Regression Models for Data Science in R: Statistical inference for data science. by MOUSAIF YASSINE

Author:MOUSAIF, YASSINE
Language: eng
Format: epub, pdf
Publisher: UNKNOWN
Published: 2022-02-01T16:00:00+00:00

Experiment 1.

Looking at this plot notices that the X variable is unrelated to treatment/group status (color). In addition, the X variable is clearly linearly related to Y, but the intercept of this relationship depends on group status. The treatment variable is also related to Y; especially look at the horizontal lines which connect the group means onto the Y axis. The third line is the what you would get if you just fit X and ignored group. Furthermore, notice that the relationship between group status and Y is constant depending on X. In other words, both the apparent relationship and our estimated model have parallel lines. (Remember, our model, by not including an interaction term, did not allow for estimated non parallel lines.)

Finally, notice that the estimated relationship between the group variable and the outcome doesn’t change much, regardless of whether X is accounted for or not. You can see this by comparing the distance between the horizontal lines and the distance between the intercepts of the fitted lines. The horizontal lines are the group averages (disregarding X). That the relationship doesn’t change much is ultimately a statement about balance. The nuisance variable (X) is well balanced between levels of the group variable. So, whether you account for X or not, you get about the same answer. Moreover, we have lots of data at every level of X to make a direct comparison of the group on Y. One way to try to achieve such balance with high probability is to randomize the group variable. This is especially useful, of course, when one doesn’t get to observe the nuisance covariate. Though be careful that as the number of unobserved covariates

Now let’s consider less ideal settings.

Experiment 2

Experiment 2.

In this experiment, the X variable is highly related to group status. That is, if you know the X variable, you could very easily predict which group they belonged to. If we disregard X, there’s an apparent strong relationship between the group variable and Y. However, if we account for X, there’s basically none. In this case, the apparent effect of group on Y is entirely explained by X. Our regression model would likely have a strong significant effect if group was included by itself and this effect would vanish if X was included.

Further notice, there’s no data to directly compare the groups at any particular value of X. (There’s no vertical overlap between the blue and red points.) Thus the adjusted effect is entirely based on the model, specifically the assumption of linearity. Try to drawing curves on this plot assuming non-linear relationships outside of their cloud of points for the blue and red groups. You quickly will conclude that many relationship are possible that would differ from this model’s conclusions. Worse still, you have no data to check the assumptions. Of course, R will churn forward without any complaints fitting this model and reporting no significant difference between the groups.

It’s worth noting at this point, that our experiments just show how the data can arrive at different effects when X is included or not.

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