R Markdown created by Joshua Polanin on August 3, 2024
In consultation with Laura Michaelson
Updated and submitted on August 13, 2024
Adapted from ClubSandwich and metafor vignettes

Script notes

Copy/paste from form, use once ready: We did not have a strong ‘a priori’ hypothesis about (a) the underlying relationships or (b) the best modeling approach. As a result, we sought to use model-building techniques that would allow for some flexibility. We limited testing only to models that advanced our understanding of the relationship, using sensitivity analyses to test the robustness of our conclusions.

Load libraries

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##     format.pval, units

Load original dataset

dat <- read.csv("ds_as_race_analysts_rev_10july24.csv", header = T)
str(dat)
## 'data.frame':    103 obs. of  9 variables:
##  $ author_year     : chr  "Guion (2009)" "Guion (2009)" "Guion (2009)" "Guion (2009)" ...
##  $ study_id        : int  100 100 100 100 100 100 200 200 300 300 ...
##  $ sample_id       : int  101 101 102 102 102 101 201 202 301 301 ...
##  $ pub_type        : chr  "dissertation" "dissertation" "dissertation" "dissertation" ...
##  $ as_inst         : chr  "CASQ-R" "CASQ-R" "CASQ-R" "CASQ-R" ...
##  $ valence         : chr  "composite" "positive" "composite" "positive" ...
##  $ corr            : num  -0.469 -0.286 -0.283 -0.186 0.309 0.485 0.257 0.271 -0.55 0.09 ...
##  $ n               : int  113 113 20 20 20 113 292 297 49 49 ...
##  $ race..0...Black.: int  1 1 0 0 0 1 1 0 1 1 ...

Conduct data processing

We start with some simple data cleaning, renaming the race levels.

dat$race..0...Black. <- as.factor(dat$race..0...Black.)
levels(dat$race..0...Black.) <- c("Black", "White")

We want the correlation to ‘mean’ the same thing, regardless of sign
So now, every correlation means that, higher ‘negative’ attribution style, higher depression

dat$corr_flip <- ifelse(dat$valence != "negative", dat$corr * -1, dat$corr)

Using metafor’s ‘escalc’ to transform the bivariate correlations to Fisher’s Z Take the ‘flipped’ correlation form above

dat <- escalc(measure = "ZCOR",
              ri = corr_flip,
              ni = n,
              data = dat,
              var.names = c("z_cor", "z_var"))

Create a numeric variable of the composite variable for correlation testing

dat$valence_num <- ifelse(dat$valence == "composite", 0,
                          ifelse(dat$valence == "positive", 1, -1))

Review updated dataset

str(dat)
## Classes 'escalc' and 'data.frame':   103 obs. of  13 variables:
##  $ author_year     : chr  "Guion (2009)" "Guion (2009)" "Guion (2009)" "Guion (2009)" ...
##  $ study_id        : int  100 100 100 100 100 100 200 200 300 300 ...
##  $ sample_id       : int  101 101 102 102 102 101 201 202 301 301 ...
##  $ pub_type        : chr  "dissertation" "dissertation" "dissertation" "dissertation" ...
##  $ as_inst         : chr  "CASQ-R" "CASQ-R" "CASQ-R" "CASQ-R" ...
##  $ valence         : chr  "composite" "positive" "composite" "positive" ...
##  $ corr            : num  -0.469 -0.286 -0.283 -0.186 0.309 0.485 0.257 0.271 -0.55 0.09 ...
##  $ n               : int  113 113 20 20 20 113 292 297 49 49 ...
##  $ race..0...Black.: Factor w/ 2 levels "Black","White": 2 2 1 1 1 2 2 1 2 2 ...
##  $ corr_flip       : num  0.469 0.286 0.283 0.186 0.309 0.485 0.257 0.271 0.55 -0.09 ...
##  $ z_cor           : num  0.509 0.294 0.291 0.188 0.319 ...
##   ..- attr(*, "ni")= int [1:103] 113 113 20 20 20 113 292 297 49 49 ...
##   ..- attr(*, "measure")= chr "ZCOR"
##  $ z_var           : num  0.00909 0.00909 0.05882 0.05882 0.05882 ...
##  $ valence_num     : num  0 1 0 1 -1 -1 -1 -1 0 0 ...
##  - attr(*, "digits")= Named num [1:9] 4 4 4 4 4 4 4 4 4
##   ..- attr(*, "names")= chr [1:9] "est" "se" "test" "pval" ...
##  - attr(*, "yi.names")= chr "z_cor"
##  - attr(*, "vi.names")= chr "z_var"

It’ll be important to conduct sensitivity analyses, so we split the dataset

dat.comp <- subset(dat, dat$valence == "composite")
dat.posneg <- subset(dat, dat$valence != "composite")

Similarly, we split the dataset into only black and only white participants

dat.black <- subset(dat, dat$race..0...Black. == "Black")
dat.white <- subset(dat, dat$race..0...Black. == "White")

Conduct descritpive analyses

Observe the distribution and test for multicolinearity among valence and sample

t1 <- table(dat$valence, dat$race..0...Black.) ## look at distribution
t1
##            
##             Black White
##   composite     8    20
##   negative     19    32
##   positive      9    15
t2 <- chisq.test(t1) ## test for differences in distribution ## result indicates no differences
t2
## 
##  Pearson's Chi-squared test
## 
## data:  t1
## X-squared = 0.68889, df = 2, p-value = 0.7086
cor(dat$valence_num, as.numeric(dat$race..0...Black.), use = "complete.obs") ## no correlation among predictors, either
## [1] 0.01411934

Conduct meta-analyses

Overall analysis w/ interaction

Notes about this model: Includes the two focal predictors plus the pub_type
Includes a random effect on the study_id
Does NOT include a random effect on sample_id so we can keep the models the same when conducting the subset sensitivity analyses
After running the rma.mv model, we run the coef_test function which adjusts the SEs for within-study correlation

m1 <- rma.mv(z_cor ~ factor(valence) + factor(race..0...Black.) + factor(dat$pub_type) + valence*race..0...Black. ,
                     V = z_var, random = list(~ 1 | study_id),
                     data = dat)
## Warning: Redundant predictors dropped from the model.
summary(m1)
## 
## Multivariate Meta-Analysis Model (k = 103; method: REML)
## 
##   logLik  Deviance       AIC       BIC      AICc   
##  -4.9996    9.9993   25.9993   46.5141   27.6545   
## 
## Variance Components:
## 
##             estim    sqrt  nlvls  fixed    factor 
## sigma^2    0.0161  0.1268     36     no  study_id 
## 
## Test for Residual Heterogeneity:
## QE(df = 96) = 282.3150, p-val < .0001
## 
## Test of Moderators (coefficients 2:7):
## QM(df = 6) = 54.8304, p-val < .0001
## 
## Model Results:
## 
##                                        estimate      se     zval    pval 
## intrcpt                                  0.3416  0.0749   4.5619  <.0001 
## factor(valence)negative                  0.0926  0.0700   1.3234  0.1857 
## factor(valence)positive                 -0.1890  0.0790  -2.3921  0.0168 
## factor(race..0...Black.)White            0.1794  0.0602   2.9813  0.0029 
## factor(dat$pub_type)journal             -0.0106  0.0564  -0.1888  0.8502 
## valencenegative:race..0...Black.White   -0.1675  0.0721  -2.3228  0.0202 
## valencepositive:race..0...Black.White    0.0271  0.0821   0.3297  0.7416 
##                                          ci.lb    ci.ub      
## intrcpt                                 0.1949   0.4884  *** 
## factor(valence)negative                -0.0446   0.2298      
## factor(valence)positive                -0.3438  -0.0341    * 
## factor(race..0...Black.)White           0.0614   0.2973   ** 
## factor(dat$pub_type)journal            -0.1211   0.0999      
## valencenegative:race..0...Black.White  -0.3088  -0.0262    * 
## valencepositive:race..0...Black.White  -0.1339   0.1880      
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
m1_ct <- coef_test(m1, "CR2")
m1_ct
##                                  Coef. Estimate     SE t-stat d.f. (Satt)
##                                intrcpt   0.3416 0.0880  3.880        6.84
##                factor(valence)negative   0.0926 0.0856  1.082        5.65
##                factor(valence)positive  -0.1890 0.0818 -2.309        4.57
##          factor(race..0...Black.)White   0.1794 0.0797  2.252        3.60
##            factor(dat$pub_type)journal  -0.0106 0.0612 -0.174       16.71
##  valencenegative:race..0...Black.White  -0.1675 0.0811 -2.064        6.00
##  valencepositive:race..0...Black.White   0.0271 0.0728  0.372        4.92
##  p-val (Satt) Sig.
##       0.00634   **
##       0.32333     
##       0.07391    .
##       0.09486    .
##       0.86405     
##       0.08458    .
##       0.72540

In addition, we conduct a Wald_test on the interaction because it is what we are interested in - any differnces in relationships across race

m1_wt <- Wald_test(m1, "CR2", constraints = constrain_zero(6:7), cluster = dat$study_id)
m1_wt
##  test Fstat df_num df_denom  p_val sig
##   HTZ  7.68      2     4.09 0.0412   *

Results indicate:

  1. Positive valence weaker relationship to depression, relative to other two measures
  2. Compared to black participants, white participants overall att-dep is stronger
  3. However, possible interaction among predictors, especailly for the negative valence
  4. This is confirmed by the Wald Test conducted on the interaction, which indicates significant differences among these predictors
  5. Therefore, further analysis by subset is required to validate and observe particulars.
  6. Also notable - no difference in published vs unpublished studies.

Subset analyses

Only composite attributional style measures

m2 <- rma.mv(z_cor ~ factor(race..0...Black.),
             V = z_var, random = list(~ 1 | study_id),
             data = dat.comp)
summary(m2)
## 
## Multivariate Meta-Analysis Model (k = 28; method: REML)
## 
##   logLik  Deviance       AIC       BIC      AICc   
##   0.7044   -1.4089    4.5911    8.3654    5.6820   
## 
## Variance Components:
## 
##             estim    sqrt  nlvls  fixed    factor 
## sigma^2    0.0171  0.1306     13     no  study_id 
## 
## Test for Residual Heterogeneity:
## QE(df = 26) = 76.3825, p-val < .0001
## 
## Test of Moderators (coefficient 2):
## QM(df = 1) = 9.1284, p-val = 0.0025
## 
## Model Results:
## 
##                                estimate      se    zval    pval   ci.lb   ci.ub 
## intrcpt                          0.3580  0.0668  5.3609  <.0001  0.2271  0.4888 
## factor(race..0...Black.)White    0.1830  0.0606  3.0213  0.0025  0.0643  0.3018 
##                                    
## intrcpt                        *** 
## factor(race..0...Black.)White   ** 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
m2_ct <- coef_test(m2, "CR2")
m2_ct
##                          Coef. Estimate     SE t-stat d.f. (Satt) p-val (Satt)
##                        intrcpt    0.358 0.0745   4.80        4.78      0.00547
##  factor(race..0...Black.)White    0.183 0.0775   2.36        3.55      0.08582
##  Sig.
##    **
##     .

Results indicate:

  1. Compared to black participants, white participants’ corr stronger (marginally sig)
  2. Overall correlation about the same from the full model

Only negative or positive attributional style measures

m3 <- rma.mv(z_cor ~ factor(race..0...Black.),
             V = z_var, random = list(~ 1 | study_id),
             data = dat.posneg)
summary(m3)
## 
## Multivariate Meta-Analysis Model (k = 75; method: REML)
## 
##   logLik  Deviance       AIC       BIC      AICc   
## -21.0915   42.1831   48.1831   55.0545   48.5309   
## 
## Variance Components:
## 
##             estim    sqrt  nlvls  fixed    factor 
## sigma^2    0.0122  0.1103     29     no  study_id 
## 
## Test for Residual Heterogeneity:
## QE(df = 73) = 247.4694, p-val < .0001
## 
## Test of Moderators (coefficient 2):
## QM(df = 1) = 4.1961, p-val = 0.0405
## 
## Model Results:
## 
##                                estimate      se    zval    pval   ci.lb   ci.ub 
## intrcpt                          0.3418  0.0376  9.0886  <.0001  0.2681  0.4155 
## factor(race..0...Black.)White    0.0700  0.0342  2.0484  0.0405  0.0030  0.1370 
##                                    
## intrcpt                        *** 
## factor(race..0...Black.)White    * 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
m3_ct<- coef_test(m3, "CR2")
m3_ct
##                          Coef. Estimate     SE t-stat d.f. (Satt) p-val (Satt)
##                        intrcpt    0.342 0.0393   8.69        9.55       <0.001
##  factor(race..0...Black.)White    0.070 0.0463   1.51        5.76        0.183
##  Sig.
##   ***
##

Results indicate:

  1. White participants relationship stronger compared to Black participants, but not quite as much
  2. Overall correlational similar to composite measure

Only Black participants

m4 <- rma.mv(z_cor ~ factor(valence),
             V = z_var, random = list(~ 1 | study_id),
             data = dat.black)
summary(m4)
## 
## Multivariate Meta-Analysis Model (k = 36; method: REML)
## 
##   logLik  Deviance       AIC       BIC      AICc   
##   5.2684  -10.5369   -2.5369    3.4492   -1.1083   
## 
## Variance Components:
## 
##             estim    sqrt  nlvls  fixed    factor 
## sigma^2    0.0058  0.0759     20     no  study_id 
## 
## Test for Residual Heterogeneity:
## QE(df = 33) = 34.1994, p-val = 0.4099
## 
## Test of Moderators (coefficients 2:3):
## QM(df = 2) = 18.3776, p-val = 0.0001
## 
## Model Results:
## 
##                          estimate      se     zval    pval    ci.lb    ci.ub 
## intrcpt                    0.3742  0.0581   6.4374  <.0001   0.2603   0.4882 
## factor(valence)negative    0.0255  0.0681   0.3743  0.7081  -0.1080   0.1590 
## factor(valence)positive   -0.2346  0.0790  -2.9716  0.0030  -0.3894  -0.0799 
##                              
## intrcpt                  *** 
## factor(valence)negative      
## factor(valence)positive   ** 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
m4_ct <- coef_test(m4, "CR2")
m4_ct
##                    Coef. Estimate     SE t-stat d.f. (Satt) p-val (Satt) Sig.
##                  intrcpt   0.3742 0.0547  6.847        3.59      0.00349   **
##  factor(valence)negative   0.0255 0.0709  0.359        5.78      0.73203     
##  factor(valence)positive  -0.2346 0.0940 -2.496        4.69      0.05802    .

Results indicate:

  1. For Black participants, negative measure is strongest predictor of depression
  2. Difference between negative and composite measure not stat. sig though
  3. Weakest predictor is positive att style and it’s relatively weak (0.14)

Only white participants

m5 <- rma.mv(z_cor ~ factor(valence),
             V = z_var, random = list(~ 1 | study_id),
             data = dat.white)
summary(m5)
## 
## Multivariate Meta-Analysis Model (k = 67; method: REML)
## 
##   logLik  Deviance       AIC       BIC      AICc   
##  -9.3227   18.6453   26.6453   35.2809   27.3233   
## 
## Variance Components:
## 
##             estim    sqrt  nlvls  fixed    factor 
## sigma^2    0.0169  0.1301     34     no  study_id 
## 
## Test for Residual Heterogeneity:
## QE(df = 64) = 261.5820, p-val < .0001
## 
## Test of Moderators (coefficients 2:3):
## QM(df = 2) = 22.4734, p-val < .0001
## 
## Model Results:
## 
##                          estimate      se     zval    pval    ci.lb    ci.ub 
## intrcpt                    0.5063  0.0363  13.9310  <.0001   0.4351   0.5776 
## factor(valence)negative   -0.0743  0.0332  -2.2388  0.0252  -0.1393  -0.0093 
## factor(valence)positive   -0.1598  0.0346  -4.6166  <.0001  -0.2276  -0.0919 
##                              
## intrcpt                  *** 
## factor(valence)negative    * 
## factor(valence)positive  *** 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
m5_ct <- coef_test(m5, "CR2")
m5_ct
##                    Coef. Estimate     SE t-stat d.f. (Satt) p-val (Satt) Sig.
##                  intrcpt   0.5063 0.0330  15.33       11.41       <0.001  ***
##  factor(valence)negative  -0.0743 0.0292  -2.55        4.94       0.0520    .
##  factor(valence)positive  -0.1598 0.0495  -3.23        4.36       0.0282    *

Results indicate:

  1. Again, positive att style is the weakest among the predictors
  2. However still a relatively strong predictor of depression (0.35)
  3. Also interesting, composite measure has the strongest relationship, ahead of negative measure
  4. This is different from what we observe among Black participants, where the negative att style had the strongest relationship

Sensitivity analysis

Adding a random effect on sample_id

m6 <- rma.mv(z_cor ~ factor(valence) + factor(race..0...Black.) + factor(dat$pub_type) + valence*race..0...Black. ,
             V = z_var, random = list(~ 1 | study_id, ~ 1 | sample_id),
             data = dat)
## Warning: Redundant predictors dropped from the model.
summary(m6)
## 
## Multivariate Meta-Analysis Model (k = 103; method: REML)
## 
##   logLik  Deviance       AIC       BIC      AICc   
##  -4.1906    8.3812   26.3812   49.4603   28.4742   
## 
## Variance Components:
## 
##             estim    sqrt  nlvls  fixed     factor 
## sigma^2.1  0.0138  0.1176     36     no   study_id 
## sigma^2.2  0.0024  0.0489     56     no  sample_id 
## 
## Test for Residual Heterogeneity:
## QE(df = 96) = 282.3150, p-val < .0001
## 
## Test of Moderators (coefficients 2:7):
## QM(df = 6) = 48.8478, p-val < .0001
## 
## Model Results:
## 
##                                        estimate      se     zval    pval 
## intrcpt                                  0.3508  0.0780   4.4986  <.0001 
## factor(valence)negative                  0.0764  0.0739   1.0335  0.3014 
## factor(valence)positive                 -0.2035  0.0828  -2.4597  0.0139 
## factor(race..0...Black.)White            0.1636  0.0660   2.4788  0.0132 
## factor(dat$pub_type)journal             -0.0077  0.0561  -0.1378  0.8904 
## valencenegative:race..0...Black.White   -0.1490  0.0773  -1.9274  0.0539 
## valencepositive:race..0...Black.White    0.0435  0.0869   0.5013  0.6162 
##                                          ci.lb    ci.ub      
## intrcpt                                 0.1980   0.5036  *** 
## factor(valence)negative                -0.0685   0.2213      
## factor(valence)positive                -0.3657  -0.0414    * 
## factor(race..0...Black.)White           0.0342   0.2930    * 
## factor(dat$pub_type)journal            -0.1177   0.1023      
## valencenegative:race..0...Black.White  -0.3004   0.0025    . 
## valencepositive:race..0...Black.White  -0.1267   0.2138      
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
tst1 <- coef_test(m6, "CR2")
tst1
##                                  Coef. Estimate     SE t-stat d.f. (Satt)
##                                intrcpt  0.35081 0.0767  4.574        6.61
##                factor(valence)negative  0.07641 0.0733  1.043        5.59
##                factor(valence)positive -0.20354 0.0835 -2.437        4.55
##          factor(race..0...Black.)White  0.16360 0.0650  2.518        4.05
##            factor(dat$pub_type)journal -0.00773 0.0606 -0.128       16.54
##  valencenegative:race..0...Black.White -0.14896 0.0672 -2.216        6.09
##  valencepositive:race..0...Black.White  0.04355 0.0635  0.686        4.98
##  p-val (Satt) Sig.
##       0.00296   **
##       0.34007     
##       0.06379    .
##       0.06478    .
##       0.89992     
##       0.06788    .
##       0.52325
Wald_test(m6, "CR2", constraints = constrain_zero(6:7), cluster = dat$study_id) 
##  test Fstat df_num df_denom p_val sig
##   HTZ  4.77      2     4.01 0.087   .

Results indicate:

  1. Very similar point estimates
  2. Some slight differences in statistical significance, but nothing substanative
  3. Biggest difference compared to first model (m1), the Wald_test is no longer significant;
  4. This indicates that the interaction might be relatively minimal

Graph representation of interaction

Plotting the model-based average correlations by race and valence

# Extract model coefficients and their standard errors
estimates <- c(
  intrcpt = m1_ct$beta[1],
  val_neg = m1_ct$beta[2],
  val_pos = m1_ct$beta[3],
  race_white = m1_ct$beta[4],
  neg_white_inter = m1_ct$beta[6],
  pos_white_inter = m1_ct$beta[7]
)
m1_ct$SE
## [1] 0.08803916 0.08562223 0.08183845 0.07965741 0.06122304 0.08113888 0.07280778
ses <- c(
  intrcpt = m1_ct$SE[1],
  val_neg = m1_ct$SE[2],
  val_pos = m1_ct$SE[3],
  race_white = m1_ct$SE[4],
  neg_white_inter = m1_ct$SE[6],
  pos_white_inter = m1_ct$SE[7]
)

# Calculate estimates and CIs for each combination
data <- data.frame(
  race = rep(c("Black", "White"), each = 3),
  valence = rep(c("composite", "negative", "positive"), 2),
  estimate = c(
    estimates["intrcpt"],
    estimates["intrcpt"] + estimates["val_neg"],
    estimates["intrcpt"] + estimates["val_pos"],
    estimates["intrcpt"] + estimates["race_white"],
    estimates["intrcpt"] + estimates["race_white"] + estimates["val_neg"] + estimates["neg_white_inter"],
    estimates["intrcpt"] + estimates["race_white"] + estimates["val_pos"] + estimates["pos_white_inter"]
  ),
  se = c(
    ses["intrcpt"],
    sqrt(ses["intrcpt"]^2 + ses["val_neg"]^2),
    sqrt(ses["intrcpt"]^2 + ses["val_pos"]^2),
    sqrt(ses["intrcpt"]^2 + ses["race_white"]^2),
    sqrt(ses["intrcpt"]^2 + ses["race_white"]^2 + ses["val_neg"]^2 + ses["neg_white_inter"]^2),
    sqrt(ses["intrcpt"]^2 + ses["race_white"]^2 + ses["val_pos"]^2 + ses["pos_white_inter"]^2)
  )
)

data$ci.lb <- data$estimate - 1.96 * data$se
data$ci.ub <- data$estimate + 1.96 * data$se

# Plotting the point estimates with their confidence intervals and connecting lines
#p1 <- ggplot(data, aes(x = valence, y = estimate, group = race, color = race)) +
#  geom_point() +
#  geom_line() +
#  geom_errorbar(aes(ymin = ci.lb, ymax = ci.ub), width = 0.1) +
#  scale_x_discrete(limits = c("composite", "negative", "positive"), 
#                   labels=c("composite" = "Composite", "negative" = "Negative", "positive" = "Positive")) +
#  scale_y_continuous(limits = c(-0.8, 0.8)) +
#  theme_minimal() +
#  labs(title = "Model-Based Correlations without Confidence Intervals",
#       x = "Valence",
#       y = "Correlation",
#       color = "Racial Group") +
#  theme(axis.text.x = element_text(angle = 45, hjust = 1))
#p1

p2 <- ggplot(data, aes(x = valence, y = estimate, group = race, color = race)) +
  geom_point() +
  geom_line() +
  #geom_errorbar(aes(ymin = ci.lb, ymax = ci.ub), width = 0.1) +
  scale_x_discrete(limits = c("composite", "negative", "positive"), 
                   labels=c("composite" = "Composite", "negative" = "Negative", "positive" = "Positive")) +
  scale_y_continuous(limits = c(-0.8, 0.8)) +
  theme_minimal() +
  labs(title = "Model-Based Correlations without Confidence Intervals",
       x = "Valence",
       y = "Correlation",
       color = "Racial Group") +
  theme(axis.text.x = element_text(angle = 45, hjust = 1))
p2

Best guess analysis

Difference between Black and White participants for each valence

Calculation is White participants - Black participants

Composite_Difference <- data[4,3] - data[1,3]
Negative_Difference <- data[5,3] - data[2,3]
Positive_Difference <- data[6,3] - data[3,3]
Composite_Difference
## [1] 0.1793604
Negative_Difference
## [1] 0.01187764
Positive_Difference
## [1] 0.2064421