Extract estimates from a fitted model

Example: Life satisfaction in Scotland

These data come from 112 people across 12 different Scottish dwellings (cities and towns). Information is captured on their ages and a measure of life satisfaction. The researchers are interested in if there is an association between age and life-satisfaction.

Data are available at https://uoepsy.github.io/data/lmm_lifesatscot.csv.

variable description
age Age (years)
lifesat Life Satisfaction score
dwelling Dwelling (town/city in Scotland)
size Size of Dwelling (> or <100k people) 
lifesatscot <- read_csv("https://uoepsy.github.io/data/lmm_lifesatscot.csv")

Our model is the following (to see how we figured out this random effect structure, see Identify possible random effects).

lifesat_mod <- lmer(
  lifesat ~ age + (1 + age | dwelling),
  data = lifesatscot
)

Extract fixed effects

The function fixef() will return the estimated values for the model’s fixed effects.

fixef(lifesat_mod)
(Intercept)         age 
     28.241       0.523

Extract all group-level adjustments

How much does each individual group (here, each dwelling) differ from that fixed intercept and that fixed slope over age?

In other words, how do we need to adjust the fixed intercept and the fixed slope over age to arrive at the best line for each group?

We can extract these adjustments using the function ranef(), which gives us a list containing one element per group. We’ll use the dollar-sign notation to pull out the data frame of adjustments for our grouping variable dwelling:

ranef(lifesat_mod)$dwelling
             (Intercept)     age
Aberdeen          -26.62  0.4875
Dumfries          -13.54  0.2986
Dundee             16.43 -0.2829
Dunfermline         7.75  0.1815
Edinburgh          -5.48  0.2180
Fort William       -6.12 -0.0275
Glasgow            24.08 -0.4348
Inverness         -12.73  0.2091
Kirkcaldy          14.50 -0.7339
Paisley            -3.32 -0.0903
Perth              -3.02  0.1901
Stirling            8.07 -0.0154

Each row is a level of the grouping variable (a dwelling in Scotland), and each column shows how that dwelling’s intercept and slope over age differ from the estimated fixed effects. So, since the fixed effects estimate the average intercept and average slope, we can observe (for example) that:

  • Aberdeen has an estimated intercept that is 26.62 points lower than average (lower because the value is negative), and an estimated slope over age that is 0.48 points higher than average (higher because the value is positive).
  • Kirkcaldy has an estimated intercept that is 14.50 points higher than average, and an estimated slope over age that is 0.73 points lower than average.

Extract group-level intercepts and slopes

To find out the actual parameters of the line estimated for each dwelling, we can use the function coef().

coef(lifesat_mod)$dwelling
             (Intercept)     age
Aberdeen            1.62  1.0110
Dumfries           14.70  0.8221
Dundee             44.67  0.2406
Dunfermline        35.99  0.7049
Edinburgh          22.76  0.7415
Fort William       22.12  0.4959
Glasgow            52.32  0.0886
Inverness          15.51  0.7325
Kirkcaldy          42.74 -0.2105
Paisley            24.92  0.4331
Perth              25.23  0.7136
Stirling           36.31  0.5081

Again, each row is a level of our grouping variable dwelling. Now each column represents the actual intercept and the actual slope of the line estimated for each dwelling.

Using Aberdeen as an example, here’s where these numbers come from:

  • Aberdeen’s intercept:
    • The fixed intercept is 28.24.
    • The adjustment to that fixed intercept is –26.62.
    • Combining those two values, we calculate 28.24 – 26.62 = 1.62, which matches the intercept of the line for Aberdeen given by coef().
  • Aberdeen’s slope over age:
    • The fixed slope is 0.52.
    • The adjustment to that fixed slope is 0.49.
    • 0.52 + 0.49 = 1.01, which matches the slope of the line over age for Aberdeen given by coef().

In short: fixef() + ranef() = coef().

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