ANOVA for intercept term in Simple Linear Regression

In an ANOVA framework, is it possible to test the hypothesis that $\beta_0 = 0$ given a "full" model of $\hat y_i = \beta_0 + \beta_1 x_i$? Or does the "full" model not contain the "small" model $\hat y_i = \beta_1 x_i$ (and are there bias-related issues)?

• It's not quite clear what you are asking about when you say "are there bias related issues" ... any time you have a smaller model being compared to a larger one, if the larger model is the correct one, the estimates in the smaller model are biased. If that's not what you're asking about you should clarify. Commented Oct 24, 2016 at 5:19
• Thanks -- I meant bias in the sense of errors not summing to zero, ans I guess my question was more along the lines of: is the soace spanned by the columns of the "small" model contained in the space of the columns spanned by the "full" model? Somehow I want to say "no" because the errors in the "small" model do not sum to zero, right? Commented Oct 24, 2016 at 14:35
• You mean the residuals? Sure, rather than their mean being zero you instead have a weighted mean of them being zero. But in relation to estimation, bias is about $E(T-\theta)$ for some estimator $T$ of some unknown quantity $\theta$. I'm still not entirely sure I see the issue. You still have $E(\hat{\beta}_1)=\beta_1$ for example. Commented Oct 24, 2016 at 21:11
• Sorry, I was referring to the sum of the residuals being nonzero, which you pointed out. Ok, so just to confirm, is it correct to think about $span(\text{small model}) \subset span(\text{full model})$ in the sense that the small model is a subspace of the columns of the larger model's design matrix? Commented Oct 25, 2016 at 14:56
• If you're asking "what makes models nested" you could try some of the questions on site -- e.g. maybe this one could be of some help: stats.stackexchange.com/questions/4717/… Commented Oct 26, 2016 at 6:15

Yes, you can test whether the intercept is 0 via ANOVA, or indeed by looking at the regression coefficient's t-value.

1. An example (in R) using ANOVA:

 full <- lm(dist~speed,cars)
noint <- lm(dist~0+speed,cars)
anova(noint,full)
Analysis of Variance Table

Model 1: dist ~ 0 + speed
Model 2: dist ~ speed
Res.Df   RSS Df Sum of Sq      F  Pr(>F)
1     49 12954
2     48 11354  1    1600.3 6.7655 0.01232   <-----


.... the p-value for the intercept term is 0.0123

1. Using the t-ratio for the intercept term:

 summary(lm(dist~speed,cars))

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -17.5791     6.7584  -2.601   0.0123   <---------
speed         3.9324     0.4155   9.464 1.49e-12
--
Residual standard error: 15.38 on 48 degrees of freedom
Multiple R-squared:  0.6511,    Adjusted R-squared:  0.6438
F-statistic: 89.57 on 1 and 48 DF,  p-value: 1.49e-12


Again the p-value for the intercept term is 0.0123