1. Biostatistics Case Studies 2006 Peter D. Christenson Biostatistician http://gcrc.LAbiomed.org/Biostat Session 2: Correlation of Time Courses of Simultaneous Measures

2. Case Study

3. Time Courses of Four Measures Fig 2AFig 1A Fig 2BFig 1B -2 hr 0-10 min 2 4 6 8 hr

4. Possible Analyses for Correlation of Time Trends 1.Entrainment Usually with known functional relations. Focus on parameters of differential equations, or known cyclic, circadian pattern: phase, amplitude, freq, e.g., melatonin, hormones. 2.Time trend for one measure, adjusted for another measure Focus on expected behavior over time if 2 nd measure had remained constant. 3.Time trend of ratio of two measures 4.Random coefficient modeling – this paper Time is used to pair measures, then ignored. Approximates correlating the two measures for each individual separately, then summarizing over them.

5. Ratios Over Time Ratio may have a biological meaning for particular measures: OK. Potential scaling problems: 20 10 Time 110 10 5.5 1 Y Z Z/Y Could use Δs, percentiles, Z-scores to obtain an appropriate ratio in some situations.

6. Ratios Over Time, Continued Even with appropriate scaling, as below, coupling corresponds to no change in ratio. Want to prove a negative pattern for the ratio. Time Y Z Y/Z Remember last session: proving equivalence vs. proving difference vs. observing minimal difference. Margin of Equivalence

7. Simulated Data We first look at some data that I generated: only 5 sheep, all treated the same. mean measures every hour for 8 hours. Data is NOT what actually occurred in this experiment. These data: have a similar overall pattern for two time courses. have a correlation of patterns that seems contradictory. show a problem too extreme to be common. show a problem that is still a problem that cannot be ignored.

8. Simulated Data: Time Pattern 1

9. Simulated Data: Time Pattern 2

10. Simulated Data: Correlation

11. Simulated Data Thus, Blood flow ↑ over time, as in the paper. PO2 ↓ over time, as in the paper. Blood flow and PO2 appear positively correlated in the graph which is supported by analysis: correlation = 0.82 with p<0.0001. blood flow ↑ by a mean 7.2 for each 1 mmHg increase in PO2 (95% CI: 5.5 to 8.9): Effect Estimate Std Error p-value Lower Upper Intercept 0.04718 6.4185 0.9942 -12.9464 13.0408 o2 7.2186 0.8312 <.0001 5.5359 8.9013 What is the explanation?

12. Simulated Data: Graphical Explanation Each individual fetus does indeed have an inverse relation:

13. Simulated Data: Statistical Agreement on the Graphical Explanation After specifying individual fetuses in the analysis: Effect Estimate Std Error p-value Lower Upper Intercept 73.8646 19.2858 0.0188 20.1867 127.54 Slope -3.3775 0.6007 <.0001 -4.5979 -2.1572 So, Blood flow and PO2 are negatively correlated within any individual, averaging: blood flow ↓ by a mean 3.4 for each 1 mmHg increase in PO2 (95% CI: 2.2 to 4.6).

14. Simulated Data: An Extreme Illustration of Real Issues My simulation is unrealistic because usually a pattern among different individuals in a population also occurs when the measures change within individuals: “Among recapitulates within”? A reminder that correlations between measures always depend on the ranges of the measures. The following slide shows the authors’ use of this method. Note that : They did not have the extreme reversal that I simulated. The bloodflow-O2 relation would have been biased to be stronger if they had ignored sheep differences.

15. Correlation of Actual Carotid Blood Flow and Cortical Tissue O 2 in the Paper Fig 3A Fig 2A Fig 1A

16. Further Comments on Individual Slopes This paper only reported R 2 =0.69, p<0.0001 for the bloodflow-O2 relation in Fig 3A. The estimate of the average slope and it’s CI is usually more informative. Some investigators use a standard regression for each individual and then find the mean and SE of these slopes. A mixed model should be used if: Some individuals have only a few pairs of data, so that their slopes are poorly estimated. All individuals have many pairs of data, but the # varies among individuals, so that individual slopes need to be weighted according to their amount of information.

17. Coupling of Cortical and Carotid Blood Flows Fig 2A Fig 2B Fig 3B Statistically, this is the same as the previous analysis with O2.

18. Time Trends for Each Measure Need to Account for Individual Sheep Recall Time Trend: Any reasonable person would say there is an obvious trend.

19. Time Trends for Each Measure Need to Account for Individual Sheep Incorrect Analysis Slope Estimate: 2.8 ± 2.0 95% CI: -1.26 to 6.84 p-value = 0.17 Repeated Measures Analysis Slope Estimate: 2.8 ± 0.15 95% CI: 2.38 to 3.20 p-value < 0.0001 Mean PatternPatterns for Individuals

20. Self Quiz 1.T or F: If two measures are recorded over time, the ratio of two measures at each time is a good way to assess whether they are correlated. 2.Explain why an X-Y scatterplot of two measures recorded for each of several individuals, but at different times for each individual, can be misleading. 3.Could the problem in (2) remain if every subject is measured at the same times? 4.Suppose you use separate regressions for each subject measured at multiple times to find the rate of change of one measure for a 1-unit change in the other for each subject. Give at least one reason why averaging these slopes over subjects could be misleading. 5.Does the authors’ analysis in Fig 3A account for the fact that fetuses have different pre-asphyxia bloodflow and O2?