Investigating Interaction Effects With Response Surface Analysis

Autor*innen: Martin Greisel & Kaley Lesperance

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In usual multiple regression analysis, researchers are often interested in interactions between predictor variables. However, these interactions are often investigated using linear interaction terms only. The response surface analysis allows for a more in-depth understanding of the actual interactions while the resulting response surface plots are still easy to interpret. We illustrate this approach with data from PISA 2012-2013 (campus file) for three different dependent variables: achievement in math, science, and reading. Therefore, we used the R-package RSA (Schönbrodt, F. D. & Humberg, S. (2021). RSA: An R package for response surface analysis (version 0.10.4). Retrieved from https://cran.r-project.org/package=RSA) which produces RSA-plots based on the graphics engine from the lattice-package.

The interaction effects differ for each dependent variable. The plots above might be interpreted as follows:

### Math
Students with a positive attitude toward school benefit from cognitively activating lessons as they perform better in math. However, this effect inverses for students with a negative attitude toward school: They perform worse in math with increasingly cognitively activating lessons.

### Science
In science, students with an average to positive attitude toward school perform best, as long as their lessons are at least a bit cognitively activating. A further increase in cognitive activation seems to have no additional effect. However, a very strong positive attitude seems to be related to a slightly lower performance in science.

### Reading
In contrast, cognitive activation plays a key role in reading achievement: In highly cognitively activating environments, students perform almost equally well independent of their attitude toward school. As cognitive activation declines, students' reading performance becomes highly dependent on their attitude toward school.

### Conclusion
Students benefit a bit from learning in cognitively activating math lessons, whereas cognitive activation does not matter very much in science lessons as long as a basic activation is ensured. On the contrary however, cognitive activation can almost completely compensate for differences in attitude toward school in reading lessons.

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In this code excerpt, we illustrate the approach of response surface analysis with data from PISA 2012-2013 (campus file) for three different dependent variables: achievement in math, science, and reading.

# Load packages ----------------------------------------------------------------
library(dplyr) # only for data preparation
library(sjmisc) # only for data preparation
library(RSA) # for the response surface analysis
library(latticeExtra) # for combining several lattice-based plots into one

# Import data ------------------------------------------------------------------
load("PISA-Plus-2012-2013_Dataset_CF.rda")


# Calculate scale means --------------------------------------------------------
pisa_2012_2013_data %>%
     as_tibble() %>% # not necessary, just for better output of data
     select((starts_with("attitud_") | starts_with("cognact_"))
             & ends_with("_t2")) %>%
     mutate(across(.cols = everything(), .fns = as.numeric)) %>% # convert factors
     rec((contains("_a_") | contains("_b_")), rec = "rev") %>% # reverse items
     row_means((starts_with("attitud_") & ends_with("_t2")),
             n = 1, var = "Attitude_Toward_School_T2") %>%
     row_means((starts_with("cognact_") & ends_with("_t2")),
             n = 1, var = "Cognitive_Activation_T2") %>%
     select(Attitude_Toward_School_T2, Cognitive_Activation_T2) %>%
     bind_cols(pisa_2012_2013_data, .) -> pisa_2012_2013_data

# Perform response surface analyses --------------------------------------------
model.results.math <- RSA(ma_wle_t2 ~ Attitude_Toward_School_T2 * Cognitive_Activation_T2,
     data = pisa_2012_2013_data,
     center = "pooled",
     missing = "listwise")

model.results.science <- RSA(sci_wle_t2 ~ Attitude_Toward_School_T2 * Cognitive_Activation_T2,
     data = pisa_2012_2013_data,
     center = "pooled",
     missing = "listwise")

model.results.reading <- RSA(rea_wle_t2 ~ Attitude_Toward_School_T2 * Cognitive_Activation_T2,
     data = pisa_2012_2013_data,
     center = "pooled",
     missing = "listwise")

# Comment:
# deleting missing values "listwise" might not be the best way, but this is a
# visualization challenge with a campus file, right? ;-)

# Calculate plots --------------------------------------------------------------
p_ma <- plot(model.results.math,
     # points = list(color = "White"),
     xlab = "Attitude Toward School",
     ylab = "Cognitive Activation",
     zlab = "Math Achievement",
     main = "Different Interaction Effects for Different Outcomes: Response Surface Analysis of PISA Data from 2013",
     cex.tickLabel = 1.5,
     cex.axesLabel = 2,
     cex.main = 3)

p_sci <- plot(model.results.science,
     # points = list(color = "White"),
     xlab = "Attitude Toward School",
     ylab = "Cognitive Activation",
     zlab = "Science Achievement",
     main = "Different Interaction Effects for Different Outcomes: Response Surface Analysis of PISA Data from 2013",
     cex.tickLabel = 1.5,
     cex.axesLabel = 2,
     cex.main = 3)

p_rea <- plot(model.results.reading,
     # points = list(color = "White"),
     xlab = "Attitude Toward School",
     ylab = "Cognitive Activation",
     zlab = "Reading Achievement",
     main = "Different Interaction Effects for Different Outcomes: Response Surface Analysis of PISA Data from 2013",
     cex.tickLabel = 1.5,
     cex.axesLabel = 2,
     cex.main = 3)

# Plot output ------------------------------------------------------------------
c(p_ma, p_sci, p_rea, layout = c(3,1))