Writing hypothesis for linear multiple regression models I struggle writing hypothesis because I get very much confused by reference groups in the context of regression models.
For my example I'm using the mtcars dataset. The predictors are wt(weight), cyl(number of cylinders), and gear(number of gears), and the outcome variable is mpg (miles per gallon).
Say all your friends think you should buy a 6 cylinder car, but before you make up your mind you want to know how 6 cylinder cars perform miles-per-gallon-wise compared to 4 cylinder cars because you think there might be a difference.
Would this be a fair null hypothesis (since 4 cylinder cars is the reference group)?:
There is no difference between 6 cylinder car miles-per-gallon performance and 4 cylinder car miles-per-gallon performance.
Would this be a fair model interpretation?:
6 cylinder vehicles travel fewer miles per gallon (p=0.010, β -4.00, CI -6.95 - -1.04) as compared to 4 cylinder vehicles when adjusting for all other predictors, thus rejecting the null hypothesis.
Sorry for troubling, and thanks in advance for any feedback!
# Data 
data(mtcars)
mtcars$cyl <- as.factor(as.character(mtcars$cyl)) 
mtcars$gear <- as.factor(as.character(mtcars$gear))

# Model   
mtcars.lm <- lm(mpg ~ wt + cyl + gear, data = mtcars)

# Model output
library(sjPlot)
tab_model(mtcars.lm)


 A: Yes, you already got the right answer to both of your questions.

*

*Your null hypothesis in completely fair. You did it the right way. When you have a factor variable as predictor, you omit one of the levels as a reference category (the default is usually the first one, but you also can change that). Then all your other levels’ coefficients are tested for a significant difference compared to the omitted category. Just like you did.

If you would like to compare 6-cylinder cars with 8-cylinder car, then you would have to change the reference category. In your hypothesis you just could had added at the end (or as a footnote): "when adjusting for weight and gear", but it is fine the way you did it.


*Your model interpretation is correct: It is perfect the way you did it. You could even had said: "the best estimate is that 6 cylinder vehicles travel 4 miles per gallon less than 4 cylinder vehicles (p-value: 0.010; CI: -6.95, -1.04), when adjusting for weight and gear, thus rejecting the null hypothesis".

Let's assume that your hypothesis was related to gears, and you were comparing 4-gear vehicles with 3-gear vehicles. Then your result would be β: 0.65; p-value: 0.67; CI: -2.5, 3.8. You would say that: "There is no statistically significant difference between three and four gear cars in fuel consumption, when adjusting for weight and engine power, thus failing to reject the null hypothesis".
