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  • Continuous predictors: centering
    • Creating centered variables
      • What’s the difference?
    • How are our variables coded?
    • Intraclass Correlation Coefficient (ICC)
    • Two-level crossed model with individual repeated measures
    • Two-level crossed model with distinguishable dyads
  • Categorical predictors: contrast coding
    • Change variable class
    • Dummy coding
    • Effects coding
    • Contrast coding in R
      • Apply contrasts (effect code)
      • Apply contrasts (dummy code)
      • Summary of intercepts
  • Two-level crossed model
    • Analysis steps
      • What is our takeaway from the ICC?
        • Output
        • Residual structure
          • What can we learn from this?
  • Your turn!
    • Contrast coding
    • Centering & Two-Level Crossed Model (with dyads!)

To demonstrate centering, contrast coding, and two-level crossed models, we’re going to use a few different datasets.

Please download the following datasets from my github:

  1. sample data_two_level (1).sav
  2. lab5_anova_contrasts.sav
  3. DyadicLongitudinal_ExampleDataset_RM3.csv
  4. ViagraCovariate.dat
  5. sample data_week4.sav

Continuous predictors: centering

Let’s say you’re interested in the effect of perceived social support on anxiety. Anxiety can vary both between- and within-person. The question is, are you interested in the within-person effect (how does receiving more social support than I typically do influence my anxiety?) or between-person (how anxious are people who received more social support on average?). We can actually look at both by centering our predictors in different ways.

  • uncentered (raw)
  • centering on a sample value (same for everyone): typically, grand-mean centered (centered on sample mean), but can center on another relevant value (e.g., centering on the first timepoint for the “time” variable).
  • person-level centering (different for everyone): typically, person-mean centering (subtracting each individual’s mean from their own scores), but could centering on another person-level variable (e.g., first Sunday for each person when participants start daily diaries on different days of the week). Only relevant for time-varying variables

Centering on a sample value (same for everyone)

  1. Fixed intercept reflects whatever you set 0 to be for the whole sample
  2. Fixed regression estimates (fixed slopes) are identical to uncentered

Person-level centering (different for everyone)

  1. Fixed intercept reflects whatever 0 is for each participant (e.g., each participant’s mean)
  2. Fixed regression estimates (fixed slopes) will be different
    • expected change in outcome corresponding to a 1 unit change in a predictor relative to each participant’s centered value (e.g., their own average)

Example: Amie slept 6 hours last night, but she usually sleeps 9 hours. Kate slept 6 hours last night and she usually sleeps 6 hours. When person-mean centering, 6 hours for Amie is a deficit of 3 hours but average (0) for Kate.

With more complex data structures:
• You can unconfound effects for any random factor
• Example: ESM data with momentary assessments nested within days within people
  • Within-day and between-day effects
    • Times when I am more angry than usual that day
    • Days when I am more angry than I usually am
  • Within-person and between-person effects (You could do this across all momentary assessments, ignoring day)
    • Times I am more angry than I usually am across the whole study
    • People who are more angry than the average person in the sample

OR you could have all 3 levels:
• Each momentary assessment deviation from that day’s mean
• Each day’s mean from the person’s overall mean
• Person’s overall mean
  • This provides all three ways of looking at the data:
    • Times when I am more angry than usual that day
    • Days when I am more angry than I usual am
    • People who are more angry than the average person in the sample

Creating centered variables

DATASET 1: Revisiting the roommate data

df_roommate<-read_sav("/Users/kareenadelrosario/Desktop/everything/2_areas/spring 2024/person perception/sample data_two_level.sav")
head(df_roommate)
## # A tibble: 6 × 658
##    Dyad partnum Participant_ID   day dayR       dorm awish sloppy_roommate
##   <dbl>   <dbl>          <dbl> <dbl> <dbl+lbl> <dbl> <dbl>           <dbl>
## 1     1       1              1    10 -1            1     1               2
## 2     1       1              1    11  1            1     2               1
## 3     6       1              6    10 -1            1    NA               1
## 4     6       1              6    11  1            1     1               1
## 5     7       1              7    10 -1            1     1               1
## 6     7       1              7    11  1            1     1               1
## # ℹ 650 more variables: sloppy_roommateC <dbl>, first_similarN <dbl>,
## #   first_similarN_low <dbl>, first_similarN_high <dbl>, genderN <dbl>,
## #   EXAMPLE_VARS_END_HERE <dbl>, dayC <dbl>, contrast1 <dbl>, contrast2 <dbl>,
## #   raceN <dbl>, praceN <dbl>, daybeg <dbl>, awhitemin_pwhitemin <dbl>,
## #   prace_think <chr>, race_think <chr>, dyad_type_JESP <dbl>, pa <dbl>,
## #   I2 <dbl>, OBS_ID <dbl>, ahurt_comp <dbl>, aphurt_comp <dbl>,
## #   pahurt_comp <dbl>, pid1 <dbl>, length <dbl>, pcheck <chr>, …

Centering roommate messiness in roommate data

df_mess <- df_roommate %>% 
  mutate(proom_mess_GMC = proom_mess - mean(proom_mess, na.rm = TRUE)) %>% # grand mean centered
  group_by(Participant_ID) %>% 
  mutate(proom_mess_PC = proom_mess - mean(proom_mess, na.rm = TRUE)) %>% # person centered
  ungroup() %>% 
  mutate(proom_mess_BPC = proom_mess_GMC - proom_mess_PC)


# dayR must be an integer
df_mess$dayR_int <- as.integer(as.factor(df_mess$dayR))
model_mess <- gls(awish ~ proom_mess_GMC + proom_mess_PC,
             data = df_mess,
             correlation = corSymm(form = ~ dayR_int | Participant_ID),
             weights=varIdent(form=~1|dayR_int),
             method = "REML",
             na.action = na.omit)

summary(model_mess)
## Generalized least squares fit by REML
##   Model: awish ~ proom_mess_GMC + proom_mess_PC 
##   Data: df_mess 
##        AIC     BIC    logLik
##   392.0479 408.672 -190.0239
## 
## Correlation Structure: General
##  Formula: ~dayR_int | Participant_ID 
##  Parameter estimate(s):
##  Correlation: 
##   1    
## 2 0.731
## Variance function:
##  Structure: Different standard deviations per stratum
##  Formula: ~1 | dayR_int 
##  Parameter estimates:
##        1        2 
## 1.000000 1.264524 
## 
## Coefficients:
##                     Value Std.Error   t-value p-value
## (Intercept)     1.5087665 0.1522062  9.912652  0.0000
## proom_mess_GMC  0.0737453 0.1028680  0.716892  0.4749
## proom_mess_PC  -0.1281800 0.1448588 -0.884862  0.3780
## 
##  Correlation: 
##                (Intr) p__GMC
## proom_mess_GMC  0.017       
## proom_mess_PC  -0.019 -0.709
## 
## Standardized residuals:
##        Min         Q1        Med         Q3        Max 
## -0.7252142 -0.3673895 -0.3324373 -0.2905358  4.5930239 
## 
## Residual standard error: 1.209577 
## Degrees of freedom: 121 total; 118 residual

What’s the difference?

Categorical predictors: contrast coding

To demonstrate contrast coding, we’re going to use a different dataset

DATASET 2: Sadness by 3-level condition and gender

df<- read_sav("/Users/kareenadelrosario/Desktop/local r code/regression_code/lab5_anova_contrasts.sav")

Predictors:

  • condition_3Level: 1, 2, 3
  • genderR: -1, 1

Change variable class

df.short <- df %>% 
  mutate(condition_3Level = as.factor(condition_3Level),
         genderR = as.factor(genderR))

# check class
class(df.short$condition_3Level)
## [1] "factor"
class(df.short$genderR)
## [1] "factor"

Dummy coding

This is the most common form of contrast coding. In dummy coding, one level of the categorical variable is chosen as a reference group. For example, if you have a variable “Race” with three levels (White, Black, Asian), you can create two dummy variables: the effect of being Black and the effect of being Asian. The reference group (White) gets coded as 0 in both new variables. The intercept represents the mean outcome for the reference group.

White = 1 Black = 2 Asian = 3

Race Dummy1 Dummy2
White 0 0
Black 1 0
Asian 0 1
Dummy 1: outcome for Black participants vs reference (White)
Dummy 2: outcome for Asian participants vs reference (White)
Intercept = average sleep quality when predictor is zero (sleep for White participants)

Effects coding

Unlike dummy coding, where the reference category is represented by all 0’s, effects coding doesn’t leave out one category as the reference. Instead, categories are coded with numbers that balance out, such that the sum of codes for each categorical level across all coded variables equals zero. This balance allows the model’s intercept to represent the grand mean. You would use this coding scheme if you wanted to compare the mean of each group to the grand mean. The reference group gets -1 for both effects.

The sum of your contrasts should = 0.

Race Effect1 Effect2
White 1 0
Black 0 1
Asian -1 -1
Effect 1: compares the effect of White to the overall mean
Effect 2: compares the effect of Black to the overall mean
Intercept = grand mean across all categories of race

tldr; Dummy coding gives us the “simple effects,” comparing different levels to a reference group while effects coding gives us the “main effects,” comparing different levels to the group mean

How are our variables coded?

contrasts(df.short$genderR)
##    1
## -1 0
## 1  1

Identifiers:

  • -1 = Female

  • 1 = Male


Contrasts:

  • 0 = Female

  • 1 = Male

How does this impact our interpretation of the intercept?

  • Average sadness after the interaction for women (when genderR = 0)
contrasts(df.short$condition_3Level)
##   2 3
## 1 0 0
## 2 1 0
## 3 0 1

Identifiers:

  • 1 = Control dyad

  • 2 = Sad actor

  • 3 = Sad partner


Contrasts:

FIRST EFFECT: what is the effect of sad actor vs control condition?

SECOND EFFECT: what is the effect of sad partner vs control condition?


Contrast coding in R

# effect-code
contr.sum(2)
##   [,1]
## 1    1
## 2   -1
contr.sum(3)
##   [,1] [,2]
## 1    1    0
## 2    0    1
## 3   -1   -1
# dummy code
contr.treatment(2)
##   2
## 1 0
## 2 1
contr.treatment(3)
##   2 3
## 1 0 0
## 2 1 0
## 3 0 1

Apply contrasts (effect code)

contrasts(df.short$genderR) <- contr.sum(2)
contrasts(df.short$condition_3Level) <- contr.sum(3)

# alternative: custom effects coding
effectContrasts <- cbind(c(1, 0, -1), c(0, 1, -1))
model_shortcut <- lm(avg_sad_int ~ genderR*condition_3Level, data = df.short)

summary(model_shortcut)
## 
## Call:
## lm(formula = avg_sad_int ~ genderR * condition_3Level, data = df.short)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -0.2738 -0.2167 -0.1576 -0.1111  2.0595 
## 
## Coefficients:
##                              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                 1.1972874  0.0309211  38.721   <2e-16 ***
## genderR1                   -0.0550935  0.0309211  -1.782   0.0762 .  
## condition_3Level1           0.0184052  0.0388069   0.474   0.6358    
## condition_3Level2           0.0149933  0.0460489   0.326   0.7450    
## genderR1:condition_3Level1 -0.0030233  0.0388069  -0.078   0.9380    
## genderR1:condition_3Level2  0.0007076  0.0460489   0.015   0.9878    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4279 on 222 degrees of freedom
##   (2 observations deleted due to missingness)
## Multiple R-squared:  0.02079,    Adjusted R-squared:  -0.001268 
## F-statistic: 0.9425 on 5 and 222 DF,  p-value: 0.4543

Apply contrasts (dummy code)

contrasts(df.short$genderR) <- contr.treatment(2)
contrasts(df.short$condition_3Level) <- contr.treatment(3)

# alternative: custom dummy coding
dummyContrasts <- cbind(c(0,1,0), c(0, 0, 1))
model_shortcut2 <- lm(avg_sad_int ~ genderR*condition_3Level, data = df.short)

summary(model_shortcut2)
## 
## Call:
## lm(formula = avg_sad_int ~ genderR * condition_3Level, data = df.short)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -0.2738 -0.2167 -0.1576 -0.1111  2.0595 
## 
## Coefficients:
##                             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                 1.157576   0.057697  20.063   <2e-16 ***
## genderR2                    0.116234   0.081230   1.431    0.154    
## condition_3Level2           0.000319   0.090261   0.004    0.997    
## condition_3Level3          -0.046465   0.089574  -0.519    0.604    
## genderR2:condition_3Level2 -0.007462   0.143426  -0.052    0.959    
## genderR2:condition_3Level3 -0.010678   0.142995  -0.075    0.941    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4279 on 222 degrees of freedom
##   (2 observations deleted due to missingness)
## Multiple R-squared:  0.02079,    Adjusted R-squared:  -0.001268 
## F-statistic: 0.9425 on 5 and 222 DF,  p-value: 0.4543

Summary of intercepts

The intercept represents the dependent variable when all predictors are zero

  • raw (no centering): 0 here can vary quite a bit depending on what your predictors are. It may not be meaningful.

  • dummy-coding: 0 is the reference group. The intercept is the average DV for the reference group.

  • effects-coding: 0 is the grand mean. The intercept is the DV when all predictors are averaged. Let’s say we coded women -1 and men 1. The intercept would be the DV regardless of gender (averaging across gender).

  • grand-mean centering: 0 is the grand mean. The intercept is the expected DV at the average level of the predictors.

  • person-mean centering: 0 is the person’s average for that predictor. So, the intercept would be ‘the expected level of anxiety (DV) when you sleep your average amount’

Two-level crossed model

DATASET 3: Intellectual humility measured 3x courtesy of Kate Thorson & Amie Gordon

# load in our example data
rm_df <- read.csv("/Users/kareenadelrosario/Downloads/DyadicLongitudinal_ExampleDataset_RM3.csv")

964 people reported on their intellectual humility (IH) at three time points: pre (before the first conversation), post (after the last conversation), and follow-up (one month after the last conversation). We are going to use these data to ask whether IH changes across the three time points.

Here are the variables we’re interested in:

  • IH: intellectual humility, reported at three time points
  • time: time, pre = 1; post = 2; follow-up = 3
  • id: unique ID number for each person
  • timecontrast1_pre_post: pre = 0, post = 1, follow-up = 0
  • timecontrast2_pre_followup: pre = 0, post = 0, follow-up = 1

TIME • 1: pre • 2: post • 3: follow-up

Our predictors are timecontrast1_pre_post, timecontrast2_pre_followup, and time.

  • Predictors:
    • timecontrast1_pre_post,
    • timecontrast2_pre_followup,
    • time,
    • timecontrast1_pre_post x time,
    • timecontrast2_pre_followup x time
  • Error variance for time1, time2, time3

Analysis steps

Before we jump into the analysis, let’s take a step back and go through some pre-analyses to figure out:

  1. whether we need to account for nonindependence in the first place
  2. build our model

Intraclass Correlation Coefficient (ICC)

Sometimes used to “justify” nonindependent analyses. ICC indexes the similarity between time points for a particular outcome. Technically, it represents the proportion of the total variance in an outcome that is explained by mean differences between subjects. An ICC of 1 would mean that all of the variability in the DV is between subjects. In repeated self-report data, it’s common to have ICCs between .2-.4.

Calculating an ICC:

  1. Run an MLM with no predictors (i.e., a null model)
  2. Use person as a random factor
  3. Using the output, calculate the ICC as the following:

Run null model with nlme

library(haven) # read SPSS and SAS files
library(reghelper) # for ICC command


null_model_lme <- lme(IH ~ 1,
             data = rm_df,
             random = ~ 1 | id,
             method = "REML",
             na.action = na.omit)

summary(null_model_lme)
## Linear mixed-effects model fit by REML
##   Data: rm_df 
##        AIC      BIC    logLik
##   5807.233 5824.256 -2900.616
## 
## Random effects:
##  Formula: ~1 | id
##         (Intercept)  Residual
## StdDev:   0.6476011 0.7498407
## 
## Fixed effects:  IH ~ 1 
##               Value  Std.Error   DF  t-value p-value
## (Intercept) 5.70632 0.02681951 1217 212.7675       0
## 
## Standardized Within-Group Residuals:
##        Min         Q1        Med         Q3        Max 
## -5.6254028 -0.3335210  0.1209683  0.5563851  2.5299155 
## 
## Number of Observations: 2154
## Number of Groups: 937
ICC(null_model_lme)
## [1] 0.4272276

Run null model with lme4

library(lme4)

null_model_lmer <- lmer(IH ~ 1 + 
                    (1 |id), 
                  data=rm_df)

summary(null_model_lmer)
## Linear mixed model fit by REML ['lmerMod']
## Formula: IH ~ 1 + (1 | id)
##    Data: rm_df
## 
## REML criterion at convergence: 5801.2
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -5.6254 -0.3335  0.1210  0.5564  2.5299 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  id       (Intercept) 0.4194   0.6476  
##  Residual             0.5623   0.7498  
## Number of obs: 2154, groups:  id, 937
## 
## Fixed effects:
##             Estimate Std. Error t value
## (Intercept)  5.70632    0.02682   212.8

Here’s the ICC equation:

ICC= between−dyad variance (random intercept variance)total variance (random intercept variance + residual variance)

id_variance <- 0.4194
residual <- 0.5623

# calculating ICC
id_variance / (id_variance + residual)
## [1] 0.4272181

What is our takeaway from the ICC?

Two-level crossed model with individual repeated measures

This will be a marginal model– we’re going to adjust for nonindependence by providing a variance-covariance structure to the residuals. This approach is useful for repeated measure with 3 or 4 time points.

In other words, this will model the nonindependence with a covariance (adjust for nonindependence by correlating the residuals between dyad members. This allows for both positive and negative nonindependence).

Output

  1. Fixed parameter estimates
  2. Error covariance: are the two time points’ errors related ot each other?
  3. Error variance: is there significant variance that is unexplained by the other components in the model?

Residual structure

We’re estimating 6 parameters:

• 3 variances
  • Variance at pre
  • Variance at post
  • Variance at followup
  
• 3 within-person covariances at different time points
  • Within-person covariance between pre and post
  • Within-person covariance between pre and follow-up
  • Within-person covariance between post and follow-up

For reference, here is the SPSS syntax:

MIXED IH WITH timecontrast1_pre_post timecontrast2_pre_followup
/FIXED = timecontrast1_pre_post timecontrast2_pre_followup |
SSTYPE(3)
/METHOD = REML
/PRINT = G TESTCOV SOLUTION
/REPEATED time | subject(Participant_ID) COVTYPE(UN).

Weighted by day (each day has its own variance)

# dayR must be an integer
rm_df$time <- as.integer(as.factor(rm_df$time))

model1 <- gls(IH ~ timecontrast1_pre_post + timecontrast2_pre_followup,
             data = rm_df,
             correlation = corSymm(form = ~ time | id), 
             weights=varIdent(form=~1|time),
             method = "REML",
             na.action = na.omit)

summary(model1)
## Generalized least squares fit by REML
##   Model: IH ~ timecontrast1_pre_post + timecontrast2_pre_followup 
##   Data: rm_df 
##        AIC      BIC    logLik
##   5648.821 5699.884 -2815.411
## 
## Correlation Structure: General
##  Formula: ~time | id 
##  Parameter estimate(s):
##  Correlation: 
##   1     2    
## 2 0.447      
## 3 0.501 0.499
## Variance function:
##  Structure: Different standard deviations per stratum
##  Formula: ~1 | time 
##  Parameter estimates:
##         1         2         3 
## 1.0000000 0.9362196 0.7794307 
## 
## Coefficients:
##                               Value  Std.Error   t-value p-value
## (Intercept)                5.498933 0.03363166 163.50465       0
## timecontrast1_pre_post     0.388216 0.03496641  11.10255       0
## timecontrast2_pre_followup 0.352448 0.04193941   8.40374       0
## 
##  Correlation: 
##                            (Intr) tmc1__
## timecontrast1_pre_post     -0.560       
## timecontrast2_pre_followup -0.489  0.435
## 
## Standardized residuals:
##        Min         Q1        Med         Q3        Max 
## -5.0706050 -0.4379075  0.1170871  0.6358563  1.4580824 
## 
## Residual standard error: 1.02948 
## Degrees of freedom: 2154 total; 2151 residual
getVarCov(model1, individual = "58")
## Marginal variance covariance matrix
##         [,1]    [,2]    [,3]
## [1,] 1.05980 0.44320 0.41387
## [2,] 0.44320 0.92895 0.38554
## [3,] 0.41387 0.38554 0.64386
##   Standard Deviations: 1.0295 0.96382 0.80241

Two-level crossed model with distinguishable dyads

We’re estimating 21 parameters:

• 3 variances
  • Variance at pre
  • Variance at post
  • Variance at followup

• 3 within-dyad covariances at the same time point
  • Within-dyad covariance between pre ratings
  • Within-dyad covariance between post ratings
  • Within-dyad covariance between follow-up ratings

• 3 within-person covariances at different time points
  • Within-person covariance between pre and post
  • Within-person covariance between pre and follow-up
  • Within-person covariance between post and follow-up

• 3 within-dyad covariances at different time points
  • Within-dyad covariance between pre and post
  • Within-dyad covariance between pre and follow-up
  • Within-dyad covariance between post and follow-up
model <- gls(IH ~ timecontrast1_pre_post + timecontrast2_pre_followup + party + 
               party*timecontrast1_pre_post +
                party*timecontrast2_pre_followup,
             data = rm_df,
             correlation = corSymm(form = ~ RM | dyad),
             weights=varIdent(form=~1|RM),
             method = "REML",
             na.action = na.omit)

# omnibus test
anova(model, type = "marginal")
## Denom. DF: 2148 
##                                  numDF   F-value p-value
## (Intercept)                          1 25041.583  <.0001
## timecontrast1_pre_post               1   115.434  <.0001
## timecontrast2_pre_followup           1    67.907  <.0001
## party                                1     0.072  0.7878
## timecontrast1_pre_post:party         1     0.367  0.5446
## timecontrast2_pre_followup:party     1     0.085  0.7702

SAS Omnibus output

SAS Covariance output

What can we learn from this?

Your turn!

Contrast coding

DATASET 4: Andy Field’s Viagra Data

v_df <- read.delim("/Users/kareenadelrosario/Desktop/everything/2_areas/spring 2024/person perception/Viagra.dat")
head(v_df)
##   person dose libido
## 1      1    1      3
## 2      2    1      2
## 3      3    1      1
## 4      4    1      1
## 5      5    1      4
## 6      6    2      5
  1. Use contrast coding so that the intercept represents the average libido across all participants.
  2. Code the variables so that the intercept represents the average libido for the control group.
  3. Code the variables so that the first contrast compares the treatment group (Low + High) to the control group and the second contrast compares the control to the low dose group.

Centering & Two-Level Crossed Model (with dyads!)

Now, we’ll follow the same steps with the Roommate data we’ve been working with. Use the roommate df to test whether people are more comfortable expressing themselves (pexpress) are happier (phappy).

Here are some other variables you’ll need dayR Dyad RM partnum

  1. Are the data nonindependent? (do we actually need to use MLM)
  2. Are people who are more comfortable expressing themselves happier?
  3. Are people happier if they feel more comfortable expressing themselves than they typically do?

DATASET 5: More Roommate Data

Note this is a different file than what we’ve been using.

library(haven)

roommate_df <- read_sav("/Users/kareenadelrosario/Desktop/everything/2_areas/spring 2024/person perception/sample data_1.sav")

In this analysis, our grouping/subject variable is “Dyad” and our repeated variable is “RM”. For reference, here is the SPSS syntax:

MIXED DV WITH IV 
 /FIXED = IV 
 /PRINT = SOLUTION TESTCOV
 /REPEATED =RM | SUBJECT (Dyad) COVTYPE (UNR).

For R, the equivalent to COVTYPE (UNR) is correlation = corSymm.

---
title: "Person Perception: Centering and contrast coding for MLM"
author: "Kareena del Rosario"
date: "`r Sys.Date()`"
output: 
  html_document:
    theme: readable
    highlight: arrow
    toc: true
    toc_float: true
    collapsed: true
    code_download: true
---


```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)

library(readr)
library(tidyr) 
library(tidyverse)
library(nlme)
library(haven)
```


To demonstrate centering, contrast coding, and two-level crossed models, we're going to use a few different datasets. 

Please download the following datasets from my github:

  1. sample data_two_level (1).sav
  2. lab5_anova_contrasts.sav
  3. DyadicLongitudinal_ExampleDataset_RM3.csv
  4. ViagraCovariate.dat
  5. sample data_week4.sav


---------------

# Continuous predictors: centering


Let's say you're interested in the effect of perceived social support on anxiety. Anxiety can vary both between- and within-person. The question is, are you interested in the within-person effect (how does receiving more social support than I typically do influence my anxiety?) or between-person (how anxious are people who received more social support on average?). We can actually look at both by centering our predictors in different ways. 

- **uncentered (raw)**
- **centering on a sample value (same for everyone):** typically, grand-mean centered (centered on sample mean), but can center on another relevant value (e.g., centering on the first timepoint for the “time” variable).
- **person-level centering (different for everyone)**: typically, person-mean centering (subtracting each individual’s mean from their own scores), but could centering on another person-level variable (e.g., first Sunday for each person when participants start daily
diaries on different days of the week). Only relevant for time-varying variables


Centering on a sample value (same for everyone) 
    
  1. Fixed intercept reflects whatever you set 0 to be for the whole sample
  2. Fixed regression estimates (fixed slopes) are identical to uncentered

Person-level centering (different for everyone)
    
  1. Fixed intercept reflects whatever 0 is for each participant (e.g., each participant’s mean)
  2. Fixed regression estimates (fixed slopes) will be different
      - expected change in outcome corresponding to a 1 unit change in a predictor relative to each participant’s centered value (e.g., their own average)

**Example: Amie slept 6 hours last night, but she usually sleeps 9 hours. Kate slept 6 hours last night and she usually sleeps 6 hours. When person-mean centering, 6 hours for Amie is a deficit of 3 hours but average (0) for Kate.**

    With more complex data structures:
    • You can unconfound effects for any random factor
    • Example: ESM data with momentary assessments nested within days within people
      • Within-day and between-day effects
        • Times when I am more angry than usual that day
        • Days when I am more angry than I usually am
      • Within-person and between-person effects (You could do this across all momentary assessments, ignoring day)
        • Times I am more angry than I usually am across the whole study
        • People who are more angry than the average person in the sample

    OR you could have all 3 levels:
    • Each momentary assessment deviation from that day’s mean
    • Each day’s mean from the person’s overall mean
    • Person’s overall mean
      • This provides all three ways of looking at the data:
        • Times when I am more angry than usual that day
        • Days when I am more angry than I usual am
        • People who are more angry than the average person in the sample
        
## Creating centered variables


*DATASET 1: Revisiting the roommate data*

```{r}
df_roommate<-read_sav("/Users/kareenadelrosario/Desktop/everything/2_areas/spring 2024/person perception/sample data_two_level.sav")
```

```{r}
head(df_roommate)
```

#### Centering roommate messiness in roommate data

```{r}
df_mess <- df_roommate %>% 
  mutate(proom_mess_GMC = proom_mess - mean(proom_mess, na.rm = TRUE)) %>% # grand mean centered
  group_by(Participant_ID) %>% 
  mutate(proom_mess_PC = proom_mess - mean(proom_mess, na.rm = TRUE)) %>% # person centered
  ungroup() %>% 
  mutate(proom_mess_BPC = proom_mess_GMC - proom_mess_PC)


# dayR must be an integer
df_mess$dayR_int <- as.integer(as.factor(df_mess$dayR))
```

```{r}
model_mess <- gls(awish ~ proom_mess_GMC + proom_mess_PC,
             data = df_mess,
             correlation = corSymm(form = ~ dayR_int | Participant_ID),
             weights=varIdent(form=~1|dayR_int),
             method = "REML",
             na.action = na.omit)

summary(model_mess)
```


### What's the difference?

# Categorical predictors: contrast coding

To demonstrate contrast coding, we're going to use a different dataset


*DATASET 2: Sadness by 3-level condition and gender*

```{r}
df<- read_sav("/Users/kareenadelrosario/Desktop/local r code/regression_code/lab5_anova_contrasts.sav")
```


Predictors:

- condition_3Level: 1, 2, 3
- genderR: -1, 1

### Change variable class
```{r}
df.short <- df %>% 
  mutate(condition_3Level = as.factor(condition_3Level),
         genderR = as.factor(genderR))

# check class
class(df.short$condition_3Level)
class(df.short$genderR)
```

### Dummy coding

This is the most common form of contrast coding. In dummy coding, one level of the categorical variable is chosen as a reference group. For example, if you have a variable "Race" with three levels (White, Black, Asian), you can create two dummy variables: the effect of being Black and the effect of being Asian. The reference group (White) gets coded as 0 in both new variables. The intercept represents the mean outcome for the reference group. 

White = 1
Black = 2
Asian = 3


| Race  | Dummy1 | Dummy2 |
|-------|--------|--------|
| White | 0      | 0      |
| Black | 1      | 0      |
| Asian | 0      | 1      |

| Dummy 1: outcome for Black participants vs reference (White)
| Dummy 2: outcome for Asian participants vs reference (White)
| Intercept = average sleep quality when predictor is zero (sleep for White participants)

### Effects coding

Unlike dummy coding, where the reference category is represented by all 0's, effects coding doesn't leave out one category as the reference. Instead, categories are coded with numbers that balance out, such that the sum of codes for each categorical level across all coded variables equals zero. This balance allows the model's intercept to represent the grand mean. You would use this coding scheme if you wanted to compare the mean of each group to the grand mean. The reference group gets -1 for both effects.

The sum of your contrasts should = 0.

| Race  |Effect1 |Effect2 |
|:------|:-------|:-------|
| White | 1      | 0      |
| Black | 0      | 1      |
| Asian | -1     | -1     |


| Effect 1: compares the effect of White to the overall mean
| Effect 2: compares the effect of Black to the overall mean
| Intercept = grand mean across all categories of race 


**tldr; Dummy coding gives us the "simple effects," comparing different levels to a *reference group* while effects coding gives us the "main effects," comparing different levels to the *group mean* **

## How are our variables coded?

```{r}
contrasts(df.short$genderR)
```

**Identifiers:**

* -1 = Female

* 1 = Male

<br>

**Contrasts:**

* 0 = Female

* 1 = Male

**How does this impact our interpretation of the intercept?**

* Average sadness after the interaction for women (when genderR = 0)

```{r}
contrasts(df.short$condition_3Level)
```


**Identifiers:**

* 1 = Control dyad

* 2 = Sad actor

* 3 = Sad partner

<br>

**Contrasts:**

FIRST EFFECT: what is the effect of sad actor vs control condition?

SECOND EFFECT: what is the effect of sad partner vs control condition?

<br>

## Contrast coding in R

```{r}
# effect-code
contr.sum(2)
contr.sum(3)
```

```{r}
# dummy code
contr.treatment(2)
contr.treatment(3)
```

### Apply contrasts (effect code)

```{r}
contrasts(df.short$genderR) <- contr.sum(2)
contrasts(df.short$condition_3Level) <- contr.sum(3)

# alternative: custom effects coding
effectContrasts <- cbind(c(1, 0, -1), c(0, 1, -1))
```

```{r}
model_shortcut <- lm(avg_sad_int ~ genderR*condition_3Level, data = df.short)

summary(model_shortcut)
```

### Apply contrasts (dummy code)

```{r}
contrasts(df.short$genderR) <- contr.treatment(2)
contrasts(df.short$condition_3Level) <- contr.treatment(3)

# alternative: custom dummy coding
dummyContrasts <- cbind(c(0,1,0), c(0, 0, 1))
```

```{r}
model_shortcut2 <- lm(avg_sad_int ~ genderR*condition_3Level, data = df.short)

summary(model_shortcut2)
```

----------------

### Summary of intercepts


The intercept represents the dependent variable when all predictors are zero

- *raw (no centering):* 0 here can vary quite a bit depending on what your predictors are. It may not be meaningful.

- *dummy-coding:* 0 is the reference group. The intercept is the average DV for the reference group.

- *effects-coding:* 0 is the grand mean. The intercept is the DV when all predictors are averaged. Let's say we coded women -1 and men 1. The intercept would be the DV regardless of gender (averaging across gender).

- *grand-mean centering:* 0 is the grand mean. The intercept is the expected DV at the average level of the predictors. 

- *person-mean centering:* 0 is the person's average for that predictor. So, the intercept would be 'the expected level of anxiety (DV) when you sleep your average amount'


------

# Two-level crossed model

*DATASET 3: Intellectual humility measured 3x courtesy of Kate Thorson & Amie Gordon*

```{r, message=FALSE}
# load in our example data
rm_df <- read.csv("/Users/kareenadelrosario/Downloads/DyadicLongitudinal_ExampleDataset_RM3.csv")
```

964 people reported on their intellectual humility (IH) at three time points: pre (before the first conversation), post (after the last conversation), and follow-up (one month after the last conversation). We are going to use these data to ask whether IH changes across the three time points.


**Here are the variables we're interested in:** 

- **IH**: intellectual humility, reported at three time points
- **time**: time, pre = 1; post = 2; follow-up = 3
- **id**: unique ID number for each person
- **timecontrast1_pre_post**: pre = 0, post = 1, follow-up = 0
- **timecontrast2_pre_followup**: pre = 0, post = 0, follow-up = 1

<span style="color:blue">TIME</span>
• 1: pre
• 2: post
• 3: follow-up


Our predictors are **timecontrast1_pre_post**, **timecontrast2_pre_followup**, and **time.**

* Predictors: 
  + timecontrast1_pre_post, 
  + timecontrast2_pre_followup, 
  + time, 
  + timecontrast1_pre_post x time, 
  + timecontrast2_pre_followup x time


* Error variance for time1, time2, time3

--------------------

### Analysis steps

Before we jump into the analysis, let's take a step back and go through some pre-analyses to figure out:

1. whether we need to account for nonindependence in the first place
2. build our model

------------------

## Intraclass Correlation Coefficient (ICC)

Sometimes used to "justify" nonindependent analyses. ICC indexes the similarity between time points for a particular outcome. Technically, it represents the proportion of the total variance in an outcome that is explained by mean differences between subjects. An ICC of 1 would mean that all of the variability in the DV is between subjects. In repeated self-report data, it's common to have ICCs between .2-.4.

Calculating an ICC:

1. Run an MLM with no predictors (i.e., a null model)
2. Use person as a random factor
3. Using the output, calculate the ICC as the following:

#### Run null model with nlme

```{r, message=FALSE, warning=FALSE}
library(haven) # read SPSS and SAS files
library(reghelper) # for ICC command


null_model_lme <- lme(IH ~ 1,
             data = rm_df,
             random = ~ 1 | id,
             method = "REML",
             na.action = na.omit)

summary(null_model_lme)

ICC(null_model_lme)
```

#### Run null model with lme4

```{r, message=FALSE, warning=FALSE}
library(lme4)

null_model_lmer <- lmer(IH ~ 1 + 
                    (1 |id), 
                  data=rm_df)

summary(null_model_lmer)
```

**Here's the ICC equation:**

$$ICC = \frac {\ between-dyad\ variance\ (random\ intercept\ variance)} {total\ variance\ (random \ intercept\ variance\ +\ residual\ variance)} $$
```{r}
id_variance <- 0.4194
residual <- 0.5623

# calculating ICC
id_variance / (id_variance + residual)
```

### What is our takeaway from the ICC?



--------------

## Two-level crossed model with individual repeated measures


This will be a marginal model– we’re going to adjust for nonindependence by providing a variance-covariance structure to the residuals. This approach is useful for repeated measure with 3 or 4 time points.


In other words, this will model the nonindependence with a **covariance** (adjust for nonindependence by correlating the residuals between dyad members. This allows for both positive and negative nonindependence). 


### Output

1. Fixed parameter estimates
2. **Error covariance**: are the two time points' errors related ot each other?
3. **Error variance**: is there significant variance that is unexplained by the other components in the model?

### Residual structure

**We're estimating 6 parameters:**

    • 3 variances
      • Variance at pre
      • Variance at post
      • Variance at followup
      
    • 3 within-person covariances at different time points
      • Within-person covariance between pre and post
      • Within-person covariance between pre and follow-up
      • Within-person covariance between post and follow-up


For reference, here is the SPSS syntax:


    MIXED IH WITH timecontrast1_pre_post timecontrast2_pre_followup
    /FIXED = timecontrast1_pre_post timecontrast2_pre_followup |
    SSTYPE(3)
    /METHOD = REML
    /PRINT = G TESTCOV SOLUTION
    /REPEATED time | subject(Participant_ID) COVTYPE(UN).
    
#### Weighted by day (each day has its own variance)

```{r}
# dayR must be an integer
rm_df$time <- as.integer(as.factor(rm_df$time))

model1 <- gls(IH ~ timecontrast1_pre_post + timecontrast2_pre_followup,
             data = rm_df,
             correlation = corSymm(form = ~ time | id), 
             weights=varIdent(form=~1|time),
             method = "REML",
             na.action = na.omit)

summary(model1)
```

```{r}
getVarCov(model1, individual = "58")
```


## Two-level crossed model with distinguishable dyads

**We're estimating 21 parameters:**

    • 3 variances
      • Variance at pre
      • Variance at post
      • Variance at followup
  
    • 3 within-dyad covariances at the same time point
      • Within-dyad covariance between pre ratings
      • Within-dyad covariance between post ratings
      • Within-dyad covariance between follow-up ratings
  
    • 3 within-person covariances at different time points
      • Within-person covariance between pre and post
      • Within-person covariance between pre and follow-up
      • Within-person covariance between post and follow-up
  
    • 3 within-dyad covariances at different time points
      • Within-dyad covariance between pre and post
      • Within-dyad covariance between pre and follow-up
      • Within-dyad covariance between post and follow-up

```{r}
model <- gls(IH ~ timecontrast1_pre_post + timecontrast2_pre_followup + party + 
               party*timecontrast1_pre_post +
                party*timecontrast2_pre_followup,
             data = rm_df,
             correlation = corSymm(form = ~ RM | dyad),
             weights=varIdent(form=~1|RM),
             method = "REML",
             na.action = na.omit)

# omnibus test
anova(model, type = "marginal")
```

#### SAS Omnibus output

```{r echo=FALSE, out.width='80%'}
knitr::include_graphics('pp2_omnibus.png')
```


#### Print covariance parameter estimates

```{r}
getVarCov(model)
getVarCov(model, individual = 2)
```

#### SAS Covariance output

```{r echo=FALSE, out.width='80%'}
knitr::include_graphics('pp2_covariance.png')
```

### What can we learn from this?



-------------

# Your turn!

## Contrast coding


*DATASET 4: Andy Field's Viagra Data*

```{r}
v_df <- read.delim("/Users/kareenadelrosario/Desktop/everything/2_areas/spring 2024/person perception/Viagra.dat")
```

```{r}
head(v_df)
```

1. Use contrast coding so that the intercept represents the average libido across all participants.
2. Code the variables so that the intercept represents the average libido for the control group.
3. Code the variables so that the first contrast compares the treatment group (Low + High) to the control group and the second contrast compares the control to the low dose group.

## Centering & Two-Level Crossed Model (with dyads!)

Now, we'll follow the same steps with the Roommate data we've been working with. Use the roommate df to test whether people are more comfortable expressing themselves (pexpress) are happier (phappy).

Here are some other variables you'll need
**dayR Dyad RM partnum** 

1. Are the data nonindependent? (do we actually need to use MLM)
2. Are people who are more comfortable expressing themselves happier?
3. Are people happier if they feel more comfortable expressing themselves than they typically do?


*DATASET 5: More Roommate Data*

Note this is a different file than what we've been using.

```{r, message = FALSE}
library(haven)

roommate_df <- read_sav("/Users/kareenadelrosario/Desktop/everything/2_areas/spring 2024/person perception/sample data_1.sav")
```

In this analysis, our grouping/subject variable is "Dyad" and our repeated variable is "RM". For reference, here is the SPSS syntax:

    MIXED DV WITH IV 
     /FIXED = IV 
     /PRINT = SOLUTION TESTCOV
     /REPEATED =RM | SUBJECT (Dyad) COVTYPE (UNR).


For R, the equivalent to COVTYPE (UNR) is correlation = corSymm. 


