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

1. Significance Testing of Differences Between Predictions I: Contrasts and Pairwise Comparisons

2. Significance Testing of Differences Between Predictions II: Comparisons of Slopes, Floodlight and Spotlight Analysis (Johnson-Neyman Intervals)

3. Significance Testing of Differences Between Predictions III: Contrasts and Comparisons for Generalized Linear Models

4. Significance Testing of Differences Between Predictions IV: Contrasts and Comparisons for Zero-Inflation Models

## Hypothesis testing for slopes of numeric predictors

For numeric focal terms, it is possible to conduct hypothesis testing for slopes, or the linear trend of these focal terms.

## Summary of most important points:

• For categorical predictors (focal terms), it is easier to define which values to compare. For continuous predictors, however, you may want to compare different (meaningful) values of that predictor, or its slope (or even compare slopes of different continuous focal terms).
• By default, when the first focal term is continuous, contrasts or comparisons are calculated for the slopes of that predictor.
• It is even more complicated to deal with interactions of (at least) two continuous predictors. In this case, the `johnson_neyman()` function can be used, which is a special case of pairwise comparisons for interactions with continuous predictors.

``````library(ggeffects)
library(parameters)
data(iris)
m <- lm(Sepal.Width ~ Sepal.Length + Species, data = iris)
model_parameters(m)
#> Parameter            | Coefficient |   SE |         95% CI | t(146) |      p
#> ----------------------------------------------------------------------------
#> (Intercept)          |        1.68 | 0.24 | [ 1.21,  2.14] |   7.12 | < .001
#> Sepal Length         |        0.35 | 0.05 | [ 0.26,  0.44] |   7.56 | < .001
#> Species [versicolor] |       -0.98 | 0.07 | [-1.13, -0.84] | -13.64 | < .001
#> Species [virginica]  |       -1.01 | 0.09 | [-1.19, -0.82] | -10.80 | < .001``````

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.

``````predict_response(m, "Sepal.Length [4,5,6,7]")
#> # Predicted values of Sepal.Width
#>
#> Sepal.Length | Predicted |     95% CI
#> -------------------------------------
#>            4 |      3.08 | 2.95, 3.20
#>            5 |      3.43 | 3.35, 3.51
#>            6 |      3.78 | 3.65, 3.90
#>            7 |      4.13 | 3.93, 4.33
#>
#> * Species = setosa``````

Consequently, in this case of a simple slope, we see the same result for the hypothesis test for the linar trend of `Sepal.Length`:

``````test_predictions(m, "Sepal.Length")
#> # (Average) Linear trend for Sepal.Length
#>
#> Slope |     95% CI |      p
#> ---------------------------
#> 0.35  | 0.26, 0.44 | < .001``````

### 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 <- predict_response(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`.

Since we don’t want to do pairwise comparisons, we set `test = NULL`. 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`.

``````test_predictions(m, c("Sepal.Length", "Species"), test = NULL)
#> # (Average) Linear trend for Sepal.Length
#>
#> Species    | Slope |     95% CI |      p
#> ----------------------------------------
#> setosa     |  0.80 | 0.58, 1.02 | < .001
#> versicolor |  0.32 | 0.17, 0.47 | < .001
#> virginica  |  0.23 | 0.11, 0.35 | < .001``````

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 the default for `test`, which is `"pairwise"`.

``````test_predictions(m, c("Sepal.Length", "Species"))
#> # (Average) Linear trend for Sepal.Length
#>
#> Species              | Contrast |      95% CI |      p
#> ------------------------------------------------------
#> setosa-versicolor    |     0.48 |  0.21, 0.74 | < .001
#> setosa-virginica     |     0.57 |  0.32, 0.82 | < .001
#> versicolor-virginica |     0.09 | -0.10, 0.28 | 0.367``````

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 = 0.367).

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`?

This difference-in-differences we’re interested in is again indicated by the purple arrow in the below plot.

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

``````test_predictions(m, c("Sepal.Length", "Species"), test = NULL)
#> # (Average) Linear trend for Sepal.Length
#>
#> Species    | Slope |     95% CI |      p
#> ----------------------------------------
#> setosa     |  0.80 | 0.58, 1.02 | < .001
#> versicolor |  0.32 | 0.17, 0.47 | < .001
#> virginica  |  0.23 | 0.11, 0.35 | < .001``````

The first difference of slopes we’re interested in is the one between `setosa` (0.8) and `versicolor` (0.32), i.e. `b1 - b2` (=0.48). The second difference is between levels `setosa` (0.8) and `virginica` (0.23), which is `b1 - b3` (=0.57). We test the null hypothesis that `(b1 - b2) = (b1 - b3)`.

``````test_predictions(m, c("Sepal.Length", "Species"), test = "(b1 - b2) = (b1 - b3)")
#> Hypothesis      | Contrast |      95% CI |     p
#> ------------------------------------------------
#> (b1-b2)=(b1-b3) |    -0.09 | -0.28, 0.10 | 0.367
#>
#> Tested hypothesis: (Species[setosa] - Species[versicolor]) = (Species[setosa] - Species[virginica])``````

The difference between the two differences is -0.09 and not statistically significant (p = 0.367).

### 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 interaction terms, adjusted predictions are usually shown at representative values. If a numeric variable is specified as second or third interaction term, representative values (see `values_at()`) are typically 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 (1.2), 1 SD above the mean (1.96) and 1 SD below the mean (0.44).

#### Predictions

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

For `test_predictions()`, these three values (mean, +1 SD and -1 SD) work in the same way as if `Petal.Width` was a categorical predictor with three levels.

First, we want to see at which value of `Petal.Width` the slopes of `Petal.Length` are significant. We do no pairwise comparison here, hence we set `test = NULL`.

``````test_predictions(pred, test = NULL)
#> # (Average) Linear trend for Petal.Length
#>
#> Petal.Width | Slope |       95% CI |      p
#> -------------------------------------------
#> 0.44        | -0.28 | -0.39, -0.16 | < .001
#> 1.20        | -0.11 | -0.23,  0.01 | 0.075
#> 1.96        |  0.06 | -0.09,  0.20 | 0.438
# same as:
# test_predictions(m, c("Petal.Length", "Petal.Width"), test = NULL)``````

#### 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.

``````test_predictions(pred)
#> # (Average) Linear trend for Petal.Length
#>
#> Petal.Width | Contrast |       95% CI |      p
#> ----------------------------------------------
#> 0.44-1.2    |    -0.17 | -0.21, -0.12 | < .001
#> 0.44-1.96   |    -0.33 | -0.43, -0.24 | < .001
#> 1.2-1.96    |    -0.17 | -0.21, -0.12 | < .001``````

#### 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, which is implemented in the `johnson_neyman()` function that creates Johnson-Neyman intervals. 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 nine different values of `Illiteracy` (there are no more colors in the default palette to show more lines).

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

myfun <- seq(0.5, 3, length.out = 9)
pr <- predict_response(m_mod, c("Murder", "Illiteracy [myfun]"))
plot(pr)``````

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

``````test_predictions(pr, test = NULL)
#> # (Average) Linear trend for Murder
#>
#> Illiteracy |   Slope |          95% CI |     p
#> ----------------------------------------------
#> 0.50       |   82.08 |   18.69, 145.47 | 0.012
#> 0.81       |   51.76 |   -1.10, 104.61 | 0.055
#> 1.12       |   21.43 |  -29.44,  72.30 | 0.401
#> 1.44       |   -8.89 |  -67.20,  49.42 | 0.760
#> 1.75       |  -39.22 | -111.55,  33.12 | 0.281
#> 2.06       |  -69.54 | -159.45,  20.37 | 0.126
#> 2.38       |  -99.87 | -209.20,   9.46 | 0.072
#> 2.69       | -130.19 | -259.97,  -0.41 | 0.049
#> 3.00       | -160.52 | -311.34,  -9.69 | 0.038``````

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 signifiant.

However, this test can be simplified using the `johnson_neyman()` function:

``````johnson_neyman(pr)
#> The association between `Murder` and `Income` is positive for values of
#> `Illiteracy` lower than 0.79 and negative for values higher than 2.67.
#> Inside the interval of [0.79, 2.67], there were no clear associations.``````

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

``````plot(johnson_neyman(pr))
#> The association between `Murder` and `Income` is positive for values of `Illiteracy` lower than 0.80 and negative for values higher than 2.67. Inside the interval of [0.80, 2.67], there were no clear associations.``````

To avoid misleading interpretations of the plot, we speak of “positive” and “negative” associations, respectively, or “no clear” associations (instead of “significant” or “non-significant”). This should prevent considering a non-significant range of values of the moderator as “accepting the null hypothesis”.

The results of the spotlight analysis suggest that values below `0.79` and above `2.67` 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 blue line.

``````pr <- predict_response(m_mod, c("Murder", "Illiteracy [0.7, 1.5, 2.8]"))
plot(pr, grid = TRUE)``````

Go to next vignette: Significance Testing of Differences Between Predictions III: 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