Biostatistics Advance Access originally published online on April 5, 2006
Biostatistics 2007 8(1):72-85; doi:10.1093/biostatistics/kxj034
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The logistic transform for bounded outcome scores
Biostatistical Centre, Catholic University of Leuven, U.Z. St. Rafaël, Kapucijnenvoer 35, B–3000 Leuven, Belgium emmanuel.lesaffre{at}med.kuleuven.be
* To whom correspondence should be addressed.
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SUMMARY |
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 TOP SUMMARY 1. INTRODUCTION2. THE LOGISTIC TRANSFORMATION...3. MODELING BOUNDED OUTCOME...4. SIMULATION STUDY5. APPLICATIONS6. ALTERNATIVE APPROACHES7. CONCLUSIONREFERENCES |
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The logistic transformation, originally suggested by Johnson (1949), is applied to analyze responses that are restricted to a finite interval (e.g.
), so-called bounded outcome scores. Bounded outcome scores often have a non-standard distribution, e.g. J- or U-shaped, precluding classical parametric statistical approaches for analysis. Applying the logistic transformation on a normally distributed random variable, gives rise to a logit-normal (LN) distribution. This distribution can take a variety of shapes on . Further, the model can be extended to correct for (baseline) covariates. Therefore, the method could be useful for comparative clinical trials. Bounded outcomes can be found in many research areas, e.g. drug compliance research, quality-of-life studies, and pain (and pain relief) studies using visual analog scores, but all these scores can attain the boundary values 0 or 1. A natural extension of the above approach is therefore to assume a latent score on having a LN distribution. Two cases are considered: (a) the bounded outcome score is a proportion where the true probabilities have a LN distribution on and (b) the bounded outcome score on 
is a coarsened version of a latent score with a LN distribution on . We also allow the variance (on the transformed scale) to depend on treatment. The usefulness of our approach for comparative clinical trials will be assessed in this paper. It turns out to be important to distinguish the case of equal and unequal variances. For a bounded outcome score of the second type and with equal variances, our approach comes close to ordinal probit (OP) regression. However, ignoring the inequality of variances can lead to highly biased parameter estimates. A simulation study compares the performance of our approach with the two-sample Wilcoxon test and with OP regression. Finally, the different methods are illustrated on two data sets.
Keywords: Barthel index; Bounded outcome scores; Compliance research; Logistic-transform, Ordinal probit regression
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1. INTRODUCTION |
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TOPSUMMARY1. INTRODUCTION2. THE LOGISTIC TRANSFORMATION...3. MODELING BOUNDED OUTCOME...4. SIMULATION STUDY5. APPLICATIONS6. ALTERNATIVE APPROACHES7. CONCLUSIONREFERENCES |
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Bounded outcome scores are measurements that are restricted to a finite interval, which can be closed, open, or half-closed. Examples of bounded outcome scores can be found in many medical disciplines. For instance, in compliance research one measures the proportion of days that patients correctly take their drug, hereafter denoted as "pdays." Another example is the Barthel index (Mahoney and Barthel, 1965
) which is an Activities on Daily Living scale that (in one version) jumps with steps of 5 from 0 (death or completely immobilized) to 100 (able to perform all daily activities independently). This scale is often used in stroke trials to measure the recovery of a patient after an acute stroke. Finally, in pain and pain-relief studies, a visual analog score (VAS) is used to measure the psychological state of the subject.
Bounded outcome scores show a variety of distributions, from unimodal to J- and U-shaped. These peculiar shapes often motivate the use of non-parametric methods, like the Wilcoxon test (Lesaffre and others, 1993) when comparing two treatments. However, possibilities for statistical modeling, e.g. when covariate adjustment is envisaged, are then limited. Alternatively, a dichotomized version of the score may be constructed and analyzed using logistic regression. For instance, the Barthel index could be split at 0.9. A value above 0.9 implies that the patients are able to perform most of their daily activities, and hence the dichotomized Barthel index has a simple interpretation. However, such an approach has two disadvantages; first the choice of the threshold is usually ad hoc and second reducing the score to a binary variable may reduce the efficiency of the comparison. Ordinal regression (McCullagh, 1980) is an alternative method to analyze bounded outcome scores although this ignores the numeric character of the data.
In this paper, we explore the use of the logistic transformation first suggested by Johnson (1949) to model the distribution of bounded outcome scores on (0,1). However, many outcome scores are defined on a closed interval. In this case, we assume here that a latent variable with range (0,1) gives rise to an observed score on [0,1]. Bounded outcomes on can be discrete or of a mixed continuous–discrete type. Here, we will concentrate on the first type and consider two cases: (1) a proportion where the true probabilities have a logit-normal (LN) distribution on and (2) a discrete score ranging from 0 to 1 interpreted as the grouped version of a continuous latent variable on . We call the first case the "binomial-logit-normal (BLN) approach" and the second case the "coarsening (CO) approach." A typical example of a continuous–discrete score on would be the visual analog scale, which takes values continuously on but can also give the extreme values 0 or 1 with a non-zero probability.
In Section 2, we indicate the usefulness of the logistic transformation for comparative clinical research with bounded outcomes on . We then present methods for analyzing bounded outcomes on and focus on the comparison of two treatments. We consider the cases of equal and unequal variances on the transformed scale. The competitor to the CO approach is the classical ordinal probit (OP) regression. We will show that OP regression is very similar to the CO approach for equal variances, but with unequal variances it may give a severely biased estimate of the treatment effect. Section 4 describes a simulation study evaluating the performance of our approaches for various distributions on in comparison with the two-sample Wilcoxon test and OP regression. Details of the simulation study are given in supplementary material available at Biostatistics online (http://www.biostatistics.oxfordjournals.org). In Section 5, we illustrate our method first on "pdays," the primary endpoint of the THAMES study, a recent compliance-enhancing intervention study performed in Belgium. Further, we re-analyzed the primary endpoint (Barthel index) of the European Cooperative Acute Stroke Study I study, an early placebo-controlled randomized clinical trial evaluating the effect of a thrombolytic drug on patients with an acute ischemic stroke. In Section 6, we look at distributions other than the LN, discuss some other approaches, and look at the goodness-of-fit of the LN distribution. Finally, in Section 7, we summarize our results and make some suggestions for further research in this area.
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2. THE LOGISTIC TRANSFORMATION AND ITS APPLICATION TO CLINICAL RESEARCH |
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TOPSUMMARY1. INTRODUCTION2. THE LOGISTIC TRANSFORMATION...3. MODELING BOUNDED OUTCOME...4. SIMULATION STUDY5. APPLICATIONS6. ALTERNATIVE APPROACHES7. CONCLUSIONREFERENCES |
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Johnson (1949)
suggested the logistic transformation 
, where 
and U is a score on the interval 
. The aim of Johnson was to achieve standard normality. In the case of proportions, 
and 
. Here we take 
and 
and assume that the logistic transformation achieves a normal density 
. In general, when Z has density 
, then U has density 
, where 
. When 
, then U has a LN distribution, denoted as 
and 
.
The LN distribution can take very different shapes depending on the choice of 
and 
, as is shown in Figure 1. Note that when changes sign this corresponds to mirroring the distribution around 
. Hence, the logistic transformation is very well suited to model a variety of distributions on . A similar property holds for the Beta family, but Aitchison and Begg (1976) indicate that the LN distribution is richer and can approximate any Beta density.
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It is clear that when the bounded outcome scores have a LN distribution, the analysis could be done on the Z-scale using classical statistical analyses assuming a Gaussian distribution. For instance, suppose we wish to compare the effects of control and new treatments based on a bounded score with distributions
and 
, respectively. When 
, a simple unpaired t-test can be calculated on the final Z-values and a 95% confidence interval can be obtained for 
(
0) on the Z-scale. The interpretation of is more difficult because it represents a location shift on the transformed scale. Since the logistic transformation is strictly monotone, 
, where 
and 
are the medians for the control and new treatment, respectively, on the original scale. Figure 2 gives an example of how the location-shift alternative on the Z-scale is translated into an alternative hypothesis on the observed U-scale, when 
. The parameter can also be interpreted in relation to the Wilcoxon–Mann–Whitney test (Lehmann and D'Abrera, 1998). Assume that 
and 
are independent random variables on the transformed scale corresponding to the control and new treatments, respectively, then 
, which is also equal to 
for the corresponding original U values. Brunner and Munzel (2000) called 
the "relative effect" of the treatment, which is therefore seen to be determined by the ratio 
. In general, the relative effect is equal to 
, where 
is the cumulative distribution function of 
or here equivalently of 
. Hence, loosely speaking, determines the proportion of individuals better off with the new treatment than with the control treatment. A 95% CI for can be obtained using the Delta method when estimates for , 
, and their covariance matrix are available. If instead transformation to a logistic distribution is envisaged, then can be directly interpreted as a log-odds ratio of cumulative distribution functions, see Section 6.
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When
, one could use the Welch test (Welch, 1951) on the transformed Z-scale. This test is also called the unpaired t-test for unequal variances. However, it is well known that ignoring the inequality of the variances, i.e. applying in this case the classical unpaired t-test, has no great impact on the type I error as long as 
(see, e.g. Wetherill, 1960; Murphy, 1967). Further, in this case the relative effect is equal to 
, which can be estimated in a similar manner as above.
The logistic transformation is useful for power and sample size calculations in a clinical trial with a bounded outcome score U as primary endpoint because the classical location-shift alternative is most often not appropriate. While power and sample size calculations are more difficult, they can be realized by first specifying the relative effect together with .
Finally, the logistic transformation is also useful in statistical modeling of bounded outcome scores on . Indeed, the logistic regression model
 | (2.1) |
with 
, has been used in various applications (Kieschnick and McCullough, 2003). This approach is especially useful in clinical trial applications when baseline covariate adjustment is envisaged. Finally, Expression (2.1) can easily be extended to allow to depend on covariates, as for example in Pourahmadi (1999).
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3. MODELING BOUNDED OUTCOME SCORES ON [0, 1] |
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TOPSUMMARY1. INTRODUCTION2. THE LOGISTIC TRANSFORMATION...3. MODELING BOUNDED OUTCOME...4. SIMULATION STUDY5. APPLICATIONS6. ALTERNATIVE APPROACHES7. CONCLUSIONREFERENCES |
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In this section, we primarily focus on bounded outcomes on
which we denote by 
to distinguish them from 
. Two different types of outcomes are considered here. First, the bounded outcome scores are observed proportions equal to 
(
), whereby 
is the ith count out of 
units. In this case, 
represents the true proportion measured with error by . For instance, in compliance research, 
, whereby is the number of days out of on which the ith patient has correctly taken the drug. Second, is a coarsened version of on . Typical examples can be found in Quality of Life research (e.g. the Barthel index) where denote sums of individual items each measuring an aspect of the subject's true quality of life . Such a score can be standardized to lie between 0 and 1 while taking a finite number of values.
When the bounded outcome score is a proportion derived from a series of conditionally (conditional on the subject) independent Bernoulli experiments, then an obvious choice is to work with a binomial distribution. Namely, in the BLN approach, we assume that
 | (3.1) |
with 
and say that has a BLN distribution. For each value of , one observes binary outcomes 
(
) summing up to . For the compliance example, the rec- orded adherence is the observed proportion of days that the patients take their medication correctly (with respect to dosage and timing) in a period of days. In this case, the could be interpreted as the (true but unobserved) latent adherence of the ith patient to the drug. Observe that this model is actually a classical measurement error model (Carroll and others, 1995), specifying the distribution 
.
Model (3.1) can be extended by replacing by 
to give a "generalized linear mixed-effects model," whereby conditional on the are assumed to be independent. As indicated above, a further extension allows to depend on covariates. Fitting such a model can be done with, e.g. the SAS procedure NLMIXED or the function "lmer()" in package "lme4" in R Development Core Team (2005).
When is a discrete random variable on (e.g. the Barthel index), but not a proportion, then it is natural to assume that is a grouped version of . As a specific example, could have been realized from a CO mechanism such that 
when 
, where 
. At the boundaries, 
when 
and 
when 
. The above grid of boundary values is equal for all subjects and is denoted below as
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However, our approach also allows a grid varying with the subjects.
The framework of coarsened data has been formalized by Heitjan and Rubin (1991) and Heitjan (1993). In their terminology, we consider here only deterministic CO. More formally, we assume that 
when is recorded. For the likelihood this implies the following expression:
 | (3.2) |
where 
is the probability density function of the LN distribution. This leads to the likelihood
 | (3.3) |
where 
, 
, and 
is the distribution function of the standard normal distribution. At the boundaries, i.e. 
and 
, the values of 
become 
and 
, respectively. When depends on covariates, the above expression needs to be adapted. For instance, in the two-group comparison Expression (3.3) splits up in two parts, one with 
(first treatment) and the other with 
(second treatment). For obvious reasons, we have called this the "CO approach" either assuming equal variances or allowing unequal variances.
The maximum likelihood estimates for this model can easily be obtained using standard numerical optimization procedures such as the Broyden, Fletcher, Goldfarb, and Shanno (BFGS) algorithm (Lange, 2004).
An interesting feature of the CO model is its resemblance to ordinal regression models. Bounded outcomes could also be regarded as ordinal response variables taking values in 
, where . The OP regression model (McCullagh, 1980) specifies that the following equation should hold:
 | (3.4) |
where 
are unknown ordered cut points that need to be estimated from the data.
Suppose that, after standardization to the interval, satisfies the CO model of Section 3.2 and does not depend on covariates. Then (3.2) and (3.3) imply that
 | (3.5) |
with 
. Consequently, with 
and 
, the CO model equates to a classical OP regression model. However, in the CO model the cut points are known up to a translation and scale factor, whereas in the OP regression model they need to be estimated under constraints. Since maximum likelihood estimators are consistent, 
for large n so that in this case the two approaches will give similar parameter estimates, up to a proportionality factor. One might expect the CO approach to be more efficient, since only two parameters need to be estimated, rather than 
cut points. We give some numerical results in Section 4.
When depends on covariates, e.g. when for the two treatment groups, the CO model as specified by Expression (3.3) will give biased parameter estimates and thus also a biased estimate of the treatment effect. The same is also true for OP regression. In fact, for a binary the true treatment effect is equal to 
, where 
is the regression coefficient of the binary treatment indicator (0 for control and 1 otherwise). On the other hand, the estimate from the OP regression model will converge to 
, where 
when no covariate adjustment is involved. When covariates are involved their regression coefficients will also be distorted. In Section 3.2, we have seen that we can adapt the CO approach quite easily to allow for unequal variances. The OP regression model can also be extended to accommodate this more general case, as has been done by Johnson and Albert (1999). Their generalization is quite similar to the CO model with a variance depending on covariates, but cannot be fitted using standard software.
When the cut points differ between individuals, as in Heitjan and Rubin (1991) and Heitjan (1993), Expression (3.3) can also be extended easily. The corresponding generalization of the OP regression model again leads to the approach of Johnson and Albert (1999).
Finally, when the bounded outcome score is a proportion, we have suggested to fit the data using the BLN model. Clearly, in this case the OP regression model is not equivalent to our approach because it does not take into account the variability with which the proportions are determined.
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4. SIMULATION STUDY |
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TOPSUMMARY1. INTRODUCTION2. THE LOGISTIC TRANSFORMATION...3. MODELING BOUNDED OUTCOME...4. SIMULATION STUDY5. APPLICATIONS6. ALTERNATIVE APPROACHES7. CONCLUSIONREFERENCES |
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In this section, we describe a simulation study that we have performed to evaluate our proposals in Sections 3.1 and 3.2 for analyzing bounded outcome scores on
in the two-group situation and under the location-shift alternative on the transformed scale. For more details, we refer to supplementary material available at Biostatistics online (http://www.biostatistics.oxfordjournals.org).
We have divided the simulation study according to the type of the bounded outcome score on . For a proportion, we have compared the Wilcoxon test and the BLN approach, and in some cases also the OP regression model, despite it being not strictly appropriate for proportions. For coarsened data, we have compared the Wilcoxon test, the OP regression model, and the CO approach. A variety of scenarios were considered, all involving two-group comparisons.
One of the main purposes of the simulation study is to show that including covariates can greatly increase the power in detecting a treatment effect when dealing with bounded outcome scores. Therefore, we considered cases with and without covariates. Further, we included a variety of distributions on . Three different treatment effects were evaluated, which could be classified as low, moderate, and large. Finally, we also varied the study size, but in all cases specified 
.
For a proportion, we compared the probability of the type I error and the power of the three approaches. For coarsened data, we additionally determined the estimated treatment effect, except of course for the Wilcoxon test.
For coarsened data on , we considered both the case of equal variances (
) and unequal variances (), specifically 
and 
. Consequently, we included two versions of the CO approach: (a) assuming equal variances (CO1 approach) and (b) allowing for unequal variances (CO2 approach). While OP regression is a popular approach in this setting, the possibility of unequal variances is often neglected. Therefore, we have included the additional case of unequal variances for coarsened data to highlight the impact of ignoring inequality of variances. On the other hand, for proportions the BLN approach is actually the only appropriate method, and can also be easily extended to the case of unequal variances. Thus, in this case extensive empirical comparison with OP regression is unnecessary.
To determine the performance of the different approaches, we used the following software—(a) Wilcoxon test: the R-function wilcox.test(), (b) OP regression: R-function polr() from package MASS, (c) BLN approach: generalized mixed-effects model (SAS proc NLMIXED and R-function lmer() from package lme4), and (d): CO approach: R-function grouped() from package grouped written by the authors (available from CRAN http://cran.r-project.org).
When no covariates are involved, the performance of the Wilcoxon test is nearly identical to that of the BLN model. The Pr(type I error) is close to
for all cases even for a sample size of 
. Further, we observed the expected positive association of the power with the sample size and the effect size . However, the power also seems to depend on the shape of the distribution. In particular, we observed that the U-shaped distribution yields in general a higher power, followed by the unimodal and the J-shaped distributions. An explanation of this phenomenon lies in the fact that the latent score (true probability of success) is not known but only the observed proportion 
. When all true proportions are relatively close to 0 or 1, the observed proportions will be relatively close to each other. A proof of this is seen in the power of the Wilcoxon test which shows a similar behavior.
When a significant baseline covariate is included, the Wilcoxon test had a much lower power than the covariate adjusted parametric models, and mainly as the effect of the covariate and increases. Finally, the BLN and OP models behaved similarly with a slight inferiority for the latter.
First we summarize the results when there are no covariates. When
, the type I error was well preserved for all approaches. Further, overall the power of the CO2-approach was less than for the other approaches, which is natural because the other approaches are developed under the assumption of equal variances. In all cases, the treatment effect was estimated without bias. When , the type I error was well preserved for the CO2-approach, but was sometimes severely increased for the other approaches. For the CO1- and the OP regression models, the reason is that the treatment effect is sometimes estimated with a large bias. The anti-conservative character of the Wilcoxon test is explained by its relationship with ordinal logistic regression (McCullagh, 1980). The power of the CO2-approach was sometimes much less than for the other approaches, but this can be explained by their anti-conservative character. Indeed, the power of the other approaches was even higher than the corresponding power obtained from the Welch test, i.e. when no CO is involved. When covariates are available, the first and obvious conclusion is that the power can be greatly improved depending on the relationship of the covariates with the response. Apart from that, the conclusions are similar to those reported.
We expected the BLN approach, and especially the CO approach, to be more powerful than OP regression since the latter requires more parameters to be estimated. However, the simulation results showed only small differences. We attribute this to the low correlation of all estimated cut points (except for
) with the estimated regression parameters in the OP regression model.
When the variances are unequal in the treatment groups, our simulations indicated that the Wilcoxon test, the CO1 approach, and classical OP regression yield seriously distorted type I errors. The two latter approaches also produced severely biased treatment effects, even with . In Wetherill (1960), theoretical calculations revealed that the type I error and the power of the Wilcoxon test are fairly insensitive to unequal variances provided . However, we observed that when the data are coarsened the performance of the Wilcoxon test is severely affected even when the sample sizes are equal, probably due to the large number of ties. The sensitivity of the CO1 approach and classical OP regression is explained by the fact that for non-linear models misspecification of the (co)variance structure has an impact on the correct estimation of the mean parameters (see, e.g. Butler and Louis, 1992).
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5. APPLICATIONS |
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Recently, an open-label, multicenter compliance-enhancing intervention (THAMES) study was completed in Belgium to measure the effect of a program of pharmaceutical care, designed to enhance adherence to atorvastatin treatment. Four well-defined districts were identified, two in Flanders (northern Belgium) and two in Wallonia (southern Belgium). In both Flanders and Wallonia, all pharmacists in one of the districts were to apply measures to improve compliance and enhance persistence, whereas in the second district no such measures were taken. There were 187 patients in the intervention group and 182 patients in the control group. All pharmacists were equipped with the Medication Electronic Monitoring System (MEMS) system, an electronically monitored pharmaceutical package designed to compile the dosing histories of ambulatory patients taking oral medications (Urquhart, 1997). The total study duration was 12 months. The number of visits to the pharmacy ranged from 5 to 13. At each visit, the patient's dosing history was checked by means of the electronic monitoring system. The period between the first and second visit was considered to be the baseline period. More details on the setup of this intervention study can be found in Vrijens and others (2006).
The primary efficacy parameter of the THAMES study was adherence to prescribed therapy in the post-baseline period, whereby adherence was defined for each patient as the proportion of days during which the MEMS record showed that the patient had opened the pill container correctly. This variable was also estimated at baseline (baseline adherence). Finally, for the calculation of the "post-baseline adherence" the post-baseline period was arbitrarily cut off at day 300.
The THAMES study could not be randomized due to practical difficulties. Therefore, we need to compare the baseline covariates of the intervention and control groups. In Table 1, we compared gender, age, weight, work status (unemployed versus employed), a cardiovascular risk score (Vrijens and others, 2006), family history of CHD, and the pdays at baseline with the appropriate statistical techniques. The adherence at baseline is of particular interest and the Wilcoxon test gives a significant result (
). The reasons for this significant difference at the start are not clear, but it requires that the imbalance at the start needs to be taken into account. We were aware of the potential dangers of correcting for baseline covariates in the presence of imbalance at baseline (Wainer and Brown, 2004). However, for illustrative purposes we still performed an analysis of covariance type of analysis in Section 5.2.
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Initially, we checked for a treatment effect without correcting for any baseline covariates. Both the Wilcoxon test and the BLN model gave a significant intervention effect with
. Figure 3 shows histograms for the two treatment groups with superimposed kernel estimates and fitted LN distributions. The LN distributions provide a good fit to the observed data. Since 
, and 
, the estimated effect size 
is equal to 
with 95% CI 
, supporting the intervention effect. The same conclusion can be drawn from 
with 95% CI 
.
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We then re-analyzed the data taking into account baseline covariates, including the logit of baseline adherence. Table 2 shows the results for the BLN model. The effect of intervention and baseline adherence are both highly significant (
). The estimated value of decreased from 
to 
after taking into account the imbalance at baseline. Further, the estimated value of decreased from 
to 
because the inclusion of covariates decreased the residual variability. As a result, the intervention effect remained at 
with 95% CI 
and with 95% CI 
. An additional analysis, allowing for unequal variances for the latent adherence score, resulted in practically identical parameter estimates and almost equal variances.
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Thus, we can conclude that the intervention significantly improves the adherence of the patients despite the fact that the two groups differed in adherence already at baseline. Finally, we did not fit an OP regression model, for reasons stated above.
The ECASS-1 stroke study is a double-blind randomized parallel study designed to compare the effect of alteplase (a thrombolytic drug) and placebo in patients with an acute ischemic stroke. In total, 545 patients were randomly allocated to the alteplase arm (268 patients) and to the control arm (277 patients). The primary outcome is patient status at 3 months assessed by the Barthel index standardized such that the values lie in [0, 1]. The score could be considered as a reflection of a latent scale which measures the ability to cope with a handicap resulting from, say, a cerebrovascular stroke. In this way, it is plausible that the latent score for most if not all patients is less than 1. This is medically supported since an observed score of 1 does not necessarily imply complete neurologic recovery of the patient. Also, patients who survived a stroke with an observed score of zero may have a true latent score close to, rather than equal to, zero, whereas non-survivors can be considered as having a true score of zero. One way to treat "death" in the analysis is to regard it as a separate class and hence to distinguish it from the zero values of survivors. However, in a clinical trial context this approach does not give a clear picture of the overall effect of the treatment. For this reason, we prefer to analyze the Barthel index using the approach outlined in Section 3.2.
In accordance with Lesaffre and others (1993), we first performed a Wilcoxon test. This gives a non-significant treatment effect (
). Observe that this analysis addresses the more practical question of how beneficial the thrombolytic treatment is overall, combining ability with mortality.
To apply the CO model, we assumed for the Barthel index the CO mechanism outlined in Section 3.2, with 
. Without baseline covariates the CO model assuming equal variances gives 
and 
yielding a small, non-significant effect size, namely 
(95% CI 
). The estimated relative effect is also close to 
, i.e. 
with 95% CI of 
. However, there is strong evidence that the two variances are unequal. A likelihood ratio test comparing the CO1 with the CO2 model is highly significant (
). The two variances are estimated as 
(control) and 
(alteplase). The treatment effect now becomes 
, with 95% CI 
, 
. We again observe that histograms for the two treatment groups (figure not shown) are well fitted by LN distributions now assuming unequal variances.
In a second step the baseline covariates gender and age are introduced into the model. Again, there is evidence for unequal variances. The p-value for the likelihood ratio test is now 
. Table 3 presents the results of the CO2 approach. The two standard deviations are now estimated as 
(control) and 
(alteplase). However, while the estimate of remained at 
, including the covariates resulted in a significant treatment effect (
). We do not wish to overemphasize the significant result of the treatment effect. There does seem to be some effect of alteplase on average, but treating acute stroke patients with alteplase may also give more variable outcomes, either very bad (completely disabled or died) or very good (completely independent of others). The estimated relative effect now becomes 
with 95% CI of 
.
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Finally, we also performed an OP regression. Nineteen cut points needed to be estimated. With the same covariates, the estimated treatment effect is equal to
with 95% CI of 
and p-value equal to 
. Hence, the results of the OP regression model and the CO1 approach are quite close, but they differ considerably from the CO2 approach. |
6. ALTERNATIVE APPROACHES |
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TOPSUMMARY1. INTRODUCTION2. THE LOGISTIC TRANSFORMATION...3. MODELING BOUNDED OUTCOME...4. SIMULATION STUDY5. APPLICATIONS6. ALTERNATIVE APPROACHES7. CONCLUSIONREFERENCES |
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In this section, we will consider only the case of equal variances. When the logistic transformation yields a logistic distribution on the z-scale, then the effect size
can be interpreted as a log-odds ratio for the cumulative distributions on the z-scale, i.e.
 | (6.1) |
where 
and 
are the cumulative logistic distributions under 
and 
, respectively. Thus, a generalization of the proportional odds model to continuous data is obtained. Further, a more robust approach would be to use the logistic t-distribution instead of the LN (Lange and others, 1989). Of course, for scores on a part of the attractiveness of the approach is lost because we would need to replace classical techniques, like the t-test, by more sophisticated ones.
Another extension to the LN distribution is obtained by changing the logistic transformation. Aitchison and Lauder (1985) suggested a Box-Cox transformation. For the analysis of clinical trial data, we suspect that this extra flexibility has little to offer. Trigonometric functions like arctan (suitably scaled to ) provide another class of transformations to normality. Dexter and Chestnut (1995) experimented with the arc sine square root transformation for the analysis of VAS pain data. This is an interesting transformation since it attains its boundary values. Finally, one referee pointed out that OP regression allows the possibility that an arbitrary transformation of the underlying distribution is normally distributed.
A rather different approach has been suggested in Quality of Life Research (Grootendorst, 2000). In this approach a classical distribution, such as the normal distribution, is assumed to be censored at the boundaries 0 and 1 on the original scale.
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7. CONCLUSION |
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TOPSUMMARY1. INTRODUCTION2. THE LOGISTIC TRANSFORMATION...3. MODELING BOUNDED OUTCOME...4. SIMULATION STUDY5. APPLICATIONS6. ALTERNATIVE APPROACHES7. CONCLUSIONREFERENCES |
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The logistic transform is one of the most used transformations in statistics. Hence, we do not claim originality when proposing the BLN and CO approaches. However, we argue that the strategy to analyze bounded outcome scores as laid down in this paper has some advantages over other well-established approaches, like OP and logistic regression. First, our strategy makes the distinction between the two types of bounded outcome scores. Secondly, we believe that explicitly drawing the relationship with a latent normal variable on the transformed scale will help practitioners in planning a clinical trial even when they decide to use an ordinal regression model for the analysis. Third, our approach is quite flexible. The extension to unequal variances could be quite important in practice. Other extensions, like allowing for a varying grid of cut points, are also easy to do. Finally, we have developed an approach to perform sample size calculation based on the CO approach, see Tsonaka and others (2005). Given that the power of the OP regression model is close to that of the CO approach in the case of equal variances, we believe that this approach is also useful for sample size calculations for ordinal regression.
With respect to further research, we believe that models for repeated bounded outcome scores either using a multivariate approach as suggested by Aitchison and Shen (1980) or incorporating random effects could be useful. In this context, it would also be useful to draw the connection with random-effects OP regression. Further, it is of interest to develop an approach which can handle a bounded outcome score as a covariate in a regression model. Recently, Liang and others (2005) have published a related paper that deals with a bounded covariate. Finally, we intend to explore various procedures to analyze VAS data, again allowing the variance to depend on covariates.
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ACKNOWLEDGMENTS |
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The authors acknowledge support from the Interuniversity Attraction Poles Program P5/24—Belgian State—Federal Office for Scientific, Technical, and Cultural Affairs. The authors also thank Pfizer Belgium for the permission to use the data of the THAMES study and Boehringer Ingelheim for the permission to use the ECASS-1 data. Finally, the authors also acknowledge the comments of a referee, the associate editor, and the editor which improved the readability of the paper considerably. Conflict of Interest: None declared.
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TOPSUMMARY1. INTRODUCTION2. THE LOGISTIC TRANSFORMATION...3. MODELING BOUNDED OUTCOME...4. SIMULATION STUDY5. APPLICATIONS6. ALTERNATIVE APPROACHES7. CONCLUSIONREFERENCES |
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Received September 30, 2004; revised June 8, 2005; revised January 17, 2006; revised February 17, 2006; accepted for publication March 16, 2006.
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