Evaluating a Variable as a Proxy for another Measure: Assessing the Step Test Exercise Prescription...

Evaluating a Variable as a Proxy for another Measure: Assessing the Step Test Exercise Prescription as a Proxy for the Maximal, High-intensity Peak Oxygen Consumption in Older Adults

Jonathan D. Mahnken, Ph.D., PStat®

Associate Professor of Biostatistics

The University of Kansas Medical Center

Director, Data Management and Statistics Core

The University of Kansas Alzheimer’s Disease Center

© 2013 Jonathan D. Mahnken. All Rights Reserved. 1

In Collaboration with…

Xueyi Chen, Ph.D. Alexandra R. Brown, M.S. Eric D. Vidoni, Ph.D. Sandra A. Billinger, Ph.D.


Acknowledgement The University of Kansas Alzheimer’s Disease

Center (P30 AG035982) for support of the Data Management and Statistics Core

Aim 2: Provide Statistical Expertise for ADC Projects

Aim 2a: Support Study Design, Oversight, and Data Analyses

Aim 2b: Participate in the Preparation of Study Presentations and Publications

Aim 2c: Develop Novel Statistical Methodology to Ensure Proper Interpretation of ADC Data


Motivating Example KU ADC thematic focus on modifiable lifestyle risk

factors and AD prevention Maximal, high-intensity peak oxygen consumption (VO2

peak) an important measure of fitness Interested in whether a lower-impact, sub-maximal Step

Test Exercise Prescription, or STEP, could serve as valid proxy

Less equipment required: staircase and stopwatch Likely to facilitate better subject recruitment for AD research

Validity of STEP as a proxy for VO2 peak in older adult population unclear


Statistical Definition


(Kotz and Johnson eds., 1986, p 323)

Valid Proxy For a variable, , to serve as a valid proxy

measure for another (the original) variable, , the relationship between these two variables would be approximately


Valid Proxy (cont.)


Valid Proxy (cont.) Using simple linear regression…

is random error, with most of its distribution falling at or near zero

More formally, where • Proxy is unbiased when • Ideally, also, small; but focus of this work on bias

only


Underlying Distributions of and

Using simple linear regression, neither nor need to be normally distributed—or even be continuous… …so long as the residual from is

approximately normally distributed


Test of Unbiased Proxy

test (herein the “linear transformation” or “2 df” approach)

Other approaches to the problem One-sample paired test

• Take differences of the original () and proxy () and used

• Two-sided inference equivalent to test (herein the “paired test” or “1 df” approach)


Linear Transformation To compare the approaches, define…

(say)

• Re-parameterization also facilitates conducting this test in SAS PROC GLM (original reason for transformation)

It can be shown that and Analogous inferences can be derived


Linear Transformation (cont.) For where

where Under , so If proxy is biased (), where

Note that , so


Linear Transformation (cont.) Power is then > …so, power depends on…

Sample size () Type I error level () True (unknown) parameters (, , and ) Design matrix (but only and needed)

…because

Power can be determined given these parameters Common parameters required for power calculations


Paired Test Again define…

(say)

• So the design matrix on the r.h.s. is (say)

It can be shown that (say ) True, unknown mean

So


Paired Test (cont.) For where

Under , so If proxy is biased (), where

Recall that Shortcomings because equivalent to


Paired Test (cont.) Power is then > …so, power depends on…

Sample size () Type I error level () True (unknown) parameters (, , and ) Design matrix (but only needed)

…because Power can be determined given these parameters

Common parameters required for power calculations Note that is not part of this power function


Comments on Comparing the Power Functions Recall that

…so • Assuming (seems reasonable) implies

Linear transformation (2 df) approach has non-centrality parameter as large or larger implies greater power for 2 df approach…

…but paired test (1 df) approach has fewer numerator df—estimates fewer parameters, which implies greater power for 1 df approach


So which approach

should we use?

Compare Power Curves Vary the parameters of the original relationship

for each figure…

Figures vary , , and ( not varied) ROC-type curves (type I error by power)



Power = type I error when unbiased


When , 2 df approach better


When , 2 df approach slightly worse


For 1 df approach, all curves overlap because not included in


Slight advantage for 1 df approach when close to zero (light-dashed)


When (only linear bias), 2 df approach better


When , 2 df approach better


When (only linear bias), 2 df approach better

Sample size had NO impact on power

Motivating Example


Motivating Example (cont.)

Linear transformation approach• , , and

– Reject ; STEP not a valid proxy for VO2 peak in this cohort (STEP is biased)

– In this example, the paired test approach yielded the same conclusion

» Not unexpected given the bias appeared to be largely due to the intercept—not the linear term

» Remember, for 1 df approach; so in isolation this approach does not as fully address the biased proxy question


Motivating Example (cont.)


Solid line represents overall fit Light gray dashed represents

younger REACH cohort Black dashed represents KU

ADC cohort

CONCLUSION: “The validity of the STEP was not supported across the… older population including those with AD. The STEP may not be appropriate for clinical measurement… in these groups.” (Vidoni et al., in press)

95% prediction interval , which translates to 4 METs

Summary and Limitations When evaluating new proxy measure, it is important to

consider the potential for linear bias Recommend 2 df approach over paired test

No intercept model—alternative approach There is only a single parameter to assess Would likely have analogous problems to paired test, but w.r.t.

parameter instead of Did not consider higher-order deviations from

relationship Did not include variance formally—results limited to bias

Likelihood ratio approach should handle all three parameters simultaneously, but ML estimates of variance parameters known to be biased downward


Future Directions Current structure of is that is a valid

proxy of …may want to start with that is NOT a valid

proxy, so rejection concludes that is a good proxy for • Akin to equivalence design

General model assessment Probability-Probability (P-P) plots


ReferencesKotz S, Johnson NL eds., Encyclopedia of Statistical Sciences, Vol. 7. John Wiley & Sons: New York, 1986.

Vidoni ED, Mattlage A, Mahnken J, Burns JM, McDonough J, Billinger SA: “Validity of the Step Test for Exercise Prescription Does Not Extend to a Larger Age Range.” Journal of Aging and Physical Activity. (Accepted in 2012.)


I S H E D


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Evaluating a Variable as a Proxy for another Measure: Assessing the Step Test Exercise Prescription...

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