# September 2010 Archives

## Plotting survey responses representing three or more dimensions

Bob Williams recently posted a question to EVALTALK about open source software for plotting survey responses in three dimensions:

I'm wondering how to display Likert scale data in three dimensions (ie the Likert scores for each subject on 3 scale dimensions). My graphic software is not that sophisticated and I haven't been able to find an open source equivalent.... that can plot something along three dimensions (X,Y,Z) like this:
          X, Y, Z
Subject A 2, 5, 8
Subject B 3, 5, 9
Subject C 3, 7, 3

Another evaluator, Bill Harris, recommended and ggplot2 in particular.

Bob's question and the mention of inspired me to write up a brief demonstration of plotting multi-dimensional data. As I have mentioned in the past (see here, here, and here), plotting or mapping three or more dimensions can be difficult and potentially misleading because plots and maps are inherently two-dimensional. I would say that plotting three survey response dimensions actually requires plotting four dimensions: X, Y, Z, and the frequency of each X, Y, and Z combination. That's because evaluators commonly have large data sets with repeated response patterns. I simulated a larger data set from Bob's example in order to illustrate plotting four dimensions.

#Open a recording window (if using Windows).
windows(record=T)

data.3D <- data.frame(Subject = c("A", "B", "C"),
X = c(2, 3, 3),
Y = c(5, 5, 7),
Z = c(8, 9, 3))
data.3D

#Larger data sets will contain frequently repeated combinations of responses.
#Simulate a larger data set with repeated responses combinations.
data.4D <- data.3D
for(i in 1:200) {
temp <- data.frame(Subject=factor(paste(as.character(data.3D[, 1]), i, sep=".")),
X=round(jitter(data.3D[, 2], 7), 0),
Y=round(jitter(data.3D[, 2], 7), 0),
Z=round(jitter(data.3D[, 4], 7), 0))
data.4D <- rbind(data.4D, temp)
}
library(plyr)
data.4D <- ddply(data.4D, c("X", "Y", "Z"), "nrow")
names(data.4D)[which(names(data.4D)=="nrow")] <- "Frequency"
data.4D <- data.4D[order(data.4D\$Frequency, decreasing=T), ]

#Scatterplot matrices
pairs(data.3D[2:4], xlim=c(0, 10), ylim=c(0, 10),
main="Responses to three Likert scale items")
library(psych)
pairs.panels(sapply(data.4D[1:3], jitter), xlim=c(0, 10), ylim=c(0, 10), ellipses=F,
hist.col="blue", pch=1, scale=T, main="Responses to three Likert scale items")


#Load ggplot2 library; set theme without gray background.
library(ggplot2)
theme_set(theme_bw())

#Create a scatterplot with size and color varying by the third dimension.
plot1 <- ggplot(data=data.3D, aes(x=X, y=Y, color=Z, size=Z)) +
geom_point() +
xlim(0, 10) +
ylim(0, 10) +
scale_colour_gradient(breaks=0:10, limits=c(0, 10), low="lightblue", high="darkblue") +
scale_size_continuous(breaks=0:10, limits=c(0, 10)) +
opts(title="Responses to three Likert scale items")
plot1

#Create a jittered scatterplot with color varying by the third dimension
#and size by response combination frequency.
plot2 <- ggplot(data=data.4D, aes(x=X, y=Y, color=Z, size=Frequency)) +
geom_jitter() +
xlim(0, 10) +
ylim(0, 10) +
scale_colour_gradient(breaks=0:10, limits=c(0, 10), low="lightblue", high="darkblue") +
scale_size_continuous() +
opts(title="Responses to three Likert scale items")
plot2


#Reshape data from wide to longer.
data.3D.longer <- melt(data.3D, id.vars=c("Subject", "Y"), variable_name="Item")
names(data.3D.longer)[which(names(data.3D.longer)=="value")] <- "Response"
data.3D.longer

#Create a scatterplot with faceted second and third dimensions.
plot3 <- ggplot(data=data.3D.longer, aes(x=Response, y=Y)) +
geom_point(color="blue") +
xlim(0, 10) +
ylim(0, 10) +
facet_grid(. ~ Item) +
opts(title="Responses to three Likert scale items")
plot3

#Reshape simulated data from wide to longer.
data.4D.longer <- melt(data.4D, id.vars=c("Y", "Frequency"), variable_name="Item")
names(data.4D.longer)[which(names(data.4D.longer)=="value")] <- "Response"

#Create a scatterplot with faceted second and third dimensions
#and size varying by response combination frequency.
plot4 <- ggplot(data=data.4D.longer, aes(x=Response, y=Y, size=Frequency)) +
geom_jitter(color="blue") +
xlim(0, 10) +
ylim(0, 10) +
facet_grid(. ~ Item) +
opts(title="Responses to three Likert scale items")
plot4


## KFAI rocks

I want to give a shout-out to KFAI, Fresh Air Community Radio. I listen to KFAI just about every day. And every day I am impressed by the station's diverse programming and skilled volunteer DJs. I urge everyone to tune into 90.3 in Minneapolis, 106.7 in Saint Paul, or on the Web at http://www.kfai.org. Then support them with a contribution. Here's a list of my favorite shows to get you started:

## Linear mixed-effects regression p-values in R: A likelihood ratio test function

Following Douglas Bates' advice for those required to produce p-values for fixed-effects terms in a mixed-effects model, I wrote a function to perform a likelihood ratio test for each term in an lmer() object. Bates has championed the notion that calculating the p-values for fixed effects is not trivial. That's because with unbalanced, multilevel data, the denominator degrees of freedom used to penalize certainty are unknown (i.e., we're uncertain about how uncertain we should be). As the author of lme4, the foremost mixed-effects modeling package in , he has practiced what he preaches by declining to approximate denominator degrees of freedom as SAS does. Bates contends that alternative inferential approaches make p-values unnecessary. I agree with that position, focusing instead on information criteria and effect sizes. However, as a program evaluator, I recognize that some stakeholders find p-values useful for understanding findings.

A likelihood ratio test can be used to test $\inline H_0:\beta=0$ if the sample size is large. According to Fitzmaurice, Laird, and Ware, twice the difference between the maximized log-likelihoods of two nested models, $\inline G^2 = 2(\hat l_\text{full}-\hat l_\text{reduced})$, represents the degree to which the reduced model is inadequate. When the sample size is large, $\inline G^2 \sim \chi^2$ with degrees of freedom equal to the difference in parameters between the full and reduced models, $\inline m_\text{full} - m_\text{reduced}$. The p-value is $\inline \text{Pr} ( \chi^2 > G^2 | H_0)$.

What if the sample size is not large? A p-value based on $\inline \chi^2$ will be too liberal (i.e., the type I error rate will exceed the nominal p-value). More conservatively, we might say that $\inline G^2 \sim F$ with $\inline {m_\text{full} - m_\text{reduced}$ numerator and $\inline N_\text{effective}-m_\text{full}-1$ denominator degrees of freedom. According to Snijders and Bosker, the effective sample size lies somewhere between $\inline M$ total micro-observations (i.e., at level one) and $\inline N$ clusters randomly sampled in earlier stages (i.e., at higher levels). Formally, the effective sample size is $\inline N_\text{effective}=\tfrac{Nn}{1+(n-1)\rho}$, where $\inline n$ observations are nested within each cluster and intraclass correlation is $\inline \rho=\tfrac{\tau^2}{\tau^2+\sigma^2}$. Even if $\inline M=Nn$ is large, $\inline N_\text{effective}$ (and statistical power) could be quite small if $\inline N$ is small and $\inline \rho}$ is large: $\inline \tfrac{5*100}{1+(100-1)0.8}=6.2$. Unbalanced designs, modeling three or more levels, and cross-level interactions add to our uncertainty about the denominator degrees of freedom.

The function I wrote chews up the lmer() model call and concatenates the frame and model matrix slots, after which it iteratively fits (via maximum likelihood instead of restricted ML) models reduced by each fixed effect and compares them to the full model, yielding a vector of p-values based on $\inline \chi^2 (1)$. As the example shows, the function can handle shortcut formulas whereby lower order terms are implied by an interaction term. The function doesn't currently handle weights, glmer() objects, or on-the-fly transformations of the dependent variable [e.g., log(dep.var) ~ ...]. The accuracy of resulting p-values depends on large sample properties, as discussed above, so I don't recommend using the function with small samples. I'm working on another function that will calculate p-values based on the effective sample size estimated from intraclass correlation. I will post that function in a future entry. I'm sure the following function could be improved, but I wanted to go ahead share it with other applied researchers whose audience likes p-values. Please let me know if you see ways to make it better.

p.values.lmer <- function(x) {
summary.model <- summary(x)
data.lmer <- data.frame(model.matrix(x))
names(data.lmer) <- names(fixef(x))
names(data.lmer) <- gsub(pattern=":", x=names(data.lmer), replacement=".", fixed=T)
names(data.lmer) <- ifelse(names(data.lmer)=="(Intercept)", "Intercept", names(data.lmer))
string.call <- strsplit(x=as.character(x@call), split=" + (", fixed=T)
var.dep <- unlist(strsplit(x=unlist(string.call)[2], " ~ ", fixed=T))[1]
vars.fixef <- names(data.lmer)
formula.ranef <- paste("+ (", string.call[[2]][-1], sep="")
formula.ranef <- paste(formula.ranef, collapse=" ")
formula.full <- as.formula(paste(var.dep, "~ -1 +", paste(vars.fixef, collapse=" + "),
formula.ranef))
data.ranef <- data.frame(x@frame[,
which(names(x@frame) %in% names(ranef(x)))])
names(data.ranef) <- names(ranef(x))
data.lmer <- data.frame(x@frame[, 1], data.lmer, data.ranef)
names(data.lmer)[1] <- var.dep
out.full <- lmer(formula.full, data=data.lmer, REML=F)
p.value.LRT <- vector(length=length(vars.fixef))
for(i in 1:length(vars.fixef)) {
formula.reduced <- as.formula(paste(var.dep, "~ -1 +", paste(vars.fixef[-i],
collapse=" + "), formula.ranef))
out.reduced <- lmer(formula.reduced, data=data.lmer, REML=F)
print(paste("Reduced by:", vars.fixef[i]))
print(out.LRT <- data.frame(anova(out.full, out.reduced)))
p.value.LRT[i] <- round(out.LRT[2, 7], 3)
}
summary.model@coefs <- cbind(summary.model@coefs, p.value.LRT)
summary.model@methTitle <- c("\n", summary.model@methTitle,
"\n(p-values from comparing nested models fit by maximum likelihood)")
print(summary.model)
}

library(lme4)
library(SASmixed)
lmer.out <- lmer(strength ~ Program * Time + (Time|Subj), data=Weights)
p.values.lmer(lmer.out)


Yields:

Linear mixed model fit by REML
(p-values from comparing nested models fit by maximum likelihood)
Formula: strength ~ Program * Time + (Time | Subj)
Data: Weights
AIC  BIC logLik deviance REMLdev
1343 1383 -661.7     1313    1323
Random effects:
Groups   Name        Variance Std.Dev. Corr
Subj     (Intercept) 9.038486 3.00641
Time        0.031086 0.17631  -0.118
Residual             0.632957 0.79559
Number of obs: 399, groups: Subj, 57

Fixed effects:
Estimate Std. Error   t value p.value.LRT
(Intercept)     79.99018    0.68578 116.64000       0.000
ProgramRI        0.07009    1.02867   0.07000       0.944
ProgramWI        1.11526    0.95822   1.16000       0.235
Time            -0.02411    0.04286  -0.56000       0.564
ProgramRI:Time   0.12902    0.06429   2.01000       0.043
ProgramWI:Time   0.18397    0.05989   3.07000       0.002

Correlation of Fixed Effects:
(Intr) PrgrRI PrgrWI Time   PrRI:T
ProgramRI   -0.667
ProgramWI   -0.716  0.477
Time        -0.174  0.116  0.125
ProgrmRI:Tm  0.116 -0.174 -0.083 -0.667
ProgrmWI:Tm  0.125 -0.083 -0.174 -0.716  0.477