Multiple regression

more than one explanatory variable, with moderndive

two models, one question + vs *

  • lm(y ~ x1 + x2)
  • lm(y ~ x1 * x2)
  • geom_parallel_slopes()
  • coef()

Fertility rate, life expectancy, and income group

Why add a variable?

One explanatory variable can mislead you when a second one is doing part of the work.

  • A slope in simple regression ignores everything else.
  • In multiple regression each slope holds the other variables fixed.
  • Adding a categorical variable gives each group its own intercept.
  • Adding an interaction gives each group its own slope too.

y ~ x1 + x2

The parallel slopes model. Each group gets its own intercept, but every group shares one common slope.

One slope for everybody, shifted up or down by group.

lm()

Example

the verb (function) gives each group its own intercept

model_no_int <- lm(fert_rate ~ life_exp + income, data = UN_data_ch6)

response ~ numerical + categorical the data frame

geom_parallel_slopes()

Draw the parallel slopes model. Same code as a scatterplot, with geom_smooth() swapped out.

Example

one color per group

ggplot(UN_data_ch6, aes(x = life_exp, y = fert_rate, color = income)) +
  geom_point() +
  geom_parallel_slopes(se = FALSE)

Scatterplot of fertility rate against life expectancy for 182 UN member states, colored by four income groups, with four parallel downward-sloping lines, one per group; low-income countries sit highest and to the left, high-income countries lowest and to the right

one line per group, all the same slope

y ~ x1 * x2

The interaction model. Each group gets its own intercept and its own slope, so the lines are free to fan out.

Use it when the effect of x differs from group to group.

lm()

The star fits an interaction: each group gets its own slope as well as its own intercept.

Example

the verb (function) interaction: slopes differ too

model_int <- lm(fert_rate ~ life_exp * income,
                data = UN_data_ch6)

the data frame

coef()

Read the coefficients. With an interaction, the extra rows are offsets from the baseline group, not slopes on their own.

Example

the fitted model

coef(model_int)
                       (Intercept)                           life_exp 
                       11.91826460                        -0.11848075 
         incomeLower middle income          incomeUpper middle income 
                       -1.50420617                        -1.89291757 
                 incomeHigh income life_exp:incomeLower middle income 
                       -6.57953685                         0.01265027 
life_exp:incomeUpper middle income         life_exp:incomeHigh income 
                        0.01117481                         0.07221696

the verb (function)

Reading the coefficients

The baseline group is the first level of the factor: here, Low income.

  • The (Intercept) entry belongs to the baseline group.
  • A group’s entry is how far its intercept sits from the baseline.
  • In an interaction model, an x:group entry shifts that group’s slope.
  • Add the offset to the baseline to get the group’s own value.

Which model?

Fit both, look at both, then choose.

  • Plot the data colored by group before fitting anything.
  • If the clouds trend at visibly different angles, try the interaction.
  • If the lines would be nearly parallel anyway, the simpler model wins.
  • Compare the two fitted models’ coefficients side by side with coef().