TLDR: We should interpret regression coefficients for continuous variables as we would descriptive dummy variables, unless we intend to make causal claims.

I am going to be teaching regression labs in the Fall, and somehow, I stumbled onto Gelman and Hill’s *Data analysis using regression and multilevel/hierarchical models*.^{1} So I started reading it and it’s a good book.

A useful piece of advice they give is to interpret regression coefficients in a predictive manner (p. 34). To see what they mean, let us consider an example.

## Predicting student performance

I’ll use a subset of the High School & Beyond dataset.

```
hsb <- read.csv("datasets/hsb_comb_full.csv")
names(hsb)
[1] "schoolid" "minority" "female" "ses" "mathach" "size" "sector"
[8] "pracad" "disclim" "himinty" "MEANSES" "N_BREAK" "sesdev" "myschool"
# Let's go with the first school, and the first 5 student-level variables
hsb <- hsb[hsb$schoolid == hsb$schoolid[1], 1:5]
summary(hsb)
schoolid minority female ses mathach
Min. :1224 Min. :0.00000 Min. :0.0000 Min. :-1.6580 Min. :-2.832
1st Qu.:1224 1st Qu.:0.00000 1st Qu.:0.0000 1st Qu.:-0.8830 1st Qu.: 3.450
Median :1224 Median :0.00000 Median :1.0000 Median :-0.4680 Median : 8.296
Mean :1224 Mean :0.08511 Mean :0.5957 Mean :-0.4344 Mean : 9.715
3rd Qu.:1224 3rd Qu.:0.00000 3rd Qu.:1.0000 3rd Qu.:-0.0330 3rd Qu.:16.370
Max. :1224 Max. :1.00000 Max. :1.0000 Max. : 0.9720 Max. :23.584
# Mathach, ses and female seem to have some variability
# Let's predict math achievement using female (dummy), ses (continuous)
lm(mathach ~ female + ses, hsb)
Call:
lm(formula = mathach ~ female + ses, data = hsb)
Coefficients:
(Intercept) female ses
12.092 -2.062 2.643
```

Now the typical approach to interpreting the coefficient for `female`

is:

Holding SES constant, there is on average, a 2.06-point difference in math achievement between males and females, with males performing better.

There is nothing wrong with this approach, however to clarify the language, we could say:

For students with the same SES, we expect a 2.06-point difference in math achievement between males and females, with males performing better.

The problem arises with the interpretation of `ses`

, it typically goes:

Holding gender constant, a point improvement in SES relates with a 2.64 increase in math achievement.

We typically claim this is a correlational statement, devoid of causal claims. However, it has causal overtones. It insinuates that within an individual, if we could raise their SES by 1 point, we can expect an increase in math achievement by 2.64 points.

Gelman and Hill advice phrasing its interpretation like this:

For students of the same gender, we expect a 2.64-point difference in math achievement between students who have a point difference in SES.

This is what they call a *predictive interpretation* of regression coefficients. It is devoid of causality, and communicates that we are making predictions for or describing the difference between different individuals.

- Gelman, A., & Hill, J. (2007).
*Data analysis using regression and multilevel/hierarchical models*. Cambridge University Press. ↩

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