Analysis of Red Urchins *(Strongylocentrotus franciscanus*) Data

From Channel Islands National Park

Paul Geissler

U.S. Geological Survey

<Paul_Geissler@usgs.gov>, 301-497-5780

DRAFT 3/3/3

## 

Year and site effects are crossed because the same year effect (e.g.
weather) applies to all plots and the same plot effects (e.g., location)
applies to all years. However, neither plots or year can be rerandomized
between years or plots, so this is a repeated measure design, but either
years or plots could be considered to be the repeated measurements. This
design is variously referred to as a "split-block", "spilt-plot in time"
or a design with both factors in strips (Steel and Torrie 1980:390-400,
Cochran and Cox 1957: 306-309).

## 

## Analysis Of Variance Approach -- Period Comparison

Another approach is to use an analysis of variance with fixed effects
for periods and islands and with random effects for years within periods
and for sites within islands. The expected mean squares and F-tests are
shown in Table 5 (rules for finding expected mean squares are available
in Neter et al. 1996: 1373-1386).

Table 5. Expected Mean Squares

  -------------------- -------------- --------------------------- --------
  **Source**           **d.f.**       **Expected Mean Square**    **F**

  I: Islands           i-1            spy~I~^2^ + py~S~^2^ +      I/S
                                      ~SY~^2^                     

  S: Sites (Island)    i(s-1)         py~S~^2^ + ~SY~^2^          

  P: Periods           p-1            isy~P~^2^ + is~Y~^2^ +      P/Y
                                      ~SY~^2^                     

  Y: Years (Period)    p(y-1)         is~Y~^2^ + ~SY~^2^          

  IP: I x P            (a-1)(p-1)     sy~IP~^2^ + ~SY~^2^         IP/SY

  SY: S x Y (I, P)     ip(s-1)(y-1)   ~SY~^2^                     
  -------------------- -------------- --------------------------- --------

This test for trends (Table 6) is very different from the within-subject
functions approach, which uses the consistency over sites of the period
differences or slopes to judge the significance. Here the variation
among years within periods is used to test for period differences. One
is using variation among sites and the other is using variation among
years to test for differences.

Table 5. Analysis of Variance

+----------------+----+----------+--------+---------------+----------+
|                |    | *        |        | **Log         |          |
|                |    | *Untrans |        | Transformed** |          |
|                |    | formed** |        |               |          |
+----------------+----+----------+--------+---------------+----------+
| **Source**     | ** | **Mean   | **P**  | **Mean        | **P**    |
|                | d. | Square** |        | Square**      |          |
|                | f. |          |        |               |          |
|                | ** |          |        |               |          |
+----------------+----+----------+--------+---------------+----------+
| I: Islands     | 4  | 60.943   | 0.715  | 4.345         | 0.727    |
+----------------+----+----------+--------+---------------+----------+
| S: Sites       | 13 | 114.822  |        | 8.449         |          |
| (Island)       |    |          |        |               |          |
+----------------+----+----------+--------+---------------+----------+
| P: Periods     | 1  | 51.457   | 0.203  | 0.005         | 0.966    |
+----------------+----+----------+--------+---------------+----------+
| Y: Years       | 18 | 29.426   |        | 2.61          |          |
| (Period)       |    |          |        |               |          |
+----------------+----+----------+--------+---------------+----------+
| IP: I x P      | 4  | 32.702   | 0.000  | 2.509         | 0.000    |
+----------------+----+----------+--------+---------------+----------+
| SY: S x Y (I,  | 2  | 3.305    |        |   ----------  |          |
| P)             | 25 |          |        |   0.277       |          |
|                |    |          |        |               |          |
|                |    |          |        |   ----------  |          |
+----------------+----+----------+--------+---------------+----------+

I used SYSTAT for the analysis, which cannot calculate type 4 sums of
squares to allow for missing cells. To calculate the IP and SY (but not
other) sums of squares, I imputed the missing values as the mean of the
row and column means for a period in Table 1 (Steel and Torrie 1980:
209-214). The degrees of freedom for SY were reduced by the number of
missing cells. SAS can calculate these sums of squares without
imputation and provides a much better analysis.

The power for an analysis of variance (Figure 9) can be obtained from
the table 8.3.1 in Cohen (1988: 273-380). The difference between two
means d = (mean 2 -- mean 1) / (square root of denominator mean square)
= 2 f, u = 1 = numerator degrees of freedom in the F ratio, and n =
\[(denominator degrees of freedom)/(u+1)\] + 1. I used the original
scale for these calculations.

![](media/image1.wmf){width="4.0in" height="2.765277777777778in"}Figure
9. Power Curves For All Islands

The significant island by period interaction indicates that a separate
analysis should be conducted for each island because the period effects
differ among islands (Table 7).

Table 7. Analysis of Variance For Each Island

+----------------------+------+------------+-------+------------+-----+
|                      |      | ### Unt    |       | **Tra      |     |
|                      |      | ransformed |       | nsformed** |     |
+----------------------+------+------------+-------+------------+-----+
| **Source**           | **   | **Mean     | **P** | **Mean     | **  |
|                      | df** | Square**   |       | Square**   | P** |
+----------------------+------+------------+-------+------------+-----+
| Anacapa              |      |            |       |            |     |
+----------------------+------+------------+-------+------------+-----+
| S: Sites             | 2    | 68.723     |       | 3.676      |     |
+----------------------+------+------------+-------+------------+-----+
| P: Periods           | 1    | 19.9       | 0.006 | 0.837      | 0.  |
|                      |      |            |       |            | 008 |
+----------------------+------+------------+-------+------------+-----+
| Y: Years (Period)    | 18   | 2.092      |       | 0.093      |     |
+----------------------+------+------------+-------+------------+-----+
| Santa Barbara        |      |            |       |            |     |
+----------------------+------+------------+-------+------------+-----+
| S: Sites             | 2    | 6.541      |       | 0.19       |     |
+----------------------+------+------------+-------+------------+-----+
| P: Periods           | 1    | 158.903    | 0.005 | 5.729      | 0.  |
|                      |      |            |       |            | 005 |
+----------------------+------+------------+-------+------------+-----+
| Y: Years (Period)    | 13   | 13.584     |       | 0.51       |     |
+----------------------+------+------------+-------+------------+-----+
| Santa Cruz           |      |            |       |            |     |
+----------------------+------+------------+-------+------------+-----+
| S: Sites             | 5    | 41.617     |       | 3.451      |     |
+----------------------+------+------------+-------+------------+-----+
| P: Periods           | 1    | 0.24       | 0.893 | 0.255      | 0.  |
|                      |      |            |       |            | 689 |
+----------------------+------+------------+-------+------------+-----+
| Y: Years (Period)    | 13   | 12.697     |       | 1.522      |     |
+----------------------+------+------------+-------+------------+-----+
| #### San Miguel      |      |            |       |            |     |
+----------------------+------+------------+-------+------------+-----+
| S: Sites             | 1    | 941.143    |       | 56.229     |     |
+----------------------+------+------------+-------+------------+-----+
| P: Periods           | 1    | 11.174     | 0.262 | 0.308      | 0.  |
|                      |      |            |       |            | 493 |
+----------------------+------+------------+-------+------------+-----+
| Y: Years (Period)    | 18   | 8.318      |       | 0.629      |     |
+----------------------+------+------------+-------+------------+-----+
| #### Santa Rosa      |      |            |       |            |     |
+----------------------+------+------------+-------+------------+-----+
| S: Sites             | 3    | 59.282     |       | 8.968      |     |
+----------------------+------+------------+-------+------------+-----+
| P: Periods           | 1    | 9.664      | 0.353 | 5.905      | 0.  |
|                      |      |            |       |            | 080 |
+----------------------+------+------------+-------+------------+-----+
| Y: Years (Period)    | 17   | 10.616     |       | 1.699      |     |
+----------------------+------+------------+-------+------------+-----+

This shows increases only for Anacapa and Santa Barbara (P\<0.05).

![](media/image2.wmf){width="4.0in" height="2.7430555555555554in"}Figure
10. Power curves for each island.

![](media/image3.wmf){width="4.0in" height="2.7430555555555554in"}

![](media/image4.wmf){width="4.0in" height="2.779166666666667in"}

![](media/image5.wmf){width="4.0in" height="2.745833333333333in"}

![](media/image6.wmf){width="4.0in" height="2.747916666666667in"}

## Analysis Of Variance Approach -- Slope

Instead of comparing the means of two periods, one can estimate the
slope of a linear regression on years. I used the general linear test
approach (Neter et al. 1996: 78-80), fitting a full model with
categorical years and a reduced model with numeric years (Table 8). The
indented effects are partitions of the above effects. I fitted main
effects before the interactions, so the main effects would not be
adjusted for the interactions.

Table 8. Analysis of Variance

+--------------+---+--------+--------+------+--------+-------+------+
|              |   | **Unt  |        |      | **Log  |       |      |
|              |   | ransfo |        |      | T      |       |      |
|              |   | rmed** |        |      | ransfo |       |      |
|              |   |        |        |      | rmed** |       |      |
+--------------+---+--------+--------+------+--------+-------+------+
| **Source**   | * | **Sum  | **     | *    | **Sum  | **M   | *    |
|              | * | of**   | Mean** | *P** | of**   | ean** | *P** |
|              | d |        |        |      |        |       |      |
|              | f | **Squ  | **Sq   |      | **Squ  | **Squ |      |
|              | * | ares** | uare** |      | ares** | are** |      |
|              | * |        |        |      |        |       |      |
+--------------+---+--------+--------+------+--------+-------+------+
| Islands      | 4 | 2      | 60.943 |      | 17.378 | 4.345 |      |
|              |   | 43.772 |        |      |        |       |      |
+--------------+---+--------+--------+------+--------+-------+------+
| Sites        | 1 | 1      | 1      |      | 1      | 8.449 |      |
| (Island)     | 3 | 492.68 | 14.822 |      | 09.839 |       |      |
+--------------+---+--------+--------+------+--------+-------+------+
| Years        | 1 | 5      | 30.770 |      | 46.982 | 2.473 |      |
|              | 9 | 84.632 |        |      |        |       |      |
+--------------+---+--------+--------+------+--------+-------+------+
| \- Slope     | 1 | 1      | 1      | 0    | 1.384  | 1.384 | 0    |
|              |   | 18.698 | 18.698 | .046 |        |       | .469 |
+--------------+---+--------+--------+------+--------+-------+------+
| \- Residual  | 1 | 4      | 25.885 |      | 45.598 | 2.533 |      |
|              | 8 | 65.935 |        |      |        |       |      |
+--------------+---+--------+--------+------+--------+-------+------+
| Islands \*   | 7 | 4      | 6.055  |      | 4      | 6.055 |      |
| Years        | 6 | 60.163 |        |      | 60.163 |       |      |
+--------------+---+--------+--------+------+--------+-------+------+
| \- Slopes    | 4 | 98.527 | 24.632 | 0    | 98.527 | 2     | 0    |
|              |   |        |        | .001 |        | 4.632 | .000 |
+--------------+---+--------+--------+------+--------+-------+------+
| \- Residual  | 7 | 3      | 5.023  |      | 38.687 | 0.537 |      |
|              | 2 | 61.636 |        |      |        |       |      |
+--------------+---+--------+--------+------+--------+-------+------+

This again shows that the island slopes differ (P\<0.01) and that
islands should be analyzed individually (Table 9).

Table 9. Analysis of Variance For Each Island

+---------+---+--------+-------+---------+--------+-------+---------+
|         |   | **Unt  |       |         | **T    |       |         |
|         |   | ransfo |       |         | ransfo |       |         |
|         |   | rmed** |       |         | rmed** |       |         |
+---------+---+--------+-------+---------+--------+-------+---------+
| **S     | * | **Sum  | **M   | **P**   | **Sum  | **M   | **P**   |
| ource** | * | of**   | ean** |         | of**   | ean** |         |
|         | d |        |       |         |        |       |         |
|         | f | **Squ  | **Squ |         | **Squ  | **Squ |         |
|         | * | ares** | are** |         | ares** | are** |         |
|         | * |        |       |         |        |       |         |
+---------+---+--------+-------+---------+--------+-------+---------+
| Anacapa |   |        |       |         |        |       |         |
+---------+---+--------+-------+---------+--------+-------+---------+
| Sites   | 2 | 1      | 6     |         | 7.351  | 3.676 |         |
|         |   | 37.445 | 8.723 |         |        |       |         |
+---------+---+--------+-------+---------+--------+-------+---------+
| Years   | 1 | 57.554 | 3.029 |         | 2.520  | 0.133 |         |
|         | 9 |        |       |         |        |       |         |
+---------+---+--------+-------+---------+--------+-------+---------+
| \-      | 1 | 28.911 | 2     |         | 1.349  | 1.349 |         |
| Slope   |   |        | 8.911 | ------- |        |       | ------- |
|         |   |        |       |   0.000 |        |       |   0.000 |
|         |   |        |       |         |        |       |         |
|         |   |        |       |         |        |       |         |
|         |   |        |       | ------- |        |       | ------- |
+---------+---+--------+-------+---------+--------+-------+---------+
| \-      | 1 | 28.643 | 1.591 |         | 1.171  | 0.065 |         |
| R       | 8 |        |       |         |        |       |         |
| esidual |   |        |       |         |        |       |         |
+---------+---+--------+-------+---------+--------+-------+---------+
| Slope   |   | 0.120  |       |         | 0.026  |       |         |
| Est.    |   |        |       |         |        |       |         |
+---------+---+--------+-------+---------+--------+-------+---------+
| Santa   |   |        |       |         |        |       |         |
| Barbara |   |        |       |         |        |       |         |
+---------+---+--------+-------+---------+--------+-------+---------+
| Sites   | 2 | 23.206 | 1     |         | 0.772  | 0.386 |         |
|         |   |        | 1.603 |         |        |       |         |
+---------+---+--------+-------+---------+--------+-------+---------+
| Years   | 1 | 3      | 2     |         | 15.146 | 0.797 |         |
|         | 9 | 98.618 | 0.980 |         |        |       |         |
+---------+---+--------+-------+---------+--------+-------+---------+
| \-      | 1 | 1      | 12    |         | 3.899  | 3.899 |         |
| Slope   |   | 29.003 | 9.003 | ------- |        |       | ------- |
|         |   |        |       |   0.009 |        |       |   0.022 |
|         |   |        |       |         |        |       |         |
|         |   |        |       |         |        |       |         |
|         |   |        |       | ------- |        |       | ------- |
+---------+---+--------+-------+---------+--------+-------+---------+
| \-      | 1 | 2      | 1     |         | 11.247 | 0.625 |         |
| R       | 8 | 69.615 | 4.979 |         |        |       |         |
| esidual |   |        |       |         |        |       |         |
+---------+---+--------+-------+---------+--------+-------+---------+
| Slope   |   | 0.278  |       |         | 0.048  |       |         |
| Est.    |   |        |       |         |        |       |         |
+---------+---+--------+-------+---------+--------+-------+---------+
| Santa   |   |        |       |         |        |       |         |
| Cruz    |   |        |       |         |        |       |         |
+---------+---+--------+-------+---------+--------+-------+---------+
| Sites   | 5 | 2      | 4     |         | 18.054 | 3.611 |         |
|         |   | 06.289 | 1.258 |         |        |       |         |
+---------+---+--------+-------+---------+--------+-------+---------+
| Years   | 1 | 2      | 1     |         | 25.999 | 1.368 |         |
|         | 9 | 25.438 | 1.865 |         |        |       |         |
+---------+---+--------+-------+---------+--------+-------+---------+
| \-      | 1 | 9.123  | 9.123 |         | 0.351  | 0.351 |         |
| Slope   |   |        |       | ------- |        |       | ------- |
|         |   |        |       |   0.395 |        |       |   0.626 |
|         |   |        |       |         |        |       |         |
|         |   |        |       |         |        |       |         |
|         |   |        |       | ------- |        |       | ------- |
+---------+---+--------+-------+---------+--------+-------+---------+
| \-      | 1 | 2      | 1     |         | 25.648 | 1.425 |         |
| R       | 8 | 16.315 | 2.018 |         |        |       |         |
| esidual |   |        |       |         |        |       |         |
+---------+---+--------+-------+---------+--------+-------+---------+
| Slope   |   | 0.056  |       |         | 0.011  |       |         |
| Est.    |   |        |       |         |        |       |         |
+---------+---+--------+-------+---------+--------+-------+---------+
| San     |   |        |       |         |        |       |         |
| Miguel  |   |        |       |         |        |       |         |
+---------+---+--------+-------+---------+--------+-------+---------+
| Sites   | 1 | 9      | 94    |         | 56.229 | 5     |         |
|         |   | 41.143 | 1.143 |         |        | 6.229 |         |
+---------+---+--------+-------+---------+--------+-------+---------+
| Years   | 1 | 1      | 8.468 |         | 11.634 | 0.612 |         |
|         | 9 | 60.901 |       |         |        |       |         |
+---------+---+--------+-------+---------+--------+-------+---------+
| \-      | 1 | 50.897 | 5     |         | 0.297  | 0.297 |         |
| Slope   |   |        | 0.897 | ------- |        |       | ------- |
|         |   |        |       |   0.010 |        |       |   0.501 |
|         |   |        |       |         |        |       |         |
|         |   |        |       |         |        |       |         |
|         |   |        |       | ------- |        |       | ------- |
+---------+---+--------+-------+---------+--------+-------+---------+
| \-      | 1 | 1      | 6.111 |         | 11.337 | 0.630 |         |
| R       | 8 | 10.004 |       |         |        |       |         |
| esidual |   |        |       |         |        |       |         |
+---------+---+--------+-------+---------+--------+-------+---------+
| Slope   |   | 0.196  |       |         | 0.015  |       |         |
| Est.    |   |        |       |         |        |       |         |
+---------+---+--------+-------+---------+--------+-------+---------+
| Santa   |   |        |       |         |        |       |         |
| Rosa    |   |        |       |         |        |       |         |
+---------+---+--------+-------+---------+--------+-------+---------+
| Sites   | 3 | 1      | 6     |         | 27.526 | 9.175 |         |
|         |   | 80.617 | 0.206 |         |        |       |         |
+---------+---+--------+-------+---------+--------+-------+---------+
| Years   | 1 | 2      | 1     |         | 37.135 | 1.954 |         |
|         | 9 | 02.285 | 0.647 |         |        |       |         |
+---------+---+--------+-------+---------+--------+-------+---------+
| \-      | 1 | 1.255  | 1.255 |         | 2.464  | 2.464 |         |
| Slope   |   |        |       | ------- |        |       | ------- |
|         |   |        |       |   0.741 |        |       |   0.273 |
|         |   |        |       |         |        |       |         |
|         |   |        |       |         |        |       |         |
|         |   |        |       | ------- |        |       | ------- |
+---------+---+--------+-------+---------+--------+-------+---------+
| \-      | 1 | 2      | 1     |         | 34.671 | 1.926 |         |
| R       | 8 | 01.030 | 1.168 |         |        |       |         |
| esidual |   |        |       |         |        |       |         |
+---------+---+--------+-------+---------+--------+-------+---------+
| Slope   |   | -0.026 |       |         | -0.036 |       |         |
| Est.    |   |        |       |         |        |       |         |
+---------+---+--------+-------+---------+--------+-------+---------+

Figure 11. Power Curves For All Islands Slope.

![](media/image7.wmf){width="4.0in" height="2.765277777777778in"}

Figure 12. Power Curves For Each Island Slopes.

![](media/image8.wmf){width="4.0in" height="2.69375in"}

![](media/image9.wmf){width="4.0in" height="2.7756944444444445in"}

![](media/image10.wmf){width="6.944444444444445e-4in"
height="6.944444444444445e-4in"}

![](media/image11.wmf){width="4.0in" height="2.779166666666667in"}

![](media/image12.wmf){width="4.0in" height="2.779166666666667in"}

![](media/image13.wmf){width="4.0in" height="2.7868055555555555in"}

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