Slopes, floodlight and spotlight analysis (Johnson-Neyman intervals)"

In modelbased: Estimation of Model-Based Predictions, Contrasts and Means

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  dev = "png",
  fig.width = 7,
  fig.height = 3.5,
  message = FALSE,
  warning = FALSE,
  package.startup.message = FALSE
)
options(width = 800)
options(modelbased_join_dots = FALSE)
ht6 <- m <- NULL

arrow_color <- "#FF00cc"

pkgs <- c("ggplot2", "see", "marginaleffects", "parameters")

if (!all(insight::check_if_installed(pkgs, quietly = TRUE))) {
  knitr::opts_chunk$set(eval = FALSE)
}

This vignette is the second in a 4-part series:

1. Contrasts and Pairwise Comparisons

2. Comparisons of Slopes, Floodlight and Spotlight Analysis (Johnson-Neyman Intervals)

3. Contrasts and Comparisons for Generalized Linear Models

4. Contrasts and Comparisons for Zero-Inflation Models

Contrasts and comparisons for slopes of numeric predictors

For numeric focal terms, it is possible to calculate contrasts for slopes, or the linear trend of these focal terms. Let's start with a simple example again.

library(modelbased)
library(parameters)
data(iris)
m <- lm(Sepal.Width ~ Sepal.Length + Species, data = iris)
model_parameters(m)

We can already see from the coefficient table that the slope for Sepal.Length is 0.35. We will thus find the same increase for the predicted values in our outcome when our focal variable, Sepal.Length increases by one unit.

estimate_means(m, "Sepal.Length=c(4,5,6,7)")

Consequently, in this case of a simple slope, we see the same result for the estimated linear trend of Sepal.Length:

estimate_slopes(m, "Sepal.Length")

Is the linear trend of Sepal.Length significant for the different levels of Species?

Let's move on to a more complex example with an interaction between a numeric and categorical variable.

Predictions

m <- lm(Sepal.Width ~ Sepal.Length * Species, data = iris)
pred <- estimate_means(m, c("Sepal.Length", "Species"))
plot(pred)

Slopes by group

We can see that the slope of Sepal.Length is different within each group of Species.

library(ggplot2)
p <- plot(pred)
dat <- as.data.frame(pred)
dat1 <- data.frame(
  x = dat$Sepal.Length[c(7, 19)],
  y = dat$Mean[c(7, 19)],
  group_col = "setosa",
  stringsAsFactors = FALSE
)
dat2 <- data.frame(
  x = dat$Sepal.Length[c(8, 20)],
  y = dat$Mean[c(8, 23)],
  group_col = "versicolor",
  stringsAsFactors = FALSE
)
dat3 <- data.frame(
  x = dat$Sepal.Length[c(15, 27)],
  y = dat$Mean[c(15, 27)],
  group_col = "virginica",
  stringsAsFactors = FALSE
)
p + annotate(
  "segment",
  x = dat1$x[1], xend = dat1$x[2], y = dat1$y[1], yend = dat1$y[1],
  color = "orange", linewidth = 1, linetype = "dashed"
) + annotate(
  "segment",
  x = dat1$x[2], xend = dat1$x[2], y = dat1$y[1], yend = dat1$y[2],
  color = "orange", linewidth = 1, linetype = "dashed"
) + annotate(
  "segment",
  x = dat1$x[1], xend = dat1$x[2], y = dat1$y[1], yend = dat1$y[2],
  color = "orange", linewidth = 1
) + annotate(
  "segment",
  x = dat2$x[1], xend = dat2$x[2], y = dat2$y[1], yend = dat2$y[1],
  color = "orange", linewidth = 1, linetype = "dashed"
) + annotate(
  "segment",
  x = dat2$x[2], xend = dat2$x[2], y = dat2$y[1], yend = dat2$y[2],
  color = "orange", linewidth = 1, linetype = "dashed"
) + annotate(
  "segment",
  x = dat2$x[1], xend = dat2$x[2], y = dat2$y[1], yend = dat2$y[2],
  color = "orange", linewidth = 1
) + annotate(
  "segment",
  x = dat3$x[1], xend = dat3$x[2], y = dat3$y[1], yend = dat3$y[1],
  color = "orange", linewidth = 1, linetype = "dashed"
) + annotate(
  "segment",
  x = dat3$x[2], xend = dat3$x[2], y = dat3$y[1], yend = dat3$y[2],
  color = "orange", linewidth = 1, linetype = "dashed"
) + annotate(
  "segment",
  x = dat3$x[1], xend = dat3$x[2], y = dat3$y[1], yend = dat3$y[2],
  color = "orange", linewidth = 1
) +
  ggtitle("Linear trend of `Sepal.Length` by `Species`")

Since we don't want to do pairwise comparisons, we still use estimate_slopes() to test whether the linear trend (by groups) is significant or not. In this case, when interaction terms are included, the linear trend (slope) for our numeric focal predictor, Sepal.Length, is tested for each level of Species.

estimate_slopes(m, "Sepal.Length", by = "Species")

As we can see, each of the three slopes is significant, i.e. we have "significant" linear trends.

Pairwise comparisons

Next question could be whether or not linear trends differ significantly between each other, i.e. we test differences in slopes, which is a pairwise comparison between slopes. To do this, we use estimate_contrasts().

estimate_contrasts(m, "Sepal.Length", by = "Species")

The linear trend of Sepal.Length within setosa is significantly different from the linear trend of versicolor and also from virginica. The difference of slopes between virginica and versicolor is not statistically significant (p = 0.366).

Is the difference linear trends of Sepal.Length in between two groups of Species significantly different from the difference of two linear trends between two other groups?

Similar to the example for categorical predictors, we can also test a difference-in-differences for this example. For instance, is the difference of the slopes from Sepal.Length between setosa and versicolor different from the slope-difference for the groups setosa and vigninica?

Let's first look at the different slopes separately again, i.e. the slopes of Sepal.Length by levels of Species:

estimate_slopes(m, "Sepal.Length", by = "Species")

The first difference of slopes we're interested in is the one between setosa (0.80) and versicolor (0.32), i.e. b1 - b2 (=0.48). The second difference is between levels setosa (0.80) and virginica (0.23), which is b1 - b3 (=0.57). We test the null hypothesis that (b1 - b2) = (b1 - b3).

estimate_contrasts(
  m,
  "Sepal.Length",
  by = "Species",
  comparison = "(b1 - b2) = (b1 - b3)"
)

The difference between the two differences is -0.09 and not statistically significant (p = 0.366).

Is the linear trend of Sepal.Length significant at different values of another numeric predictor?

When we have two numeric terms in an interaction, the comparison becomes more difficult, because we have to find meaningful (or representative) values for the moderator, at which the associations between the predictor and outcome are tested. We no longer have distinct categories for the moderator variable.

Spotlight analysis, floodlight analysis and Johnson-Neyman intervals

The following examples show interactions between two numeric predictors. In case of numeric interaction terms, it makes sense to calculate adjusted predictions for representative values, e.g. mean +/- SD. This is sometimes also called "spotlight analysis" (Spiller et al. 2013).

In the next example, we have Petal.Width as second interaction term, thus we see the predicted values of Sepal.Width (our outcome) for Petal.Length at three different, representative values of Petal.Width: Mean (r round(mean(iris$Petal.Width), 2)), 1 SD above the mean (r round(mean(iris$Petal.Width) + sd(iris$Petal.Width), 2)) and 1 SD below the mean (r round(mean(iris$Petal.Width) - sd(iris$Petal.Width), 2)).

Predictions

m <- lm(Sepal.Width ~ Petal.Length * Petal.Width, data = iris)
pred <- estimate_means(m, c("Petal.Length", "Petal.Width=[sd]"))
plot(pred)

First, we want to see at which value of Petal.Width the slopes of Petal.Length are significant. We do no pairwise comparison for now, hence we use estimate_slopes().

estimate_slopes(m, "Petal.Length", by = "Petal.Width=[sd]")

Pairwise comparisons

The results of the pairwise comparison are shown below. These tell us that all linear trends (slopes) are significantly different from each other, i.e. the slope of the green line is significantly different from the slope of the red line, and so on.

estimate_contrasts(m, "Petal.Length", by = "Petal.Width=[sd]", digits = 1)

Floodlight analysis and Johnson-Neyman intervals

Another way to handle models with two numeric variables in an interaction is to use so-called floodlight analysis, a spotlight analysis for all values of the moderator variable. These intervals indicate the values of the moderator at which the slope of the predictor is significant (cf. Johnson et al. 1950, McCabe et al. 2018).

Let's look at an example. We first plot the predicted values of Income for Murder at different values of Illiteracy.

states <- as.data.frame(state.x77)
states$HSGrad <- states$`HS Grad`
m_mod <- lm(Income ~ HSGrad + Murder * Illiteracy, data = states)

pr <- estimate_means(m_mod, c("Murder", "Illiteracy"))
plot(pr)

It's difficult to say at which values from Illiteracy, the association between Murder and Income might be statistically significant. We still can use estimate_slopes():

estimate_slopes(m_mod, "Murder", by = "Illiteracy")

As can be seen, the results might indicate that at the lower and upper tails of Illiteracy, i.e. when Illiteracy is roughly smaller than 0.8 or larger than 2.6, the association between Murder and Income is statistically significant.

However, this test can be simplified using the summary() function. This will show us in detail at which values for Illiteracy the interaction term is statistically significant, and whether the association between Murder and the outcome is positive or negative.

# we will force to calculate slopes at 200 values for "Illiteracy" using `length`
slopes <- estimate_slopes(m_mod, "Murder", by = "Illiteracy", length = 200)
summary(slopes)

Furthermore, it is possible to create a spotlight-plot.

plot(slopes)

The results of the spotlight analysis suggest that values below 0.82 and above 2.57 are significantly different from zero, while values in between are not. We can plot predictions at these values to see the differences. The red and the green line represent values of Illiteracy at which we find clear positive resp. negative associations between Murder and Income, while we find no clear (positive or negative) association for the red line.

Here an example, using values from the three "ranges" of the Johnson-Neyman-Interval: the red and blue lines are significantly positive and negative associated with the outcome, while the green line is not significant.

pr <- estimate_means(m_mod, c("Murder", "Illiteracy=c(0.7,1.5,2.8)"))
plot(pr) + ggplot2::facet_wrap(~Illiteracy)

Go to next vignette: Contrasts and Comparisons for Generalized Linear Models

References

Johnson, P.O. & Fay, L.C. (1950). The Johnson-Neyman technique, its theory and application. Psychometrika, 15, 349-367. doi: 10.1007/BF02288864

McCabe CJ, Kim DS, King KM. (2018). Improving Present Practices in the Visual Display of Interactions. Advances in Methods and Practices in Psychological Science, 1(2):147-165. doi:10.1177/2515245917746792

Spiller, S. A., Fitzsimons, G. J., Lynch, J. G., & McClelland, G. H. (2013). Spotlights, Floodlights, and the Magic Number Zero: Simple Effects Tests in Moderated Regression. Journal of Marketing Research, 50(2), 277–288. doi:10.1509/jmr.12.0420

modelbased documentation built on April 12, 2025, 2:22 a.m.