Quadrat Counts and Goodness-of-fit

Three point patterns (Japanese black pines, redwood seedlings, and biological cells) have each been reduced to quadrat counts through rectangular grids of size 2x2, 3x3, and 4x4. The figures and listings that follow show the original point patterns, the count listings for each grid size, and then a plot showing the counts within their respective quadrats.

To each data set (point pattern and grid size):

Fit binomial, Poisson, and negative binomial distributions.
Test for goodness-of-fit to the Poisson model using the Pearson chi-square statistic, and use the Pearson chi-square and Deviance statistics to help judge fit of the other models.
Specify which distribution you think best describes the quadrat counts.
To summarize over grid sizes, state whether the distribution selected appears consistent over the grid sizes or whether the distribution selected depends upon the grid size.

Japanese Black Pines

2 x 2 Grid

Quadrat	X	Y	COUNT
1	0.25	0.25	13
2	0.75	0.25	13
3	0.25	0.75	22
4	0.75	0.75	17

Japanese Black Pines

2 x 2 Grid

                    Plot of Y*X$COUNT.  Symbol used is ' '.                     
                                                                                
    --+-------------+-------------+-------------+-------------+-------------+---
  Y |                                    |                                     |
1.0 +                                    |                                     +
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
0.8 +                                    |                                     +
    |                                    |                                     |
    |                   22               |                 17                  |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
0.6 +                                    |                                     +
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |------------------------------------+-------------------------------------|
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
0.4 +                                    |                                     +
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                   13               |                 13                  |
    |                                    |                                     |
0.2 +                                    |                                     +
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
0.0 +                                    |                                     +
    |                                    |                                     |
    --+-------------+-------------+-------------+-------------+-------------+---
     0.0           0.2           0.4           0.6           0.8           1.0  
                                                                                
                                         X

Japanese Black Pines

3 x 3 Grid

Quadrat	X	Y	COUNT
1	0.16667	0.16667	4
2	0.50000	0.16667	8
3	0.83333	0.16667	8
4	0.16667	0.50000	10
5	0.50000	0.50000	4
6	0.83333	0.50000	3
7	0.16667	0.83333	6
8	0.50000	0.83333	15
9	0.83333	0.83333	7

Japanese Black Pines

3 x 3 Grid

                    Plot of Y*X$COUNT.  Symbol used is ' '.                     
                                                                                
    --+-------------+-------------+-------------+-------------+-------------+---
  Y |                        |                       |                         |
1.0 +                        |                       |                         +
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |             6          |           15          |          7              |
0.8 +                        |                       |                         +
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |------------------------+-----------------------+-------------------------|
    |                        |                       |                         |
    |                        |                       |                         |
0.6 +                        |                       |                         +
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |             10         |           4           |          3              |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
0.4 +                        |                       |                         +
    |                        |                       |                         |
    |                        |                       |                         |
    |------------------------+-----------------------+-------------------------|
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
0.2 +                        |                       |                         +
    |             4          |           8           |          8              |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
0.0 +                        |                       |                         +
    |                        |                       |                         |
    --+-------------+-------------+-------------+-------------+-------------+---
     0.0           0.2           0.4           0.6           0.8           1.0  
                                                                                
                                         X

Japanese Black Pines

4 x 4 Grid

Quadrat	X	Y	COUNT
1	0.125	0.125	2
2	0.375	0.125	6
3	0.625	0.125	6
4	0.875	0.125	5
5	0.125	0.375	4
6	0.375	0.375	1
7	0.625	0.375	2
8	0.875	0.375	0
9	0.125	0.625	5
10	0.375	0.625	5
11	0.625	0.625	5
12	0.875	0.625	4
13	0.125	0.875	4
14	0.375	0.875	8
15	0.625	0.875	4
16	0.875	0.875	4

Japanese Black Pines

4 x 4 Grid

                    Plot of Y*X$COUNT.  Symbol used is ' '.                     
                                                                                
    --+-------------+-------------+-------------+-------------+-------------+---
  Y |                   |                |                 |                   |
1.0 +                   |                |                 |                   +
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |          4        |       8        |        4        |       4           |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
0.8 +                   |                |                 |                   +
    |                   |                |                 |                   |
    |-------------------+----------------+-----------------+-------------------|
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |          5        |       5        |        5        |       4           |
0.6 +                   |                |                 |                   +
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |-------------------+----------------+-----------------+-------------------|
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
0.4 +                   |                |                 |                   +
    |          4        |       1        |        2        |       0           |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |-------------------+----------------+-----------------+-------------------|
    |                   |                |                 |                   |
0.2 +                   |                |                 |                   +
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |          2        |       6        |        6        |       5           |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
0.0 +                   |                |                 |                   +
    |                   |                |                 |                   |
    --+-------------+-------------+-------------+-------------+-------------+---
     0.0           0.2           0.4           0.6           0.8           1.0  
                                                                                
                                         X

Redwood Seedlings

2 x 2 Grid

Quadrat	X	Y	COUNT
1	0.25	0.25	9
2	0.75	0.25	18
3	0.25	0.75	21
4	0.75	0.75	14

Redwood Seedlings

2 x 2 Grid

                    Plot of Y*X$COUNT.  Symbol used is ' '.                     
                                                                                
    --+-------------+-------------+-------------+-------------+-------------+---
  Y |                                    |                                     |
1.0 +                                    |                                     +
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
0.8 +                                    |                                     +
    |                                    |                                     |
    |                   21               |                 14                  |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
0.6 +                                    |                                     +
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |------------------------------------+-------------------------------------|
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
0.4 +                                    |                                     +
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                   9                |                 18                  |
    |                                    |                                     |
0.2 +                                    |                                     +
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
0.0 +                                    |                                     +
    |                                    |                                     |
    --+-------------+-------------+-------------+-------------+-------------+---
     0.0           0.2           0.4           0.6           0.8           1.0  
                                                                                
                                         X

Redwood Seedlings

3 x 3 Grid

Quadrat	X	Y	COUNT
1	0.16667	0.16667	0
2	0.50000	0.16667	6
3	0.83333	0.16667	13
4	0.16667	0.50000	13
5	0.50000	0.50000	8
6	0.83333	0.50000	2
7	0.16667	0.83333	5
8	0.50000	0.83333	9
9	0.83333	0.83333	6

Redwood Seedlings

3 x 3 Grid

                    Plot of Y*X$COUNT.  Symbol used is ' '.                     
                                                                                
    --+-------------+-------------+-------------+-------------+-------------+---
  Y |                        |                       |                         |
1.0 +                        |                       |                         +
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |             5          |           9           |          6              |
0.8 +                        |                       |                         +
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |------------------------+-----------------------+-------------------------|
    |                        |                       |                         |
    |                        |                       |                         |
0.6 +                        |                       |                         +
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |             13         |           8           |          2              |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
0.4 +                        |                       |                         +
    |                        |                       |                         |
    |                        |                       |                         |
    |------------------------+-----------------------+-------------------------|
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
0.2 +                        |                       |                         +
    |             0          |           6           |          13             |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
0.0 +                        |                       |                         +
    |                        |                       |                         |
    --+-------------+-------------+-------------+-------------+-------------+---
     0.0           0.2           0.4           0.6           0.8           1.0  
                                                                                
                                         X

Redwood Seedlings

4 x 4 Grid

Quadrat	X	Y	COUNT
1	0.125	0.125	0
2	0.375	0.125	4
3	0.625	0.125	2
4	0.875	0.125	9
5	0.125	0.375	5
6	0.375	0.375	0
7	0.625	0.375	7
8	0.875	0.375	0
9	0.125	0.625	8
10	0.375	0.625	2
11	0.625	0.625	5
12	0.875	0.625	2
13	0.125	0.875	2
14	0.375	0.875	9
15	0.625	0.875	2
16	0.875	0.875	5

Redwood Seedlings

4 x 4 Grid

                    Plot of Y*X$COUNT.  Symbol used is ' '.                     
                                                                                
    --+-------------+-------------+-------------+-------------+-------------+---
  Y |                   |                |                 |                   |
1.0 +                   |                |                 |                   +
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |          2        |       9        |        2        |       5           |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
0.8 +                   |                |                 |                   +
    |                   |                |                 |                   |
    |-------------------+----------------+-----------------+-------------------|
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |          8        |       2        |        5        |       2           |
0.6 +                   |                |                 |                   +
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |-------------------+----------------+-----------------+-------------------|
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
0.4 +                   |                |                 |                   +
    |          5        |       0        |        7        |       0           |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |-------------------+----------------+-----------------+-------------------|
    |                   |                |                 |                   |
0.2 +                   |                |                 |                   +
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |          0        |       4        |        2        |       9           |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
0.0 +                   |                |                 |                   +
    |                   |                |                 |                   |
    --+-------------+-------------+-------------+-------------+-------------+---
     0.0           0.2           0.4           0.6           0.8           1.0  
                                                                                
                                         X

Biological Cells

2 x 2 Grid

Quadrat	X	Y	COUNT
1	0.25	0.25	11
2	0.75	0.25	10
3	0.25	0.75	10
4	0.75	0.75	11

Biological Cells

2 x 2 Grid

                    Plot of Y*X$COUNT.  Symbol used is ' '.                     
                                                                                
    --+-------------+-------------+-------------+-------------+-------------+---
  Y |                                    |                                     |
1.0 +                                    |                                     +
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
0.8 +                                    |                                     +
    |                                    |                                     |
    |                   10               |                 11                  |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
0.6 +                                    |                                     +
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |------------------------------------+-------------------------------------|
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
0.4 +                                    |                                     +
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                   11               |                 10                  |
    |                                    |                                     |
0.2 +                                    |                                     +
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
    |                                    |                                     |
0.0 +                                    |                                     +
    |                                    |                                     |
    --+-------------+-------------+-------------+-------------+-------------+---
     0.0           0.2           0.4           0.6           0.8           1.0  
                                                                                
                                         X

Biological Cells

3 x 3 Grid

Quadrat	X	Y	COUNT
1	0.16667	0.16667	3
2	0.50000	0.16667	6
3	0.83333	0.16667	4
4	0.16667	0.50000	4
5	0.50000	0.50000	7
6	0.83333	0.50000	6
7	0.16667	0.83333	3
8	0.50000	0.83333	6
9	0.83333	0.83333	3

Biological Cells

3 x 3 Grid

                    Plot of Y*X$COUNT.  Symbol used is ' '.                     
                                                                                
    --+-------------+-------------+-------------+-------------+-------------+---
  Y |                        |                       |                         |
1.0 +                        |                       |                         +
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |             3          |           6           |          3              |
0.8 +                        |                       |                         +
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |------------------------+-----------------------+-------------------------|
    |                        |                       |                         |
    |                        |                       |                         |
0.6 +                        |                       |                         +
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |             4          |           7           |          6              |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
0.4 +                        |                       |                         +
    |                        |                       |                         |
    |                        |                       |                         |
    |------------------------+-----------------------+-------------------------|
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
0.2 +                        |                       |                         +
    |             3          |           6           |          4              |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
    |                        |                       |                         |
0.0 +                        |                       |                         +
    |                        |                       |                         |
    --+-------------+-------------+-------------+-------------+-------------+---
     0.0           0.2           0.4           0.6           0.8           1.0  
                                                                                
                                         X

Biological Cells

4 x 4 Grid

Quadrat	X	Y	COUNT
1	0.125	0.125	2
2	0.375	0.125	3
3	0.625	0.125	3
4	0.875	0.125	2
5	0.125	0.375	2
6	0.375	0.375	4
7	0.625	0.375	2
8	0.875	0.375	3
9	0.125	0.625	2
10	0.375	0.625	1
11	0.625	0.625	4
12	0.875	0.625	3
13	0.125	0.875	3
14	0.375	0.875	4
15	0.625	0.875	2
16	0.875	0.875	2

Biological Cells

4 x 4 Grid

                    Plot of Y*X$COUNT.  Symbol used is ' '.                     
                                                                                
    --+-------------+-------------+-------------+-------------+-------------+---
  Y |                   |                |                 |                   |
1.0 +                   |                |                 |                   +
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |          3        |       4        |        2        |       2           |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
0.8 +                   |                |                 |                   +
    |                   |                |                 |                   |
    |-------------------+----------------+-----------------+-------------------|
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |          2        |       1        |        4        |       3           |
0.6 +                   |                |                 |                   +
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |-------------------+----------------+-----------------+-------------------|
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
0.4 +                   |                |                 |                   +
    |          2        |       4        |        2        |       3           |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |-------------------+----------------+-----------------+-------------------|
    |                   |                |                 |                   |
0.2 +                   |                |                 |                   +
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |          2        |       3        |        3        |       2           |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
    |                   |                |                 |                   |
0.0 +                   |                |                 |                   +
    |                   |                |                 |                   |
    --+-------------+-------------+-------------+-------------+-------------+---
     0.0           0.2           0.4           0.6           0.8           1.0  
                                                                                
                                         X