Evaluating a Variable as a Proxy for another Measure: Assessing the Step Test Exercise Prescription...
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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.
© 2013 Jonathan D. Mahnken. All Rights Reserved. 2
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
© 2013 Jonathan D. Mahnken. All Rights Reserved. 3
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
© 2013 Jonathan D. Mahnken. All Rights Reserved. 4
Statistical Definition
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(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
© 2013 Jonathan D. Mahnken. All Rights Reserved. 6
Valid Proxy (cont.)
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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
© 2013 Jonathan D. Mahnken. All Rights Reserved. 8
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
© 2013 Jonathan D. Mahnken. All Rights Reserved. 9
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)
© 2013 Jonathan D. Mahnken. All Rights Reserved. 11
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
© 2013 Jonathan D. Mahnken. All Rights Reserved. 12
Linear Transformation (cont.) For where
where Under , so If proxy is biased (), where
Note that , so
© 2013 Jonathan D. Mahnken. All Rights Reserved. 13
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
© 2013 Jonathan D. Mahnken. All Rights Reserved. 14
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
© 2013 Jonathan D. Mahnken. All Rights Reserved. 15
Paired Test (cont.) For where
Under , so If proxy is biased (), where
Recall that Shortcomings because equivalent to
© 2013 Jonathan D. Mahnken. All Rights Reserved. 16
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
© 2013 Jonathan D. Mahnken. All Rights Reserved. 17
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
© 2013 Jonathan D. Mahnken. All Rights Reserved. 18
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)
© 2013 Jonathan D. Mahnken. All Rights Reserved. 19
© 2013 Jonathan D. Mahnken. All Rights Reserved. 20
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Power = type I error when unbiased
© 2013 Jonathan D. Mahnken. All Rights Reserved. 22
When , 2 df approach better
© 2013 Jonathan D. Mahnken. All Rights Reserved. 23
When , 2 df approach slightly worse
© 2013 Jonathan D. Mahnken. All Rights Reserved. 25
© 2013 Jonathan D. Mahnken. All Rights Reserved. 26
For 1 df approach, all curves overlap because not included in
© 2013 Jonathan D. Mahnken. All Rights Reserved. 27
When , 2 df approach slightly worse
© 2013 Jonathan D. Mahnken. All Rights Reserved. 28
Slight advantage for 1 df approach when close to zero (light-dashed)
© 2013 Jonathan D. Mahnken. All Rights Reserved. 29
When (only linear bias), 2 df approach better
© 2013 Jonathan D. Mahnken. All Rights Reserved. 31
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When , 2 df approach better
© 2013 Jonathan D. Mahnken. All Rights Reserved. 33
When , 2 df approach slightly worse
© 2013 Jonathan D. Mahnken. All Rights Reserved. 34
When (only linear bias), 2 df approach better
Sample size had NO impact on power
Motivating Example
© 2013 Jonathan D. Mahnken. All Rights Reserved. 36
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
© 2013 Jonathan D. Mahnken. All Rights Reserved. 37
Motivating Example (cont.)
© 2013 Jonathan D. Mahnken. All Rights Reserved. 38
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
© 2013 Jonathan D. Mahnken. All Rights Reserved. 39
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
© 2013 Jonathan D. Mahnken. All Rights Reserved. 40
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.)
© 2013 Jonathan D. Mahnken. All Rights Reserved. 41
I S H E D
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Questions?