14  Transformations, Measuring Change, and Regression to the Mean

14.1 Transformations

  • Need to transform continuous variables properly for parametric statistical methods to work well
  • Normality assumption often does not hold and the central limit theorem is irrelevant for non-huge datasets
    • Skewed distribution
    • Different standard deviation in different groups
  • Transforming data can be a simple method to overcome these problems
  • Non-parametric methods (Wilcoxon signed rank, Wilcoxon rank sum) another good option
  • Transformations are also needed when you combine variables
    • subtraction as with change scores
    • addition (e.g. don’t add ratios that are not proportions)

14.2 Logarithmic Transformation

  • Replace individual value by its logarithm
    • \(u = \log(x)\)
  • In statistics, always use the logarithm (base \(e\); \(ln(x)\))
  • Algebra reminders
    • \(\log(ab) = \log(a) + \log(b)\)
    • \(\log\left(\frac{a}{b}\right) = \log(a) - \log(b)\)
    • Inverse of the log function is \(\exp(u) = x\), where \(\exp(u) = e^{u}\) and \(e\) is a constant (\(e = 2.718282 ...\))

14.2.1 Example Dataset

  • From Essential Medical Statistics, 13.2 (pre data only)
  • Response: Urinary \(\beta\)-thromboglobulin (\(\beta\)-TG) excretion in 24 subjects
  • 24 total subjects: 12 diabetic, 12 normal
d <- rbind(
             btg=c(4.1, 6.3, 7.8, 8.5, 8.9, 10.4, 11.5, 12.0, 13.8,
                   17.6, 24.3, 37.2)),
             btg=c(11.5, 12.1, 16.1, 17.8, 24.0, 28.8, 33.9, 40.7,
                   51.3, 56.2, 61.7, 69.2)))
d <- data.table(d)
meds <- d[, .(btg = median(btg)), by = status]
p1 <-
  ggplot(d, aes(x=status, y=btg)) +
  geom_dotplot(binaxis='y', stackdir='center', position='dodge') +
  geom_errorbar(aes(ymin=..y.., ymax=..y..), width=.25, size=1.3, data=meds) +
   xlab('') + ylab(expression(paste(beta-TG, ' (ng/day/100 ml creatinine)'))) +
p2 <- ggplot(d, aes(x=status, y=btg)) +
  scale_y_log10(breaks=c(4,5,10,15,20,30,40,60,80)) +
  geom_dotplot(binaxis='y', stackdir='center', position='dodge') +
  xlab('') + ylab(expression(paste(beta-TG, ' (ng/day/100 ml creatinine)'))) +
arrGrob(p1, p2, ncol=2)

Figure 14.1: \(\beta\)-TG levels by diabetic status with a median line. The left plot is on the original (non-transformed) scale and includes median lines. The right plot displays the data on a log scale.

  • Original scale
    • Normal: \(\overline{x}_1 = 13.53\), \(s_1 = 9.194\), \(n_1 = 12\)
    • Diabetic: \(\overline{x}_2 = 35.28\), \(s_2 = 20.27\), \(n_2 = 12\)
  • Logarithm scale
    • Normal: \(\overline{x}_1^* = 2.433\), \(s_1^* = 0.595\), \(n_1 = 12\)
    • Diabetic: \(\overline{x}_2^* = 3.391\), \(s_2^* = 0.637\), \(n_2 = 12\)
  • \(t\)-test on log-transformed data
    • \(s_{pool} = \sqrt{\frac{11 \times .595^2 + 11 \times .637^2}{22}} = 0.616\)
    • \(t = \frac{2.433-3.391}{0.616\sqrt{1/12 + 1/12)}} = -3.81\), \(\textrm{df} = 22\), \(p = 0.001\)
  • Confidence Intervals (0.95 CI)
    • Note that \(t_{.975,22} = 2.074\)

    • For Normal subjects, a CI for the mean log \(\beta\)-TG is \[\begin{array}{ccc} 0.95 \textrm{ CI } & = & 2.433 - 2.074 \times \frac{0.595}{\sqrt{12}} \textrm{ to } 2.433 + 2.074 \frac{0.595}{\sqrt{12}} \\ & = & 2.08 \textrm{ to } 2.79 \end{array}\]

    • Can transform back to original scale by using the antilog function \(\exp(u)\) to estimate medians \[\begin{array}{ccc} \textrm{Geometric mean} & = & e^{2.433} = 11.39 \\ 0.95 \textrm{ CI } & = & e^{2.08} \textrm{ to } e^{2.79} \\ & = & 7.98 \textrm{ to } 16.27 \\ \end{array}\]

t.test(btg ~ status, data=d)

    Welch Two Sample t-test

data:  btg by status
t = 3.3838, df = 15.343, p-value = 0.003982
alternative hypothesis: true difference in means between group diabetic and group normal is not equal to 0
95 percent confidence interval:
  8.07309 35.41024
sample estimates:
mean in group diabetic   mean in group normal 
              35.27500               13.53333 
t.test(log(btg) ~ status, data=d)

    Welch Two Sample t-test

data:  log(btg) by status
t = 3.8041, df = 21.9, p-value = 0.0009776
alternative hypothesis: true difference in means between group diabetic and group normal is not equal to 0
95 percent confidence interval:
 0.4352589 1.4792986
sample estimates:
mean in group diabetic   mean in group normal 
              3.390628               2.433349 
  • Could also use a non-parametric test (Wilcoxon rank sum)
wilcox.test(btg ~ status, data=d)

    Wilcoxon rank sum test with continuity correction

data:  btg by status
W = 125.5, p-value = 0.002209
alternative hypothesis: true location shift is not equal to 0
wilcox.test(log(btg) ~ status, data=d)

    Wilcoxon rank sum test with continuity correction

data:  log(btg) by status
W = 125.5, p-value = 0.002209
alternative hypothesis: true location shift is not equal to 0
  • Note that non-parametric test is the same for the log-transformed outcomes

14.2.2 Limitations of log transformations

  • Can only be used on positive numbers
    • Sometimes use \(u = \log(x+1)\)
  • Is very arbitrary to the choice of the origin
  • Not always useful or the best transformation
  • Sometimes use a dimensionality argument, e.g., take cube root of volume measurements or per unit of volume counts like blood counts
  • Cube and square roots are fine with zeros

14.3 Analysis of Paired Observations

  • Frequently one makes multiple observations on same experimental unit
  • Can’t analyze as if independent
  • When two observations made on each unit (e.g., pre–post), it is common to summarize each pair using a measure of effect \(\rightarrow\) analyze effects as if (unpaired) raw data
  • Most common: simple difference, ratio, percent change
  • Can’t take effect measure for granted
  • Subjects having large initial values may have largest differences
  • Subjects having very small initial values may have largest post/pre ratios

14.4 What’s Wrong with Change in General?

14.4.1 Change from Baseline in Randomized Studies

Many authors and pharmaceutical clinical trialists make the mistake of analyzing change from baseline instead of making the raw follow-up measurements the primary outcomes, covariate-adjusted for baseline. The purpose of a parallel-group randomized clinical trial is to compare the parallel groups, not to compare a patient with herself at baseline. The central question is for two patients with the same pre measurement value of x, one given treatment A and the other treatment B, will the patients tend to have different post-treatment values? This is exactly what analysis of covariance assesses. Within-patient change is affected strongly by regression to the mean and measurement error. When the baseline value is one of the patient inclusion/exclusion criteria, the only meaningful change score requires one to have a second baseline measurement post patient qualification to cancel out much of the regression to the mean effect. It is the second baseline that would be subtracted from the follow-up measurement.

The savvy researcher knows that analysis of covariance is required to “rescue” a change score analysis. This effectively cancels out the change score and gives the right answer even if the slope of post on pre is not 1.0. But this works only in the linear model case, and it can be confusing to have the pre variable on both the left and right hand sides of the statistical model. And if \(Y\) is ordinal but not interval-scaled, the difference in two ordinal variables is no longer even ordinal. Think of how meaningless difference from baseline in ordinal pain categories are. A major problem in the use of change score summaries, even when a correct analysis of covariance has been done, is that many papers and drug product labels still quote change scores out of context. Patient-reported outcome scales with floor or ceiling effects are particularly problematic. Analysis of change loses the opportunity to do a robust, powerful analysis using a covariate-adjusted ordinal response model such as the proportional odds or proportional hazards model. Such ordinal response models do not require one to be correct in how to transform \(Y\).

Not only is it very problematic to analyze change scores in parallel group designs; it is problematic to even compute them. To compute change scores requires many assumptions to hold, e.g.:

  1. the variable is not used as an inclusion/exclusion criterion for the study, otherwise regression to the mean will be strong
  2. if the variable is used to select patients for the study, a second post-enrollment baseline is measured and this baseline is the one used for all subsequent analysis
  3. the post value must be linearly related to the pre value
  4. the variable must be perfectly transformed so that subtraction “works” and the result is not baseline-dependent
  5. the variable must not have floor and ceiling effects
  6. the variable must have a smooth distribution
  7. the slope of the pre value vs. the follow-up measurement must be close to 1.0 when both variables are properly transformed (using the same transformation on both)

References may be found here, and also see this tutorial.

Regarding 3. above, if pre is not linearly related to post, there is no transformation that can make a change score work. Two unpublished examples illustrate this problem. In a large degression study using longitudinal measurements of the Hamilton-D depression scale, there was a strong nonlinear relationship between baseline Hamilton D and the final measurement. The form of the relationship was a flattening of the effect for larger Ham D, indicating there are patients with severe depression at baseline who can achieve much larger reductions in depression than an average change score would represent (and likewise, patients with mild to moderate depression at baseline would have a reduction in depression symptoms that is far less than the average change indicates). Doing an ordinal ANCOVA adjusting for a smooth nonlinear effect of baseline using a spline function would have been a much better analysis than the change from baseline analysis done by study leaders. In the second example, a similar result was found for a quality of life measure, KCCQ. Again, a flattening relationship for large KCCQ indicated that subtracting from baseline provided a nearly meaningless average change score.

Regarding 7. above, often the baseline is not as relevant as thought and the slope will be less than 1. When the treatment can cure every patient, the slope will be zero. When the baseline variable is irrelevant, ANCOVA will estimate a slope of approximately zero and will effectively ignore the baseline. Change will baseline will make the change score more noisy than just analyzing the final raw measurement. In general, when the relationship between pre and post is linear, the correlation between the two must exceed 0.5 for the change score to have lower variance than the raw measurement.

Bland & Altman (2011) have an excellent article about how misleading change from baseline is for clinical trials. This blog article has several examples of problematic change score analyses in the clinical trials literature.

The following R code exemplifies how to do a powerful, robust analysis without computing change from baseline. This analysis addresses the fundamental treatment question posed at the top of this section. This example uses a semiparametric ANCOVA that utilizes only the ranks of \(Y\) and that provides the same test statistic no matter how \(Y\) is transformed. It uses the proportional odds model, with no binning of \(Y\), using the rms package orm function. This is a generalization of the Wilcoxon test—it would be almost identical to the Wilcoxon test had the baseline effect been exactly flat. Note that the proportional odds assumption is more likely to be satisfied than the normal residual, equal variance assumption of the ordinary linear model.

# Fit a smooth flexible relationship with baseline value y0
# (restricted cubic spline function with 4 default knots)
f <- orm(y ~ rcs(y0, 4) + treatment)
# Estimate mean y as a function of y0 and treatment
M <- Mean(f)   # creates R function to compute the mean
plot(Predict(f, treatment, fun=M))  # y0 set to median since not varied
# To allow treatment to interact with baseline value in a general way:
f <- orm(y ~ rcs(y0, 4) * treatment)
plot(Predict(f, y0, treatment))  # plot y0 on x-axis, 2 curves for 2 treatments
# The above plotted the linear predictor (log odds); can also plot mean

Kristoffer Magnusson has an excellent paper Change over time is not “treatment response”

Senn (2008) Chapter 7 has a very useful graphical summary of the large-sample variances (inefficiency) of change scores vs. ANCOVA vs. ignoring baseline completely, as a function of the (assumed linear) correlation between baseline and outcome.

14.4.2 Special Problems Caused By Non-Monotonic Relationship With Ultimate Outcome

Besides the strong assumptions made about how variables are transformed before a difference is calculated, there are many types of variables for which change can never be interpreted without reference to the starting point, because the relationship between the variable in question and an ultimate outcome is not even monotonic. A good example is the improper definitions of acute kidney injury (AKI) that have been accepted without question. Many of the definitions are based on change in serum creatinine (SCr). Problems with the definitions include

  1. The non-monotonic relationship between SCr and mortality demonstrates that it is not healthy to have very low SCr. This implies that increases in SCr for very low starting SCr may not be harmful as assumed in definitions of AKI.
  2. Given an earlier SCr measurement and a current SCr, the earlier measurement is not very important in predicting mortality once one adjusts for the last measurement. Hence a change in SCr is not very predictive and the current SCr is all-important. As an example consider the estimated relationship between baseline SCr and mortality in critically ill ICU patients.
combined <- subset(combined,
  select=c(id, death, d.time, hospdead, dzgroup, age, raceh, sex))
combphys <- subset(combphys, !is.na(crea1+crea3),
w <- merge(combined, combphys, by='id')
u <- 'mg/dl'
w <- upData(w, labels=c(crea1='Serum Creatinine, Day 1',
                 crea3='Serum Creatinine Day 3',
                 crea14='Serum Creatinine Day 14'),
            units=c(crea1=u, crea3=u, crea7=u, crea14=u, crea25=u),

w <- subset(w, crea1 < 2)
dd <- datadist(w); options(datadist='dd')
h <- lrm(hospdead ~ rcs(crea1, 5) + rcs(crea3, 5), data=w)
Wald Statistics for hospdead
χ2 d.f. P
crea1 19.52 4 0.0006
Nonlinear 15.60 3 0.0014
crea3 108.11 4 <0.0001
Nonlinear 49.98 3 <0.0001
TOTAL NONLINEAR 132.06 6 <0.0001
TOTAL 217.11 8 <0.0001
h <- lrm(hospdead ~ sex * rcs(crea3, 5), data=w)
p <- Predict(h, crea3, sex, fun=plogis)
ggplot(p, ylab='Risk of Hospital Death')

Figure 14.2: Estimated risk of hospital death as a function of day 3 serum creatinine and sex for 7772 critically ill ICU patients having day 1 serum creatinine \(< 2\) and surviving to the start of day 3 in the ICU

We see that the relationship is very non-monotonic so that it is impossible for change in SCr to be relevant by itself unless the study excludes all patients with SCr \(< 1.05\). To put this in perspective, in the NHANES study of asymptomatic subjects, a very significant proportion of subjects have SCr \(< 1\).

14.5 What’s Wrong with Percent Change?

  • Definition

\[ \% \textrm{ change } = \frac{\textrm{first value} - \textrm{second value}}{\textrm{second value}} \times 100 \]

  • The first value is often called the new value and the second value is called the old value, but this does not fit all situations
  • Example
    • Treatment A: 0.05 proportion having stroke
    • Treatment B: 0.09 proportion having stroke
  • The point of reference (which term is used in the denominator?) will impact the answer
    • Treatment A reduced proportion of stroke by 44
    • Treatment B increased proportion by 80
  • Two increases of 50% result in a total increase of 125%, not 100
    • Math details: If \(x\) is your original amount, two increases of 50% is \(x\times 1.5\times 1.5\). Then, % change = \((1.5\times 1.5\times x - x) / x = x\times (1.5\times 1.5 - 1) / x = 1.25\), or a 125% increase
  • Percent change (or ratio) not a symmetric measure
    • A 50% increase followed by a 50% decrease results in an overall decrease (not no change)

      • Example: 2 to 3 to 1.5
    • A 50% decrease followed by a 50% increase results in an overall decrease (not no change)

      • Example: 2 to 1 to 1.5
  • Unless percents represent proportions times 100, it is not appropriate to compute descriptive statistics (especially the mean) on percents.
    • For example, the correct summary of a 100% increase and a 50% decrease, if they both started at the same point, would be 0% (not 25%).
  • Simple difference or log ratio are symmetric

See also fharrell.com/post/percent

14.6 Objective Method for Choosing Effect Measure

  • Goal: Measure of effect should be as independent of baseline value as possible1
  • Plot difference in pre and post values vs. the pre values. If this shows no trend, the simple differences are adequate summaries of the effects, i.e., they are independent of initial measurements.
  • If a systematic pattern is observed, consider repeating the previous step after taking logs of both the pre and post values. If this removes any systematic relationship between the baseline and the difference in logs, summarize the data using logs, i.e., take the effect measure as the log ratio.
  • Other transformations may also need to be examined
  • 1 Because of regression to the mean, it may be impossible to make the measure of change truly independent of the initial value. A high initial value may be that way because of measurement error. The high value will cause the change to be less than it would have been had the initial value been measured without error. Plotting differences against averages rather than against initial values will help reduce the effect of regression to the mean.

  • 14.7 Example Analysis: Paired Observations

    14.7.1 Dataset description

    • Dataset is an extension of the diabetes dataset used earlier in this chapter
    • Response: Urinary \(\beta\)-thromboglobulin (\(\beta\)-TG) excretion in 24 subjects
    • 24 total subjects: 12 diabetic, 12 normal
    • Add a “post” measurement (previous data considered the “pre” measurement)

    14.7.2 Example Analysis

    # Now add simulated some post data to the analysis of beta TG data
    # Assume that the intervention effect (pre -> post effect) is
    # multiplicative (x 1/4) and that there is a multiplicative error
    # in the post measurements
    #| label: fig-change-tgtrans
    d$pre  <- d$btg
    d$post <- exp(log(d$pre) + log(.25) + rnorm(24, 0, .5))
    # Make plots on the original and log scales
    p1 <- ggplot(d, aes(x=pre, y=post - pre, color=status)) +
      geom_point() + geom_smooth() + theme(legend.position='bottom')
    # Use problematic asymmetric % change
    p2 <- ggplot(d, aes(x=pre, y=100*(post - pre)/pre,
                        color=status)) + geom_point() + geom_smooth() +
          xlab('pre') + theme(legend.position='none') +
          ylim(-125, 0)
    p3 <- ggplot(d, aes(x=pre, y=log(post / pre),
                        color=status)) + geom_point() + geom_smooth() +
          xlab('pre') + theme(legend.position='none') + ylim(-2.5, 0)
    arrGrob(p1, p2, p3, ncol=2)

    Difference vs. baseline plots for three transformations

    with(d, {
         print(t.test(post - pre))
         print(t.test(100*(post - pre) / pre))       # improper
         print(t.test(log(post / pre)))
         print(wilcox.test(post - pre))
         print(wilcox.test(100*(post - pre) / pre))  # improper
         print(wilcox.test(log(post / pre)))
         } )
        One Sample t-test
    data:  post - pre
    t = -5.9366, df = 23, p-value = 4.723e-06
    alternative hypothesis: true mean is not equal to 0
    95 percent confidence interval:
     -22.68768 -10.96215
    sample estimates:
    mean of x 
        One Sample t-test
    data:  100 * (post - pre)/pre
    t = -23.864, df = 23, p-value < 2.2e-16
    alternative hypothesis: true mean is not equal to 0
    95 percent confidence interval:
     -75.19217 -63.19607
    sample estimates:
    mean of x 
        One Sample t-test
    data:  log(post/pre)
    t = -13.147, df = 23, p-value = 3.5e-12
    alternative hypothesis: true mean is not equal to 0
    95 percent confidence interval:
     -1.483541 -1.080148
    sample estimates:
    mean of x 
        Wilcoxon signed rank exact test
    data:  post - pre
    V = 0, p-value = 1.192e-07
    alternative hypothesis: true location is not equal to 0
        Wilcoxon signed rank exact test
    data:  100 * (post - pre)/pre
    V = 0, p-value = 1.192e-07
    alternative hypothesis: true location is not equal to 0
        Wilcoxon signed rank exact test
    data:  log(post/pre)
    V = 0, p-value = 1.192e-07
    alternative hypothesis: true location is not equal to 0

    Note: In general, the three Wilcoxon signed-rank statistics will not agree on each other. They depend on the symmetry of the difference measure.

    14.8 Regression to the Mean

    • One of the most important of all phenomena regarding data and estimation
    • Occurs when subjects are selected because of their values
    • Examples:
      1. Intersections with frequent traffic accidents will have fewer accidents in the next observation period if no changes are made to the intersection
      2. The surgeon with the highest operative mortality will have a significant decrease in mortality in the following year
      3. Subjects screened for high cholesterol to qualify for a clinical trial will have lower cholesterol once they are enrolled
    • Observations from a randomly chosen subject are unbiased for that subject
    • But subjects selected because they are running high or low are selected partially because their measurements are atypical for themselves (i.e., selected because of measurement error)
    • Future measurements will “regress to the mean” because measurement errors are random
    • For a classic misattribution of regression to the mean to a treatment effect see this2.
  • 2 In their original study, the social workers enrolled patients having 10 or more hospital admissions in the previous year and showed that after their counseling, the number of admissions in the next year was less than 10. The same effect might have been observed had the social workers given the patients horoscopes or weather forecasts. This was reported in an abstract for the AHA meeting that has since been taken down from circ.ahajournals.org.

  • Classic paper on shrinkage: Efron & Morris (1977)

    • Shrinkage is a way of discounting observed variation that accounts for regression to the mean
    • In their example you can see that the variation in batting averages for the first 45 at bats is unrealistically large
    • Shrunken estimates (middle) have too little variation but this discounting made the estimates closer to the truth (final batting averages at the end of the season)
    • You can see the regression to the mean for the apparently very hot and very cold hitters
    nam <- c('Roberto Clemente','Frank Robinson','Frank Howard','Jay Johnstone',
         'Ken Berry','Jim Spencer','Don Kessinger','Luis Alvarado',
             'Ron Santo','Ron Swoboda','Del Unser','Billy Williams',
             'George Scott','Rico Petrocelli','Ellie Rodriguez',
             'Bert Campaneris','Thurman Munson','Max Alvis')
    initial <- c(18,17,16,15,14,14,13,12,11,11,10,10,10,10,10,9,8,7)/45
    season  <- c(345,297,275,220,272,270,265,210,270,230,265,258,306,265,225,
    initial.shrunk <- c(294,288,280,276,275,275,270,265,262,263,258,256,
    n  <- 18
    x1 <- .5
    x2 <- .75
    x3 <- 1
    points(rep(x1,n), initial)
    points(rep(x2,n), initial.shrunk)
    points(rep(x3,n), season)
    for(i in 1:n) lines(c(x1,x2,x3),c(initial[i],initial.shrunk[i],season[i]),
                        col=i, lwd=2.5)
    text(c(x1,x2+.01, x2+.25),rep(.12,3),c('First 45 ABs','Shrunken\nEstimates',
         'Rest of\nSeason'))
    for(a in unique(initial)) {
      s <- initial==a
      w <- if(sum(s) < 4) paste(nam[s],collapse=', ') else {
        j <- (1:n)[s]
        paste(nam[j[1]],', ',nam[j[2]],', ',nam[j[3]],'\n',
              nam[j[4]],', ',nam[j[5]],sep='')
      text(x1-.02, a, w, adj=1, cex=.9)

    Figure 14.3: Initial batting averages as estimates of final batting averages for players, along with shrunken estimates that account for regression to the mean

    Bland, J. M., & Altman, D. G. (2011). Comparisons against baseline within randomised groups are often used and can be highly misleading. Trials, 12(1), 264. https://doi.org/10.1186/1745-6215-12-264
    Efron, B., & Morris, C. (1977). Stein’s paradox in statistics. Sci Am, 236(5), 119–127.
    Senn, S. (2008). Statistical Issues in Drug Development (Second). Wiley.