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Hypothesis testing for categorical predictors

A reason to compute adjusted predictions (or estimated marginal means) is to help understanding the relationship between predictors and outcome of a regression model. In particular for more complex models, for example, complex interaction terms, it is often easier to understand the associations when looking at adjusted predictions instead of the raw table of regression coefficients.

The next step, which often follows this, is to see if there are statistically significant differences. These could be, for example, differences between groups, i.e. between the levels of categorical predictors or whether trends differ significantly from each other.

The ggeffects package provides a function, hypothesis_test(), which does exactly this: testing differences of adjusted predictions for statistical significance. This is usually called contrasts or (pairwise) comparisons. This vignette shows some examples how to use the hypothesis_test() function and how to test wheter differences in predictions are statistically significant.

Within episode, do levels differ?

We start with a toy example, where we have a linear model with two categorical predictors. No interaction is involved for now.

We display a simple table of regression coefficients, created with model_parameters() from the parameters package.

library(ggeffects)
library(parameters)
library(ggplot2)

set.seed(123)
n <- 200
d <- data.frame(
  outcome = rnorm(n),
  grp = as.factor(sample(c("treatment", "control"), n, TRUE)),
  episode = as.factor(sample(1:3, n, TRUE)),
  sex = as.factor(sample(c("female", "male"), n, TRUE, prob = c(0.4, 0.6)))
)
model1 <- lm(outcome ~ grp + episode, data = d)
model_parameters(model1)
#> Parameter       | Coefficient |   SE |        95% CI | t(196) |     p
#> ---------------------------------------------------------------------
#> (Intercept)     |       -0.08 | 0.13 | [-0.33, 0.18] |  -0.60 | 0.552
#> grp [treatment] |       -0.17 | 0.13 | [-0.44, 0.09] |  -1.30 | 0.197
#> episode [2]     |        0.36 | 0.16 | [ 0.03, 0.68] |   2.18 | 0.031
#> episode [3]     |        0.10 | 0.16 | [-0.22, 0.42] |   0.62 | 0.538

Predictions

Let us look at the adjusted predictions.

mydf <- ggpredict(model1, "episode")
mydf
#> # Predicted values of outcome
#> 
#> episode | Predicted |        95% CI
#> -----------------------------------
#> 1       |     -0.08 | [-0.33, 0.18]
#> 2       |      0.28 | [ 0.02, 0.54]
#> 3       |      0.02 | [-0.24, 0.28]
#> 
#> Adjusted for:
#> * grp = control

plot(mydf)

We now see that, for instance, the predicted outcome when espisode = 2 is 0.28.

Pairwise comparisons

We could now ask whether the predicted outcome for episode = 1 is significantly different from the predicted outcome at episode = 2.

To do this, we use the hypothesis_test() function. This function, like ggpredict(), accepts the model object as first argument, followed by the focal predictors of interest, i.e. the variables of the model for which contrasts or pairwise comparisons should be calculated.

By default, when all focal terms are categorical, a pairwise comparison is performed. You can specify other hypothesis tests as well, using the test argument (which defaults to "pairwise", see ?hypothesis_test). For now, we go on with the simpler example of contrasts or pairwise comparisons.

hypothesis_test(model1, "episode") # argument `test` defaults to "pairwise"
#> # Pairwise comparisons
#> 
#> episode | Contrast |         95% CI |     p
#> -------------------------------------------
#> 1-2     |    -0.36 | [-0.68, -0.03] | 0.031
#> 1-3     |    -0.10 | [-0.42,  0.22] | 0.538
#> 2-3     |     0.26 | [-0.06,  0.58] | 0.112

For our quantity of interest, the contrast between episode 1-2, we see the value -0.36, which is exactly the difference between the predicted outcome for episode = 1 (-0.08) and episode = 2 (0.28). The related p-value is 0.031, indicating that the difference between the predicted values of our outcome at these two levels of the factor episode is indeed statistically significant.

In this simple example, the contrasts of both episode = 2 and episode = 3 to episode = 1 equals the coefficients of the regression table above (same applies to the p-values), where the coefficients refer to the difference between the related parameter of episode and its reference level, episode = 1.

To avoid specifying all arguments used in a call to ggpredict() again, we can also pass the objects returned by ggpredict() directly into hypothesis_test().

pred <- ggpredict(model1, "episode")
hypothesis_test(pred)
#> # Pairwise comparisons
#> 
#> episode | Contrast |         95% CI |     p
#> -------------------------------------------
#> 1-2     |    -0.36 | [-0.68, -0.03] | 0.031
#> 1-3     |    -0.10 | [-0.42,  0.22] | 0.538
#> 2-3     |     0.26 | [-0.06,  0.58] | 0.112

Does same level of episode differ between groups?

The next example includes a pairwise comparison of an interaction between two categorical predictors.

model2 <- lm(outcome ~ grp * episode, data = d)
model_parameters(model2)
#> Parameter                     | Coefficient |   SE |        95% CI | t(194) |     p
#> -----------------------------------------------------------------------------------
#> (Intercept)                   |        0.03 | 0.15 | [-0.27, 0.33] |   0.18 | 0.853
#> grp [treatment]               |       -0.42 | 0.23 | [-0.88, 0.04] |  -1.80 | 0.074
#> episode [2]                   |        0.20 | 0.22 | [-0.23, 0.63] |   0.94 | 0.350
#> episode [3]                   |       -0.07 | 0.22 | [-0.51, 0.37] |  -0.32 | 0.750
#> grp [treatment] × episode [2] |        0.36 | 0.33 | [-0.29, 1.02] |   1.09 | 0.277
#> grp [treatment] × episode [3] |        0.37 | 0.32 | [-0.27, 1.00] |   1.14 | 0.254

Predictions

First, we look at the predicted values of outcome for all combinations of the involved interaction term.

mydf <- ggpredict(model2, c("episode", "grp"))
mydf
#> # Predicted values of outcome
#> 
#> # grp = control
#> 
#> episode | Predicted |        95% CI
#> -----------------------------------
#> 1       |      0.03 | [-0.27, 0.33]
#> 2       |      0.23 | [-0.08, 0.54]
#> 3       |     -0.04 | [-0.36, 0.28]
#> 
#> # grp = treatment
#> 
#> episode | Predicted |         95% CI
#> ------------------------------------
#> 1       |     -0.39 | [-0.74, -0.04]
#> 2       |      0.18 | [-0.18,  0.53]
#> 3       |     -0.09 | [-0.39,  0.21]

plot(mydf)

Pairwise comparisons

We could now ask whether the predicted outcome for episode = 2 is significantly different depending on the level of grp? In other words, do the groups treatment and control differ when episode = 2?

Again, to answer this question, we calculate all pairwise comparisons, i.e. the comparison (or test for differences) between all combinations of our focal predictors. The focal predictors we’re interested here are our two variables used for the interaction.

# we want "episode = 2-2" and "grp = control-treatment"
hypothesis_test(model2, c("episode", "grp"))
#> # Pairwise comparisons
#> 
#> episode |                 grp | Contrast |         95% CI |     p
#> -----------------------------------------------------------------
#> 1-2     |     control-control |    -0.20 | [-0.63,  0.23] | 0.350
#> 1-3     |     control-control |     0.07 | [-0.37,  0.51] | 0.750
#> 1-1     |   control-treatment |     0.42 | [-0.04,  0.88] | 0.074
#> 1-2     |   control-treatment |    -0.15 | [-0.61,  0.32] | 0.529
#> 1-3     |   control-treatment |     0.12 | [-0.30,  0.54] | 0.573
#> 2-3     |     control-control |     0.27 | [-0.17,  0.72] | 0.225
#> 2-1     |   control-treatment |     0.62 | [ 0.16,  1.09] | 0.009
#> 2-2     |   control-treatment |     0.06 | [-0.41,  0.52] | 0.816
#> 2-3     |   control-treatment |     0.32 | [-0.10,  0.75] | 0.137
#> 3-1     |   control-treatment |     0.35 | [-0.13,  0.82] | 0.150
#> 3-2     |   control-treatment |    -0.22 | [-0.70,  0.26] | 0.368
#> 3-3     |   control-treatment |     0.05 | [-0.39,  0.49] | 0.821
#> 1-2     | treatment-treatment |    -0.57 | [-1.06, -0.07] | 0.026
#> 1-3     | treatment-treatment |    -0.30 | [-0.76,  0.16] | 0.203
#> 2-3     | treatment-treatment |     0.27 | [-0.19,  0.73] | 0.254

For our quantity of interest, the contrast between groups treatment and control when episode = 2 is 0.06. We find this comparison in row 8 of the above output.

As we can see, hypothesis_test() returns pairwise comparisons of all possible combinations of factor levels from our focal variables. If we’re only interested in a very specific comparison, we could directly formulate this comparison as test. To achieve this, we first need to create an overview of the adjusted predictions, which we get from ggpredict() or hypothesis_test(test = NULL).

# adjusted predictions, formatted table
ggpredict(model2, c("episode", "grp"))
#> # Predicted values of outcome
#> 
#> # grp = control
#> 
#> episode | Predicted |        95% CI
#> -----------------------------------
#> 1       |      0.03 | [-0.27, 0.33]
#> 2       |      0.23 | [-0.08, 0.54]
#> 3       |     -0.04 | [-0.36, 0.28]
#> 
#> # grp = treatment
#> 
#> episode | Predicted |         95% CI
#> ------------------------------------
#> 1       |     -0.39 | [-0.74, -0.04]
#> 2       |      0.18 | [-0.18,  0.53]
#> 3       |     -0.09 | [-0.39,  0.21]

# adjusted predictions, compact table
hypothesis_test(model2, c("episode", "grp"), test = NULL)
#> episode |       grp | Predicted |         95% CI |     p
#> --------------------------------------------------------
#> 1       |   control |      0.03 | [-0.27,  0.33] | 0.853
#> 2       |   control |      0.23 | [-0.08,  0.54] | 0.139
#> 3       |   control |     -0.04 | [-0.36,  0.28] | 0.793
#> 1       | treatment |     -0.39 | [-0.74, -0.04] | 0.028
#> 2       | treatment |      0.18 | [-0.18,  0.53] | 0.328
#> 3       | treatment |     -0.09 | [-0.39,  0.21] | 0.540

In the above output, each row is considered as one coefficient of interest. Our groups we want to include in our comparison are rows two (grp = control and episode = 2) and five (grp = treatment and episode = 2), so our “quantities of interest” are b2 and b5. Our null hypothesis we want to test is whether both predictions are equal, i.e. test = "b2 = b5". We can now calculate the desired comparison directly:

# compute specific contrast directly
hypothesis_test(model2, c("episode", "grp"), test = "b2 = b5")
#> Hypothesis | Contrast |        95% CI |     p
#> ---------------------------------------------
#> b2=b5      |     0.06 | [-0.41, 0.52] | 0.816
#> 
#> Tested hypothesis: episode[2],grp[control] = episode[2],grp[treatment]

The reason for this specific way of specifying the test argument is because hypothesis_test() is a small, convenient wrapper around predictions() and slopes() of the great marginaleffects package. Thus, test is just passed to the hypothesis argument of those functions.

Do different episode levels differ between groups?

We can repeat the steps shown above to test any combination of group levels for differences.

Pairwise comparisons

For instance, we could now ask whether the predicted outcome for episode = 1 in the treatment group is significantly different from the predicted outcome for episode = 3 in the control group.

The contrast we are interested in is between episode = 1 in the treatment group and episode = 3 in the control group. These are the predicted values in rows three and four (c.f. above table of predicted values), thus we test whether "b4 = b3".

hypothesis_test(model2, c("episode", "grp"), test = "b4 = b3")
#> Hypothesis | Contrast |        95% CI |     p
#> ---------------------------------------------
#> b4=b3      |    -0.35 | [-0.82, 0.13] | 0.150
#> 
#> Tested hypothesis: episode[1],grp[treatment] = episode[3],grp[control]

Another way to produce this pairwise comparison, we can reduce the table of predicted values by providing specific values or levels in the terms argument:

ggpredict(model2, c("episode [1,3]", "grp"))
#> # Predicted values of outcome
#> 
#> # grp = control
#> 
#> episode | Predicted |        95% CI
#> -----------------------------------
#> 1       |      0.03 | [-0.27, 0.33]
#> 3       |     -0.04 | [-0.36, 0.28]
#> 
#> # grp = treatment
#> 
#> episode | Predicted |         95% CI
#> ------------------------------------
#> 1       |     -0.39 | [-0.74, -0.04]
#> 3       |     -0.09 | [-0.39,  0.21]

episode = 1 in the treatment group and episode = 3 in the control group refer now to rows two and three, thus we also can obtain the desired comparison this way:

pred <- ggpredict(model2, c("episode [1,3]", "grp"))
hypothesis_test(pred, test = "b3 = b2")
#> Hypothesis | Contrast |        95% CI |     p
#> ---------------------------------------------
#> b3=b2      |    -0.35 | [-0.82, 0.13] | 0.150
#> 
#> Tested hypothesis: episode[1],grp[treatment] = episode[3],grp[control]

Does difference between two levels of episode in the control group differ from difference of same two levels in the treatment group?

The test argument also allows us to compare difference-in-differences. For example, is the difference between two episode levels in one group significantly different from the difference of the same two episode levels in the other group?

As a reminder, we look at the table of predictions again:

hypothesis_test(model2, c("episode", "grp"), test = NULL)
#> episode |       grp | Predicted |         95% CI |     p
#> --------------------------------------------------------
#> 1       |   control |      0.03 | [-0.27,  0.33] | 0.853
#> 2       |   control |      0.23 | [-0.08,  0.54] | 0.139
#> 3       |   control |     -0.04 | [-0.36,  0.28] | 0.793
#> 1       | treatment |     -0.39 | [-0.74, -0.04] | 0.028
#> 2       | treatment |      0.18 | [-0.18,  0.53] | 0.328
#> 3       | treatment |     -0.09 | [-0.39,  0.21] | 0.540

The first difference of episode levels 1 and 2 in the control group refer to rows one and two in the above table (b1 and b2). The difference for the same episode levels in the treatment group refer to the difference between rows four and five (b4 and b5). Thus, we have b1 - b2 and b4 - b5, and our null hypothesis is that these two differences are equal: test = "(b1 - b2) = (b4 - b5)".

hypothesis_test(model2, c("episode", "grp"), test = "(b1 - b2) = (b4 - b5)")
#> Hypothesis      | Contrast |        95% CI |     p
#> --------------------------------------------------
#> (b1-b2)=(b4-b5) |     0.36 | [-0.29, 1.02] | 0.277
#> 
#> Tested hypothesis: (episode[1],grp[control] - episode[2],grp[control]) = (episode[1],grp[treatment] - episode[2],grp[treatment])

Let’s replicate this step-by-step:

  1. Predicted value of outcome for episode = 1 in the control group is 0.03.
  2. Predicted value of outcome for episode = 2 in the control group is 0.23.
  3. The first difference is -0.2
  4. Predicted value of outcome for episode = 1 in the treatment group is -0.39.
  5. Predicted value of outcome for episode = 2 in the treatment group is 0.18.
  6. The second difference is -0.57
  7. Our quantity of interest is the difference between these two differences, which is 0.36. This difference is not statistically significant (p = 0.277).

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.

Let’s start with a simple example again.

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.

ggpredict(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]
#> 
#> Adjusted for:
#> * 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:

hypothesis_test(m, "Sepal.Length")
#> # 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 <- ggpredict(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.

hypothesis_test(m, c("Sepal.Length", "Species"), test = NULL)
#> # 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".

hypothesis_test(m, c("Sepal.Length", "Species"))
#> # 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:

hypothesis_test(m, c("Sepal.Length", "Species"), test = NULL)
#> # 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).

hypothesis_test(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 last 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 <- ggpredict(m, c("Petal.Length", "Petal.Width"))
plot(pred)

For hypothesis_test(), 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.

hypothesis_test(pred, test = NULL)
#> # 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:
# hypothesis_test(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.

hypothesis_test(pred)
#> # 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)
states$HSGrad <- states$`HS Grad`
m_mod <- lm(Income ~ HSGrad + Murder * Illiteracy, data = states)

myfun <- seq(0.5, 3, length.out = 9)
pr <- ggpredict(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 hypothesis_test():

hypothesis_test(pr, test = NULL)
#> # Linear trend for Murder
#> 
#> Illiteracy |   Slope |            95% CI |     p
#> ------------------------------------------------
#> 0.50       |   82.08 | [  18.70, 145.47] | 0.012
#> 0.81       |   51.76 | [  -1.10, 104.61] | 0.055
#> 1.12       |   21.43 | [ -29.43,  72.30] | 0.401
#> 1.44       |   -8.89 | [ -67.20,  49.41] | 0.760
#> 1.75       |  -39.22 | [-111.55,  33.12] | 0.281
#> 2.06       |  -69.54 | [-159.46,  20.38] | 0.126
#> 2.38       |  -99.87 | [-209.20,   9.47] | 0.072
#> 2.69       | -130.19 | [-259.98,  -0.40] | 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 0.79 and negative for values higher than 2.67. Inside the interval of [0.79, 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”.

Contrasts and comparisons for GLM - logistic regression example

Lastly, we show an example for non-Gaussian models. For GLM models with (non-Gaussian) link-functions, ggpredict() always returns predcted values on the response scale. For example, predicted values for logistic regression models are shown as probabilities.

Let’s look at a simple example

set.seed(1234)
dat <- data.frame(
  outcome = rbinom(n = 100, size = 1, prob = 0.35),
  x1 = as.factor(sample(1:3, size = 100, TRUE, prob = c(0.5, 0.2, 0.3))),
  x2 = rnorm(n = 100, mean = 10, sd = 7)
)

m <- glm(outcome ~ x1 + x2, data = dat, family = binomial())
ggpredict(m, "x1")
#> # Predicted probabilities of outcome
#> 
#> x1 | Predicted |       95% CI
#> -----------------------------
#> 1  |      0.24 | [0.14, 0.39]
#> 2  |      0.16 | [0.06, 0.37]
#> 3  |      0.34 | [0.19, 0.54]
#> 
#> Adjusted for:
#> * x2 = 10.29

Contrasts and comparisons for categorical focal terms

Contrasts or comparisons - like predictions (see above) - are by default on the response scale, i.e. they’re represented as difference between probabilities (in percentage points).

p <- ggpredict(m, "x1")
hypothesis_test(p)
#> # Pairwise comparisons
#> 
#> x1  | Contrast |        95% CI |     p
#> --------------------------------------
#> 1-2 |     0.08 | [-0.11, 0.27] | 0.397
#> 1-3 |    -0.10 | [-0.32, 0.11] | 0.359
#> 2-3 |    -0.18 | [-0.41, 0.05] | 0.119
#> 
#> Contrasts are presented as probabilities.

The difference between the predicted probability of x1 = 1 (24.4%) and x1 = 2 (16.1%) is roughly 8.3% points. This difference is not statistically significant (p = 0.397).

The scale argument in hypothesis_test() can be used to return contrasts or comparisons on a differen scale. For example, to transform contrasts to odds ratios, we can use scale = "exp".

hypothesis_test(p, scale = "exp")
#> # Pairwise comparisons
#> 
#> x1  | Contrast |       95% CI |     p
#> -------------------------------------
#> 1-2 |     1.09 | [0.90, 1.32] | 0.397
#> 1-3 |     0.90 | [0.73, 1.12] | 0.359
#> 2-3 |     0.83 | [0.66, 1.05] | 0.119
#> 
#> Contrasts are presented as odds ratios.

Contrasts or comparisons can also be represented on the link-scale, in this case as log-odds. To do so, use scale = "link".

hypothesis_test(p, scale = "link")
#> # Pairwise comparisons
#> 
#> x1  | Contrast |        95% CI |     p
#> --------------------------------------
#> 1-2 |     0.52 | [-0.76, 1.80] | 0.427
#> 1-3 |    -0.49 | [-1.52, 0.54] | 0.350
#> 2-3 |    -1.01 | [-2.36, 0.34] | 0.142
#> 
#> Contrasts are presented as log-odds.

Contrasts and comparisons for numerical focal terms

For numeric focal variables, where the slopes (linear trends) are estimated, transformed scales (like scale = "exp") are not supported. However, scale = "link" can be used to return untransformed contrasts or comparisons on the link-scale.

hypothesis_test(m, "x2", scale = "link")
#> # Linear trend for x2
#> 
#> Slope |        95% CI |     p
#> -----------------------------
#> -0.06 | [-0.12, 0.01] | 0.093
#> 
#> Slopes are presented as log-odds.

Be aware whether and which back-transformation to use, as it affects the resulting p-values. A detailed overview of transformations can be found in this vignette.

Conclusion

Thanks to the great marginaleffects package, it is now possible to have a powerful function in ggeffects that allows to perform the next logical step after calculating adjusted predictions and to conduct hypothesis tests for contrasts and pairwise comparisons.

While the current implementation in hypothesis_test() already covers many common use cases for testing contrasts and pairwise comparison, there still might be the need for more sophisticated comparisons. In this case, I recommend using the marginaleffects package directly. Some further related recommended readings are the vignettes about Comparisons or Hypothesis Tests, Equivalence Tests, and Custom Contrasts.

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