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Epidemiology:
doi: 10.1097/EDE.0b013e31823029dd
Letters

Splines for Trend Analysis and Continuous Confounder Control

Howe, Chanelle J.; Cole, Stephen R.; Westreich, Daniel J.; Greenland, Sander; Napravnik, Sonia; Eron, Joseph J. Jr

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Department of Epidemiology; Gillings School of Global Public Health; University of North Carolina; Chapel Hill, NC; cjhowe@email.unc.edu (Howe, Cole)

Department of Obstetrics and Gynecology and Duke Global Health Institute; Duke University Durham, NC (Westreich)

Department of Epidemiology and Department of Statistics; University of California, Los Angeles; Los Angeles, CA (Greenland)

Department of Epidemiology; Gillings School of Global Public Health; University of North Carolina; Chapel Hill, NC; Division of Infectious Diseases; Department of Medicine; University of North Carolina School of Medicine; Chapel Hill, NC (Napravnik)

Division of Infectious Diseases; Department of Medicine; University of North Carolina School of Medicine; Chapel Hill, NC (Eron)

Supported by National Institutes of Health grants P30 AI50410 and K99 HD063961.

Supplemental digital content is available through direct URL citations in the HTML and PDF versions of this article (www.epidem.com).

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To the Editor:

Spline regression often represents a less biased and more efficient alternative to standard linear, curvilinear, or categorical analyses of continuous exposures and confounders. Benefits of restricted cubic and quadratic splines have been described in the epidemiologic and biomedical literature.1–2 Analogous to the SAS (SAS Institute, Inc., Cary, NC) code provided by Harrell3 for estimating restricted cubic splines, we present straightforward SAS code for estimating restricted quadratic splines. Using data from the HIV clinical cohort at the University of North Carolina Center for AIDS Research,4 we illustrate use of restricted quadratic splines in regression modeling for trend analysis and control of a continuous confounder.

Details regarding the functional form of restricted quadratic splines as well as SAS code for estimating restricted quadratic spline functions are provided in the eAppendix (http://links.lww.com/EDE/A520). The data and SAS code used to generate the results included in this letter are also in the eAppendix.

First, we illustrate the use of a restricted quadratic splines when estimating the association between log10 HIV-1 viral load centered at 2.301 log10 copies/mL and mortality. The Figure shows the unadjusted association between centered log10 HIV-1 viral load at therapy initiation and the relative hazard of death estimated from several Cox proportional hazards models that (a) assume a log-linear relationship, (b) use indicators corresponding to quartiles of centered log10 HIV-1 viral load, or (c) include a restricted quadratic spline with 4 equal knots based on the case distribution.

FIGURE. Unadjusted a...
FIGURE. Unadjusted a...
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Based on the Akaike information criterion (AIC),5 presented in the Figure, the restricted quadratic spline model provides the best fit to the data. The P-value for a joint Wald test of the 3 restricted quadratic spline-basis functions included in the model was 0.010. The restricted quadratic spline model suggests a nonlog-linear relationship between centered log10 HIV-1 viral load at therapy initiation and the relative hazard of death.

Second, we illustrate the use of a restricted quadratic spline when controlling for centered log10 HIV-1 viral load as a confounder using a Cox model. The Table shows the hazard ratios for the association between an indicator of CD4 cell count ≤350 cells/mm3 at therapy initiation and the hazard of death, both unadjusted and adjusted for confounding by viral load at therapy initiation. Adjusting for viral load using a log-linear term attenuated the point estimate corresponding to the CD4 cell count indicator by 26%. Adjustment using a restricted quadratic spline with 4 equal knots based on the case distribution attenuated the point estimate by 30%. Attenuation upon control for viral load is expected given that higher viral load was associated with lower CD4 cell count (eAppendix, http://links.lww.com/EDE/A520), and an elevated risk of subsequent mortality. For both examples, a similar result was observed when a restricted cubic spline was used instead of restricted quadratic splines with the same degrees of freedom and comparable knot locations (eAppendix, http://links.lww.com/EDE/A520).

TABLE. Hazard Ratio ...
TABLE. Hazard Ratio ...
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For the first example, use of a restricted quadratic spline rather than a linear term or indicators provided a better fit, revealing a nonlinear relationship that otherwise may have not been apparent. In the second example, use of a restricted quadratic spline resulted in stronger attenuation of a crude association, which likely represents better control of confounding by viral load.

The macro presented here offers users a straightforward SAS option for implementing restricted quadratic splines regression. This code is intended to aid in model selection as well as assessing robustness of inferences when comparing various modeling strategies.3,6–7 Furthermore, we hope the examples and code will facilitate the use of splines among researchers hesitant to use less intuitive but largely equivalent modeling strategies,3,7 and in turn broaden the use of splines in applied epidemiologic research.

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ACKNOWLEDGMENTS

We thank Elizabeth Yanik for assistance with the UNC CFAR data, Petra Sander for reviewing the macro and example code, as well as participants, clinicians, investigators, and staff involved with the UNC CFAR HIV clinical cohort.

Chanelle J. Howe

Stephen R. Cole

Department of Epidemiology

Gillings School of Global Public Health

University of North Carolina

Chapel Hill, NC

cjhowe@email.unc.edu

Daniel J. Westreich

Department of Obstetrics and Gynecology and Duke Global Health Institute

Duke University Durham, NC

Sander Greenland

Department of Epidemiology and Department of Statistics

University of California, Los Angeles

Los Angeles, CA

Sonia Napravnik

Department of Epidemiology

Gillings School of Global Public Health

University of North Carolina

Chapel Hill, NC

Division of Infectious Diseases

Department of Medicine

University of North Carolina School of Medicine

Chapel Hill, NC

Joseph J. Eron, Jr

Division of Infectious Diseases

Department of Medicine

University of North Carolina School of Medicine

Chapel Hill, NC

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REFERENCES

1. Harrell FE Jr, Lee KL, Pollock BG. Regression models in clinical studies: determining relationships between predictors and response. J Natl Cancer Inst. 1988;80:1198–1202.

2. Greenland S. Dose-response and trend analysis in epidemiology: alternatives to categorical analysis. Epidemiology. 1995;6:356–365.

3. Harrell FE Jr. DASPLINE Macro. Available at: http://biostat.mc.vanderbilt.edu/twiki/pub/Main/SasMacros/survrisk.txt. Accessed April 12, 2011.

4. Howe CJ, Cole SR, Napravnik S, Eron JJ. Enrollment, retention, and visit attendance in the University of North Carolina Center for AIDS Research HIV clinical cohort, 2001-2007. AIDS Res Hum Retroviruses. 2010;26:875–881.

5. Akaike H. New Look at Statistical-Model Identification. Ieee Transactions on Automatic Control. 1974;Ac19:716–723.

6. Royston P, Ambler G, Sauerbrei W. The use of fractional polynomials to model continuous risk variables in epidemiology. Int J Epidemiol. 1999;28:964–974.

7. Desquilbet L, Mariotti F. Dose-response analyses using restricted cubic spline functions in public health research. Stat Med. 2010;29:1037–1057.

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© 2011 Lippincott Williams & Wilkins, Inc.

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