3830 MATH Handout 3 Page 1
MATH 3830 Handout 3 Dependence of shape on size
Overview
This handout studies the dependence of shape on size for the Globorotalia data. The following R program computes the centroid size for each configuration and labels the shapes with that size in Figure 1. Also a multivariate regression analysis is carried out.
Graphical interpretation
Recall the basic properties of Bookstein coordinates with respect to baseline (1,3) for this dataset. Landmarks 1 and 3 are fixed at (0, 0)T and (1, 0)T , and the overall distribution for the two coordinates of landmark 2 follow a roughly elliptical pattern with the long axis of the ellipse in roughly the vertical direction. When size information is superimposed (see Figure 1(a); note part(b) is a blow-up of part (a)), we see that the vertical position of landmark 2 tends to increase with size, whereas the horizontal direction changes little with size.
Statistical analysis
We can fit and assess this pattern statistically using multivariate regression to regress shape on size using the lm command. The shape information for the two coordinates of the second landmark is stored in a 20 2 matrix y, and the centroid size information is stored in the vector sz, of length 20. 1. To begin with, this command fits a linear regression on centroid size sz for each of the two columns of y. The summary command shows the fitted coefficients for each column. 2. Looking at the t-values, we see that the regression coefficient for the first column of y is not significant. However, we shall not remove it from the analysis, because the vertical axis has no special interpretation in Bookstein coordinates. Indeed if we were to change the baseline, the vertical direction for this baseline would not generally lie in a special direction with repsect to the other baseline. 3. An overall test of the importance of "size" in the regression is given by the anova command. In this setting (one explanatory variable) the approximate F test is exact (F = 13.7 F2,17 ), and we see that the regression is highly signficant.
MATH 3830 Handout 3 Page 2 4. The coefficients in the model can be extracted using the command coef, in this case a 2 2 matrix whose first row contains the intercept terms and whose second row ( -0.0001164253, 0.0008427426) contains the regression parameters on size. 5. Let 1 and 2 denote these two regression coefficients. The line through the sample mean coordinates for landmarks 1 and 2 pointing in the direction (1 , 2 )T (with slope 2 /1 ) represents the regression line of shape on size, and shows the expected shape for different values of size. More specifically, write the fitted regression lines of y1 and y2 on centroid size (denoted here by s) as y1 = 1 + 1 s, y2 = 2 + 2 s.
Recall the estimated intercepts are given by 1 = y1;av - 1 sav , and similarly for y2 , where y1;av denotes the sample mean of y1 in the data, etc. Thus these regression lines can also be written in the form,
y1 = 1 s , y2 = 2 s .
The dash means "mean-corrected"; that is, y1 = y1 = y1;av , etc. Taking the ratio of these two equations removes the explicit dependence on s, and yields the line in the y1 - y2 plane, 2 2 y2 = , i.e. y2 = y1 , y1 1 1
as reported in the previous paragraph.
Comments on R
1. The Bookstein coordinates are stored in a 3220 array xb. . The matrix y (202 and and array ya (1 2 20) pick out the information on the second landmark. Note the use of the option drop=FALSE when defining ya. Otherwise R would drop the irrelevant first dimension in ya, making ya a 2 20 matrix instead of a 3-way array. We need the array form for plot.configs and we need the matrix form for lm. 2. The lm command is familiar from linear regression. Usually the response variable is a vector (of length n, say. But the command can also be used when y is an n p matrix, in which case p multiple regressions are carried out, but where the error terms are allowed be to correlated across the different regressions. Allowing this correlation does not affect the least squares estimates, but it does affect any hypothesis tests.
MATH 3830 Handout 3 Page 3 3. For example, in our case the anova command tests whether the two regression coefficients on centorid size (for the two columns of y) can be taken equal to 0. 4. In Figure 1(a) the whole data set is included (including the baseline points). The points are not plotted, but space is allowed for them in the graph. The text command is used to plot a version of centroid size at each location for landmark 2. Notice the division by 10 and rounding of centroid size to make the plot more digestible. Note the use of the title command to put a main title and a subtitle on the graph. 5. In Figure 1(b) only landmark 2 is included, which results in a blow-up of the graph. Otherwise the same information is provided as in part (a). 6. When copying graphs from R into Word, say, for writing reports, note the aspect ratio (ratio of horizontal to vertical lengths) is important. The graphs have been constructed so that if they look square, the aspect ratio is correct. Hence always make sure the graphs look square.
R program
> xb=bk.coords(glob.dat,baseline=c(1,3)) > y=t(xb[2,,]) # 20 x 2 matrix > ya=xb[2,,,drop=FALSE] # 1 x 2 x 20 array (for just landmark 2) > sz=size.configs(glob.dat) > sz [1] 699.3132 509.9510 804.7745 703.2034 692.2240 763.2103 672.8573 679.4409 [9] 594.3459 672.3325 590.0288 583.9709 604.6834 579.3591 557.3006 687.3718 [17] 590.0424 540.8299 544.9982 502.5883 > szr=round(sz/10) > szr [1] 70 51 80 70 69 76 67 68 59 67 59 58 60 58 56 69 59 54 54 50 > > mod1=lm(y~sz) > summary(mod1) Response Y1 : Call: lm(formula = Y1 ~ sz) Residuals: Min
1Q
Median
3Q
Max
MATH 3830 Handout 3 Page 4 -0.062018 -0.007782 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 4.695e-01 4.994e-02 9.401 2.29e-08 *** sz -1.164e-04 7.879e-05 -1.478 0.157 --Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 Residual standard error: 0.02868 on 18 degrees of freedom Multiple R-Squared: 0.1082, Adjusted R-squared: 0.05865 F-statistic: 2.184 on 1 and 18 DF, p-value: 0.1568 0.009025 0.017689 0.036739
Response Y2 : Call: lm(formula = Y2 ~ sz) Residuals: Min 1Q Median -0.08906 -0.03461 -0.00440
3Q 0.04492
Max 0.12409
Coefficient...