# Linear model vs. linear regression

I have a question that I find really confusing regarding linear modelling and linear regression. I have expectation regarding the way some dependent variable (DV) are going to evolve with an independent variable (IV).

In order to check for a relationship between IV and DV, on several participants, I just computed the linear model by calculating Y as follow:

Y = XB + E

Therefore I used the weight of my linear model as B and my DV as X. Finally I just calculated a weighted sum. Then I tested for an effect by using a one sample t test on the various Y.

Well I'm confused because I don't see the difference between doing that and computing a linear regression by ordinary least square and calculating the slopes.

According to the two methods (weighted sum to the predictive X) or linear regression, I get different numerical values, but these values are correlated between them to 1.

If anyone can enlighten me about the difference, on the theoretical ground, between using one of these two methods, thank you!

I don't know how do you define a "linear model" but in general this term is used as synonym for linear regression (e.g. on Wikipedia). Also from your definition:

$$y = X\beta + \epsilon$$

it appears that this is linear regression. So you computed regression twice and that is a reason why the results are equivalent. On the other hand, you say that the estimated values are different (while being highly correlated). There could be two reasons for that:

• Generally, if you compute regression, you include intercept in the model so it is:

$$y = \beta_0 + X\beta_1 + \epsilon$$

so if you computed regression without the intercept and then on the second time with the intercept, the results could be a little bit different.

• You didn't describe the way how you estimated both models. If you used different algorithms for estimating both models, the results should be similar but there could be slight differences.
• Well I'm ashamed to confess that by linear model I was implying linear contrast (C) link so I just computed the weighted sum of DV by associated weights. So finally I have similar results when I calculate (C) and when I estimate Betas (with the weights I use for the calculation of C as IV), i.e. I have a correlation of 1. – user5084 Dec 15 '14 at 17:01
• Sorry but I don't fully understand. Maybe you could edit your question and describe your procedure in greater detail..? – Tim Dec 15 '14 at 17:12