Unequal variances, Welch test, Tryon adjustment, and ``superb``"

cat("this will be hidden; use for general initializations.\n")
library(superb)
library(ggplot2)

When the group's variances are heterogeneous, it may results in error bars of differing length. It is then possible than one of the error bar includes the other mean, but that the second error bar does not. What can we conclude in such circumstances? This vignette explores this questions, examining Welch test solution, and Tryon adjustment along the way.

A simple example

Consider this example where two groups have been examined. Herein, the sample sizes are equal, but what follows does not depends on this.

grp1 <- c(56., 54., 73., 46., 59., 62., 55., 53., 77., 60., 69., 66., 63., 62., 53., 82., 74., 70., 65., 70., 72., 65., 56., 58., 83.)
grp2 <- c(51., 99., 194., 123., 40., 83., 87., 117., 46., 89., 61., 81., 53., 141., 52., 53., 39., 96., 14., 81., 63., 66., 80., 113., 82.)

These two groups have means of r round(mean(grp1),3) and r round(mean(grp2),3). However, they also have quite differing standard deviations, r round(sd(grp1),3) and r round(sd(grp2),3). The second is over four times larger than the first, which denotes a very heterogeneous variance situation.

Let's reunite the scores of these two datasets in a single data frame with a group indicator so that we can do a first plot:

dtaHetero <- data.frame( cbind(
    id    = 1:(length(grp1)+length(grp2)),
    group = c(rep(1,length(grp1)), rep(2, length(grp2)) ),
    score = c(grp1, grp2)
))
head(dtaHetero)

(just to make the plots look nicer, I prepare some directives that will be added to these:

library(superb)             # to make the summary plot
library(ggplot2)            # for all the graph directives
library(grid)               # for grid.text
library(gridExtra)          # for grid.arrange

ornate = list(
    xlab("Difference"), 
    scale_x_discrete(labels=c("Pre-\nTreatment","Post-\nTreatment")),
    ylab("Score"),
    coord_cartesian( ylim = c(40,+110) ),
    theme_light(base_size = 24) +
    theme( plot.subtitle = element_text(size=24))
)

). We are now ready to make a first plot. The adjustment purpose = "difference" is necessary as we plan to compare the two results.

pt <- superbPlot(dtaHetero, 
                 BSFactors    = "group", 
                 variables   = "score", 
                 adjustments = list(purpose = "difference"),
                 gamma       = 0.95,
                 statistic   = "mean", 
                 errorbar    = "CI",
                 plotStyle   = "line",
                 lineParams  = list(alpha = 0) #the line is made transparent
) + ornate
pt

As seen, the two error bars deliver a different message: the left one suggests a huge difference between the means, the right one does not. How do we settle this?

Averaging the error bars, sort of...

Error bars are depiction of an error variance. However, they are actually based on the standard errors which are the square root of the error variances.

Standard errors cannot be sumed (or averaged), only error variances can. Thus, when averaging the two error bars, it is their squares that must be averaged (followed by a square root). This operation is called the mean in the square sense (it is actually the Pythagorean theorem). It is denoted mean* hereafter.

Given that the first error bar is given by $\bar{X}_1 \pm \sqrt{2} \;t \; SE_1$ and the second, by $\bar{X}_2 \pm \sqrt{2} \;t \; SE_2$, where $\bar{X}_i$ is the average of the $i^{\text{th}}$ group and $SE_i$ is the standard error of the the $i^{\text{th}}$ group, averaging them can be simplified to

$$ \DeclareMathOperator{\mean}{Mean} \begin{split} \mean{^}( \text{ the two confidence intervals }) &= \mean{^} \left( \sqrt{2} \;t \; SE_1, \sqrt{2} \; t \; SE_2 \right) \ &= \sqrt{\mean \left( 2 \;t^2 \;SE_1^2, 2 \;t^2 \;SE_2^2 \right) } \ &= \sqrt{2} \;t\; \sqrt{\mean( SE_1^2, SE_2^2) } \ &= \sqrt{2} \;t\; \sqrt{ \frac{SE_1^2 + SE_2^2}{2} } \ &= t \;\sqrt{SE_1^2 + SE_2^2} \end{split} $$

Hence, asking if the lag between the two groups is significant is actually asking

$$ \DeclareMathOperator{\mean}{Mean} \begin{split} \text{Reject $H_0$ if }& {\bar{X}_1-\bar{X}_2} > \mean^*( \text{ of the two confidence intervals })\ \text{Reject $H_0$ if }& {\bar{X}_1-\bar{X}_2} > t\; {\sqrt{SE_1^2 + SE_2^2}} \ \text{Reject $H_0$ if }& \frac{\bar{X}_1-\bar{X}_2} {\sqrt{SE_1^2 + SE_2^2}} > t \ \end{split} $$

which is precisely the definition of a Welch test [@w47].

In other word, when appraising the general length of the error bars, you are actually performing a Welch test. This test is arguably the only mean test we need [@dll17].

A real Welch test with your eyes

Actually, the above is not entirely true, as the degrees of freedom of the $t$ multiplier is only based on the number of observations in each group ($n-1$).

To really have a Welch test, it is necessary to (1) pool the degrees of freedom of the groups together; (2)rectify this degree of freedom based on the amount of heterogeneity in the data.

To get the rectified degrees of freedom according to Welch, you can issue

# Welch's rectified degrees of freedom
wdf <- WelchDegreeOfFreedom(dtaHetero, "score","group")

in which "score" is the column(s) containing the dependent variable(s) and "group" is the column(s) containing the group identifiers. Here, the result is r round(wdf,4) (compared to $n_1+n_2-2$, that is r length(grp1)+length(grp2)-2, indicating a fairly large reduction in degree of freedom caused by a fairly important heterogeneity.

In superb, it is possible to override the default degree of freedom using the error bar estimator errorbar = CIwithDF" and then in the gamma argument, use a vector with both the confidence level desired and the rectified degree of freedom, as in gamma = c(0.95, 26.9965). Lets have a small plot of this:

pw <- superbPlot(dtaHetero, 
                 BSFactors   = "group", 
                 variables   = "score", 
                 adjustments = list(purpose = "difference"),
                 gamma       = c(0.95, wdf),         # new! 
                 statistic   = "mean", 
                 errorbar    = "CIwithDF",           # new!
                 plotStyle   = "halfwidthline",
                 lineParams  = list(alpha = 0)
) + ornate
pw

In this plot, I used a different layout, the "halfwidthline" which shows the mid-point of the error bar with a small gap. We will see the reason for that in a minute.

Averaging in the square sense, do you count on me to do that visually?

Averaging visually two lengths is already difficult, if it needs to be done in the square sense, that may be impossible... When averaging in the square sense, the longer lines have a stronger influence on the means so that ignoring this fact, we would tend to underestimate the mean length.

The first solution is not to bother. The usual average and the average in the square sense returns sensibly the same length. Therefore, if your results are not ambiguous, there is no need to worry any more.

The second solution is to adjust the error bar length so that they are amenable to the usual average. This is actually what Tryon's adjustment does [@t01]. When the variances are truly homogeneous, the correction for examining pair-wise comparision is an increase by a factor of $\sqrt{2}$. When we deviate from homogeneity, Tryon's adjustment tends to be a little larger than 1.41 to compensate for the underestimation introduced by regular averaging. This transformation is right on so that the usual mean of the two error bars, following a Tryon transformation, will precisely indicate whether the means are included in the mean error bar length.

pwt <- superbPlot(dtaHetero, 
                  BSFactors   = "group", 
                  variables   = "score", 
                  adjustments = list(purpose = "tryon"), #new!
                  gamma       = c(0.95, wdf),
                  statistic   = "mean", 
                  errorbar    = "CIwithDF",
                  plotStyle   = "halfwidthline",
                  lineParams  = list(alpha = 0)
)+ ornate
pwt

Finally, are the groups different or not?

Let's return to the whole question that motivated this vignette.

If we used the difference-adjusted error bars, then we need to average in the square sense the error bars. Lets do that:

# get the summary statistics with superbData
t <- superbData(dtaHetero, 
                BSFactors   = "group", 
                variables   = "score", 
                adjustments = list(purpose = "difference"),
                gamma       = 0.95,
                statistic   = "mean", errorbar = "CI"
)
# keep only the summary statistics, not the rawData structure:
t2 <- t$summaryStatistics

# the length is in column "upperwidth", for lines 1 and 2,
# so lets do the mean in the square sense:
tmean2 <- sqrt((t2$upperwidth[1]^2+t2$upperwidth[2]^2)/2)

which equals r round(tmean2,3). By comparison, here is the regular mean:

tmean    <- (t2$upperwidth[1] + t2$upperwidth[2] )/2

which equals r round(tmean,3).

We'll add a line segment of that length in the subsequent plot. Let's do the same following Tryon's adjusment and Welch rectified degrees of freedom:

# get the summary statistics with superbData
wt <- superbData(dtaHetero, 
                BSFactors   = "group", 
                variables   = "score", 
                adjustments = list(purpose = "tryon"),
                gamma       = c(0.95, wdf),
                statistic   = "mean", errorbar = "CIwithDF"
)
wt2 <- wt$summaryStatistics
wmean <- (wt2$upperwidth[1]+wt2$upperwidth[2]) / 2

As seen, both length are very similar, r round(tmean2,3) for the first, and r round(wmean,3) for the second.

In the case of Tryon's adjustment, the length can be averaged, which means that half of each length totalise the length: this is where the halfwidth error bars come handy: we see the half of their lenght. Hence, if they touch, their some is exactly equal to the lag between the two means.

Lets represent all this:

# showing all three plots, with reference lines in red
grid.arrange(
    pt + labs(subtitle="Difference-adjusted 95% CI\n with default degree of freedom") + 
        geom_text( x = 1.15, y = mean(grp1)+(mean(grp2)-mean(grp1))/2, label = "power-2 mean", angle = 90) +
        geom_text( x = 1.55, y = mean(grp1)+(mean(grp2)-mean(grp1))/2, label = "regular mean", angle = 90) +
        geom_hline(yintercept = mean(grp1)+(mean(grp2)-mean(grp1))/2-tmean2/2, colour = "red", size = 0.5, linetype=2) + 
        geom_hline(yintercept = mean(grp1)+(mean(grp2)-mean(grp1))/2+tmean2/2, colour = "red", size = 0.5, linetype=2) + 
        # arrow for the power-2 mean
        geom_segment(arrow = arrow(length =unit(0.4,"cm")), x=1.33, y=mean(grp1)+(mean(grp2)-mean(grp1))/2, 
                xend=1.33, yend=mean(grp1)+(mean(grp2)-mean(grp1))/2+tmean2/2) +
        geom_segment(arrow = arrow(length =unit(0.4,"cm")), x=1.333, y=mean(grp1)+(mean(grp2)-mean(grp1))/2, 
                xend=1.33, yend=mean(grp1)+(mean(grp2)-mean(grp1))/2-tmean2/2) + 
        # arrow for the regular mean
        geom_segment(arrow = arrow(length =unit(0.4,"cm")), x=1.66, y=mean(grp1)+(mean(grp2)-mean(grp1))/2, 
                xend=1.66, yend=mean(grp1)+(mean(grp2)-mean(grp1))/2+tmean/2) +
        geom_segment(arrow = arrow(length =unit(0.4,"cm")), x=1.66, y=mean(grp1)+(mean(grp2)-mean(grp1))/2, 
                xend=1.66, yend=mean(grp1)+(mean(grp2)-mean(grp1))/2-tmean/2), 
    pw  + labs(subtitle="Difference-adjusted 95% CI\nwith df from Welch"),

    pwt + labs(subtitle="Tryon-adjusted 95% CI\nwith df from Welch") +
        geom_hline(yintercept = mean(grp1)+wt2$upperwidth[1]/2, colour = "red", size = 0.5, linetype=2) + 
        geom_hline(yintercept = mean(grp2)+wt2$lowerwidth[2]/2, colour = "red", size = 0.5, linetype=2),
 ncol=3)

In other word, the two groups are borderline significantly different. Still not convinced? Let's do a Welch test:

t.test(grp1, grp2, alternative="two.sided", var.equal=FALSE)

The difference is right on the limit of significance at the .05 level, mirroring the 95% confidence level used in the plot.

In summary

superb is making error bars from separate error estimations. Hence, whenever the sample sizes are quite unequal, the variances are quite heterogeneous, or both, the inference performed by eyes are actually quite close to a Welch test of means. When both samples sizes are equal and variances are homogeneouse, there is no difference between a Welch test and a t-test.

When the degree of freedom are rectified and Tryon's ajustment is used, the center of both confidence intervals should be exactly aligned to indicate that their sum fills exactly the gap.

Although the Tryon/Welch pair lets you make very precise inferences, visually and practically speaking, the difference between the rectified Welch degree of freedom adjustments and a plot with the default degree of freedom are undetectable (as evidence by the first plot and the second plot of Figure 4 just above).

If you can perform average in the square sense, the difference adjustment is perfect. In doubt, consider applying the Tryon adjustment when the variances are cleary heterogeneous.

References



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superb documentation built on April 22, 2021, 1:06 a.m.