The above sections analyzed data consisting only of individuals that were measured at 20◦C, but there are also individuals measured at 25◦C. Figure 5.5 shows the growth rate
curves for both temperatures up to the peak mass value. Individuals with 6 and 7 original developmental stages, represented by cyan and red curves respectively, were measured at both temperatures. The solid cyan and red curves are the individuals measured at 20◦C,
while the dashed cyan and red curves are the individuals measured at 25◦C. This graphic
shows that temperature has a significant effect on the overall slope of the curves. But if the overall slope difference due to temperature differences is adjusted for, then how do the slopes of the cyan and red curves relate?
The objective of the adjustment is for the solid and dashed cyan curves to lie on top of each other. Also the dotted and dashed red curves should lie on top of each other. One way to do this adjustment is to multiply all of mass measurements of the cyan individuals measured at 20◦C by a constant. Also the time measurements of these individuals are
multiplied by another constant. But the constants are chosen, such that the peak mass and time values of the mean curve of the cyan individuals measured at 20◦C is shifted
to be exactly on top of the peak mass and time values of the mean curve of the cyan individuals measured at 25◦C. Then a similar adjustment is performed on the red curves.
An adjustment with overall means, i.e. both colors included, was performed and similar results were obtained. The results of this adjustment can be seen at Gaydos (2007) in the folder overallMeanAdjustment. The files of interest are NonExtra, Extra, and All, which shows the cyan curves, red curves and all curves adjusted by the overall means.
0 10 20 30 40 50 60 70 0 1 2 3 4 5 6 7 8 9 10
Mass vs Time View of Curves
Time
log(mass)
Figure 5.5: Manduca sexta growth trajectories up to peak are shown. Cyan are field caterpillars with 6 original stages and red are field caterpillars with 7 original stages. Asterisks represent
developmental stages. Solid lines are individuals measured at 20◦C, while dashed lines are
individuals measured at 25◦C.
It was decided to perform the adjustment color by color, since that way within color temperature effect would be removed. This would not necessarily be true with an overall mean adjustment. Also it was decided to multiply the landmarks by a constant, since we are dealing with slope and the curves are all assumed to start at the origin. Therefore the multiplication is easily interpreted in the mass vs time view, as a rotation and enlarging or shrinking of the curves. If mass and time of the landmarks were subtracted by a constants then the curves would no longer start at zero. Another option is to subtract a constant from all of the slopes of line segments, but then a separate adjustment is needed when length of line segments is considered, see Section 5.5. The multiplication adjustment provides a way to adjust both the slope and length of the line segments at once.
The data curves after this adjustment is performed are shown in Figure 5.6. The solid and dashed cyan curves lie roughly in the same area, as do the solid and dashed red curves. Also notice that the horizontal axis in Figure 5.5 includes values 0 to 70, while the axis in Figure 5.6 includes only values from 0 to 40. This is because the time values of the solid lines landmarks were multiplied by a constant which shifted them to the left. Also the mass values were multiplied by a constant. But the effect of the multiplication on mass was much less than for the time values, since the original mass values of the solid and dashed lines are similar.
0 5 10 15 20 25 30 35 40 0 1 2 3 4 5 6 7 8 9 10
Mass vs Time View of Curves
Time
log(mass)
Figure 5.6: Manduca sexta growth trajectories adjusted for temperature effect on overall slope are shown. Cyan are field caterpillars with 6 original stages and red are field caterpillars with 7 original stages. Asterisks represent developmental stages. Solid lines are individuals measured
at 20◦C, while dashed lines are individuals measured at 25◦C.
Once this adjustment is completed a question arises as to how do the slope structures between the 20 and 25◦C individuals relate? Ideally if we wish to compare between red
similar slope structures. Also the same should be true of the red curves. The tools described in earlier sections of this chapter will be used to analyze if the slope structures between temperatures within each color are similar.
First we will restrict our analysis to the cyan curves. The same t-test procedure will be performed but the between group comparison is now between temperatures, i.e. comparing dotted and dashed curves, rather than differing number of stages. The results are shown in Figure 5.7. The bottom left of the figure displays the adjusted curves and their group means. Notice that the last landmarks of the two mean curves lie directly on top of each other, due to the adjustment.
The upper left and lower right are the comparisons for within groups, i.e. line types (temperatures). Notice that the lower right matrix of Figure 5.7 and the upper left of Figure 5.3 are the same. This is because although the slopes of the individuals have changed, all of the slopes have only been multiplied by the same constant. Therefore by the nature of the t-test all of the p-values for the tests are the exact same.
The matrix in the upper left is the comparison of slope structure within the dashed, i.e. 25◦, curves. The dashed curves were never multiplied by a constant in the combination
of the two line types. Therefore this matrix is the same as in the upper left of Ttest25 at Gaydos (2007).
The matrix in the upper right is the t-test p-values for between group comparisons. If the adjustment is good then the two temperatures should have similar slope structures, i.e. sets of line segments in the same order along the curves will have the same mean slope. A way to tell if they have similar slope structure is to view the p-values along the diagonal of the matrix in the upper right, from lower left to upper right. If the adjustment is good then the p-values will be large along the diagonal, i.e green boxes. In this case the adjustment works well, except that the last set of line segment mean slopes are significantly different. The mean curves in the bottom left visually display this result. The two curves are relatively parallel, except the last line segments. The last line
25C Data Non−Extra Stage Data
25C Data Non−Extra Stage Data
Pvalues for Differences Of Slopes
1 0 0 0 0 1 0.567 0.728 0 0.567 1 0.713 0 0.728 0.713 1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
20C Non−Extra Stage Data
25C Data Non−Extra Stage Data
Pvalues for Differences Of Slopes
0.5 0 0 0.006 0.003 0.407 0.307 0.024 0.001 0.758 0.594 0.006 0.001 0.472 0.33 0.002 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
20C Non−Extra Stage Data
20C Non−Extra Stage Data
Pvalues for Differences Of Slopes
1 0.001 0.001 0.029 0.001 1 0.82 0.004 0.001 0.82 1 0.003 0.029 0.004 0.003 1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 5 10 15 20 25 0 2 4 6 8 10
Mass vs Time View of Curves
Time
log(mass)
Figure 5.7: Bottom left displays data along with mean curves of solid and dashed cyan curves added. Matrix in top left shows results of t-tests for within dashed curves group. Matrix in bottom right shows results of t-tests for within solid curves group. Matrix in top right displays results of t-test between solid and dashed cyan groups. Notice that diagonal from bottom right to upper left is green boxes with high p-values, except for top right box. This indicates that the adjust is good.
segments start at different points, but end at the same point. These mean curves both end at the same point due to the adjustment performed earlier in this section. Forcing the mean curves to end at the same point may be the cause of the last sets of line segments having differing slopes.
A similar analysis for the red curves was performed and a similar result was rendered. Again only the last set of line segments did not have equal mean slopes, i.e. the top right box in the matrix was yellow with a small p-value. The analysis can be viewed by opening the file Extra at Gaydos (2007) in the folder groupadjusted.
By using the tools of this section it is judged that the adjustment works well. There- fore we will proceed with the adjusted data. All cyan curves are considered to be one
group and all red curves are considered the other group. Now insight into the structure of all the cyan curves and red curves can be gained. Also how the slope structures are related between the red and cyan curves can be investigated. The next section shows the results of the t-tests for this combined data set.