Overview

This tutorial illustrates calculation of intraindividual variability metrics. Specifically, this tutorial demonstrates calculation of a few univariate metrics, a few bivariate metrics, and a few multivariate metrics - as well as how these metrics are related to other individual differences.

Outline

A. Univariate IIV Metrics

iCount, iMean, iSD, iSkew, iKurtosis, iEntropy

B. Bivariate IIV Metrics

iCor, iCov, iReg

C. Multivariate IIV Metrics

Determinant of a matrix, Frobenius Norm

D. Selected Readings

Benson, L., Ram, N., Almeida, D. M., Zautra, A. J., & Ong, A. D. (2017). Fusing biodiversity metrics into investigations of daily life: Illustrations and recommendations with emodiversity. The Journals of Gerontology: Series B, 73(1), 75-86.
Deboeck, P. R., Montpetit, M. A., Bergeman, C. S., & Boker, S. M. (2009). Using derivative estimates to describe intraindividual variability at multiple time scales. Psychological methods, 14(4), 367.
Jahng, S., Wood, P. K., & Trull, T. J. (2008). Analysis of affective instability in ecological momentary assessment: Indices using successive difference and group comparison via multilevel modeling. Psychological methods, 13(4), 354.
Ram, N., Benson, L., Brick, T. R., Conroy, D. E., & Pincus, A. L. (2017). Behavioral landscapes and earth mover’s distance: A new approach for studying individual differences in density distributions. Journal of research in personality, 69, 191-205.
Ram, N., & Gerstorf, D. (2009). Time-structured and net intraindividual variability: tools for examining the development of dynamic characteristics and processes. Psychology and aging, 24(4), 778.

Preliminaries

Loading Libraries

Loading libraries used in this script.

library(psych) # for describing the data
library(plyr) #for data manipulation
library(ggplot2) # for data visualization
library(entropy) #for entropy calculation
library(nFactors) #factor analysis

Loading Data

Our example makes use of the person-level and interaction-level (EMA-type) AMIB data files. We make use of person-level personality variables and a small set of interaction-level variables.

Loading person-level file and subsetting to variables of interest

#set filepath for data file
filepath <- "https://quantdev.ssri.psu.edu/sites/qdev/files/AMIBshare_persons_2019_0501.csv"
#read in the .csv file using the url() function
AMIB_persons <- read.csv(file=url(filepath),header=TRUE)

#subsetting to variables of interest
AMIB_persons <- AMIB_persons[ ,c("id","bfi_o","bfi_c","bfi_e","bfi_a","bfi_n")]

Loading day-level file (T = 21) and subsetting to variables of interest.

#set filepath for data file
filepath <- "https://quantdev.ssri.psu.edu/sites/qdev/files/AMIBshare_phase2_daily_2019_0501.csv"
#read in the .csv file using the url() function
AMIB_daily <- read.csv(file=url(filepath),header=TRUE)

1. Univariate IIV Metrics

There are several types of univariate IIV metrics that differ based on the type of data (continuous, binary, count), and conceptual meanings. For example, the iMean, iSD, iSkew, and iKurtosis are the first four distribution moments, utilize continuous data, and are unstructured with respect to time. iMSSD and iPAC also utilize continuous data, but do acount for time dependencies in the data. iEntropy and iTurbulence pertain to categorical data.

We illustrate just a few. The general principle is to use the functions in the plyr package to split the data by “id” and summarize each person’s data using a specifci fucntion (e.g., mean, sd). We will use this same approach when preparing data in later analyses.

Calculating univariate IIV metrics.

istats_univariate <- ddply(AMIB_daily, "id", summarize,
                            icount_posaff = sum(!is.na(posaff)),  #count of observations
                            imean_posaff  = mean(posaff, na.rm=TRUE), #imean (continuous)
                            isd_posaff    = sd(posaff, na.rm=TRUE),   #isd   (continuous)
                            iskew_posaff  = skew(posaff, na.rm=TRUE), #iskew (continuous)
                            ikurt_posaff  = kurtosi(posaff, na.rm=TRUE), # ikurtosis (continuous)
                            ientropy_stress= entropy(table(stress,
                                                             useNA="no")))  #(categorical)

Look at univariate IIV metrics data

round(head(istats_univariate),2)

##    id icount_posaff imean_posaff isd_posaff iskew_posaff ikurt_posaff
## 1 203            22         4.42       0.84        -0.70         0.13
## 2 204            22         3.10       1.17        -0.58        -1.19
## 3 205            22         4.32       0.68        -0.73        -0.53
## 4 208            22         3.92       0.67         0.11        -0.86
## 5 211            22         2.81       0.85         0.70        -0.15
## 6 214            15         4.16       1.01        -0.46        -0.82
##   ientropy_stress
## 1            1.00
## 2            1.49
## 3            1.67
## 4            1.02
## 5            1.36
## 6            1.90

Describe the bivariate IIV metrics

describe(istats_univariate[-1])

##                 vars  n  mean   sd median trimmed  mad   min   max range
## icount_posaff      1 30 20.90 2.58  22.00   21.58 0.00 14.00 22.00  8.00
## imean_posaff       2 30  3.78 0.71   3.91    3.76 0.56  2.61  5.62  3.01
## isd_posaff         3 30  0.88 0.26   0.87    0.87 0.26  0.32  1.52  1.20
## iskew_posaff       4 30 -0.15 0.46  -0.24   -0.19 0.46 -0.74  0.97  1.71
## ikurt_posaff       5 30 -0.79 0.50  -0.88   -0.84 0.47 -1.43  0.58  2.01
## ientropy_stress    6 30  1.47 0.28   1.50    1.48 0.28  0.84  1.90  1.06
##                  skew kurtosis   se
## icount_posaff   -2.02     2.28 0.47
## imean_posaff     0.29    -0.17 0.13
## isd_posaff       0.17    -0.19 0.05
## iskew_posaff     0.69    -0.45 0.08
## ikurt_posaff     0.94     0.01 0.09
## ientropy_stress -0.39    -0.77 0.05

Examine correlations

round(cor(istats_univariate[ ,-1], use="pairwise.complete.obs"),2)

##                 icount_posaff imean_posaff isd_posaff iskew_posaff
## icount_posaff            1.00         0.08      -0.32         0.16
## imean_posaff             0.08         1.00      -0.45        -0.38
## isd_posaff              -0.32        -0.45       1.00         0.21
## iskew_posaff             0.16        -0.38       0.21         1.00
## ikurt_posaff             0.24         0.14      -0.07         0.29
## ientropy_stress         -0.06        -0.24       0.29        -0.21
##                 ikurt_posaff ientropy_stress
## icount_posaff           0.24           -0.06
## imean_posaff            0.14           -0.24
## isd_posaff             -0.07            0.29
## iskew_posaff            0.29           -0.21
## ikurt_posaff            1.00           -0.11
## ientropy_stress        -0.11            1.00

Merge the univariate IIV metrics with other person-level data.

AMIB_univariate <- merge(istats_univariate, AMIB_persons, by="id")

Examine relation between positive affect variability and neuroticism.

Look at association between isd_posaff and bfi_n:

#pdf("UnivariateIIV.pdf",height=5.75, width=9)
ggplot(data=AMIB_univariate, aes(x=bfi_n, y=isd_posaff)) +
  geom_point(colour="gray40") +
  geom_smooth(aes(group=1), method=lm, se=TRUE, fullrange=TRUE, lty=1, size=2, color="#9A23DE") +
  scale_y_continuous(breaks=c(0.4, 0.8, 1.2, 1.6), limits=c(0.32,1.6)) +
  xlab("Neuroticism") + ylab("Positive Affect Variability") +
  theme_classic() +
  theme(axis.title=element_text(size=16),
        axis.text=element_text(size=12),
        plot.title=element_text(size=16, hjust=.5, face="bold")) +
  ggtitle("Between-Person Association Univariate IIV")

## Warning: Removed 1 rows containing non-finite values (stat_smooth).

## Warning: Removed 1 rows containing missing values (geom_point).

#dev.off()

And the formal model test (cross-sectional regression).

reg_univariate <- lm(isd_posaff ~ bfi_n, 
                     data=AMIB_univariate,
                     na.action = na.exclude) 
summary(reg_univariate)

## 
## Call:
## lm(formula = isd_posaff ~ bfi_n, data = AMIB_univariate, na.action = na.exclude)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.49597 -0.14662 -0.00169  0.18189  0.49217 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.59383    0.15329   3.874 0.000589 ***
## bfi_n        0.08654    0.04499   1.923 0.064654 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.252 on 28 degrees of freedom
## Multiple R-squared:  0.1167, Adjusted R-squared:  0.08516 
## F-statistic: 3.699 on 1 and 28 DF,  p-value: 0.06465

For the prototypical person, positive affect variability is expected to be 0.59
Individuals who are higher on neuroticism tend to have more variability in their positive affect. For every unit increase in neuroticism, characteristic positive affect is expected to increase by 0.09 (however, this association is not statistically significantly different from zero p = 0.06).

B. Bivariate IIV Metrics

There are are also several types of bivariate IIV metrics, including the iCorr, iCov, iPulse, iSpin, and iEntropy. The slope parameter from a within-person regression can, along with the iCorr and iCov also be considered a measure of intraindividual covariation.

We illustrate just a few of these. Again, the general principle is to use the functions in the plyr package to split the data by “id” and summarize each person’s data using a specific fucntion (e.g., cor, lm).

Calculate intraindiviudal correlation (iCor) and intraindividual covariance (iCov) and intraindividual regression (iReg):

istats_bivariate <- ddply(AMIB_daily, "id", summarize, 
                        icor_affect = cor(x=posaff,y=negaff,           
                                         use="pairwise.complete.obs", method="pearson"),
                        icov_affect = cov(x=posaff,y=negaff,          
                                         use="pairwise.complete.obs", method="pearson"),
                        ireg_affect = coef(lm(posaff ~ negaff, na.action=na.exclude))[2])

Look at bivariate IIV metrics data

round(head(istats_bivariate),2)

##    id icor_affect icov_affect ireg_affect
## 1 203       -0.63       -0.29       -0.98
## 2 204       -0.77       -1.18       -0.70
## 3 205       -0.44       -0.17       -0.53
## 4 208       -0.35       -0.06       -0.88
## 5 211       -0.08       -0.02       -0.17
## 6 214       -0.85       -0.83       -0.89

Describe the bivariate IIV metrics

describe(istats_bivariate[ ,-1])

##             vars  n  mean   sd median trimmed  mad   min  max range  skew
## icor_affect    1 30 -0.49 0.27  -0.54   -0.50 0.32 -0.90 0.01  0.91  0.30
## icov_affect    2 30 -0.39 0.41  -0.25   -0.32 0.31 -1.73 0.01  1.74 -1.61
## ireg_affect    3 30 -0.58 0.35  -0.62   -0.58 0.43 -1.21 0.03  1.24  0.03
##             kurtosis   se
## icor_affect    -1.24 0.05
## icov_affect     2.28 0.07
## ireg_affect    -1.20 0.06

Correlations among the bivariate IIV metrics

round(cor(istats_bivariate[ ,-1], use="pairwise.complete.obs"),2)

##             icor_affect icov_affect ireg_affect
## icor_affect        1.00        0.73        0.84
## icov_affect        0.73        1.00        0.51
## ireg_affect        0.84        0.51        1.00

Note the overlap between the iCor, iCov, and iReg metrics. This will be relevant when extending the iReg into the multilevel modeling framework - to examine between-person differences in intraindividual covariation.

Merge the bivariate IIV metrics with other person-level data.

AMIB_bivariate <- merge(istats_bivariate, AMIB_persons, by="id")

Examine relation between Affect Bipolarity and neuroticism.

Look at association between isd_posaff and bfi_e:

#pdf("BivariateIIV.pdf",height=3, width=9)
ggplot(data=AMIB_bivariate, aes(x=bfi_n, y=icor_affect)) +
  geom_point(colour="gray40") +
  geom_smooth(aes(group=1), method=lm, se=TRUE, fullrange=TRUE, lty=1, size=2, color="#9A23DE") +
  xlab("Neuroticism") + ylab("Affect Bipolarity") +
  theme_classic() +
  theme(axis.title=element_text(size=16),
        axis.text=element_text(size=12),
        plot.title=element_text(size=16, hjust=.5, face="bold")) +
  ggtitle("Between-Person Association Bivariate IIV")

#dev.off()

Fit model with the neuroticism predicting the individual slope score for the association between positive and negative affect.

reg_bivariate <- lm(icor_affect ~ bfi_n, 
                     data=AMIB_bivariate,
                     na.action = na.exclude) 
summary(reg_bivariate)

## 
## Call:
## lm(formula = icor_affect ~ bfi_n, data = AMIB_bivariate, na.action = na.exclude)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.42499 -0.20524 -0.05017  0.24066  0.51684 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept) -0.42619    0.16959  -2.513    0.018 *
## bfi_n       -0.01979    0.04978  -0.398    0.694  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2788 on 28 degrees of freedom
## Multiple R-squared:  0.005616,   Adjusted R-squared:  -0.0299 
## F-statistic: 0.1581 on 1 and 28 DF,  p-value: 0.6939

For the prototypical person, affect bipolarity score for the sample is -0.43
Individuals who are higher on neuroticism tend to have stronger affect bipolarity (scores further from zero). For every unit increase in neuroticism, affect bipolarity is expected to change by -0.02 (however, this association is not statistically significantly different from zero p = 0.69).

C. Multivariate IIV Metrics

Calculation of intraindividual variability metrics can be extended to multivariate time-series. Continuous measures can be summarized as a correlation/covariance matrix (and vector of means) - which in turn is summarized using metrics such as the determinant, Frobenius norm, or number of eigenvalues > 1 (Kaiser rule). Categorical measures can be summarized as a frequency table - which is, in turn, summarized using metrics such as Gini coefficient, richness, Shannon’s entropy, and Simpson’s index.

Calculating multivariate IIV metrics.

First, set the variables of interest (be sure the id variable is included in the first position)

vars <- c("id", "enthusiastic", "happy", "proud", "relaxed", "calm",
          "sluggish", "sad", "disappointed", "angry", "nervous")

Calculate the metrics. Each one takes a few steps

# remove ID 218 (due to zero variance in several items)
AMIB_daily_sub <- AMIB_daily[AMIB_daily$id!=218, ]

#calculating the determinant for each individual
istat_idet <- ddply(AMIB_daily_sub[ ,vars], "id",
                  function(x) {
                    det(cor(x[,-1],use="pairwise.complete.obs",
                            method="pearson"))} )
#renaming columns
colnames(istat_idet) <- c("id","idet")


#calculating Frobenius norm for each individual
istat_iFnorm <- ddply(AMIB_daily_sub[ ,vars], "id",
                  function(x) {
                    norm(cor(x[,-1],use="pairwise.complete.obs",
                             method="pearson"),type="F")} )
#renaming columns
colnames(istat_iFnorm) <- c("id","iFnorm")

Merging all together

istats_multivar <- merge(istat_idet,istat_iFnorm, by="id")

Look at multivariate IIV metrics data

round(head(istats_multivar),2)

##    id idet iFnorm
## 1 203 0.00   5.64
## 2 204 0.00   5.11
## 3 205 0.03   4.01
## 4 208 0.00   4.43
## 5 211 0.00   4.72
## 6 214 0.00   4.92

Describe the bivariate IIV metrics.

describe(istats_multivar[ ,-1])

##        vars  n mean   sd median trimmed  mad  min  max range skew kurtosis
## idet      1 29 0.01 0.01   0.00    0.00 0.00 0.00 0.04  0.04 2.00     2.94
## iFnorm    2 29 4.89 0.65   4.89    4.83 0.56 3.97 6.98  3.01 1.04     1.66
##          se
## idet   0.00
## iFnorm 0.12

Correlations among the bivariate IIV metrics

round(cor(istats_multivar[ ,-1], use="pairwise.complete.obs"),2)

##         idet iFnorm
## idet    1.00  -0.64
## iFnorm -0.64   1.00

Merge the multivariate IIV metrics with other person-level data.

AMIB_multivariate <- merge(istats_multivar, AMIB_persons, by="id")

Examine relation between emotion system variability and neuroticism.

Look at association between idet and bfi_n:

pdf("MultivariateIIV.pdf",height=3, width=5)
ggplot(data=AMIB_multivariate, aes(x=bfi_n, y=idet)) +
  geom_point(colour="gray40") +
  geom_smooth(aes(group=1), method=lm, se=TRUE, fullrange=TRUE, lty=1, size=2, color="#9A23DE") +
  xlab("Neuroticism") + ylab("Emotion System\nVariability") +
  theme_classic() +
  theme(axis.title=element_text(size=16),
        axis.text=element_text(size=12),
        plot.title=element_text(size=16, hjust=.5)) +
  ggtitle("Between-Person Association Multivariate IIV")
dev.off()

## quartz_off_screen 
##                 2

And the formal model test using (cross-sectional) regression.

reg_multivariate <- lm(idet ~ bfi_n, 
                     data=AMIB_multivariate,
                     na.action = na.exclude) 
summary(reg_multivariate)

## 
## Call:
## lm(formula = idet ~ bfi_n, data = AMIB_multivariate, na.action = na.exclude)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -0.0064492 -0.0046937 -0.0038566  0.0003153  0.0305352 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)
## (Intercept)  0.0082545  0.0064399   1.282    0.211
## bfi_n       -0.0009027  0.0018611  -0.485    0.632
## 
## Residual standard error: 0.009515 on 27 degrees of freedom
## Multiple R-squared:  0.008638,   Adjusted R-squared:  -0.02808 
## F-statistic: 0.2353 on 1 and 27 DF,  p-value: 0.6316

For the prototypical person, emotion system variability score for the sample is 0.01
Individuals who are higher on neuroticism tend to have lower emotion system variability. For every unit increase in neuroticism, emotion system variability is expected to change by 0 (however, this association is not statistically significantly different from zero p = 0.63).

Conclusion

This tutorial illustrated a general approach for calculting intraindividual variability metrics for univariate, bivariate, and multivariate data - metrics that can be used as indicators of specific constructs (e.g., lability, complexity, etc.).

We hope this foundational material prompts consideration of a larger set of intraindivdual variability constructs and the metrics used to operationalize those constructs.

Have fun exploring!

Intraindividual Variability Metrics Computation