# A brief manual

#### Joshua N. Pritikin

# Overview

The pcFactorStan package for R provides convenience functions and pre-programmed Stan models related to analysis of paired comparison data. Its purpose is to make fitting models using Stan easy and easy to understand. pcFactorStan relies on the rstan package, which should be installed first. See here for instructions on installing rstan.

One situation where a factor model might be useful is when there are people that play in tournaments of more than one game. For example, the computer player AlphaZero (Silver et al. 2018) has trained to play chess, shogi, and Go. We can take the tournament match outcome data for each of these games and find rankings among the players. We may also suspect that there is a latent board game skill that accounts for some proportion of the variance in the per-board game rankings. This proportion can be recovered by the factor model.

Our goal may be to fit a factor model, but it is necessary to build up the model step-by-step. There are essentially three models: ‘unidim’, ‘correlation’, and ‘factor’. ‘unidim’ analyzes a single item. ‘correlation’ is suitable for two or more items. Once you have vetted your items with the ‘unidim’ and ‘correlation’ models, then you can try the ‘factor’ model. There is also a special model ‘unidim_adapt’. Except for this model, the other models require scaling constants. To find appropriate scaling constants, we will fit ‘unidim_adapt’ to each item separately.

# Brief tutorial

## Physical activity flow propensity

The R code below first loads rstan and pcFactorStan. We load loo for extra diagnostics, and qgraph and ggplot2 for visualization.

    library(rstan)
    library(pcFactorStan)
    library(loo)
    library(qgraph)
    library(ggplot2)

Next we take a peek at the data.

    head(phyActFlowPropensity)

pa1 | pa2 | skill | predict | novelty | creative | complex | goal1 | feedback1 | chatter | waiting | body | control | present | spont | stakes | evaluated | reward
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
mountain biking | tennis | 1 | -1 | -2 | 1 | 1 | 1 | 1 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 0
mountain biking | tennis | 1 | 2 | -1 | -1 | -1 | 0 | 2 | 1 | 2 | 0 | 1 | 0 | 0 | 1 | 2 | -1
ice skating | running | -2 | 1 | -1 | -2 | -1 | 1 | 1 | -2 | -2 | -1 | 0 | 0 | -1 | -1 | -1 | 0
climbing | rowing | -2 | 2 | -2 | -2 | -2 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 0 | 0 | 0
card game | gardening | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | 2 | 1 | 0 | 0 | 2 | -2 | 2
dance | table tennis | 0 | -2 | -1 | -1 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1

These data consist of paired comparisons of 87 physical activities on 16 flow-related facets. Participants submitted two activities using free-form input. These activities were substituted into item templates. For example, Item predict consisted of the prompt, “How predictable is the action?” with response options:

- `A1` is much more predictable than `A2`.
- `A1` is somewhat more predictable than `A2`.
- Both offer roughly equal predictability.
- `A2` is somewhat more predictable than `A1`.
- `A2` is much more predictable than `A1`.

If the participant selected ‘golf’ and ‘running’ for activities then ‘golf’ was substituted into `A1` and ‘running’ into `A2`. Duly prepared, the item was presented and the participant asked to select the most plausible statement.

A somewhat more response is scored 1 or -1 and much more scored 2 or -2. A tie (i.e. roughly equal) is scored as zero. We will need to analyze each item separately before we analyze them together. Therefore, we will start with Item skill.

Data must be fed into Stan in a partially digested form. The next block of code demonstrates how a suitable data list may be constructed using the `prepData()` function. This function automatically determines the number of threshold parameters based on the range observed in your data. One thing it does not do is pick a `varCorrection` factor. The `varCorrection` determines the degree of adaption in the model. Usually some choice between 2.0 to 4.0 will obtain optimal results.

    dl <- prepData(phyActFlowPropensity[,c(paste0('pa',1:2), 'skill')])
    dl$varCorrection <- 2.0

Next we fit the model using the `pcStan()` function, which is a wrapper for `stan()` from rstan. We also choose the number of chains. As is customary Stan procedure, the first half of each chain is used to estimate the sampler’s weight matrix (i.e. warm up) and excluded from inference.

    fit1 <- pcStan("unidim_adapt", data=dl)

A variety of diagnostics are available to check whether the sampler ran into trouble.

    check_hmc_diagnostics(fit1)
    #>
    #> Divergences:
    #> 0 of 4000 iterations ended with a divergence.
    #>
    #> Tree depth:
    #> 0 of 4000 iterations saturated the maximum tree depth of 10.
    #>
    #> Energy:
    #> E-BFMI indicated no pathological behavior.

Everything looks good, but there are a few more things to check. We want \(\widehat R\) < 1.015 and effective sample size greater than 100 times the number of chains (Vehtari et al., 2019).

    allPars <- summary(fit1, probs=c())$summary
    print(min(allPars[,'n_eff']))
    #> [1] 675.4
    print(max(allPars[,'Rhat']))
    #> [1] 1.005

Again, everything looks good. If the target values were not reached then we would sample the model again with more iterations. Time for a plot,

    theta <- summary(fit1, pars=c("theta"), probs=c())$summary[,'mean']

    ggplot(data.frame(x=theta, activity=dl$nameInfo$pa, y=0.47)) +
      geom_point(aes(x=x),y=0) +
      geom_text(aes(label=activity, x=x, y=y),
                angle=85, hjust=0, size=2,
                position = position_jitter(width = 0, height = 0.4)) + ylim(0,1) +
      theme(legend.position="none",
            axis.title.x=element_blank(),
            axis.title.y=element_blank(),
            axis.text.y=element_blank(),
            axis.ticks.y=element_blank())

Intuitively, this seems like a fairly reasonable ranking for skill. As pretty as the plot is, the main reason that we fit this model was to find a scaling constant to produce a standard deviation close to 1.0,

    s50 <- summary(fit1, pars=c("scale"), probs=c(.5))$summary[,'50%']
    print(s50)
    #> [1] 0.5877

We use the median instead of the mean because `scale` is not likely to have a symmetric marginal posterior distribution. We obtained 0.5877, but that value is just for one item. We have to perform the same procedure for every item. Wow, that would be really tedious … if we did not have a function to do it for us! Fortunately, `calibrateItems` takes care of it and produces a table of the pertinent data,

    result <- calibrateItems(phyActFlowPropensity, iter=1000L)

    print(result)

item | iter | divergent | treedepth | low_bfmi | n_eff | Rhat | scale | thetaVar
--- | --- | --- | --- | --- | --- | --- | --- | ---
skill | 1000 | 0 | 0 | 0 | 492.46 | 1.007 | 0.5901 | 1.0213
predict | 1000 | 0 | 0 | 0 | 641.66 | 1.003 | 0.5686 | 0.9963
novelty | 1000 | 0 | 0 | 0 | 774.62 | 1.004 | 0.4728 | 0.8811
creative | 1000 | 0 | 0 | 0 | 690.77 | 1.004 | 0.4692 | 0.8765
complex | 1000 | 0 | 0 | 0 | 467.95 | 1.007 | 0.5349 | 0.9566
goal1 | 1000 | 0 | 0 | 1 | 94.72 | 1.030 | 0.0869 | 0.2848
feedback1 | 3375 | 0 | 0 | 0 | 435.95 | 1.002 | 0.1567 | 0.4220
chatter | 1000 | 0 | 0 | 0 | 1024.80 | 1.002 | 0.2468 | 0.5712
waiting | 1000 | 0 | 0 | 0 | 532.23 | 1.003 | 0.4969 | 0.9106
body | 1000 | 0 | 0 | 0 | 960.07 | 1.003 | 0.3628 | 0.7385
control | 1000 | 0 | 0 | 0 | 1030.68 | 1.004 | 0.3149 | 0.6719
present | 1000 | 0 | 0 | 0 | 1059.55 | 1.005 | 0.2365 | 0.5551
spont | 1000 | 0 | 0 | 0 | 1231.19 | 1.001 | 0.2622 | 0.5947
stakes | 1000 | 0 | 0 | 0 | 889.16 | 1.001 | 0.2801 | 0.6214
evaluated | 1500 | 0 | 0 | 0 | 988.22 | 1.004 | 0.4423 | 0.8427
reward | 1000 | 0 | 0 | 0 | 898.49 | 1.012 | 0.2165 | 0.5234

Item goal1 ran into trouble. A nonzero count of divergent transitions or low_bfmi means that the item contained too little signal to estimate. Item feedback1 is also prone to failure. We could try again with `varCorrection=1.0`, but we are going to exclude these items instead. The model succeeded on the rest of the items. I requested `iter=1000L` to demonstrate how `calibrateItems` will resample the model until the `n_eff` is large enough and the `Rhat` small enough. As demonstrated in the iter column, some items needed more than 1000 samples to converge.

Next we will fit the correlation model. We exclude the Cholesky factor of the correlation matrix `rawThetaCorChol` because the regular correlation matrix is also output.

    pafp <- phyActFlowPropensity
    excl <- match(c('goal1','feedback1'), colnames(pafp))
    pafp <- pafp[,-excl]
    dl <- prepData(pafp)
    dl$scale <- result[-excl,'scale']

    fit2 <- pcStan("correlation", data=dl, include=FALSE, pars=c('rawTheta', 'rawThetaCorChol'))

    check_hmc_diagnostics(fit2)
    #>
    #> Divergences:
    #> 0 of 4000 iterations ended with a divergence.
    #>
    #> Tree depth:
    #> 0 of 4000 iterations saturated the maximum tree depth of 10.
    #>
    #> Energy:
    #> E-BFMI indicated no pathological behavior.

    allPars <- summary(fit2, probs=0.5)$summary
    print(min(allPars[,'n_eff']))
    #> [1] NaN
    print(max(allPars[,'Rhat']))
    #> [1] NaN

The HMC diagnostics look good, but … oh dear! Something is wrong with the `n_eff` and \(\widehat R\). Let us look more carefully,

    head(allPars[order(allPars[,'sd']),])
    #>               mean   se_mean        sd 50%  n_eff  Rhat
    #> thetaCor[1,1]    1       NaN 0.000e+00   1    NaN   NaN
    #> thetaCor[3,3]    1 1.264e-18 6.303e-17   1 2487.0 0.999
    #> thetaCor[2,2]    1 1.030e-18 6.347e-17   1 3798.2 0.999
    #> thetaCor[4,4]    1 4.778e-18 6.925e-17   1  210.1 0.999
    #> thetaCor[5,5]    1 1.468e-18 7.042e-17   1 2302.7 0.999
    #> thetaCor[7,7]    1 1.512e-18 7.725e-17   1 2609.9 0.999

Ah ha! It looks like all the entries of the correlation matrix are reported, including the entries that are not stochastic but are fixed to constant values. We need to filter those out to get sensible results.

    allPars <- allPars[allPars[,'sd'] > 1e-6,]
    print(min(allPars[,'n_eff']))
    #> [1] 971.2
    print(max(allPars[,'Rhat']))
    #> [1] 1.003

Ah, much better. Now we can inspect the correlation matrix. There are many ways to visualize a correlation matrix. One of my favorite ways is to plot it using the qgraph package,

    covItemNames <- dl$nameInfo$item
    tc <- summary(fit2, pars=c("thetaCor"), probs=c(.5))$summary[,'50%']
    tcor <- matrix(tc, length(covItemNames), length(covItemNames))
    dimnames(tcor) <- list(covItemNames, covItemNames)

    qgraph(tcor, layout = "spring", graph = "cor", labels=colnames(tcor),
           legend.cex = 0.3,
           cut = 0.3, maximum = 1, minimum = 0, esize = 20,
           vsize = 7, repulsion = 0.8, negDashed=TRUE, theme="colorblind")

Based on this plot and theoretical considerations, I decided to exclude spont, control, evaluated, and waiting from the factor model. A detailed rationale for why these items, and not others, are excluded will be presented in a forthcoming article. For now, let us focus on the mechanics of data analysis. Here are item response curves,

    df <- responseCurve(dl, fit2,
      item=setdiff(dl$nameInfo$item, c('spont','control','evaluated','waiting')),
      responseNames=c("much more","somewhat more", 'equal',
                      "somewhat less", "much less"))
    ggplot(df) +
      geom_line(aes(x=worthDiff,y=prob,color=response,linetype=response,
                    group=responseSample), size=.2, alpha=.2) +
      xlab("difference in latent worths") + ylab("probability") +
      ylim(0,1) + facet_wrap(~item) +
        guides(color=guide_legend(override.aes=list(alpha = 1, size=1)))

These response curves are a function of the `thresholds`, `scale`, and `alpha` parameters. A detailed description of the item response model can be found in the man page for `responseCurve`. A large `alpha` (>1) can mean that the item discriminates among objects well. However, it can also mean that the model predicts all responses will be equal. If most observed responses are indeed equal then this can result in good model fit, but we can also infer that the item is useless.

    alpha <- summary(fit2, pars=c("alpha"), probs=c(.5))$summary
    rownames(alpha) <- covItemNames

    print(alpha[alpha[,'sd']>.25,,drop=FALSE])

| mean | se_mean | sd | 50% | n_eff | Rhat
--- | --- | --- | --- | --- | --- | ---
chatter | 2.046 | 0.0084 | 0.4039 | 2.016 | 2309 | 1.002
waiting | 2.905 | 0.0076 | 0.3972 | 2.873 | 2766 | 1.000

I already decided to exclude waiting by inspection of the correlation matrix. Item chatter is on the borderline, but we can retain it.

We will enter the `alpha` parameters into the factor model as non-stochastic data. Trying to estimate `alpha` in the factor model causes bias, at least in the models that I have tried. The factor model is prone to increase both `alpha` and the magnitude of factor proportions at the expense of `threshold` accuracy. To treat `alpha` as non-stochastic reduces variability in the factor model, but not by much. Simulations indicate that the posterior distribution remains well calibrated.

I will fit model ‘factor_ll’ instead of ‘factor’ so I can use the loo package to look for outliers. We also need to take care that the data `pafp` matches, one-to-one, the data seen by Stan so we can map back from the model to the data. Hence, we update `pafp` using the usual the data cleaning sequence of `filterGraph` and `normalizeData` and pass the result to `prepCleanData`.

Up until version 1.0.2, only a single factor model was available. As of 1.1.0, the factor model supports an arbitrary number of factors and arbitrary factor-to-item structure. In this example, we will keep with the simplest factor model, a single factor that predicts all items. We need to set a prior on the factor loadings. The prior is given as the standard deviation of the normal prior for the logit transformed factor proportion. Here’s an example:

    factorProportion <- .3
    factorScalePrior <- .9
    dnorm(qlogis(0.5 + factorProportion/2), sd=factorScalePrior)
    #> [1] 0.3499

Enough data are available that we can use a factor scale prior of 1.0. If the model exhibited sampling difficulties (i.e. divergences) then a smaller prior could help by constraining the factor proportion closer to zero.

    pafp <- pafp[,c(paste0('pa',1:2),
                 setdiff(covItemNames, c('spont','control','evaluated','waiting')))]
    pafp <- normalizeData(filterGraph(pafp))
    dl <- prepCleanData(pafp)
    dl <- prepSingleFactorModel(dl, 1.0)
    dl$scale <- result[match(dl$nameInfo$item, result$item), 'scale']
    dl$alpha <- alpha[match(dl$nameInfo$item, rownames(alpha)), 'mean']

    fit3 <- pcStan("factor1_ll", data=dl, include=FALSE,
                   pars=c('rawUnique', 'rawUniqueTheta', 'rawPerComponentVar',
               'rawFactor', 'rawLoadings', 'rawFactorProp', 'rawNegateFactor', 'rawSeenFactor',
               'residual'))

To check the fit diagnostics, we have to take care to examine only the parameters of interest. The factor model outputs many parameters that should not be interpreted. For example, we do not care about `unique` or `uniqueTheta`, the unique scores.

    check_hmc_diagnostics(fit3)
    #>
    #> Divergences:
    #> 0 of 4000 iterations ended with a divergence.
    #>
    #> Tree depth:
    #> 0 of 4000 iterations saturated the maximum tree depth of 10.
    #>
    #> Energy:
    #> E-BFMI indicated no pathological behavior.

    interest <- c("threshold", "pathProp", "factor", "lp__")

    allPars <- summary(fit3, pars=interest)$summary
    print(min(allPars[,'n_eff']))
    #> [1] 600.3
    print(max(allPars[,'Rhat']))
    #> [1] 1.01

Looks good! Let us see which data are the most unexpected by the model. We create a `loo` object and inspect the summary output.

    options(mc.cores=1)  # otherwise loo consumes too much RAM
    kThreshold <- 0.1
    l1 <- toLoo(fit3)
    print(l1)
    #>
    #> Computed from 4000 by 9893 log-likelihood matrix
    #>
    #>          Estimate    SE
    #> elpd_loo -13747.2  62.1
    #> p_loo       283.8   5.2
    #> looic     27494.4 124.3
    #> ------
    #> Monte Carlo SE of elpd_loo is 0.3.
    #>
    #> All Pareto k estimates are good (k < 0.5).
    #> See help('pareto-k-diagnostic') for details.

The estimated Pareto \(k\) estimates are particularly noisy due to the many activities with a small sample size. Sometimes all \(k<0.5\) and sometimes not. We can look at `p_loo`, the effective number of parameters. In well behaving cases, `p_loo` is less than the sample size and the number of parameters. This looks good. There are 610 parameters just for the unique scores.

To connect \(k\) statistics with observations, we pass the `loo` object to `outlierTable` and use a threshold of 0.1 instead of 0.5 to ensure that we get enough lines. Activities with small sample sizes are retained by `filterGraph` if they connect other activities because they contribute information to the model. When we look at outliers, we can limit ourselves to activities with a sample size of at least 11.

    pa11 <- levels(filterGraph(pafp, minDifferent=11L)$pa1)
    ot <- outlierTable(dl, l1, kThreshold)
    ot <- subset(ot, pa1 %in% pa11 & pa2 %in% pa11)

    print(ot[1:6,])

| pa1 | pa2 | item | pick | k
--- | --- | --- | --- | --- | ---
32 | hockey | snow skiing | complex | -1 | 0.3631
35 | bowling | baseball | body | 2 | 0.3599
44 | martial arts | aerobic exercise | creative | 2 | 0.3519
45 | skipping rope | volleyball | reward | 0 | 0.3517
49 | mountain biking | climbing | reward | 2 | 0.3411
52 | running on a treadmill | aerobic exercise | creative | 2 | 0.3375

    xx <- which(ot[,'pa1'] == 'mountain biking' & ot[,'pa2'] == 'climbing' & ot[,'item'] == 'predict' & ot[,'pick'] == -2)

| pa1 | pa2 | item | pick | k
--- | --- | --- | --- | --- | ---
1170 | mountain biking | climbing | predict | -2 | 0.1519

We will take a closer look at row 1170. What does a `pick` of -2 mean? `Pick` numbers are converted to response categories by adding the number of thresholds plus one. There are two thresholds (much and somewhat) so 3 + -2 = 1. Looking back at our item response curve plot, the legend gives the response category order from top (1) to bottom (5). The first response category is much more. Putting it all together we obtain an endorsement of mountain biking is much more predictable than climbing. Specifically what about that assertion is unexpected? We can examine how other participants have responded,

    pafp[pafp$pa1 == ot[xx,'pa1'] & pafp$pa2 == ot[xx,'pa2'],
         c('pa1','pa2', as.character(ot[xx,'item']))]
    #>                 pa1      pa2 predict
    #> 563 mountain biking climbing      -2
    #> 564 mountain biking climbing      -2

Hm, both participants agreed. Let us look a little deeper to understand why this response was unexpected.

    loc <- sapply(ot[xx,c('pa1','pa2','item')], unfactor)
    exam <- summary(fit3, pars=paste0("theta[",loc[paste0('pa',1:2)],
                              ",", loc['item'],"]"))$summary
    rownames(exam) <- c(as.character(ot[xx,'pa1']), as.character(ot[xx,'pa2']))

| mean | se_mean | sd | 2.5% | 25% | 50% | 75% | 97.5% | n_eff | Rhat
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
mountain biking | -0.1298 | 0.0091 | 0.4015 | -0.9238 | -0.3902 | -0.1364 | 0.1408 | 0.6493 | 1937 | 1.001
climbing | -0.7217 | 0.0105 | 0.5147 | -1.7352 | -1.0631 | -0.7220 | -0.3640 | 0.2738 | 2389 | 1.002

Here we find that mountain biking was estimated 0.5919 units more predictable than climbing, but apparently this ranking was contradicted by most other observations. What sample sizes are associated with these activities?

    sum(c(pafp$pa1 == ot[xx,'pa1'], pafp$pa2 == ot[xx,'pa1']))
    #> [1] 25
    sum(c(pafp$pa1 == ot[xx,'pa2'], pafp$pa2 == ot[xx,'pa2']))
    #> [1] 13

Hm, the predictability 95% uncertainty interval for climbing is from -1.7352 to 0.2738. So there is little information. We could continue our investigation by looking at which responses justified these predict estimates. However, let us move on and plot the marginal posterior distributions of the factor proportions,

    pi <- parInterval(fit3, 'pathProp', dl$nameInfo$item, label='item')
    pi <- pi[order(abs(pi$M)),]
    pi$item <- factor(pi$item, levels=pi$item)

    ggplot(pi) +
      geom_vline(xintercept=0, color="green") +
      geom_jitter(data=parDistributionFor(fit3, pi),
                  aes(value, item), height = 0.35, alpha=.05) +
      geom_segment(aes(y=item, yend=item, x=L, xend=U),
                   color="yellow", alpha=.5) +
      geom_point(aes(x=M, y=item), color="red", size=1) +
      theme(axis.title.y=element_blank())

Finally, we can plot the factor scores.

    pick <- paste0('factor[',match(pa11, dl$nameInfo$pa),',1]')
    pi <- parInterval(fit3, pick, pa11, label='activity')
    pi <- pi[order(pi$M),]
    pi$activity <- factor(pi$activity, levels=pi$activity)

    ggplot(pi) +
      geom_vline(xintercept=0, color="green") +
      geom_jitter(data=parDistributionFor(fit3, pi, samples=250),
                  aes(value, activity), height = 0.35, alpha=.05) +
      geom_segment(aes(y=activity, yend=activity, x=L, xend=U),
                   color="yellow", alpha=.5) +
      geom_point(aes(x=M, y=activity), color="red", size=1) +
      theme(axis.title.y=element_blank())

If this factor model is a good fit to the data then the residual item activity scores should be uncorrelated. Let us examine the residual item correlation matrix.

    ric <- summary(fit3, pars=c('residualItemCor'), prob=c(.2,.5,.8))$summary
    r20 <- matrix(ric[,'20%'], length(dl$nameInfo$item), length(dl$nameInfo$item),
           dimnames=list(dl$nameInfo$item,dl$nameInfo$item))
    r50 <- matrix(ric[,'50%'], length(dl$nameInfo$item), length(dl$nameInfo$item),
                  dimnames=list(dl$nameInfo$item,dl$nameInfo$item))
    r80 <- matrix(ric[,'80%'], length(dl$nameInfo$item), length(dl$nameInfo$item),
           dimnames=list(dl$nameInfo$item,dl$nameInfo$item))
    ric5 <- fivenum(r50[lower.tri(r50)])
    mask <- sign(r20) == -sign(r80) # small, not statistically significant
    r50[mask] <- 0
    diag(r50) <- 2*max(abs(r50[lower.tri(r50)]))  # rescale for better visualization

    qgraph(r50, layout = "spring", graph = "cor", labels=colnames(r50),
           legend.cex = 0.3,
           cut = 0.3, maximum = 1, minimum = 0, esize = 20,
           vsize = 7, repulsion = 0.8, negDashed=TRUE, theme="colorblind")

Median residual correlations ranged from -0.1703 to 0.1834. To reduce visual clutter, small correlations with 60% posterior intervals crossing zero were set to zero. This is conservative; Almost none of the correlations would remain if a 90% interval was used instead. Stakes is an ideal item, uncorrelated with other items. There seem to be two main clusters. Body, reward, and chatter seem more emotional while novelty, creative, and present seem more cognitive. That the items faintly cluster along a cognitive-emotional spectrum, with skill in between, is not necessarily problematic. It seems likely that any survey measure is going to exhibit some faint correlations of this nature.

And there you have it. If you have not done so already, go find a dojo and commence study of martial arts!

# Technical notes

Given that my background is more in software than math, I am not a fan of the greek letters used with such enthusiasm by mathematicians. When I name variables, I favor the expressive over the succinct.

If you read through the Stan models included with this package, you will find some variables prefixed with `raw`. These are special variables internal to the model. In particular, you should not try to evaluate the \(\widehat R\) or effective sample size of `raw` parameters. These parameters are best excluded from the sampling output.

## Unidim Adapt

parameter | prior | purpose
--- | --- | ---
threshold | normal(0,2) | item response thresholds
theta | normal(0,sigma) | latent score
sigma | lognormal(1,1) | latent score scale
scale | N/A | latent score scaling constant

The ‘unidim_adapt’ model has a `varCorrection` constant that is used to calibrate the `scale`. For all other models, per-item `scale` must be passed in as data. `scale` has no substantive interpretation, but it is used to partition signal between object variance and item discrimination. While object variance has no substantive interpretation, item discrimination is interpretable.

## Unidim

parameter | prior | purpose
--- | --- | ---
threshold | normal(0,2) | item response thresholds
alpha | exponential(0.1) | item discrimination
theta | normal(0,1) | latent score

## Correlation

parameter | prior | purpose
--- | --- | ---
threshold | normal(0,2) | item response thresholds
alpha | exponential(0.1) | item discrimination
thetaCor | lkj(2) | correlations between items
theta | see below | latent score

Thresholds for all items are combined into a single vector. The prior for `theta` is multivariate normal with correlations `thetaCor` and scale 1.0. Exclude `rawTheta` and `rawThetaCorChol` from sampling reports.

## Factor

parameter | prior | purpose
--- | --- | ---
threshold | normal(0,2) | item response thresholds
pathLoadings | normal(0,5) | signed scale of factor scores
factor | normal(0,1) | factor scores
pathProp | see below | signed factor variance proportion
Psi | see below | factor correlations
sigma | N/A | relative item scale
residualItemCor | N/A | residual item correlation matrix

Thresholds for all items are combined into a single vector. `pathProp` is computed based on Equation 3 of Gelman et al. (in press). The prior on `pathProp` is `normal(logit(0.5 + pathProp/2), pathScalePrior)` where `pathScalePrior` is provided as data (see `prepSingleFactorModel()`). If you have more than one factor then `Psi` is available to estimate correlations among factors. Similar to `pathScalePrior`, the prior on entries of `Psi` is `normal(logit(0.5 + Psi/2), psiScalePrior)`. The `unique` and `pathLoadings` parameters are only relatively interpretable because the scale is arbitrary. The most interpretable parameter is `pathProp`. `pathProp` is a signed proportion bounded between -1 and 1. Exclude `rawUnique`, `rawUniqueTheta`, `rawFactor`, `rawLoadings`, `rawPerComponentVar`, `rawPathProp`, `rawNegateFactor`, `rawSeenFactor`, and `residual` from sampling.

The idea of putting a prior on `pathProp` was inspired by Gelman (2019, Aug 23).

# References

Gelman, A. (2019, Aug 23). Yes, you can include prior information on quantities of interest, not just on parameters in your model [Blog post]. Retrieved from https://statmodeling.stat.columbia.edu/2019/08/23/yes-you-can-include-prior-information-on-quantities-of-interest-not-just-on-parameters-in-your-model/.

Gelman, A., Goodrich, B., Gabry, J., & Vehtari, A. (in press). R-squared for Bayesian regression models. The American Statistician. DOI: 10.1080/00031305.2018.1549100

Silver, D., Hubert, T., Schrittwieser, J., Antonoglou, I., Lai, M., Guez, A., … & Lillicrap, T. (2018). A general reinforcement learning algorithm that masters chess, shogi, and Go through self-play. Science, 362(6419), 1140-1144.

Vehtari, A., Gelman, A., Simpson, D., Carpenter, B., & Bürkner, P. C. (2019). Rank-normalization, folding, and localization: An improved \(\widehat R\) for assessing convergence of MCMC. arXiv preprint arXiv:1903.08008.

# R Session Info

    sessionInfo()
    #> R version 3.6.1 (2019-07-05)
    #> Platform: x86_64-pc-linux-gnu (64-bit)
    #> Running under: Ubuntu 19.10
    #>
    #> Matrix products: default
    #> BLAS:   /usr/lib/x86_64-linux-gnu/atlas/libblas.so.3.10.3
    #> LAPACK: /usr/lib/x86_64-linux-gnu/atlas/liblapack.so.3.10.3
    #>
    #> locale:
    #>  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C
    #>  [3] LC_TIME=en_US.UTF-8        LC_COLLATE=C
    #>  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8
    #>  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C
    #>  [9] LC_ADDRESS=C               LC_TELEPHONE=C
    #> [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
    #>
    #> attached base packages:
    #> [1] stats     graphics  grDevices utils     datasets  methods   base
    #>
    #> other attached packages:
    #> [1] ggplot2_3.2.1      qgraph_1.6.3       loo_2.1.0          pcFactorStan_1.4.0
    #> [5] Rcpp_1.0.3.7       rstan_2.19.1       StanHeaders_2.18.9 knitr_1.28
    #>
    #> loaded via a namespace (and not attached):
    #>  [1] splines_3.6.1       gtools_3.8.1        Formula_1.2-3
    #>  [4] assertthat_0.2.1    highr_0.8           BDgraph_2.59
    #>  [7] stats4_3.6.1        latticeExtra_0.6-28 d3Network_0.5.2.1
    #> [10] yaml_2.2.1          pbivnorm_0.6.0      pillar_1.4.3
    #> [13] backports_1.1.5     lattice_0.20-38     glue_1.3.1
    #> [16] digest_0.6.23       RColorBrewer_1.1-2  checkmate_1.9.3
    #> [19] ggm_2.3             colorspace_1.4-1    htmltools_0.3.6
    #> [22] Matrix_1.2-17       plyr_1.8.4          psych_1.8.12
    #> [25] pkgconfig_2.0.3     purrr_0.3.3         corpcor_1.6.9
    #> [28] mvtnorm_1.0-11      scales_1.0.0        processx_3.4.1
    #> [31] whisker_0.4         glasso_1.10         jpeg_0.1-8
    #> [34] fdrtool_1.2.15      huge_1.3.2          tibble_2.1.3
    #> [37] htmlTable_1.13.1    withr_2.1.2         pbapply_1.4-0
    #> [40] nnet_7.3-12         lazyeval_0.2.2      cli_2.0.1
    #> [43] mnormt_1.5-5        survival_2.44-1.1   magrittr_1.5
    #> [46] crayon_1.3.4        evaluate_0.14       ps_1.3.0
    #> [49] fansi_0.4.1         MASS_7.3-51.4       nlme_3.1-141
    #> [52] foreign_0.8-72      pkgbuild_1.0.6      tools_3.6.1
    #> [55] data.table_1.12.2   prettyunits_1.1.1   matrixStats_0.54.0
    #> [58] stringr_1.4.0       munsell_0.5.0       cluster_2.1.0
    #> [61] callr_3.4.1         compiler_3.6.1      rlang_0.4.4
    #> [64] grid_3.6.1          rstudioapi_0.10     rjson_0.2.20
    #> [67] htmlwidgets_1.3     igraph_1.2.4.1      labeling_0.3
    #> [70] lavaan_0.6-4        base64enc_0.1-3     rmarkdown_1.13
    #> [73] gtable_0.3.0        codetools_0.2-16    abind_1.4-5
    #> [76] inline_0.3.15       reshape2_1.4.3      R6_2.4.1
    #> [79] gridExtra_2.3       dplyr_0.8.3         Hmisc_4.2-0
    #> [82] stringi_1.4.5       parallel_3.6.1      rpart_4.1-15
    #> [85] acepack_1.4.1       png_0.1-7           tidyselect_0.2.5
    #> [88] xfun_0.12