# How to interpret an ANOVA table comparing full vs reduced OLS models?

I found this online and it is a good example of the problem I am having. In a case like this I am unsure what the first line is referring to. It is saying there are 15 DF and the RSS is 8.18. Is this line interpreted as the 'Between ' model variation? and what of the second line? How does 'between' variation apply in the situation where we are comparing models to one another?

I find understanding ANOVA in the context of comparing fitted models somewhat more difficult, so I am trying to understand it in a 'between' and 'within' context.

I will answer your question with an example that (I hope) you can follow in [R]. If you don't use [R] you can still follow the results on this post.

I'll use the data set mtcars. You can find documentation of what it is about here. But just remember that there are 32 models, and for each one the miles-per-gallon, horse-power, and other variables are recorded. This is the beginning of it:

                   mpg  cyl  disp   hp    drat    wt     qsec   vs  am gear carb
Mazda RX4         21.0   6   160    110   3.90    2.620  16.46  0   1    4    4
Mazda RX4 Wag     21.0   6   160    110   3.90    2.875  17.02  0   1    4    4
Datsun 710        22.8   4   108    93    3.85    2.320  18.61  1   1    4    1


### MODELS:

We'll run two almost random OLS regressions as follows:

fit1 <- lm(mpg ~ wt, mtcars)          #mpg regressed on weight of the car
fit2 <- lm(mpg ~ wt + qsec, mtcars)   #mpg regressed on weight and qsec


Notice that fit1 is a constrained model in the way that we force the regression coefficient for qsec in fit2 to be zero. fit2, conversely, is unconstrained.

### ANOVA:

anova(fit1, fit2)

Analysis of Variance Table

Model 1: mpg ~ wt
Model 2: mpg ~ wt + qsec

Res.Df    RSS      Df    Sum of Sq      F            Pr(>F)
1     30      278.32
2     29      195.46    1     82.858       12.293         0.0015 **


I won't enter into a lengthy explanation of what these values signify, but seeing where they come from will probably help you.

DEGREES OF FREEDOM:

1. Error or Residual Degrees of Freedom: We see them in the output of the anova call as Res. Df 30 and Res. Df 29. They are calculated as:

$\text{no. observations} - \text{no. indepen't variables} - 1 = 32 - 1 - 1 = \color{red}{30}$ for fit1, and $32-2-1 = \color{red}{29}$ for fit2. Remember that we have 32 car models.

2. Model Degrees of Freedom: It is equal to $\text{no. inepen't variables}.$

3. Total Degrees of Freedom: $\text{no. observations} -1.$

4. Constraints: The unconstrained model (fit2) has two independent variable, and hence, it is a model with $2$ degrees of freedom. In contrast, the constrained model (fit1) has only $1$ degree of freedom. The difference between $\text{model unconstrained df} - \text{model constrained df} =\color{red} 1$ is the number of constraints, shown on the output of the anova table as Df 1.

RESIDUAL SUM OF SQUARES & R SQUARED:

Let's calculate the RSS (residual sum of squares), also known as sum of squared errors (SSE), and the F value. To do so these are the pertinent manual calculations:

Mean dependent variable: $\bar y$

mu_mpg <- mean(mtcars$mpg) # Mean mpg in dataset  Total Sum of Squares (TSS):$\sum_1^n(y_i - \bar y)^2$TSS <- sum((mtcars$mpg - mu_mpg)^2)             # Total sum of squares


Model Sum of Squares (MSS): $\sum_1^n (\hat y_i-\bar y)^2$

MSS_fit1 <- sum((fitted(fit1) - mu_mpg)^2)      # Variation accounted for by model
MSS_fit2 <- sum((fitted(fit2) - mu_mpg)^2)      # Variation accounted for by model


Residual Sum of Squares (RSS, also SSE): $\sum_1^n(y_i - \hat y)^2$

RSS_fit1 <- sum((mtcars$mpg - fitted(fit1))^2) # Error sum of squares fit1  RSS_fit1$\color{red}{278.3219}$RSS_fit2 <- sum((mtcars$mpg - fitted(fit2))^2)  # Error sum of squares fit2


RSS_fit2 $\color{red}{195.4636}$

Notice that the RSS column in the ANOVA table correspond to RSS_fit1 = 278.3219 and RSS_fit2 = 195.4636 of the manual calculations above.

In the ANOVA table we also have the difference in RSS: sum(residuals(fit1)^2)-sum(residuals(fit2)^2) = 82.85831, or calculated as indicated above:

$\text{RSS_fit1 - RSS_fit2} = \color{red}{82.85831}$, indicated in the anova table as Sum of Sq.

Frac_RSS_fit1 <- RSS_fit1 / TSS                 # % Variation secndry to residuals fit1
Frac_RSS_fit2 <- RSS_fit2 / TSS                 # % Variation secndry to residuals fit2


R-squared of the model: $1 - RSS/TSS$

R.sq_fit1 <- 1 - Frac_RSS_fit1                  # % Variation secndry to Model fit1


R.sq_fit1 $\color{blue}{0.7528328}$ Compare to summary(fit1)$r.square 0.7528328 R.sq_fit2 <- 1 - Frac_RSS_fit2 # % Variation secndry to Model fit2  R.sq_fit2$\color{blue}{0.8264161}$Compare to summary(fit2)$r.square 0.8264161

F VALUE:

n <- nrow(mtcars)                               # Number of subjects or observations

Constraints <- 1                 # Constraints imposed or difference in iv's fit2 vs. fit1
UnConstrained <- 2               # Independent variables uncontrained model (fit2)


$\Large F = \frac{(R^2_{\text{mod.2}}-R^2_{\text{mod.1}})\,\times\, (N\,-\,\text{no. unconstrained}_{\text{mod.2}}\,-\,1)}{((1 - R^2_{\text{mod.2}})\,\times\, \text{no. constraints})}$

with $N$ corresponding to the number of observations; $\text{no. unconstrained}$, the number of independent variable in the full model; and $\text{no. constraints}$, the difference in independent variables between the full and the reduced model.

F_value=(R.sq_fit2 - R.sq_fit1) * (n - UnConstrained - 1) / ((1 - R.sq_fit2) * Constraints)


F_value # $\color{red}{12.29329}$

And the p-value, which in this case is 0.0015, which is significant. [R] has a system of stars to point out the level of significance, in this case p < 0.01.

In terms of a more graphical interpretation of the ANOVA of an OLS regression, we can visualize the model squared variation (MSS) for fit1 as the green lines in the plot below (equivalent to the "between groups" variance or signal). The RSS is exactly the sum of the length of the red segments separating the individual points from the fitted regression line (and corresponds to the "within group" variance or noise):

• coef(fit1) produces the regression line of mpg ~ 37.2851 - 5.3445wt. But, in the above plot, mpg is about 30 when wt=0 instead of what it should be (37.2851). How can you explain this dilemma? – Erdogan CEVHER Nov 20 '16 at 11:49
• I just figured out the issue. It would be better you add xlim=c(0,6), ylim=c(0, 40) to the plot. – Erdogan CEVHER Nov 20 '16 at 14:24

The first line refers to Model 1, the model that includes $x_1, x_2, x_3, x_5$. This model has a residual degrees of freedom of 15.

The second line refers to Model 2, the model that adds $x_4$. Adding this additional independent variable decreases the residual d.f. to 14.

• thanks. I think Im more confused now haha. I dont understand why there isnt a mean sum of squares for both models . I think Im gettign muddled up with situations where we are comparing models that are based on categorical data as well. – B.Miller Sep 12 '15 at 2:57
• Without seeing the source, my guess is the author chose to show only enough of the ANOVA table to make a point and didn't want to confuse the reader with distracting information. – BChan Sep 12 '15 at 3:01