Type I Error Rates Continuous by Continuous Interactions
Moderation
- Spotlight Analysis: Compare the mean of the dependent of the two groups (treatment and control) at every value (Simple Slopes Analysis)
- Floodlight Analysis: is spotlight analysis on the whole range of the moderator (Johnson-Neyman intervals)
Other Resources:
-
BANOVAL: floodlight analysis on Bayesian ANOVA models -
cSEM:doFloodlightAnalysisin SEM model -
(Spiller et al. 2013)
Terminology:
-
Main effects (slopes): coefficients that do no involve interaction terms
-
Simple slope: when a continuous independent variable interact with a moderating variable, its slope at a particular level of the moderating variable
-
Simple effect: when a categorical independent variable interacts with a moderating variable, its effect at a particular level of the moderating variable.
Example:
\[ Y = \beta_0 + \beta_1 X + \beta_2 M + \beta_3 X \times M \]
where
-
\(\beta_0\) = intercept
-
\(\beta_1\) = simple effect (slope) of \(X\) (independent variable)
-
\(\beta_2\) = simple effect (slope) of \(M\) (moderating variable)
-
\(\beta_3\) = interaction of \(X\) and \(M\)
Three types of interactions:
- Continuous by continuous
- Continuous by categorical
- Categorical by categorical
emmeans package
Data set is from UCLA seminar where gender and prog are categorical
Continuous by continuous
contcont <- lm ( loss ~ hours * effort,data= dat ) summary ( contcont ) #> #> Call: #> lm(formula = loss ~ hours * effort, data = dat) #> #> Residuals: #> Min 1Q Median 3Q Max #> -29.52 -10.60 -1.78 11.13 34.51 #> #> Coefficients: #> Estimate Std. Error t value Pr(>|t|) #> (Intercept) 7.79864 11.60362 0.672 0.5017 #> hours -9.37568 5.66392 -1.655 0.0982 . #> effort -0.08028 0.38465 -0.209 0.8347 #> hours:effort 0.39335 0.18750 2.098 0.0362 * #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> #> Residual standard error: 13.56 on 896 degrees of freedom #> Multiple R-squared: 0.07818, Adjusted R-squared: 0.07509 #> F-statistic: 25.33 on 3 and 896 DF, p-value: 9.826e-16 Simple slopes for a continuous by continuous model
Spotlight analysis (Aiken and West 2005): usually pick 3 values of moderating variable:
-
Mean Moderating Variable + \(\sigma \times\) (Moderating variable)
-
Mean Moderating Variable
-
Mean Moderating Variable - \(\sigma \times\) (Moderating variable)
# specify list of points mylist <- list (effort= c ( effbr,effr,effar ) ) # get the estimates emtrends ( contcont, ~ effort, var= "hours",at= mylist ) #> effort hours.trend SE df lower.CL upper.CL #> 24.5 0.261 1.352 896 -2.392 2.91 #> 29.7 2.307 0.915 896 0.511 4.10 #> 34.8 4.313 1.308 896 1.745 6.88 #> #> Confidence level used: 0.95 # plot mylist <- list (hours= seq ( 0,4,by= 0.4 ),effort= c ( effbr,effr,effar ) ) emmip ( contcont,effort ~ hours,at= mylist, CIs= TRUE ) 
# statistical test for slope difference emtrends ( contcont, pairwise ~ effort, var= "hours",at= mylist, adjust= "none" ) #> $emtrends #> effort hours.trend SE df lower.CL upper.CL #> 24.5 0.261 1.352 896 -2.392 2.91 #> 29.7 2.307 0.915 896 0.511 4.10 #> 34.8 4.313 1.308 896 1.745 6.88 #> #> Results are averaged over the levels of: hours #> Confidence level used: 0.95 #> #> $contrasts #> contrast estimate SE df t.ratio p.value #> 24.5 - 29.7 -2.05 0.975 896 -2.098 0.0362 #> 24.5 - 34.8 -4.05 1.931 896 -2.098 0.0362 #> 29.7 - 34.8 -2.01 0.956 896 -2.098 0.0362 #> #> Results are averaged over the levels of: hours The 3 p-values are the same as the interaction term.
For publication, we use
library ( ggplot2 ) # data ( mylist <- list (hours= seq ( 0,4,by= 0.4 ),effort= c ( effbr,effr,effar ) ) ) #> $hours #> [1] 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 #> #> $effort #> [1] 24.5 29.7 34.8 contcontdat <- emmip ( contcont,effort ~ hours,at= mylist, CIs= TRUE, plotit= FALSE ) contcontdat $ feffort <- factor ( contcontdat $ effort ) levels ( contcontdat $ feffort ) <- c ( "low","med","high" ) # plot p <- ggplot (data = contcontdat, aes (x = hours, y = yvar, color = feffort ) ) + geom_line ( ) p1 <- p + geom_ribbon ( aes (ymax= UCL, ymin= LCL, fill= feffort ), alpha= 0.4 ) p1 + labs (x= "Hours", y= "Weight Loss", color= "Effort", fill= "Effort" ) 
Continuous by categorical
# use Female as basline dat $ gender <- relevel ( dat $ gender, ref= "female" ) contcat <- lm ( loss ~ hours * gender, data = dat ) summary ( contcat ) #> #> Call: #> lm(formula = loss ~ hours * gender, data = dat) #> #> Residuals: #> Min 1Q Median 3Q Max #> -27.118 -11.350 -1.963 10.001 42.376 #> #> Coefficients: #> Estimate Std. Error t value Pr(>|t|) #> (Intercept) 3.335 2.731 1.221 0.222 #> hours 3.315 1.332 2.489 0.013 * #> gendermale 3.571 3.915 0.912 0.362 #> hours:gendermale -1.724 1.898 -0.908 0.364 #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> #> Residual standard error: 14.06 on 896 degrees of freedom #> Multiple R-squared: 0.008433, Adjusted R-squared: 0.005113 #> F-statistic: 2.54 on 3 and 896 DF, p-value: 0.05523 Get simple slopes by each level of the categorical moderator
emtrends ( contcat, ~ gender, var= "hours" ) #> gender hours.trend SE df lower.CL upper.CL #> female 3.32 1.33 896 0.702 5.93 #> male 1.59 1.35 896 -1.063 4.25 #> #> Confidence level used: 0.95 # test difference in slopes emtrends ( contcat, pairwise ~ gender, var= "hours" ) # which is the same as the interaction term #> $emtrends #> gender hours.trend SE df lower.CL upper.CL #> female 3.32 1.33 896 0.702 5.93 #> male 1.59 1.35 896 -1.063 4.25 #> #> Confidence level used: 0.95 #> #> $contrasts #> contrast estimate SE df t.ratio p.value #> female - male 1.72 1.9 896 0.908 0.3639 # plot ( mylist <- list (hours= seq ( 0,4,by= 0.4 ),gender= c ( "female","male" ) ) ) #> $hours #> [1] 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 #> #> $gender #> [1] "female" "male" emmip ( contcat, gender ~ hours, at= mylist,CIs= TRUE ) 
Categorical by categorical
# relevel baseline dat $ prog <- relevel ( dat $ prog, ref= "read" ) dat $ gender <- relevel ( dat $ gender, ref= "female" ) catcat <- lm ( loss ~ gender * prog, data = dat ) summary ( catcat ) #> #> Call: #> lm(formula = loss ~ gender * prog, data = dat) #> #> Residuals: #> Min 1Q Median 3Q Max #> -19.1723 -4.1894 -0.0994 3.7506 27.6939 #> #> Coefficients: #> Estimate Std. Error t value Pr(>|t|) #> (Intercept) -3.6201 0.5322 -6.802 1.89e-11 *** #> gendermale -0.3355 0.7527 -0.446 0.656 #> progjog 7.9088 0.7527 10.507 < 2e-16 *** #> progswim 32.7378 0.7527 43.494 < 2e-16 *** #> gendermale:progjog 7.8188 1.0645 7.345 4.63e-13 *** #> gendermale:progswim -6.2599 1.0645 -5.881 5.77e-09 *** #> --- #> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 #> #> Residual standard error: 6.519 on 894 degrees of freedom #> Multiple R-squared: 0.7875, Adjusted R-squared: 0.7863 #> F-statistic: 662.5 on 5 and 894 DF, p-value: < 2.2e-16 Simple effects
emcatcat <- emmeans ( catcat, ~ gender * prog ) # differences in predicted values contrast ( emcatcat, "revpairwise", by = "prog", adjust = "bonferroni" ) #> prog = read: #> contrast estimate SE df t.ratio p.value #> male - female -0.335 0.753 894 -0.446 0.6559 #> #> prog = jog: #> contrast estimate SE df t.ratio p.value #> male - female 7.483 0.753 894 9.942 <.0001 #> #> prog = swim: #> contrast estimate SE df t.ratio p.value #> male - female -6.595 0.753 894 -8.762 <.0001 Plot
emmip ( catcat, prog ~ gender,CIs= TRUE ) 
Bar graph
catcatdat <- emmip ( catcat, gender ~ prog, CIs = TRUE, plotit = FALSE ) p <- ggplot (data = catcatdat, aes (x = prog, y = yvar, fill = gender ) ) + geom_bar (stat = "identity", position = "dodge" ) p1 <- p + geom_errorbar ( position = position_dodge ( .9 ), width = .25, aes (ymax = UCL, ymin = LCL ), alpha = 0.3 ) p1 + labs (x = "Program", y = "Weight Loss", fill = "Gender" ) 
probmod package
- Not recommend: package has serious problem with subscript.
library ( probemod ) myModel <- lm ( loss ~ hours * gender, data = dat %>% select ( loss, hours, gender ) ) jnresults <- jn ( myModel, dv= 'loss', iv= 'hours', mod= 'gender' ) pickapoint ( myModel, dv= 'loss', iv= 'hours', mod= 'gender', alpha= .01 ) plot ( jnresults ) interactions package
- Recommend
Continuous interaction
- (at least one of the two variables is continuous)
| Observations | 50 |
| Dependent variable | Income |
| Type | OLS linear regression |
| F(4,45) | 10.65 |
| R² | 0.49 |
| Adj. R² | 0.44 |
| Est. | S.E. | t val. | p | |
|---|---|---|---|---|
| (Intercept) | 1414.46 | 737.84 | 1.92 | 0.06 |
| Illiteracy | 753.07 | 385.90 | 1.95 | 0.06 |
| Murder | 130.60 | 44.67 | 2.92 | 0.01 |
HS Grad | 40.76 | 10.92 | 3.73 | 0.00 |
| Illiteracy:Murder | -97.04 | 35.86 | -2.71 | 0.01 |
| Standard errors: OLS |
For continuous moderator, the three values chosen are:
-
-1 SD above the mean
-
The mean
-
-1 SD below the mean
interact_plot ( fiti, pred = Illiteracy, modx = Murder, # centered = "none", # if you don't want the plot to mean-center # modx.values = "plus-minus", # exclude the mean value of the moderator # modx.values = "terciles" # split moderator's distribution into 3 groups plot.points = T, # overlay data point.shape = T, # different shape for differennt levels of the moderator jitter = 0.1, # if two data points are on top one another, this moves them apart by little # other appearance option x.label = "X label", y.label = "Y label", main.title = "Title", legend.main = "Legend Title", colors = "blue", # include confidence band interval = TRUE, int.width = 0.9, robust = TRUE # use robust SE ) 
To include weights from the regression inn the plot
fiti <- lm ( Income ~ Illiteracy * Murder, data = states, weights = Population ) interact_plot ( fiti, pred = Illiteracy, modx = Murder, plot.points = TRUE ) 
Partial Effect Plot
| Observations | 234 |
| Dependent variable | cty |
| Type | OLS linear regression |
| F(16,217) | 99.73 |
| R² | 0.88 |
| Adj. R² | 0.87 |
| Est. | S.E. | t val. | p | |
|---|---|---|---|---|
| (Intercept) | -200.98 | 47.01 | -4.28 | 0.00 |
| year | 0.12 | 0.02 | 5.03 | 0.00 |
| cyl | -1.86 | 0.28 | -6.69 | 0.00 |
| displ | -3.56 | 0.66 | -5.41 | 0.00 |
| classcompact | -2.60 | 0.93 | -2.80 | 0.01 |
| classmidsize | -2.63 | 0.93 | -2.82 | 0.01 |
| classminivan | -4.41 | 1.04 | -4.24 | 0.00 |
| classpickup | -4.37 | 0.93 | -4.68 | 0.00 |
| classsubcompact | -2.38 | 0.93 | -2.56 | 0.01 |
| classsuv | -4.27 | 0.87 | -4.92 | 0.00 |
| fld | 6.34 | 1.69 | 3.74 | 0.00 |
| fle | -4.57 | 1.66 | -2.75 | 0.01 |
| flp | -1.92 | 1.59 | -1.21 | 0.23 |
| flr | -0.79 | 1.57 | -0.50 | 0.61 |
| drvf | 1.40 | 0.40 | 3.52 | 0.00 |
| drvr | 0.49 | 0.46 | 1.06 | 0.29 |
| cyl:displ | 0.36 | 0.08 | 4.56 | 0.00 |
| Standard errors: OLS |
interact_plot ( fitc, pred = displ, modx = cyl, partial.residuals = TRUE, # the observed data is based on displ, cyl, and model error modx.values = c ( 4, 5, 6, 8 ) ) 
Check linearity assumption in the model
Plot the lines based on the subsample (red line), and whole sample (black line)
x_2 <- runif (n = 200, min = - 3, max = 3 ) w <- rbinom (n = 200, size = 1, prob = 0.5 ) err <- rnorm (n = 200, mean = 0, sd = 4 ) y_2 <- 2.5 - x_2 ^ 2 - 5 * w + 2 * w * ( x_2 ^ 2 ) + err data_2 <- as.data.frame ( cbind ( x_2, y_2, w ) ) model_2 <- lm ( y_2 ~ x_2 * w, data = data_2 ) summ ( model_2 ) | Observations | 200 |
| Dependent variable | y_2 |
| Type | OLS linear regression |
| F(3,196) | 1.36 |
| R² | 0.02 |
| Adj. R² | 0.01 |
| Est. | S.E. | t val. | p | |
|---|---|---|---|---|
| (Intercept) | -0.21 | 0.45 | -0.47 | 0.64 |
| x_2 | 0.43 | 0.27 | 1.61 | 0.11 |
| w | 0.78 | 0.66 | 1.18 | 0.24 |
| x_2:w | -0.52 | 0.39 | -1.35 | 0.18 |
| Standard errors: OLS |
interact_plot ( model_2, pred = x_2, modx = w, linearity.check = TRUE, plot.points = TRUE ) 
Simple Slopes Analysis
-
continuous by continuous variable interaction (still work for binary)
-
conditional slope of the variable of interest (i.e., the slope of \(X\) when we hold \(M\) constant at a value)
Using sim_slopes it will
-
mean-center all variables except the variable of interest
-
For moderator that is
-
Continuous, it will pick mean, and plus/minus 1 SD
-
Categorical, it will use all factor
-
sim_slopes requires
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A regression model with an interaction term)
-
Variable of interest (
pred =) -
Moderator: (
modx =)
sim_slopes ( fiti, pred = Illiteracy, modx = Murder, johnson_neyman = FALSE ) #> SIMPLE SLOPES ANALYSIS #> #> Slope of Illiteracy when Murder = 5.420973 (- 1 SD): #> #> Est. S.E. t val. p #> -------- -------- -------- ------ #> -71.59 268.65 -0.27 0.79 #> #> Slope of Illiteracy when Murder = 8.685043 (Mean): #> #> Est. S.E. t val. p #> --------- -------- -------- ------ #> -437.12 175.82 -2.49 0.02 #> #> Slope of Illiteracy when Murder = 11.949113 (+ 1 SD): #> #> Est. S.E. t val. p #> --------- -------- -------- ------ #> -802.66 145.72 -5.51 0.00 # plot the coefficients ss <- sim_slopes ( fiti, pred = Illiteracy, modx = Murder, modx.values = c ( 0, 5, 10 ) ) plot ( ss ) 
| Value of Murder | Slope of Illiteracy |
| Value of Murder | slope |
|---|---|
| 0.00 | 535.50 (458.77) |
| 5.00 | -24.44 (282.48) |
| 10.00 | -584.38 (152.37)*** |
Johnson-Neyman intervals
To know all the values of the moderator for which the slope of the variable of interest will be statistically significant, we can use the Johnson-Neyman interval (P. O. Johnson and Neyman 1936)
Even though we kind of know that the alpha level when implementing the Johnson-Neyman interval is not correct (Bauer and Curran 2005), not until recently that there is a correction for the type I and II errors (Esarey and Sumner 2017).
Since Johnson-Neyman inflates the type I error (comparisons across all values of the moderator)
sim_slopes ( fiti, pred = Illiteracy, modx = Murder, johnson_neyman = TRUE, control.fdr = TRUE, # correction for type I and II # cond.int = TRUE, # include conditional intecepts robust = "HC3", # rubust SE # centered = "none", # don't mean-centered non-focal variables jnalpha = 0.05 ) #> JOHNSON-NEYMAN INTERVAL #> #> When Murder is OUTSIDE the interval [-11.70, 8.75], the slope of Illiteracy #> is p < .05. #> #> Note: The range of observed values of Murder is [1.40, 15.10] #> #> Interval calculated using false discovery rate adjusted t = 2.33 #> #> SIMPLE SLOPES ANALYSIS #> #> Slope of Illiteracy when Murder = 5.420973 (- 1 SD): #> #> Est. S.E. t val. p #> -------- -------- -------- ------ #> -71.59 256.60 -0.28 0.78 #> #> Slope of Illiteracy when Murder = 8.685043 (Mean): #> #> Est. S.E. t val. p #> --------- -------- -------- ------ #> -437.12 191.07 -2.29 0.03 #> #> Slope of Illiteracy when Murder = 11.949113 (+ 1 SD): #> #> Est. S.E. t val. p #> --------- -------- -------- ------ #> -802.66 178.75 -4.49 0.00 For plotting, we can use johnson_neyman
johnson_neyman ( fiti, pred = Illiteracy, modx = Murder, control.fdr = TRUE, # correction for type I and II alpha = .05 ) #> JOHNSON-NEYMAN INTERVAL #> #> When Murder is OUTSIDE the interval [-22.57, 8.52], the slope of Illiteracy #> is p < .05. #> #> Note: The range of observed values of Murder is [1.40, 15.10] #> #> Interval calculated using false discovery rate adjusted t = 2.33 
Note:
- y-axis is the conditional slope of the variable of interest
3-way interaction
# fita3 <- # lm(rating ~ privileges * critical * learning, data = attitude) # # probe_interaction( # fita3, # pred = critical, # modx = learning, # mod2 = privileges, # alpha = .1 # ) mtcars $ cyl <- factor ( mtcars $ cyl, labels = c ( "4 cylinder", "6 cylinder", "8 cylinder" ) ) fitc3 <- lm ( mpg ~ hp * wt * cyl, data = mtcars ) interact_plot ( fitc3, pred = hp, modx = wt, mod2 = cyl ) + theme_apa (legend.pos = "bottomright" ) 
Johnson-Neyman 3-way interaction
library ( survey ) data ( api ) dstrat <- svydesign ( id = ~ 1, strata = ~ stype, weights = ~ pw, data = apistrat, fpc = ~ fpc ) regmodel3 <- survey :: svyglm ( api00 ~ avg.ed * growth * enroll, design = dstrat ) sim_slopes ( regmodel3, pred = growth, modx = avg.ed, mod2 = enroll, jnplot = TRUE ) #> ############### While enroll (2nd moderator) = 153.0518 (- 1 SD) ############## #> #> JOHNSON-NEYMAN INTERVAL #> #> When avg.ed is OUTSIDE the interval [2.75, 3.82], the slope of growth is p #> < .05. #> #> Note: The range of observed values of avg.ed is [1.38, 4.44] #> #> SIMPLE SLOPES ANALYSIS #> #> Slope of growth when avg.ed = 2.085002 (- 1 SD): #> #> Est. S.E. t val. p #> ------ ------ -------- ------ #> 1.25 0.32 3.86 0.00 #> #> Slope of growth when avg.ed = 2.787381 (Mean): #> #> Est. S.E. t val. p #> ------ ------ -------- ------ #> 0.39 0.22 1.75 0.08 #> #> Slope of growth when avg.ed = 3.489761 (+ 1 SD): #> #> Est. S.E. t val. p #> ------- ------ -------- ------ #> -0.48 0.35 -1.37 0.17 #> #> ################ While enroll (2nd moderator) = 595.2821 (Mean) ############### #> #> JOHNSON-NEYMAN INTERVAL #> #> When avg.ed is OUTSIDE the interval [2.84, 7.83], the slope of growth is p #> < .05. #> #> Note: The range of observed values of avg.ed is [1.38, 4.44] #> #> SIMPLE SLOPES ANALYSIS #> #> Slope of growth when avg.ed = 2.085002 (- 1 SD): #> #> Est. S.E. t val. p #> ------ ------ -------- ------ #> 0.72 0.22 3.29 0.00 #> #> Slope of growth when avg.ed = 2.787381 (Mean): #> #> Est. S.E. t val. p #> ------ ------ -------- ------ #> 0.34 0.16 2.16 0.03 #> #> Slope of growth when avg.ed = 3.489761 (+ 1 SD): #> #> Est. S.E. t val. p #> ------- ------ -------- ------ #> -0.04 0.24 -0.16 0.87 #> #> ############### While enroll (2nd moderator) = 1037.5125 (+ 1 SD) ############## #> #> JOHNSON-NEYMAN INTERVAL #> #> The Johnson-Neyman interval could not be found. Is the p value for your #> interaction term below the specified alpha? #> #> SIMPLE SLOPES ANALYSIS #> #> Slope of growth when avg.ed = 2.085002 (- 1 SD): #> #> Est. S.E. t val. p #> ------ ------ -------- ------ #> 0.18 0.31 0.58 0.56 #> #> Slope of growth when avg.ed = 2.787381 (Mean): #> #> Est. S.E. t val. p #> ------ ------ -------- ------ #> 0.29 0.20 1.49 0.14 #> #> Slope of growth when avg.ed = 3.489761 (+ 1 SD): #> #> Est. S.E. t val. p #> ------ ------ -------- ------ #> 0.40 0.27 1.49 0.14 
Report
ss3 <- sim_slopes ( regmodel3, pred = growth, modx = avg.ed, mod2 = enroll ) plot ( ss3 ) 
| enroll = 153 | |
| Value of avg.ed | Slope of growth |
| Value of avg.ed | slope |
|---|---|
| 2.09 | 1.25 (0.32)*** |
| 2.79 | 0.39 (0.22)# |
| enroll = 595.28 | |
| Value of avg.ed | Slope of growth |
| 3.49 | -0.48 (0.35) |
| 2.09 | 0.72 (0.22)** |
| 2.79 | 0.34 (0.16)* |
| enroll = 1037.51 | |
| Value of avg.ed | Slope of growth |
| 3.49 | -0.04 (0.24) |
| 2.09 | 0.18 (0.31) |
| 2.79 | 0.29 (0.20) |
| 3.49 | 0.40 (0.27) |
Categorical interaction
library ( ggplot2 ) mpg2 <- mpg %>% mutate (cyl = factor ( cyl ) ) mpg2 [ "auto" ] <- "auto" mpg2 $ auto [ mpg2 $ trans %in% c ( "manual(m5)", "manual(m6)" ) ] <- "manual" mpg2 $ auto <- factor ( mpg2 $ auto ) mpg2 [ "fwd" ] <- "2wd" mpg2 $ fwd [ mpg2 $ drv == "4" ] <- "4wd" mpg2 $ fwd <- factor ( mpg2 $ fwd ) ## Drop the two cars with 5 cylinders (rest are 4, 6, or 8) mpg2 <- mpg2 [ mpg2 $ cyl != "5",] ## Fit the model fit3 <- lm ( cty ~ cyl * fwd * auto, data = mpg2 ) library ( jtools ) # for summ() summ ( fit3 ) | Observations | 230 |
| Dependent variable | cty |
| Type | OLS linear regression |
| F(11,218) | 61.37 |
| R² | 0.76 |
| Adj. R² | 0.74 |
| Est. | S.E. | t val. | p | |
|---|---|---|---|---|
| (Intercept) | 21.37 | 0.39 | 54.19 | 0.00 |
| cyl6 | -4.37 | 0.54 | -8.07 | 0.00 |
| cyl8 | -8.37 | 0.67 | -12.51 | 0.00 |
| fwd4wd | -2.91 | 0.76 | -3.83 | 0.00 |
| automanual | 1.45 | 0.57 | 2.56 | 0.01 |
| cyl6:fwd4wd | 0.59 | 0.96 | 0.62 | 0.54 |
| cyl8:fwd4wd | 2.13 | 0.99 | 2.15 | 0.03 |
| cyl6:automanual | -0.76 | 0.90 | -0.84 | 0.40 |
| cyl8:automanual | 0.71 | 1.18 | 0.60 | 0.55 |
| fwd4wd:automanual | -1.66 | 1.07 | -1.56 | 0.12 |
| cyl6:fwd4wd:automanual | 1.29 | 1.52 | 0.85 | 0.40 |
| cyl8:fwd4wd:automanual | -1.39 | 1.76 | -0.79 | 0.43 |
| Standard errors: OLS |
cat_plot ( fit3, pred = cyl, modx = fwd, plot.points = T ) 
#line plots cat_plot ( fit3, pred = cyl, modx = fwd, geom = "line", point.shape = TRUE, # colors = "Set2", # choose color vary.lty = TRUE ) 
# bar plot cat_plot ( fit3, pred = cyl, modx = fwd, geom = "bar", interval = T, plot.points = TRUE ) 
interactionR package
- For publication purposes
- Following
-
(Knol and VanderWeele 2012) for presentation
-
(Hosmer and Lemeshow 1992) for confidence intervals based on the delta method
-
(Zou 2008) for variance recovery "mover" method
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(Longnecker et al. 1996) for bootstrapping
-
sjPlot package
-
For publication purposes (recommend, but more advanced)
-
link
Source: https://bookdown.org/mike/data_analysis/moderation.html
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