# Interpreting the coefficients when predictors are standarised

Let's say I have a model of the form:

mdl <- lm(y ~ x1 + x2 + x3, data=dat)


I am interested in knowing the relationship between x3 and y when x1 and x2 remain unchanged. To do this, I do the following:

dat$x1 <- mean(dat$x1)
dat$x2 <- mean(dat$x2)
dat$x3 <- dat$x3

pred <- predict(mdl, data=dat)
mean(pred)


If I interpret mean(pred) correctly, it represents average change in y with a unit change in x3 while keeping x2 and x1 constant. Please correct me if I am wrong.

Now imagine a second model wherex1, x2 and x3 are standardised.

dat$z.x1 <- scale(dat$x1,scale = T,center = T)
dat$z.x2 <- scale(dat$x2,scale = T,center = T)
dat$z.x3 <- scale(dat$x3,scale = T,center = T)

mdl1 <- lm(y ~ z.x1 + z.x2 + z.x3, data=dat)

# keeping x1 and x2 at their mean values (which is zero since the variables
are standardised)

dat$z.x1 <- 0 dat$z.x2 <- 0
dat$z.x3 <- dat$z.x3

pred.z <- predict(mdl1, data=dat)

mean(pred.z)


This is where my question is. Does mean(pred.z) has the same interpretation as mean(pred) in the previous model? Can mean(pred.z) be interpreted as mean change in y with a unit change in x3 while keeping x2 and x1 constant. OR

Does mean(pred.z) represent mean change in y when x3 changes by one standard deviation since I had standardised x3 in this model?

Thank you

## 1 Answer

You are doing more work than necessary to know "the relationship between x3 and y when x1 and x2 remain unchanged," and at first glance you seem to be making an error in your approach (if I understand it properly).

The regression coefficient for x3 in your first model is precisely the change in y per unit change in x3 with the other predictors held constant. In your second model, with x3 standardized before the regression, the regression coefficient for z.x3 is the change in y per one-standard-deviation change in x3, again with the other predictors held constant.

Using predict() and mean():

pred <- predict(mdl, data=dat)
mean(pred)


after setting all the other predictors to their mean values doesn't seem, at first glance, to give you what you are looking for. It gives you the mean y prediction for a data set that kept the original x3 values but kept all of the other predictors at their individual means. The distribution of predicted y values would thus be very dependent on the distribution x3 values and wouldn't necessarily bear any relation to "the relationship between x3 and y when x1 and x2 remain unchanged." It's easy to come up with a scenario in which the mean of predicted y values would be 0 even with a strong linear relation between y and x3. If I'm missing something in your approach, please clarify.

In your second model, the pre-scaling of x3 will lead to a corresponding change in the regression coefficient for z.x3 so it seems that the final predicted y values with

pred.z <- predict(mdl1, data=dat)
mean(pred.z)


should be the same numerically as in the first model, although I haven't checked that directly.

Again, the regression coefficients themselves provide directly your answer.

• Great. Thank you. What the background of the question is that I am trying to do produce a response function of y and x3. So I plotted the predicted value of y against x3 when x1 and x2 are held constant. Jul 5, 2017 at 19:22
• @user53020 plotting predicted y values against x3 values should be OK; the issue is with taking the mean of the predicted y values. Again, you shouldn't have to work so hard to make a plot; to plot a response function in R you could just use the abline() function with the slope equal to the coefficient for x3 and the intercept chosen appropriately based on x1 and x2 values. No need to go back to the original data once the regression is done. You could even plot a family of response functions based on sets of {x1,x2} values.
– EdM
Jul 5, 2017 at 19:29