What does "1-unit change" mean when centering predictors in regression?

Question

I know that when we standardize a predictor, "one unit change" becomes one standard deviation in the predictor, but what if we only center the data on the mean (i.e. only subtract all values by the mean) ?

• edit below: bounty

This example is from Winter (2019: 139):

mutate(SER_c = SER - mean(SER, na.rm = T)

term  estimate
intercept 0.66
SER_c     0.11
POSVerb   0.72
SER_c:POSVerb 0.50

Obs: SER is a continuous predictor and POSVerb is a categorical predictor with two levels (0: Noun, 1: Verb)

The author concludes the following: "POSVerb is the difference between nouns and verbs for words with average SER, rather than some arbitrary 0". All right, that makes sense. The intercept should be, then, the value of the outcome variable for the mean of SER. Ok.

But, still, what does one-unit change in SER mean? If it wasn't centered, I'd say "one unit change in SER implies + 0.11 points to the outcome variable for the noun category", but I now that the intercept means the value at SER mean, how do I interpret this?

this post was helpful, but not enough. Thanks in advance!

• edit : bounty:

Hello, again. Now I have a follow-up question:

• 1: My model has a 2-level categorical predictor and a continuous one centered around its mean (no interaction). Should I say "one-unit change in the continuons predictor" or "one-unit change in the mean of the continous predictor" adds X to the mean of each categorical level?

Answer 1

As mentioned in the article that you linked, centering a feature does not impact its scale, so it should not change the interpretation of your model coefficients.

For example, consider the following linear model, where $x_1$ is continuous and $\delta_1$ is a dummy variable {0, 1}:

$$=B_0 + B_1x_1 + B_2\delta_1 + B_3 x_1 \delta_1 + u$$ In the equation above, the partial effect of $x_1$ on $y$ is

$$ \frac{\Delta y}{\Delta x_1} = B_1 + B_3\delta_1 $$ So, one unit of change in $x_1$ leads to an increase of $B1+B3$ in $y$ if $d_1 = 1$ and an increase of $B_1$ otherwise.

Suppose that we now center $x_1$ around its mean $\bar{x_1}$, let's call it $\hat{x}_1$, with the following operation:

$$ \hat{x_1} = x_1 - \bar{x_1} $$

Notice that we can replace $x_1$ in the initial model by $\hat{x_1} + \bar{x_1}$, such that $$ =B_0 + B_1(\hat{x_1} + \bar{x_1}) + B_2\delta_1 + B_3 (\hat{x_1} + \bar{x_1}) \delta_1 + u $$ Which you can expand to

$$ =B_0 + B_1\hat{x_1} + B_1\bar{x_1} + B_2\delta_1 + B_3\hat{x_1}\delta_1 + B_3\bar{x_1}\delta_1 + u $$ So now, the partial effect of $\hat{x_1}$ on $y$ is:

$$ \frac{\Delta y}{\Delta \hat{x_1}} = B_1 + B_3\delta_1 $$

Which is the same as before. In your particular case:

$$ \frac{\Delta y}{\Delta SER_c} = \frac{\Delta y}{\Delta SER} = 0.11 + 0.50 POSVerb $$

In the case of noun ($POSVerb = 0$) one unit change of $SER$ or $SER_c$ yields a 0.11 increase to the outcome. In the case of a verb ($POSVerb = 1$), then one unit of change in $SER$ or $SER_c$ yields a 0.11 + 0.50 = 0.61 increase to the outcome.

Hope this helps!

Answer 2

I know that when we standardize a predictor, "one unit change" becomes one standard deviation in the predictor, but what if we only center the data on the mean (i.e. only subtract all values by the mean) ?

If you don't standardize then one unit change will relate to whatever the units are that are used to measure the predictor variable.

For instance if the predictor is temperature measured in Kelvin, then on unit change means an increase by one Kelvin.

In your example you have 'sensory experience ratings' as predictor and one-unit change will relate to whatever the scale is used to obtain those ratings.

Possibly your sensory experience rating is the one from

Juhasz, B.J., Yap, M.J. Sensory experience ratings for over 5,000 mono- and disyllabic words. Behav Res 45, 160–168 (2013). https://doi.org/10.3758/s13428-012-0242-9

Please rate each word on a 1 to 7 scale, with 1 meaning the word evokes no sensory experience for you, 4 meaning the word evokes a moderate sensory experience, and 7 meaning the word evokes a strong sensory experience.

In that case the ratings are based on averages of questionnaires with a 7 point scale.

The meaning of one-unit is the change of the score on that questionnaire by one.