# Regression: how to interpret different linear relations?

I have three datasets, let's call them X and Y1 and Y2. A scatterplot is produced out of them, with Y1 and Y2 sharing them same X dataset.

My question: if the two regression lines are different in both slope and intercept, is there a way to evaluate if the X dataset has more influence on Y1 or Y2?

Based on the image below, this is to say - which Y dataset is more influenced by the X dataset?
Is there any metrics for measuring this?

• Blue slope (Y1): -112
• Red slope (Y2): -90
• When x=100 must y be 0? – Glen_b -Reinstate Monica Dec 2 '15 at 11:03
• Let's focus on (ii) to begin with because that impacts the other two. If x=100 implies y=0, then the relationship is constrained to go through (100,0) and you only have one free parameter. If we consider $x_2=100-x$ then it's effectively regression through the origin in the new variable (i.e. your model for the relationship is actually $E(y)=b (100-x)$). Are you just fitting plain two-parameter linear regression with equal weight to all data points? – Glen_b -Reinstate Monica Dec 2 '15 at 11:36
• No, the free parameter is the slope parameter - "b" in $E(y)=b(100-x)=100b-bx$. As soon as you know the slope, the intercept is known. You can enforce that by letting $x_2=100-x$ and then fitting regression through the origin, but that ignores the fact that the variance will reduce as you approach that fixed point; indeed with your red data points it looks like you have spread proportional to mean. If that's a reasonable assumption, there's a few possible choices with how to incorporate that. ... ctd – Glen_b -Reinstate Monica Dec 2 '15 at 11:52
• ctd... One you can easily do in Excel is as follows: if you believe spread is proportional to mean you're fitting $y = b x_2 + x_2 e$ for $\text{Var}(e)$ constant. Dividing through by $x_2$ gives: $y/x_2 = y/(100-x) = b + e$. This converts it from a regression through the origin with changing spread to fitting only a constant term, with constant spread (but it does rely on the spread assumption being close to right). Note that this $b$ is effectively the negative of your slope. – Glen_b -Reinstate Monica Dec 2 '15 at 11:59
• Note that your datasets would usually be called variables (as @Glen_b has done throughout). – Nick Cox Dec 2 '15 at 13:30

If by "influence" you mean the magnitude of output change for a unit change in the input, then slope is your measure.

R^2 gives you a measure of "explained variance" (as Ivo already noted), that is - how well your model describes the output variations against input variations.

"which Y dataset is more influenced by the X dataset?"

Y1, because it has a higher average difference in output magnitude for the same levels of change in input magnitude.

Y1 accidentally has also a higher R^2 value, but even if it was lower than for Y2, with the same slope, the answer would be the same.

Hope I helped.

• I would just that it's important to use correct terminology: A dataset cannot "influence" anything -- it's just data. Are Y1 and Y2 different outputs/output features or just different datasets of the same input/output model? – Nemanja Radojković Dec 2 '15 at 12:05
• They are two different outcome datasets of the same model. If "influence" is not correct terminology, what would be? – FaCoffee Dec 2 '15 at 12:15
• Well, it depends a what is the difference between the datasets. If they, for example, describe the response (Yx) of two different groups of subjects on a same treatment drug (X), then you could say: "In group 1, the treatment has a higher effect, compared to group 2 -- where effect size is taken as the slope of the regression line" – Nemanja Radojković Dec 2 '15 at 13:26

This plot gives you two things: First, X is more strongly correlated with Y1 than with Y2, since R2 is higher for Y1. R2 is a quantification of the explained variance. Second, the slope of Y1 is more steep than Y2 meaning that it changes more as X changes. Does that answer your question?

• Kind of. So, is the slope the only metric I can refer to or are there other metrics which are the result of more complicated procedures (e.g., ANCOVA)? – FaCoffee Dec 2 '15 at 10:23
• Well, in this case it could be either slope or R2 (they are not the same). For other procedures there are similar measures (e.g. R2 or group mean differences). – Ivo Dec 2 '15 at 10:29