R help - Apr 2005 - ANOVA with both discreet and continuous variable

Hi all,

I have dataset with 2 independent variable, one (x1)
is continuous, the other (x2) is a categorical
variable with 2 levels. The dependent variable (y) is
continuous. When I run linear regression y~x1*x2, I
found that the p value for the continuous independent
variable x1 changes when different contrasts was used
(helmert vs. treatment), while the p values for the
categorical x2 and interaction are independent of the
contrasts used. Can anyone explain why? I guess the p
value for x1 is testing different hypothesis under
different contrasts? If the interaction is NOT
significant, what contrast should I use to test the
hypothesis that x1 is not significantly related with
y?


x1<-rnorm(50,9,2)
x2<-as.factor(as.numeric(runif(50)<0.35))
y<-rnorm(50,30,5)

options(contrasts=c('contr.treatment','contr.poly'))
summary(lm(y~x1*x2))

options(contrasts=c('contr.helmert','contr.poly'))
summary(lm(y~x1*x2))

An anonymous enquirer asks:

: -----Original Message-----
: From: r-help-bounces at stat.math.ethz.ch 
: [mailto:r-help-bounces at stat.math.ethz.ch] On Behalf Of array chip
: Sent: Saturday, 23 April 2005 2:32 PM
: To: r-help at stat.math.ethz.ch
: Subject: [R] ANOVA with both discreet and continuous variable
: 
: 
: Hi all,
: 
: I have dataset with 2 independent variable, one (x1)
: is continuous, the other (x2) is a categorical
: variable with 2 levels. The dependent variable (y) is
: continuous. When I run linear regression y~x1*x2, I
: found that the p value for the continuous independent
: variable x1 changes when different contrasts was used
: (helmert vs. treatment), while the p values for the
: categorical x2 and interaction are independent of the
: contrasts used. Can anyone explain why? 

Because the hypotheses the corresponding test statistics
are testing are invariant with respect to the choice of
contrast matrices you have considered.  (This is NOT true
if your factor has more than two levels, by the way.)

: I guess the p
: value for x1 is testing different hypothesis under
: different contrasts? 

The tests are for different null hypotheses, yes.

: If the interaction is NOT
: significant, what contrast should I use to test the
: hypothesis that x1 is not significantly related with
: y?

There is no choice of contrast matrix that will give the
test statistic associated with the linear term x1 this 
meaning.  Your question only specifies a null hypothesis,
a significance test requires a null and an alternative
hypothesis.  Both matter.  In the context you have set
up below the way I would go about addressing what I 
think is your question would be something like:

M0 <- lm(y ~ x2)     ## Null hypthesis with no x1
M1 <- lm(y ~ x1*x2)	  ## outer hypothesis as below
anova(M0, M1)
 
: 
: 
: x1<-rnorm(50,9,2)
: x2<-as.factor(as.numeric(runif(50)<0.35))
: y<-rnorm(50,30,5)
: 
: options(contrasts=c('contr.treatment','contr.poly'))
: summary(lm(y~x1*x2))
: 
: options(contrasts=c('contr.helmert','contr.poly'))
: summary(lm(y~x1*x2))
: 
: ______________________________________________
: R-help at stat.math.ethz.ch mailing list
: https://stat.ethz.ch/mailman/listinfo/r-help
: PLEASE do read the posting guide! 
: http://www.R-project.org/posting-guide.html
:

If the treatment contrast is used, the p value for x1
is testing whether the slope at the reference level of
x2 is equal to 0 (think about the model y~x1*x2 as
fitting 2 straight lines, one for the reference level
of x2, and one for the other level of x2). I am not
quite sure about what it tests when the helmert
contrast is used, my guess is it tests whether the
slope at the mid-level (?) of the x2 is equal to 0.
maybe other experts can comment on this.

Steve


: -----Original Message-----
: From: r-help-bounces at stat.math.ethz.ch 
: [mailto:r-help-bounces at stat.math.ethz.ch] On Behalf
Of array chip
: Sent: Saturday, 23 April 2005 2:32 PM
: To: r-help at stat.math.ethz.ch
: Subject: [R] ANOVA with both discreet and continuous
variable
: 
: 
: Hi all,
: 
: I have dataset with 2 independent variable, one (x1)
: is continuous, the other (x2) is a categorical
: variable with 2 levels. The dependent variable (y)
is
: continuous. When I run linear regression y~x1*x2, I
: found that the p value for the continuous
independent
: variable x1 changes when different contrasts was
used
: (helmert vs. treatment), while the p values for the
: categorical x2 and interaction are independent of
the
: contrasts used. Can anyone explain why? 

Because the hypotheses the corresponding test
statistics
are testing are invariant with respect to the choice
of
contrast matrices you have considered.  (This is NOT
true
if your factor has more than two levels, by the way.)

: I guess the p
: value for x1 is testing different hypothesis under
: different contrasts? 

The tests are for different null hypotheses, yes.

: If the interaction is NOT
: significant, what contrast should I use to test the
: hypothesis that x1 is not significantly related with
: y?

There is no choice of contrast matrix that will give
the
test statistic associated with the linear term x1 this

meaning.  Your question only specifies a null
hypothesis,
a significance test requires a null and an alternative
hypothesis.  Both matter.  In the context you have set
up below the way I would go about addressing what I 
think is your question would be something like:

M0 <- lm(y ~ x2)     ## Null hypthesis with no x1
M1 <- lm(y ~ x1*x2)	  ## outer hypothesis as below
anova(M0, M1)
 
: 
: 
: x1<-rnorm(50,9,2)
: x2<-as.factor(as.numeric(runif(50)<0.35))
: y<-rnorm(50,30,5)
: 
: options(contrasts=c('contr.treatment','contr.poly'))
: summary(lm(y~x1*x2))
: 
: options(contrasts=c('contr.helmert','contr.poly'))
: summary(lm(y~x1*x2))
: