1 Results

1.1 Demographics

# demographics 

# one participant indicated 3 for age. Needs to be excluded for the age analysis.
tdata_age <- tdata 

min(tdata_age$Age)
## [1] 18
max(tdata_age$Age)
## [1] 60
mean(tdata_age$Age)
## [1] 32.16667
sd(tdata_age$Age)
## [1] 10.82118
# 1 = male, 2 = female, 3 = other
table(tdata$Sex)
## 
##  1  2  3 
## 45 71  4

2 Graphs

myTheme <- theme(plot.title = element_text(face="bold", size = 22),
        axis.title.x = element_blank(),
        axis.title.y = element_text(face = "bold", size = 20),
        axis.text.x = element_text(size = 18, angle = 0), 
        axis.text.y = element_text(size = 16, angle = 0),
        legend.text = element_text(size = 18),
        legend.title = element_text(face = "bold", size = 18),
        strip.text.x = element_text(size = 18),
        #panel.grid.major = element_blank(), 
        panel.grid.minor = element_blank(), 
        panel.background = element_blank(), 
        axis.line.x = element_line(colour = "black"), 
        axis.line.y = element_line(colour = "black"),
        axis.text = element_text(colour ="black"), 
        axis.ticks = element_line(colour ="black"))


library(see)
## first, turn sID into a factor
tdata_sub$sID <- factor(tdata_sub$sID)

pd <- position_dodge(width = 0.3)

tdata_sub$valueJitter <- jitter(tdata_sub$value, factor = 1, amount = 0.04)

theme_set(theme_light(base_size = 20, base_family = "Poppins"))

# new labes for the facets 

g <- ggplot(tdata_sub, aes(x=variable, y=valueJitter, group = sID)) +
  guides(fill=FALSE)+
  #facet_grid( ~ Side + Q_order)+
  #ggtitle("Subjects' causal srength ratings") +
  scale_y_continuous(limits = c(-0.05, 1.05), breaks=seq(0, 1, 0.1), expand = c(0,0)) +
  scale_x_discrete(labels=c("single-effect \n cause", "common \n cause")) +
  #stat_summary(fun.y = mean, geom = "bar", position = "dodge", colour = "black", alpha =0.5) +
  geom_violinhalf(aes(y = value, group = variable, fill = variable), color = NA, position=position_dodge(1), alpha = 0.3)+
  geom_line(position = pd, color = "black", size = 1, alpha=0.07) +
  geom_point(aes(color = variable), position = pd, alpha = 0.4, size = 2) +
  stat_summary(aes(y = value,group=1), fun.data = mean_cl_boot, geom = "errorbar", width = 0, size = 1) +
  stat_summary(aes(y = value,group=1), fun.y=mean, colour="black", geom="line",group=1, size = 1.5, linetype = "solid", alpha = 1)+
  stat_summary(aes(y = value,group=1, fill = variable), fun.y=mean, geom="point", color = "black", shape = 22, size = 5, group=1, alpha = 1)+
  stat_summary(aes(y = value,group=1), fun.y=median, geom="point", color = "black", shape = 3, size = 4, group=1, alpha = 1, position = position_dodge(width = 0.5))+
  labs(x = "Target Cause", y = "Causal Strength Rating") +
  scale_color_manual(name = "Entity",values=c("#fc9272", "#3182bd"))+
  scale_fill_manual(name = "Entity",values=c("#fc9272", "#3182bd"))+
  theme(legend.position = "none")+
  myTheme
## Warning: `guides(<scale> = FALSE)` is deprecated. Please use `guides(<scale> =
## "none")` instead.
## Warning: `fun.y` is deprecated. Use `fun` instead.
## `fun.y` is deprecated. Use `fun` instead.
## `fun.y` is deprecated. Use `fun` instead.
g

#ggsave("results_lines_main.svg",width=6,height=5)
#ggsave("results_lines_main.pdf",width=4.5,height=4.3)

Overall, a strong dilution effect. But what about the different counterbalancing conditions?

myTheme <- theme(plot.title = element_text(face="bold", size = 22),
        axis.title.x = element_text(face = "bold", size = 20),
        axis.title.y = element_text(face = "bold", size = 20),
        axis.text.x = element_text(size = 18, angle = 0), 
        axis.text.y = element_text(size = 14, angle = 0),
        legend.text = element_text(size = 18),
        legend.title = element_text(face = "bold", size = 18),
        strip.text.x = element_text(size = 18),
        #panel.grid.major = element_blank(), 
        panel.grid.minor = element_blank(), 
        panel.background = element_blank(), 
        axis.line.x = element_line(colour = "black"), 
        axis.line.y = element_line(colour = "black"),
        axis.text = element_text(colour ="black"), 
        axis.ticks = element_line(colour ="black"))


library(see)
## first, turn sID into a factor
tdata_sub$sID <- factor(tdata_sub$sID)

pd <- position_dodge(width = 0.3)

tdata_sub$valueJitter <- jitter(tdata_sub$value, factor = 1, amount = 0.04)

theme_set(theme_light(base_size = 20, base_family = "Poppins"))

# new labes for the facets 

g <- ggplot(tdata_sub, aes(x=variable, y=valueJitter, group = sID)) +
  guides(fill=FALSE)+
  facet_grid( ~ Side + Q_order)+
  #ggtitle("Subjects' causal srength ratings") +
  scale_y_continuous(limits = c(-0.05, 1.05), breaks=seq(0, 1, 0.1), expand = c(0,0)) +
  #scale_x_discrete(labels=c("Single-effect \n cause", "Multiple-effects \n cause")) +
  #stat_summary(fun.y = mean, geom = "bar", position = "dodge", colour = "black", alpha =0.5) +
  geom_violinhalf(aes(y = value, group = variable, fill = variable), color = NA, position=position_dodge(1), alpha = 0.2)+
  geom_line(position = pd, color = "black", size = 1, alpha=0.04) +
  geom_point(aes(color = variable), position = pd, alpha = 0.2) +
  stat_summary(aes(y = value,group=1), fun.data = mean_cl_boot, geom = "errorbar", width = 0, size = 1) +
  stat_summary(aes(y = value,group=1), fun.y=mean, colour="black", geom="line",group=1, size = 1.5, linetype = "solid", alpha = 1)+
  stat_summary(aes(y = value,group=1, fill = variable), fun.y=mean, geom="point", color = "black", shape = 22, size = 5, group=1, alpha = 1)+
  stat_summary(aes(y = value,group=1), fun.y=median, geom="point", color = "black", shape = 3, size = 4, group=1, alpha = 1, position = position_dodge(width = 0.5))+
  labs(x = "Number of Cause's Effects", y = "Causal Strength Rating") +
  scale_color_manual(name = "Entity",values=c("#fc9272", "#3182bd"))+
  scale_fill_manual(name = "Entity",values=c("#fc9272", "#3182bd"))+
  theme(legend.position = "none")+
  myTheme
## Warning: `guides(<scale> = FALSE)` is deprecated. Please use `guides(<scale> =
## "none")` instead.
## Warning: `fun.y` is deprecated. Use `fun` instead.
## `fun.y` is deprecated. Use `fun` instead.
## `fun.y` is deprecated. Use `fun` instead.
g

#ggsave("results_lines.svg",width=15.5,height=9)
#ggsave("results_lines.pdf",width=15.5,height=9)

A quite pronounced dilution effect in all conditions.

3 Descriptive Stats

## : one
##       median         mean      SE.mean CI.mean.0.95          var      std.dev 
##   0.99000000   0.87141667   0.02020736   0.04001259   0.04900050   0.22136056 
##     coef.var 
##   0.25402378 
## ------------------------------------------------------------ 
## : three
##       median         mean      SE.mean CI.mean.0.95          var      std.dev 
##   0.50000000   0.56425000   0.02425404   0.04802543   0.07059103   0.26568972 
##     coef.var 
##   0.47087235

4 Statistical Test

library(afex)
## ************
## Welcome to afex. For support visit: http://afex.singmann.science/
## - Functions for ANOVAs: aov_car(), aov_ez(), and aov_4()
## - Methods for calculating p-values with mixed(): 'S', 'KR', 'LRT', and 'PB'
## - 'afex_aov' and 'mixed' objects can be passed to emmeans() for follow-up tests
## - NEWS: emmeans() for ANOVA models now uses model = 'multivariate' as default.
## - Get and set global package options with: afex_options()
## - Set orthogonal sum-to-zero contrasts globally: set_sum_contrasts()
## - For example analyses see: browseVignettes("afex")
## ************
## 
## Attache Paket: 'afex'
## Das folgende Objekt ist maskiert 'package:lme4':
## 
##     lmer
library(emmeans)

a1 <- aov_car(value ~ Side*Q_order*Target_effect + Error(sID/(variable)), tdata_sub)
## Contrasts set to contr.sum for the following variables: Side, Q_order, Target_effect
a1
## Anova Table (Type 3 tests)
## 
## Response: value
##                                 Effect     df  MSE         F   ges p.value
## 1                                 Side 1, 108 0.06    3.84 +  .017    .053
## 2                              Q_order 1, 108 0.06      0.28  .001    .597
## 3                        Target_effect 2, 108 0.06      0.71  .006    .495
## 4                         Side:Q_order 1, 108 0.06      0.93  .004    .338
## 5                   Side:Target_effect 2, 108 0.06      1.22  .011    .300
## 6                Q_order:Target_effect 2, 108 0.06      0.24  .002    .785
## 7           Side:Q_order:Target_effect 2, 108 0.06      0.99  .009    .376
## 8                             variable 1, 108 0.06 92.09 ***  .307   <.001
## 9                        Side:variable 1, 108 0.06      0.19 <.001    .666
## 10                    Q_order:variable 1, 108 0.06      1.03  .005    .312
## 11              Target_effect:variable 2, 108 0.06      0.98  .009    .380
## 12               Side:Q_order:variable 1, 108 0.06      1.05  .005    .307
## 13         Side:Target_effect:variable 2, 108 0.06    2.49 +  .023    .088
## 14      Q_order:Target_effect:variable 2, 108 0.06      1.07  .010    .346
## 15 Side:Q_order:Target_effect:variable 2, 108 0.06      0.83  .008    .437
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1

Only a main effect for “variable” (i.e., causal scope)

############### 
# a conditional analysis 

ls2 <- lsmeans(a1, c("variable")) # group means by between-condition
ls2
##  variable lsmean     SE  df lower.CL upper.CL
##  one       0.871 0.0197 108    0.832    0.910
##  three     0.564 0.0245 108    0.516    0.613
## 
## Results are averaged over the levels of: Side, Q_order, Target_effect 
## Confidence level used: 0.95
# simple main effects 
t <- pairs(ls2) # compares rep-measure differences separately for each between-factor level
t
##  contrast    estimate    SE  df t.ratio p.value
##  one - three    0.307 0.032 108   9.596  <.0001
## 
## Results are averaged over the levels of: Side, Q_order, Target_effect
confint(t, level = 0.95)
##  contrast    estimate    SE  df lower.CL upper.CL
##  one - three    0.307 0.032 108    0.244    0.371
## 
## Results are averaged over the levels of: Side, Q_order, Target_effect 
## Confidence level used: 0.95

A clear dilution effect.

Make a difference plot:

t <- qt(0.975, 108, lower.tail = TRUE, log.p = FALSE)
#t

effect <- "Mdiff"
Mdiff <- 0.307 
SE <- 0.032
CI <- SE*t
CI_low <- Mdiff - CI
CI_up <- Mdiff + CI

Mdiff
## [1] 0.307
CI_low
## [1] 0.2435704
CI_up
## [1] 0.3704296
# Plot 

myTheme <- theme(plot.title = element_text(face="bold", size = 22),
        axis.title.x = element_text(face = "bold", size = 20),
        axis.title.y = element_blank(),
        axis.text.x = element_text(size = 18, angle = 0), 
        axis.text.y = element_text(size = 40, angle = 0),
        legend.text = element_text(size = 18),
        legend.title = element_text(size = 22),
        strip.text.x = element_text(size = 18),
        #panel.grid.major = element_blank(), 
        #panel.grid.minor = element_blank(), 
        #panel.background = element_blank(), 
        axis.line.x = element_line(colour = "black"), 
        axis.line.y = element_line(colour = "black"),
        axis.text = element_text(colour ="black"), 
        axis.ticks = element_line(colour ="black"))

theme_set(theme_light(base_size = 30, base_family = "Poppins"))



barchart <- ggplot()+
  myTheme+
  #guides(fill=FALSE)+
  #facet_wrap(~Latency + SampleSize, ncol=2)+
  #ggtitle("Mean difference (95% CI)") +
  #coord_cartesian(ylim=c(-0.1,2)) + 
  scale_y_continuous(limits = c(-0.1, 0.5), breaks=seq(-0.1, 0.5, 0.1), expand = c(0,0)) +
  scale_x_discrete(labels=c("r")) +
  #annotate("rect", xmin=1.7, xmax=2.3, ymin=0.95, ymax=1.05, color="#31a354", fill = "white", size = 1) +
  #stat_summary(fun.y=mean, colour="grey20", geom="point", shape = 21, size = 3)+
  #stat_summary(fun.y = mean, geom = "bar", position = "dodge", colour = "black")+
  #stat_summary(fun.data = mean_cl_boot, geom = "errorbar", position = position_dodge(width = 0.90), width = 0.2) +
  #geom_jitter(width = 0.3, height = 0.02, alpha = 0.6, colour = "red") +
  #ggtitle("Means (95% bootstr. CIs)") +
  #theme(axis.text.x = element_text(size = 10, angle = 0, hjust = 0.5))+
  labs(x= "", y = "Mean change") +
  #scale_color_manual(values=c("#005083",  "#f0b64d"))# +
  #scale_fill_manual(values=c("#969696",  "#969696"))
  #annotate("point", x = 1, y = 100, colour = "firebrick", size = 2)+
  #annotate(xmin = -Inf, xmax = Inf, ymin = 4.77-1.96*0.297, ymax = 4.77+1.96*0.297, geom = "rect", alpha = 0.2, fill = "firebrick")+
  #annotate(xmin = -Inf, xmax = Inf, ymin = 5.02-1.96*0.372, ymax = 5.02+1.96*0.372, geom = "rect", alpha = 0.2, fill = "blue")+
  #annotate(geom = "hline",yintercept = 100, y = 100, color = "red")+
  annotate("pointrange", x = 1, y = Mdiff, ymin = CI_low, ymax = CI_up, colour = "black", size = 2, shape = 24, fill = "darkgrey")+
  #annotate("pointrange", x = 2, y = 5.02, ymin = 5.02-1.96*0.372, ymax = 5.02+1.96*0.372, colour = "blue", size = 0.8, shape = 15)+
  #annotate("text", x = 0.5, y = 2.6, family = "Poppins", size = 6, color = "gray20", label = "Impfeffekt")+
  #geom_curve(aes(x = 0.5, y = 3, xend = 0.9, yend = 4),arrow = arrow(length = unit(0.03, "npc")),color = "gray20", curvature = +0.2)+
  #annotate("text", x = 1.8, y = 2.6, family = "Poppins", size = 6, color = "gray20", label = "Dosierungseffekt")+
  #geom_curve(aes(x = 1.8, y = 3, xend = 2, yend = 4),arrow = arrow(length = unit(0.03, "npc")),color = "gray20", curvature = +0.2)+
  annotate(geom = "hline",yintercept = 0, y = 0, color = "red", size = 1.2)+
  theme(plot.background = element_rect(
    fill = "white",
    colour = "white",
    size = 1
  ))
## Warning: Ignoring unknown aesthetics: y
barchart

#ggsave("delta_1st_vs_4th.svg",width=2.5,height=4)
#ggsave("delta.pdf",width=2.5,height=4)

What value for Cohen’s d is this?

library(effsize)
cohen.d(tdata_sub$value, tdata_sub$variable, paired = T)
## 
## Cohen's d
## 
## d estimate: 1.256598 (large)
## 95 percent confidence interval:
##     lower     upper 
## 0.9086239 1.6045715

A large effect.

5 Cluster Analysis

data_cluster <- tdata[-c(7:13)]

# append absolut deviations as new columns
data_cluster$Delta_Rating <- data_cluster$single - data_cluster$multiple

crit = 3

data_cluster$Change[data_cluster$Delta_Rating > crit] <- "Dilution"
data_cluster$Change[data_cluster$Delta_Rating >= -crit & data_cluster$Delta_Rating <= crit] <- "Invariance"
data_cluster$Change[data_cluster$Delta_Rating < -crit] <- "Strengthening"

data_cluster %>% count(Change)
##          Change  n
## 1      Dilution 82
## 2    Invariance 26
## 3 Strengthening 12
# now append the clustering results to the main data frame 
data_cluster <- subset(data_cluster, select = c(1,8))
tdata_sub <- merge(tdata_sub, data_cluster, by = c("sID"))
# get CIs for the proportions 

prop.test(82,120,correct=FALSE)
## 
##  1-sample proportions test without continuity correction
## 
## data:  82 out of 120, null probability 0.5
## X-squared = 16.133, df = 1, p-value = 5.904e-05
## alternative hypothesis: true p is not equal to 0.5
## 95 percent confidence interval:
##  0.5955213 0.7597717
## sample estimates:
##         p 
## 0.6833333
prop.test(26,120,correct=FALSE)
## 
##  1-sample proportions test without continuity correction
## 
## data:  26 out of 120, null probability 0.5
## X-squared = 38.533, df = 1, p-value = 5.383e-10
## alternative hypothesis: true p is not equal to 0.5
## 95 percent confidence interval:
##  0.1523672 0.2985436
## sample estimates:
##         p 
## 0.2166667
prop.test(12,120,correct=FALSE)
## 
##  1-sample proportions test without continuity correction
## 
## data:  12 out of 120, null probability 0.5
## X-squared = 76.8, df = 1, p-value < 2.2e-16
## alternative hypothesis: true p is not equal to 0.5
## 95 percent confidence interval:
##  0.05813359 0.16668174
## sample estimates:
##   p 
## 0.1
## Warning: `guides(<scale> = FALSE)` is deprecated. Please use `guides(<scale> =
## "none")` instead.
## Warning: `fun.y` is deprecated. Use `fun` instead.
## `fun.y` is deprecated. Use `fun` instead.