Yolanda Munoz Maldonado Department of Statistics Texas A&M University E-mail: ymunoz@stat.tamu

1. A Similarity Analysis of Curves: A Comparison of the Distribution of Gangliosides in Brains of Old and Young Rats. Yolanda Munoz Maldonado Department of Statistics Texas A&M University E-mail: ymunoz@stat.tamu.edu Dr. Joan Staniswalis Department of Mathematical Sciences University of Texas at El Paso E-mail: joan@math.utep.edu This project was partially supported by RCMI grant 5G12-RR08124 from the National Institute of Health.

2. Overview • Introduction of the Biological Problem • Methodology • Simulation • Data Analysis • Summary

3. Thin Silica Gel Plate

4. Ganglioside Standards

5. Standard Curves

6. Functional Object The intensity of the gangliosides is considered as a function of distance, so the first step in the analysis is to reconstruct the entire profile on a closed interval so that it can be evaluated at any point (Ramsay and Silverman, 1998). Regression splines are used for this purpose (Eubank, 1988).

7. Regression Splines The sampled curve Y(t), t in G, is interpolated by fitting a linear combination of B-splines . This involves the minimization of over .

8. Splines A spline of order K with knots , is any function of the form:

9. B-splines The i th normalized B-spline of orderKfor the knot sequence is denoted by and satisfy the properties:

10. Cubic B-Splines

11. Ganglioside Profiles

12. Warping Functions • The registration of the curves requires: • a monotone transformation w for each curve Y(t) such that the registered curves have more or less identical argument values for any of the characteristic features.

13. Individual Curves

14. Properties

15. Warping function The warping functions were estimated using the Penalized Least-Squares Error Criterion by minimizing The minimizer of this is expression is a natural cubic spline (Shoenberg 1946). Since we want to preserve the area under the curve, the registered curve is given by

16. Warping Functions

17. Aligned Curves

18. Similarity Similarity is based upon comparison of the functions evaluated on a common grid G . The index of similarity between two curves uses the Pearson’s sample correlation coefficient.

19. Test Statistics • Three test statistics were considered: • The pooled mean similarity within groups: • The pooled variance similarity within groups: • . • 3. The ratio of the pooled-mean to the square-root of the pooled variance:

20. Permutation Distribution • The permutation distribution of each test statistic under the null hypothesis is obtained by permuting the 10 curves, and then dividing them into two groups “old” and “young”. • The p-value for the pooled-mean and the ratio is given by the number of permutations which yield a value of the test statistic greater than the observed value. • The p-value for the pooled-variance is obtained by the number of permutations which yield a value of the test statistic that is less than the observed value.

21. Simulation • The noisy data were simulated according to • is the normal pdf. • is generated following a • is the vector of the center of the peaks of the original data. • is the variance-covariance matrix of these points.

22. Simulation • The follow a chi-square distribution with the following degreesof freedom: 20, 45, 30, 20, 20. • The are normally distributed with mean 0 and covariance . • The are independent, uniformly distributed coefficients on the intervals: • min = ( 0.175, 0.25, .0.2, .0.08, 0.1) • max = (0.5 ,0.7, 0.5, 0.417, 0.4)

23. Simulated Curves under OLD YOUNG

24. Significance Level Proportion of Rejections Pooled Mean Pooled Variance Ratio Pooled Mean/Variance 0.1 0.080 0.104 0.108 0.05 0.064 0.068 0.068 0.01 0.024 * 0.008 0.032 * Size of the Test

25. Simulation under

26. Power function at a=0.05

27. Data Analysis • Three data sets are studied: • Medulla • Locus Coeruleus • Hippocampus The last data set was expected to show no differences between old and young rats.

28. Registered, Cut and Normalized Profiles

29. Brain Region Test Statistic P-value Medulla Mean 0.84114 0.046 Variance 0.0024 0.027 Ratio of Means/Variances 588.8824 0.027 Locus Coeruleus Mean 0.736441 0.009 Variance 0.00987 0.028 Ratio of Means/Variances 109.815 0.014 Hippocampus Mean 0.8347 0.805 Variance 0.0059 0.706 Ratio of Means/Variances 235.4329 0.333 Analysis Result

30. Conclusions • The result confirms the biologists expectations of differences in ganglioside concentration in the Medulla region and no difference for Hippocampus. • The result for Locus Coeruleus region provides new evidence for a significant development shift in ganglioside pattern.

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