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Interactions Between Treatment and Continuous Covariates: A Step Toward Individualizing Therapy

Cancer and Statistical Methodology Groups, Medical Research Council Clinical Trials Unit, London, United Kingdom

Willi Sauerbrei

Institute of Medical Biometry and Informatics, University Medical Center Freiburg, Freiburg, Germany

Interactions between treatment and covariates, such as prognostic factors, in randomized oncology trials are important because they are essential ingredients of individualized cancer treatments rather than assuming that one size fits all. Such factors are often called predictive. When the covariate is continuous (such as age or hormone receptor level), such interactions are often unfortunately sought by crude statistical methods, typically involving dichotomizing the continuous covariate. Sometimes, treatment effects in subgroups are compared; the results may depend on the cut point chosen. A formal test of interaction is needed, but often omitted.¹ Methods that keep all of the information in the covariate are more powerful than dichotomization. We therefore welcome the use of one such method, Subpopulation Treatment Effect Pattern Plot (STEPP), reported in this issue of the Journal of Clinical Oncology by Viale et al.² At the same time, we would like to draw readers' attention to an alternative approach, multivariable fractional polynomial interaction (MFPI), that we feel has much to offer.

Viale et al² investigate the predictive value of estrogen receptor (ER) and progesterone receptor (PgR) levels in node-negative breast cancer patients treated either with chemoendocrine or endocrine adjuvant therapy alone, in the randomized controlled trials VIII and IX of the International Breast Cancer Study Group (IBCSG). ER and PgR levels, measured by immunohistochemistry, were available for 2,248 patients. The authors used the Subpopulation Treatment Effect Pattern Plot (STEPP)^3,4 to explore the association between hormone receptor levels and disease-free survival (DFS) at 5 years. The 5-year DFS rate in each subpopulation was plotted against receptor levels for each treatment arm. If a covariate has no effect, 5-year DFS rates should be similar for all subpopulations, with any differences resulting from chance alone.

The plots used by Viale et al² are one way of exploring prognostic effects of continuous variables nonparametrically, but a more relevant question is that of an interaction between treatment and a continuous prognostic factor. In such a situation, the difference in 5-year DFS rates between treatments would vary according to the value of the covariate. One example of an interaction is when the DFS rate for treatment A increases for larger ER values (ER has a prognostic effect in these patients), whereas for treatment B, no such change occurs (ER has no prognostic effect, and the corresponding STEPP plot is roughly a horizontal line). The magnitude of the treatment effect (A –B), expressed as the difference in 5-year DFS rates, increases with the value of the covariate. A test for interaction is available and can give better guidance as to whether the differences from the graph represent a statistically significant interaction. The authors present six comparisons in different situations, one of which indicates a significant interaction between ER and the effect of CMF (cyclophosphamide, methotrexate, fluorouracil) in trial IX. In this trial, postmenopausal patients received 5 years of tamoxifen treatment. Additional therapy with three cycles of CMF improved 5-year DFS in patients with low ER values, whereas in patients with high ER values, 5-year DFS rates were similar whether or not CMF was administered.

How does STEPP work? STEPP is mainly intended to identify interactions between treatment and a continuous covariate of interest, avoiding dichotomization of the covariate. The observations are divided into many subpopulations based on the covariate. As illustrated in Figure 1, the treatment effect is estimated separately within each subpopulation. To increase the number of patients that contribute to each treatment-effect estimate, subpopulations overlap. This makes the individual estimates more precise.

View larger version (13K): [in this window] [in a new window] [PowerPoint Slide for Teaching]   Fig 1. Schematic depiction of the two types of subpopulations used in the Subpopulation Treatment Effect Pattern Plot (STEPP): (A) sliding window and (B) tail-oriented. The horizontal axis indicates the various subpopulations for which treatment or covariate effects are estimated. The vertical axis shows the range of covariate values used to define the cohort of patients included in each subpopulation (shaded areas). The tail-oriented variant has the overall population as the center group. Adapted from Sauerbrei et al.8

Subpopulations in the SW variant have an overlapping portion and a part that differs between neighboring subpopulations. The numbers of subpopulations and of overlapping patients are important parameters of this variant, and must be chosen by the investigator. Viale et al evidently used the SW variant for their analyses, taking 90 and 80 patients in each subpopulation and 15 and 10 overlapping patients in their ER analyses of trials VIII and IX, respectively.

The entire sample forms the central group in the TO variant. With increasing distance from the center, more and more patients with high covariate values (to the left side) or low covariate values (to the right side) are removed. The TO variant has a parameter called g, giving g –1 subpopulations from which patients with larger values are eliminated and the same number from which patients with smaller values are excluded. The total number of subpopulations is 1 + 2(g –1) = 2g –1. In the graph g = 5.

If a covariate has no prognostic influence on the outcome in any treatment arm (therefore also no predictive effect), the outcome is similar in all subpopulations. The STEPP plot then consists of a set of joined-up, roughly horizontal lines for all treatments, with small steps between them resulting from chance. The vertical difference between the lines for two treatments in each subpopulation is an estimate of the treatment effect for the corresponding range of covariate values, and can itself be plotted as a roughly horizontal line (this difference plot was not provided by Viale et al²). If an interaction is present, the STEPP plots for the prognostic effect in the different treatment arms differ in shape. A prognostic effect has to be present in at least one of the treatment arms for the plot showing differences between the treatments to be no longer horizontal. Plots with CIs showing the estimated treatment effect in the subpopulations are suggested for graphical investigation of an interaction. Tests of interaction, based on the deviation of treatment effects in the subpopulations from those in the overall population, are available.

The main selling points of STEPP are that it avoids dichotomization of the covariate and makes no assumptions about the nature of the relationship between the outcome and the covariate in each treatment group. The nonparametric nature of STEPP also implies that the relationship between the size of the treatment effect and the value of the covariate is modeled flexibly. A test of interaction, that is of whether the treatment effect depends on the covariate, is available.⁴ Viale et al used 5-year DFS rates as the outcome, but any reasonable summary statistic for survival data (eg, hazard ratios from a Cox model, with or without adjustment for other covariates) can be used. In essence, STEPP is an exploratory technique whose main output is graphical. But it has several disadvantages. There are two variants to choose between (described in more detail later in this editorial). The size of the subpopulations is critical to the performance of the method and, hence, to the interpretation of the results. Probably most important, as a graphical method, it does not provide a usable, quantitative estimate of a treatment-covariate interaction.

What alternatives or adjuncts to STEPP would we recommend? Definitely not simplistic approaches based on dichotomization. A method we recently proposed to detect and estimate treatment-covariate interactions is the MFPI procedure.⁵ First, MFPI estimates for each treatment group a fractional polynomial function (eg, Sauerbrei et al⁶) representing the prognostic effect of the covariate, optionally adjusting for other covariates. Second, the difference between the functions for the treatment groups is calculated and tested for significance. A plot of the difference (eg, log hazard ratio) against the covariate, together with a 95% CI, is termed a "treatment-effect plot." A treatment-effect plot for a continuous covariate not interacting with treatment would be a straight line parallel to the x-axis, whereas a treatment-covariate interaction would be indicated by an increasing or a decreasing line or curve.

Using MFPI, we identified an interaction between white cell count and treatment in the Medical Research Council (MRC) RE01 trial for patients with metastatic renal carcinoma (Fig 2). According to the model, patients with low white cell counts benefit from interferon-, whereas those with higher white cell counts do not. Is this result an artifact of the modeling approach, or is it really in the data? Comparing Kaplan-Meier estimates of survival in each treatment arm in four subgroups defined by white cell count (Fig 3 of Royston et al⁷) is a simple and effective check, and confirms the interpretation of the treatment-effect plot. A trend of estimated unadjusted and adjusted hazard ratios across the subgroups ordered by the covariate values provides further confirmation.

View larger version (9K): [in this window] [in a new window] [PowerPoint Slide for Teaching]   Fig 2. Treatment-effect plot for white cell count in the Medical Research Council (MRC) RE01 renal cancer trial. The curve shows the relative hazard (on a logarithmic scale) for interferon- versus medroxyprogesterone acetate treatment at different values of white cell count, with pointwise 95% CI (shaded area), derived by multivariable fractional polynomial interaction (MFPI). Squares denote estimates of the treatment effect in subpopulations of WCC, derived by Subpopulation Treatment Effect Pattern Plot (STEPP; tail-oriented with g = 6, ie, 11 subpopulations). Adapted from Sauerbrei et al.8

The main advantage of MFPI compared with STEPP is that it produces a smooth curve with CI for the treatment-covariate interaction that is easy to interpret and to describe mathematically for others to use. The software can search several covariates for interactions, and can produce treatment-effect plots and tests of interaction.

In conclusion, we view the search for interactions between treatments and (continuous) prognostic factors in clinical trials to be an important activity of definite clinical relevance. It therefore requires the best available statistical methods, which excludes dichotomization.⁹ Quantitative description and reporting of interactions requires suitable methodology. We believe MFPI is a good candidate, while recognizing that further refinement and exploration of its properties in a wide range of data sets and statistical simulation studies would be beneficial. Use of STEPP is a move in the right direction because it is flexible and avoids dichotomization. We believe an appropriate role for STEPP is as an exploratory technique, and as a method to check interactions detected by MFPI.

Of course, general issues of research into treatment-covariate interactions, such as increased type I error from multiple testing or difference in interpretation of the results of preplanned and exploratory analyses, also apply here. In this regard, the white cell count interaction in the MRC RE01 trial generates an interesting hypothesis needing confirmation from similar trials.

AUTHORS' DISCLOSURES OF POTENTIAL CONFLICTS OF INTEREST

The author(s) indicated no potential conflicts of interest.

AUTHOR CONTRIBUTIONS

Manuscript writing: Patrick Royston, Willi Sauerbrei

Final approval of manuscript: Patrick Royston, Willi Sauerbrei

REFERENCES

1. Assmann SF, Pocock SJ, Enos LE, et al: Subgroup analysis and other (mis)uses of baseline data in clinical trials. Lancet 355:1064-1069, 2000[CrossRef][Medline]

2. Viale G, Regan MM, Maiorano E, et al: Chemoendocrine versus endocrine adjuvant therapies for node-negative breast cancer: Predictive value of centrally reviewed expression of estrogen and progesterone receptors—International Breast Cancer Study Group. J Clin Oncol 26:1404-1410, 2008[Abstract/Free Full Text]

3. Bonetti M, Gelber RD: A graphical method to assess treatment-covariate interactions using the Cox model on subsets of the data. Stat Med 19:2595-2609, 2000[CrossRef][Medline]

4. Bonetti M, Gelber RD: Patterns of treatment effects in subsets of patients in clinical trials. Biostatistics 5:465-481, 2004[Abstract]

5. Royston P, Sauerbrei W: A new approach to modelling interactions between treatment and continuous covariates in clinical trials by using fractional polynomials. Stat Med 23:2509-2525, 2004[CrossRef][Medline]

6. Sauerbrei W, Royston P, Bojar H, et al: Modelling the effects of standard prognostic factors in node positive breast cancer. Br J Cancer 79:1752-1760, 1999[CrossRef][Medline]

7. Royston P, Sauerbrei W, Ritchie A: Is treatment with interferon-alpha effective in all patients with metastatic renal carcinoma? A new approach to the investigation of interactions. Br J Cancer 90:794-799, 2004[CrossRef][Medline]

8. Sauerbrei W, Royston P, Zapien K: Detecting an interaction between treatment and a continuous covariate: A comparison of two approaches. Comp Statist Data Anal 51:4054-4063, 2007[CrossRef]

9. Royston P, Altman DG, Sauerbrei W: Dichotomizing continuous predictors in multiple regression: A bad idea. Stat Med 25:127-141, 2006[CrossRef][Medline]

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This article has been cited by other articles:

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