Rather than saying the relationship is stronger, I think it's more precise to say that weight increases significantly more quickly with height for males than for females. Strength of relationship would be measured by measures like $R^2$, and these are affected not only by the rate of increase of one variable with another, but by the amount of noise in the data. e.g. if the data were something like this:
maleheight <- rnorm(1000, 70, 3)
femaleheight <- rnorm(1000, 65, 2.5)
maleweight <- maleheight*2.2 + rnorm(1000, 0, 20)
femaleweight <- femaleheight*1.3 + rnorm(1000, 0, 10)
height <- c(maleheight, femaleheight)
weight <- c(maleweight, femaleweight)
male <- c(rep(1, 1000), rep(0, 1000))
data <- data.frame(cbind(height, weight, male))
and the model
m1 <- with(data, lm(weight~height + male + height*male))
summary(m1)
shows your pattern, but the relationship looks stronger for women
It is certainly a valid way to run a regression. The interpretation of the coefficients in your example ridge regression is simple:
- If you are a male, then your predicted value is $\widehat{y}=\beta_0+\beta_m$
- If you are a female, then your predicted value is $\widehat{y}=\beta_0+\beta_f$
- The predicted difference between males and females is $\beta_f-\beta_m$
Then the interpretation is completely analogous if you have more than one categorical value, or if your variable has more than two categories. For example, if gender had a "not given" category that you wanted to include in your model with $x_n$, then you would simply add that:
- If gender is not given, then your predicted value is $\widehat{y}=\beta_0+\beta_n$
- The predicted difference between males and "not given"'s is $\beta_m-\beta_n$
- The predicted difference between females and "not given"'s is $\beta_f-\beta_n$
And you can keep adding similar examples. There's no limitation on the interpretation of the coefficients because of the intercept/dummy issue.
Best Answer
You should say, that you could find no evidence (or better yet, "no strong evidence") for statistically significant differences in the average outcomes between males and females at the $X$% confidence level, when controlling for the other variables in the model. Certainly there are significant differences: their genitalia for example and other minor variations that don't matter to you. It also does not make sense to talk about statistically significant differences if you don't refer to the Probability of making a Type I error somehow.