 
> Residual standard error: 1.11 on 572 degrees of freedom
> Multiple R-Squared: 0.04974,    Adjusted R-squared: 0.04475
> F-statistic: 9.979 on 3 and 572 DF,  p-value: 2.026e-06
> 
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> 

-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
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