# Why do I not get a p-value from this ANOVA in R?

Here's the data:

> tires <- data.frame(Wear  = c(17, 14, 12, 13, 14, 14, 12, 11,
13, 13, 10, 11, 13, 8, 9, 9),
Brand = rep(LETTERS[1:4], 4),
Car   = as.character(as.roman(rep(1:4, each = 4))))
> tires
Wear Brand Car
1    17     A   I
2    14     B   I
3    12     C   I
4    13     D   I
5    14     A  II
6    14     B  II
7    12     C  II
8    11     D  II
9    13     A III
10   13     B III
11   10     C III
12   11     D III
13   13     A  IV
14    8     B  IV
15    9     C  IV
16    9     D  IV


Now I fit a two-way ANOVA with interaction:

two.way <- aov(Wear ~ Brand + Car + Brand:Car, data = tires)


Finally, no p-values:

> summary(two.way)
Df Sum Sq Mean Sq
Brand        3  30.69  10.229
Car          3  38.69  12.896
Brand:Car    9  11.56   1.285


A regular two-way ANOVA (i.e., Wear ~ Brand + Car) gives me p-values:

> summary(aov(Wear ~ Brand + Car, data = tires))
Df Sum Sq Mean Sq F value  Pr(>F)
Brand        3  30.69  10.229   7.962 0.00668 **
Car          3  38.69  12.896  10.038 0.00313 **
Residuals    9  11.56   1.285
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


Is there a way to interpret this? The interaction plot shows me that there is definitely interaction between Brand and Car so I am hoping to incorporate this into my model.

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