Hi:

On Tue, Jul 20, 2010 at 2:41 AM, StatWM <wmusial@gmx.de> wrote:

>
> Dear R community,
>
> is there a way to get correct t- and p-values and R squared for linear
> regression models specified without an intercept?
>
> example model:
> summary(lm(y ~ 0 + x))
>
> This gives too low p-values and too high R squared. Is there a way to
> correct it? Or should I specify with intercept to get the correct values?
>

How do you know that the p-value is too low and R^2 is too high? Too low or
too high compared to what? You've constrained the intercept of the model to
pass through zero, which affects several features of a simple linear
regression model. For example, sum the residuals from your no-intercept
model - I'll bet they don't add to zero. Do you think that might affect  a
few things? Here's an example:

# Generate some data; notice that the true y-intercept is 2 and the true
slope is 2
dd <- data.frame(x = 1:10, y = 2 + 2 * 1:10 + rnorm(10))
plot(y ~ x, data = dd, xlim = c(0, 10), ylim = c(0, 25))
m1 <- lm(y ~ x, data = dd)
abline(coef(m1))
m2 <- lm(y ~ x + 0, data = dd)
abline(c(0, coef(m2)), lty = 'dotted')

# As you noted, the no-intercept model has a higher R^2,
# even though the 'usual' simple linear regression (SLR)
# model provided a better visual fit. Why?
summary(m1)$r.squared
[1] 0.982328
summary(m2)$r.squared
[1] 0.9946863

# The p-value for the F-test on the slope is higher in the
# no-intercept model is lower than in the SLR model. Why?
anova(m1)

Analysis of Variance Table

Response: y
          Df Sum Sq Mean Sq F value    Pr(>F)
x          1 385.22  385.22  444.69 2.686e-08 ***
Residuals  8   6.93    0.87
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

anova(m2)

Analysis of Variance Table

Response: y
          Df  Sum Sq Mean Sq F value    Pr(>F)
x          1 2164.07 2164.07  1684.7 1.507e-11 ***
Residuals  9   11.56    1.28


Look at the differences in sums of squares between the two models, both in
terms of model SS and error SS. What is responsible for those differences?
Once you understand that, it becomes clear why the apparent anomalies in R^2
and in the F-test occur by applying the definitions. Also try

sum(m1$resid)
sum(m2$resid)

Why is there a difference? Why dies m2$resid not have to sum to zero?

(Hint: The output in each case is correct, so it's not an R problem. You
need to derive the differences among the various quantities in regression
modeling between the intercept and no-intercept models to understand the
paradox.)

HTH,
Dennis

Thank you in advance!
>
> Wojtek Musial
> --
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> Sent from the R help mailing list archive at Nabble.com.
>
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