What’s in a latent trait score?

Author

Affiliation

Magnus Johansson

RISE Research Institutes of Sweden

We frequently use questionnaires to measure latent variables based on multiple items/indicators, but the connection between item responses and a latent trait score is often unclear. Here, I will present a way to visualize item responses and their corresponding latent score and measurement uncertainty that hopefully makes what lies behind the latent score much easier to understand.

It should be noted that the term “latent trait score” is not entirely established. The intent is to refer to what in Item Response Theory and Rasch modeling is labelled “theta” - 𝜃 (also known as “person ability”, “person location”, or “person score”). For each participant, IRT/Rasch models make it possible to both estimate a latent trait score on an interval scale and a measurement error specific for each participant’s score. The measurement error depends on the item properties and is not a constant value that is the same for all participants/scores.

In classical test theory, latent trait scores may be estimated using factor analysis and used in structural equation models. However, I have never seen anyone “export” the actual latent trait scores for each participant in a dataset, based on a confirmatory factor analysis (CFA). What is usually done, based on classical test theory/CFA, is to use the raw ordinal sum score as a representation of the latent variable.

Jump ahead to Section 5 to see the visualization and skip the psychometric background.

1 Background

The Mental Health Continuum Short Form (Ryff, 1989; Ryff, 2014) will be used as an example. I have conducted a Rasch analysis and preliminary results indicate that some modifications are needed for the response categories, and some items need to be removed. This will be reported elsewhere in greater detail. Here, only a summary of the changes and the measurement properties of the final set of items will be reported.

The analysis makes use of my R package for Rasch psychometric analysis, which in turn depends on (and automatically loads) a lot of packages. See the RISEkbmRasch vignette and GitHub package repository for more details.

Code

library(RISEkbmRasch)
library(arrow)
library(ggdist)
library(ggpp)

### some commands exist in multiple packages, here we define preferred ones that are frequently used
select <- dplyr::select
count <- dplyr::count
recode <- car::recode
rename <- dplyr::rename

# import item information
itemlabels <- read_excel("/Volumes/magnuspjo/RegionUppsala/data/RegUaItemlabelsEng.xls", sheet = 1) %>% 
  filter(str_detect(itemnr,"mhc"))
itemresponses <- read_excel("/Volumes/magnuspjo/RegionUppsala/data/RegUaItemlabelsEng.xls", sheet = 3) %>% 
  rename(`How often during the past month did you feel...` = item)

# import recoded data
df.all <- read_parquet("/Volumes/magnuspjo/RegionUppsala/data/RegUaLHUdata.parquet")
df <- df.all

# recode response categories to numerics
df <- df %>% 
  mutate(across(starts_with("I1_"), ~ recode(.x,"'Aldrig'=0;
                    'En eller två gånger'=1;
                    '1 gång/vecka'=2;
                    '2-3 gånger/vecka'=3;
                    'Nästan dagligen'=4;
                    'Dagligen'=5",
                    as.factor = FALSE)
  ))

df <- df %>% 
  filter(inars %in% c(2019,2021)) %>% 
  mutate(Age = recode(arskurs,"1='13 yo';2='15 yo';3=' 17 yo';9=NA;99=NA", as.factor = TRUE),
         Gender = recode(kon,"99=NA;9=NA;2='Girls';1='Boys'", as.factor = T),
         ) %>% 
  filter(Age %in% c("15 yo","17 yo")) %>% 
  select(starts_with("I1_"),Age,Gender) %>% 
  na.omit()

# create variables for analysis of Differential Item Functioning
dif.age <- df$Age
dif.gender <- df$Gender
df$Age <- NULL
df$Gender <- NULL

# set variable names to match
names(df) <- itemlabels$itemnr

Questions in the MHC-SF are prefixed with How often during the past month did you feel…, and the items are listed below in Section 2 in the right-hand side margin.

Six response categories were used, as specified in the original MHC-SF: every day, almost every day, about 2 or 3 times a week, about once a week, once or twice, never and the Rasch analysis found them disordered for all items. This means that some response categories do not contribute meaningful differential information to the latent score, compared to an adjacent response category - the categories are too similar to the respondents. After merging never with once or twice, and about once a week with about 2 or 3 times a week, the disordering was resolved. However, there are still quite small distances between item category thresholds, as shown in the ICC plots. This leaves us with 4 response categories for each item.

The sample consists of 4243 respondents.

Code

for (i in itemlabels$itemnr) {
  df[[i]] <- recode(df[[i]],"1=0;2=1;3=1;4=2;5=3",
                    as.factor = FALSE)
}

Tileplot

Code

RItileplot(df)

ICC plots

Code

ermOut <- PCM(df)
plotICC(ermOut)

2 Rasch analysis summary

Five items were removed, mostly due to issues of multidimensionality and large residual correlations. Here, the analysis of remaining items is presented.

Code

removed.items <- c("mhc2","mhc3","mhc4","mhc8","mhc7")
df <- df %>% 
  select(!any_of(removed.items))

itemnr	item
mhc1	happy
mhc5	hat you belonged to a community
mhc6	that our society is becoming a better place for people like you
mhc9	that you liked most parts of your personality
mhc10	good at managing the responsibilities of your daily life
mhc11	that you had warm and trusting relationships with others
mhc12	that you had experiences that challenged you to grow and become a better person
mhc13	confident to think or express your own ideas and opinions
mhc14	that your life has a sense of direction or meaning to it

Item fit

Code

RIitemfitPCM2(df, 300, 32, 8)

	OutfitMSQ	InfitMSQ	OutfitZSTD	InfitZSTD
mhc1	1.025	0.996	0.416	0.069
mhc5	1.192	1.125	1.889	1.604
mhc6	1.336	1.228	1.757	2.222
mhc9	0.727	0.746	-3.542	-3.697
mhc10	0.974	0.95	-0.092	-0.562
mhc11	0.779	0.794	-2.439	-2.94
mhc12	0.947	0.927	-0.805	-1.026
mhc13	0.882	0.876	-1.466	-1.678
mhc14	0.693	0.704	-3.497	-3.964

PCA

Code

RIpcmPCA(df)

PCA of Rasch model residuals

Eigenvalues
1.51
1.27
1.20
1.13
1.09

Residual correlations

Code

RIresidcorr(df, cutoff = 0.2)

	mhc1	mhc5	mhc6	mhc9	mhc10	mhc11	mhc12	mhc13	mhc14
mhc1
mhc5	-0.02
mhc6	-0.11	-0.05
mhc9	-0.07	-0.23	-0.11
mhc10	-0.19	-0.22	-0.12	-0.07
mhc11	-0.04	0.01	-0.19	-0.13	-0.07
mhc12	-0.24	-0.26	-0.11	-0.15	-0.04	-0.08
mhc13	-0.16	-0.18	-0.17	0.01	-0.15	-0.21	-0.04
mhc14	-0.07	-0.24	-0.14	-0.04	-0.09	-0.16	-0.02	-0.04
Note:
Relative cut-off value (highlighted in red) is 0.084, which is 0.2 above the average correlation.

1st contrast loadings

Code

RIloadLoc(df)

Targeting

Code

# increase fig-height above as needed, if you have many items
RItargeting(df)

Item hierarchy

Code

RIitemHierarchy(df)

DIF

Interaction between age and gender.

Code

df.tree <- data.frame(matrix(ncol = 0, nrow = nrow(df)))
df.tree$difdata <- as.matrix(df)
df.tree$dif.gender <- dif.gender
df.tree$dif.age<- dif.age

pctree.out <- pctree(difdata ~ dif.gender + dif.age, data = df.tree)
plot(pctree.out)

Code

cutoff <- 0.5
diffig <- itempar(pctree.out) %>%
  as.data.frame() %>%
  t() %>%
  as.data.frame() %>%
  mutate(
    `Mean location` = rowMeans(.),
    StDev = rowSds(as.matrix(.))
  ) %>%
  rowwise() %>%
  mutate(MaxDiff = (max(c_across(c(1:(ncol(.) - 2))))) -
    min(c_across(c(1:(ncol(.) - 2))))) %>%
  ungroup() %>%
  mutate(across(where(is.numeric), round, 3)) %>%
  rownames_to_column(var = "Item") %>%
  mutate(Item = names(df)) %>%
  relocate(MaxDiff, .after = last_col())

  formattable(diffig,
    list(MaxDiff = formatter("span",
      style = ~ style(color = ifelse(MaxDiff < -cutoff,
        "red", ifelse(MaxDiff > cutoff, "red", "black")
      ))
    )
    ),
    table.attr = "class=\"table table-striped\" style=\"font-size: 15px; font-family: Lato\""
  )

Item	2	3	Mean location	StDev	MaxDiff
mhc1	-0.836	-0.770	-0.803	0.047	0.067
mhc5	-0.276	-0.390	-0.333	0.081	0.114
mhc6	1.774	2.021	1.897	0.175	0.247
mhc9	-0.148	0.109	-0.019	0.182	0.257
mhc10	-0.010	-0.275	-0.143	0.187	0.265
mhc11	-0.356	-0.718	-0.537	0.256	0.362
mhc12	0.238	0.103	0.170	0.096	0.135
mhc13	-0.299	-0.127	-0.213	0.122	0.172
mhc14	-0.086	0.047	-0.020	0.094	0.133

Only DIF between genders found, and none large enough to cause problems.

Reliability

Code

RItif(df)

Item fit shows some minor issues, but we’ll leave them as good enough for the primary goal of visualization. Do have a look at the item hierarchy and give it some thought?

3 Estimating latent scores

First, we need the item threshold locations to later estimate person scores.

Code

RIitemparams(df, "mhcItemParams.csv")

	Threshold 1	Threshold 2	Threshold 3	Item location
mhc1	-2.17	-0.38	1.19	-0.45
mhc5	-0.60	-0.01	0.66	0.02
mhc6	1.17	2.34	3.14	2.22
mhc9	-0.78	0.55	1.25	0.34
mhc10	-1.16	0.49	1.30	0.21
mhc11	-1.49	0.11	0.82	-0.19
mhc12	-0.81	1.06	1.32	0.52
mhc13	-1.00	0.53	0.91	0.15
mhc14	-0.57	0.64	0.92	0.33

Code

MHCitemParameters <- as.matrix(read.csv("mhcItemParams.csv"))

Actually, the function used to estimate person scores can automatically estimate the item parameters. But since this is just a practical use case, and you might be interested in applying the item parameters on your own dataset (since I cannot share this data), I thought it worth mentioning and displaying the parameters.

Code

df$MHCscore <- RIestThetas2(df, itemParams = MHCitemParameters, cpu = 8)

3.1 Plotting the distribution

Code

# for visualization, we add Gender back to the df
df$Gender <- dif.gender

ggplot(df,aes(x = MHCscore, y = Gender, fill = Gender)) +
  stat_slab(
    side = "right", show.legend = F,
    scale = 0.7, 
    aes(fill_ramp = after_stat(level)),
    .width = c(.50, .75, 1)
  ) +
  stat_summary(fun.data = "mean_cl_normal",show.legend = F, size = .4,
               position = position_dodge2nudge(x=.05,width = .8)) +
scale_fill_ramp_discrete(from='black', aesthetics = "fill_ramp") +
      scale_fill_viridis_d(begin = 0.4, direction = -1) +

  xlab("MHC-SF latent scores") +
  ylab("") +
  theme_minimal() +
  theme_rise() +
  theme(axis.text.y = element_text(size = 13))

4 Preparing visualization

Let’s use some sample values, based on distribution quintiles, as a starting point.

Code

quintiles <- quantile(df$MHCscore, probs = c(0.2,0.4,0.6,0.8))
quintiles

       20%        40%        60%        80% 
-0.5148720  0.1545279  0.7598396  1.6389371 

Now we find the mode responses for each item based on the respondents in each quintile.

First, lets make groupings based on the quintile cutoff values.

Code

df <- df %>% 
  mutate(Q_group = case_when(MHCscore < quintiles[1] ~ "0-20",
                             MHCscore >= quintiles[1] & MHCscore < quintiles[2] ~ "20-40",
                             MHCscore >= quintiles[2] & MHCscore < quintiles[3] ~ "40-60",
                             MHCscore >= quintiles[3] & MHCscore < quintiles[4] ~ "60-80",
                             MHCscore >= quintiles[4] ~ "80-100",
                             TRUE ~ NA)
         )

df %>% 
  count(Q_group) %>% 
  ggplot(aes(x = Q_group, y = n, fill = Q_group)) +
  geom_col() +
  scale_fill_viridis_d(begin = 0.4, direction = -1) +
  theme_minimal() +
  theme_rise()

Identifying mode response categories for each item for each group is the next step

Code

Mode <- function(x) {
  ux <- unique(x)
  ux[which.max(tabulate(match(x, ux)))]
}

# a df with one row per group and one column per item
modeResponses <- data.frame(matrix(ncol = 9, nrow = 5))
names(modeResponses) <- df %>% 
  select(starts_with("mhc")) %>% 
  select(!MHCscore) %>% 
  names()

for (i in names(modeResponses)) {
  q0 <- Mode(df %>% filter(Q_group == "0-20") %>% select(i) %>% na.omit() %>% pull())
  q1 <- Mode(df %>% filter(Q_group == "20-40") %>% select(i) %>% na.omit() %>% pull())
  q2 <- Mode(df %>% filter(Q_group == "40-60") %>% select(i) %>% na.omit() %>% pull())
  q3 <- Mode(df %>% filter(Q_group == "60-80") %>% select(i) %>% na.omit() %>% pull())
  q4 <- Mode(df %>% filter(Q_group == "80-100") %>% select(i) %>% na.omit() %>% pull())
  modeResponses[[i]] <- rbind(q0,q1,q2,q3,q4)
}

# transform to dataframe with numeric variables for later extraction as vectors
modeResponses <- modeResponses %>% 
  t() %>% 
  as.data.frame()

Estimating the latent score and SEM for each group’s mode response pattern.

Code

# sorry for the copy+paste code here...
avg1 <- thetaEst(MHCitemParameters, modeResponses$V1, model = "PCM", method = "WL")
sem1 <- semTheta(thEst = avg1, it = MHCitemParameters, x = modeResponses$V1, model = "PCM", method = "WL")
avg2 <- thetaEst(MHCitemParameters, modeResponses$V2, model = "PCM", method = "WL")
sem2 <- semTheta(thEst = avg2, it = MHCitemParameters, x = modeResponses$V2, model = "PCM", method = "WL")
avg3 <- thetaEst(MHCitemParameters, modeResponses$V3, model = "PCM", method = "WL")
sem3 <- semTheta(thEst = avg3, it = MHCitemParameters, x = modeResponses$V3, model = "PCM", method = "WL")
avg4 <- thetaEst(MHCitemParameters, modeResponses$V4, model = "PCM", method = "WL")
sem4 <- semTheta(thEst = avg4, it = MHCitemParameters, x = modeResponses$V4, model = "PCM", method = "WL")
avg5 <- thetaEst(MHCitemParameters, modeResponses$V5, model = "PCM", method = "WL")
sem5 <- semTheta(thEst = avg5, it = MHCitemParameters, x = modeResponses$V5, model = "PCM", method = "WL")

5 Visualization

Code

respTable <- function(qLevel) {
    # typical response patterns for different risk levels
    if (qLevel == "0-20") {
      p.resp <- modeResponses$V1 
    } else if (qLevel == "20-40") {
      p.resp <- modeResponses$V2 
    } else if (qLevel == "40-60") {
      p.resp <- modeResponses$V3 
    } else if (qLevel == "60-80") {
      p.resp <- modeResponses$V4 
    } else if (qLevel == "80-100") {
      p.resp <- modeResponses$V5 
    }

    itemresponses %>%
      formattable(.,
        align = c("r", "l", "c", "c", "c", "c", "c"), list(
          formattable::area(row = 1, col = 3 + p.resp[1]) ~ color_tile("lightblue", "lightpink"),
          formattable::area(row = 2, col = 3 + p.resp[2]) ~ color_tile("lightblue", "lightpink"),
          formattable::area(row = 3, col = 3 + p.resp[3]) ~ color_tile("lightblue", "lightpink"),
          formattable::area(row = 4, col = 3 + p.resp[4]) ~ color_tile("lightblue", "lightpink"),
          formattable::area(row = 5, col = 3 + p.resp[5]) ~ color_tile("lightblue", "lightpink"),
          formattable::area(row = 6, col = 3 + p.resp[6]) ~ color_tile("lightblue", "lightpink"),
          formattable::area(row = 7, col = 3 + p.resp[7]) ~ color_tile("lightblue", "lightpink"),
          formattable::area(row = 8, col = 3 + p.resp[8]) ~ color_tile("lightblue", "lightpink"),
          formattable::area(row = 9, col = 3 + p.resp[9]) ~ color_tile("lightblue", "lightpink")
        ),
        table.attr = 'class=\"table table-striped\" style="font-size: 15px; font-family: Lato"'
      )
}

respDist <- function(qLevel) {
  if (qLevel == "0-20") {
      score <- avg1
      sem <- sem1
    } else if (qLevel == "20-40") {
      score <- avg2
      sem <- sem2
    } else if (qLevel == "40-60") {
      score <- avg3
      sem <- sem3
    } else if (qLevel == "60-80") {
      score <- avg4
      sem <- sem4
    } else if (qLevel == "80-100") {
      score <- avg5
      sem <- sem5
    }

  qlabel <- qLevel
  ggplot(df, aes(x = MHCscore, y = Gender, fill = Gender)) +
  stat_slab(
    side = "right", show.legend = F,
    scale = 0.8, justification = 0,
    aes(fill_ramp = after_stat(level)),
    .width = c(.50, .75, 1)
    
  ) +
  stat_summary(
    fun.data = "mean_cl_normal", show.legend = F, size = .4,
    position = position_dodge2nudge(x = .05, width = .8)
  ) +
  scale_fill_ramp_discrete(from = "black", aesthetics = "fill_ramp") +
  scale_fill_viridis_d(begin = 0.4, direction = -1) +
  geom_vline(xintercept = score, color = "red", linetype = 2) +
  annotate("rect", ymin = 0, ymax = Inf, xmin = score - sem, xmax = score + sem, alpha = .15) +
  #annotate("text", label = qlabel, x = score-0.12, y = 1.5, 
  #         angle = 90, size = 4, family = "Lato") +
  xlab("MHC-SF latent scores") +
  ylab("") +
  theme_minimal() +
  theme_rise(axissize = 11) +
  theme(axis.text.y = element_text(size = 11)) +
  scale_y_discrete(expand = c(0,0))
}

0-20

Code

respTable("0-20")

itemnr	How often during the past month did you feel…	Category0	Category1	Category2	Category3
mhc1	happy	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc5	that you belonged to a community	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc6	that our society is becoming a better place for people like you	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc9	that you liked most parts of your personality	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc10	good at managing the responsibilities of your daily life	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc11	that you had warm and trusting relationships with others	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc12	that you had experiences that challenged you to grow and become a better person	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc13	confident to think or express your own ideas and opinions	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc14	that your life has a sense of direction or meaning to it	Never/once or twice	1-3 times a week	Almost every day	Every day

Code

respDist("0-20")

20-40

Code

respTable("20-40")

itemnr	How often during the past month did you feel…	Category0	Category1	Category2	Category3
mhc1	happy	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc5	that you belonged to a community	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc6	that our society is becoming a better place for people like you	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc9	that you liked most parts of your personality	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc10	good at managing the responsibilities of your daily life	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc11	that you had warm and trusting relationships with others	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc12	that you had experiences that challenged you to grow and become a better person	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc13	confident to think or express your own ideas and opinions	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc14	that your life has a sense of direction or meaning to it	Never/once or twice	1-3 times a week	Almost every day	Every day

Code

respDist("20-40")

40-60

Code

respTable("40-60")

itemnr	How often during the past month did you feel…	Category0	Category1	Category2	Category3
mhc1	happy	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc5	that you belonged to a community	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc6	that our society is becoming a better place for people like you	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc9	that you liked most parts of your personality	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc10	good at managing the responsibilities of your daily life	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc11	that you had warm and trusting relationships with others	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc12	that you had experiences that challenged you to grow and become a better person	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc13	confident to think or express your own ideas and opinions	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc14	that your life has a sense of direction or meaning to it	Never/once or twice	1-3 times a week	Almost every day	Every day

Code

respDist("40-60")

60-80

Code

respTable("60-80")

itemnr	How often during the past month did you feel…	Category0	Category1	Category2	Category3
mhc1	happy	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc5	that you belonged to a community	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc6	that our society is becoming a better place for people like you	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc9	that you liked most parts of your personality	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc10	good at managing the responsibilities of your daily life	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc11	that you had warm and trusting relationships with others	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc12	that you had experiences that challenged you to grow and become a better person	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc13	confident to think or express your own ideas and opinions	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc14	that your life has a sense of direction or meaning to it	Never/once or twice	1-3 times a week	Almost every day	Every day

Code

respDist("60-80")

80-100

Code

respTable("80-100")

itemnr	How often during the past month did you feel…	Category0	Category1	Category2	Category3
mhc1	happy	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc5	that you belonged to a community	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc6	that our society is becoming a better place for people like you	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc9	that you liked most parts of your personality	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc10	good at managing the responsibilities of your daily life	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc11	that you had warm and trusting relationships with others	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc12	that you had experiences that challenged you to grow and become a better person	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc13	confident to think or express your own ideas and opinions	Never/once or twice	1-3 times a week	Almost every day	Every day
mhc14	that your life has a sense of direction or meaning to it	Never/once or twice	1-3 times a week	Almost every day	Every day

Code

respDist("80-100")

All together

Code

ggplot(df, aes(x = MHCscore, y = Gender, fill = Gender)) +
  stat_slab(
    side = "right", show.legend = F,
    scale = 0.7,
    aes(fill_ramp = after_stat(level)),
    .width = c(.50, .75, 1)
  ) +
  stat_summary(
    fun.data = "mean_cl_normal", show.legend = F, size = .4,
    position = position_dodge2nudge(x = .05, width = .8)
  ) +
  scale_fill_ramp_discrete(from = "black", aesthetics = "fill_ramp") +
  scale_fill_viridis_d(begin = 0.4, direction = -1) +
  geom_vline(xintercept = avg1, color = "orange", linetype = 2) +
  annotate("rect", ymin = 0, ymax = Inf, xmin = avg1 - sem1, xmax = avg1 + sem1, alpha = .1) +
  annotate("text", label = "0-20", x = avg1-0.1, y = 1.5, angle = 90, size = 3) +
  geom_vline(xintercept = avg2, color = "orange", linetype = 2) +
  annotate("rect", ymin = 0, ymax = Inf, xmin = avg2 - sem2, xmax = avg2 + sem2, alpha = .1) +
  annotate("text", label = "20-40", x = avg2-0.1, y = 1.5, angle = 90, size = 3) +
  geom_vline(xintercept = avg3, color = "orange", linetype = 2) +
  annotate("rect", ymin = 0, ymax = Inf, xmin = avg3 - sem3, xmax = avg3 + sem3, alpha = .1) +
  annotate("text", label = "40-60", x = avg3-0.1, y = 1.5, angle = 90, size = 3) +
  geom_vline(xintercept = avg4, color = "orange", linetype = 2) +
  annotate("text", label = "60-80", x = avg4-0.1, y = 1.5, angle = 90, size = 3) +
  annotate("rect", ymin = 0, ymax = Inf, xmin = avg4 - sem4, xmax = avg4 + sem4, alpha = .1) +
  geom_vline(xintercept = avg5, color = "orange", linetype = 2) +
  annotate("text", label = "80-100", x = avg5-0.1, y = 1.5, angle = 90, size = 3) +
  annotate("rect", ymin = 0, ymax = Inf, xmin = avg5 - sem5, xmax = avg5 + sem5, alpha = .1) +
  xlab("MHC-SF latent scores") +
  ylab("") +
  theme_minimal() +
  theme_rise() +
  theme(axis.text.y = element_text(size = 12))

6 Other visualizations

We can use the same set of scores and combine them with the previously reported figure for item hierarchy.

6.1 Targeting example

Code

RIitemHierarchy(df[,c(1:9)]) +
  geom_hline(aes(yintercept = avg4), linetype = 2, color = "red") +
  annotate("rect", xmin = 0, xmax = Inf, ymin = avg4 - sem4, ymax = avg4 + sem4, alpha = .1) +
  annotate("text", label = "60-80", y = avg4-0.1, x = 9, angle = 90, size = 3, color = "darkorange") +
  geom_hline(aes(yintercept = avg2), linetype = 2, color = "red") +
  annotate("rect", xmin = 0, xmax = Inf, ymin = avg2 - sem2, ymax = avg2 + sem2, alpha = .1) +
  annotate("text", label = "20-40", y = avg2-0.1, x = 9, angle = 90, size = 3, color = "darkorange") 

7 Software used

Code

library(grateful)
pkgs <- cite_packages(cite.tidyverse = TRUE, 
                      output = "table",
                      bib.file = "grateful-refs.bib",
                      include.RStudio = TRUE)
formattable(pkgs, 
            table.attr = 'class=\"table table-striped\" style="font-size: 14px; font-family: Lato; width: 80%"')

Package	Version	Citation
arrow	10.0.0	Richardson et al. (2022)
base	4.2.2	R Core Team (2022)
car	3.1.1	Fox & Weisberg (2019)
formattable	0.2.1	Ren & Russell (2021)
ggdist	3.2.0	Kay (2022)
ggpp	0.4.5	Aphalo (2022)
grateful	0.1.11	Rodríguez-Sánchez et al. (2022)
RISEkbmRasch	0.1.9	Johansson (2023)
rmarkdown	2.20	Xie et al. (2018); Xie et al. (2020); Allaire et al. (2023)
tidyverse	1.3.2	Wickham et al. (2019)

8 References

Allaire, J., Xie, Y., McPherson, J., Luraschi, J., Ushey, K., Atkins, A., Wickham, H., Cheng, J., Chang, W., & Iannone, R. (2023). Rmarkdown: Dynamic documents for r. https://github.com/rstudio/rmarkdown

Aphalo, P. J. (2022). Ggpp: Grammar extensions to ’ggplot2’. https://CRAN.R-project.org/package=ggpp

Fox, J., & Weisberg, S. (2019). An R companion to applied regression (Third). Sage. https://socialsciences.mcmaster.ca/jfox/Books/Companion/

Johansson, M. (2023). RISEkbmRasch: Psychometric analysis in r with rasch measurement theory. https://github.com/pgmj/RISEkbmRasch

Kay, M. (2022). ggdist: Visualizations of distributions and uncertainty. https://doi.org/10.5281/zenodo.3879620

R Core Team. (2022). R: A language and environment for statistical computing. R Foundation for Statistical Computing. https://www.R-project.org/

Ren, K., & Russell, K. (2021). Formattable: Create ’formattable’ data structures. https://CRAN.R-project.org/package=formattable

Richardson, N., Cook, I., Crane, N., Dunnington, D., François, R., Keane, J., Moldovan-Grünfeld, D., Ooms, J., & Apache Arrow. (2022). Arrow: Integration to ’apache’ ’arrow’. https://CRAN.R-project.org/package=arrow

Rodríguez-Sánchez, F., Jackson, C. P., & Hutchins, S. D. (2022). Grateful: Facilitate citation of r packages. https://github.com/Pakillo/grateful

Ryff, C. D. (1989). Happiness Is Everything, or Is It? Explorations on the Meaning of Psychological Well-Being. Journal of Personality and Social Psychology, 57(6), 1069–1081. https://doi.org/http://dx.doi.org/10.1037/0022-3514.57.6.1069

Ryff, C. D. (2014). Psychological Well-Being Revisited: Advances in the Science and Practice of Eudaimonia. Psychotherapy and Psychosomatics, 83(1), 10–28. https://doi.org/10.1159/000353263

Wickham, H., Averick, M., Bryan, J., Chang, W., McGowan, L. D., François, R., Grolemund, G., Hayes, A., Henry, L., Hester, J., Kuhn, M., Pedersen, T. L., Miller, E., Bache, S. M., Müller, K., Ooms, J., Robinson, D., Seidel, D. P., Spinu, V., … Yutani, H. (2019). Welcome to the tidyverse. Journal of Open Source Software, 4(43), 1686. https://doi.org/10.21105/joss.01686

Xie, Y., Allaire, J. J., & Grolemund, G. (2018). R markdown: The definitive guide. Chapman; Hall/CRC. https://bookdown.org/yihui/rmarkdown

Xie, Y., Dervieux, C., & Riederer, E. (2020). R markdown cookbook. Chapman; Hall/CRC. https://bookdown.org/yihui/rmarkdown-cookbook

Reuse

https://creativecommons.org/licenses/by/4.0/

Citation

BibTeX citation:

@online{johansson2023,
  author = {Magnus Johansson},
  title = {What’s in a Latent Trait Score?},
  date = {2023-02-24},
  url = {https://pgmj.github.io/LatentResponse.html},
  langid = {en}
}

For attribution, please cite this work as:

Magnus Johansson. (2023, February 24). What’s in a latent trait score? https://pgmj.github.io/LatentResponse.html