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head(mtcars)
                   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
Hornet 4 Drive    21.4   6  258 110 3.08 3.215 19.44  1  0    3    1
Hornet Sportabout 18.7   8  360 175 3.15 3.440 17.02  0  0    3    2
Valiant           18.1   6  225 105 2.76 3.460 20.22  1  0    3    1

unique(mtcars$cyl)
[1] 6 4 8

unique(mtcars$gear)
[1] 4 3 5


summary(lm(mpg ~ as.factor(cyl) * as.factor(gear),data=mtcars))

Call:
lm(formula = mpg ~ as.factor(cyl) * as.factor(gear), data = mtcars)

Residuals:
   Min     1Q Median     3Q    Max 
-5.525 -1.800  0.075  1.425  6.975 

Coefficients: (1 not defined because of singularities)
                                 Estimate Std. Error t value Pr(>|t|)    
(Intercept)                        21.500      3.349   6.421 1.22e-06 ***
as.factor(cyl)6                    -1.750      4.101  -0.427   0.6734    
as.factor(cyl)8                    -6.450      3.485  -1.851   0.0766 .  
as.factor(gear)4                    5.425      3.552   1.527   0.1397    
as.factor(gear)5                    6.700      4.101   1.634   0.1154    
as.factor(cyl)6:as.factor(gear)4   -5.425      4.585  -1.183   0.2483    
as.factor(cyl)8:as.factor(gear)4       NA         NA      NA       NA    
as.factor(cyl)6:as.factor(gear)5   -6.750      5.800  -1.164   0.2559    
as.factor(cyl)8:as.factor(gear)5   -6.350      4.833  -1.314   0.2013    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.349 on 24 degrees of freedom
Multiple R-squared:  0.761, Adjusted R-squared:  0.6913 
F-statistic: 10.92 on 7 and 24 DF,  p-value: 3.921e-06

What are the correct values for the main and interaction effects of the reference categories? If listed, I imagine they would hypothetically be:

                                  Estimate Std. Error t value Pr(>|t|)
as.factor(cyl)4                       0         0       0         0
as.factor(gear)3                      0         0       0         0
as.factor(cyl)4:as.factor(gear)3      0         0       0         0

Thank you!

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