Discussion for Marine Diversity

Simpson’s Diversity Index

Simpson’s Diversity Index is a measure both of species richness (i.e. the number of different species present) and species evenness (i.e. how evenly distributed each species is).

\(D = \frac{(N\;\times\;(N\;-\;1))}{(Σn\;\times\;(n\;-\;1))}\)

\(D\) = Simpson’s Diversity Index
\(n\) = the number of individuals of each species
\(N\) = the total number of individuals

Mann Whitney U test

Mann Whitney U is a statistical test that is used either to test whether there is a significant difference between the medians of two sets of data.

The Mann Whitney U test can only be used if there are at least 6 pairs of data. It does not require a normal distribution.

There are 3 steps to take when using the Mann Whitney U test

Step 1. State the null hypothesis

There is no significant difference between _______ and _______

Step 2. Calculate the Mann Whitney U statistic

[llatex]U_1= n_1 \times n_2 + 0.5 n_2 (n_2 + 1)\;- ∑ R_2[/latex]

[llatex]U_2 = n_1 \times n_2 + 0.5 n_1 (n_1 + 1)\;- ∑ R_1[/latex]

`\(n_1\) is the number of values of \(x_1\)
\(n_2\) is the number of values of \(x_2\)
\(R_1\) is the ranks given to \(x_1\)
\(R_2\) is the ranks given to \(x_2[latex]

Step 3. Test the significance of the result

Compare the value of U against the critical value for U at a confidence level of 95% / significance value of P = 0.05.

If U is equal to or smaller than the critical value (p=0.05) the REJECT the null hypothesis. There is a SIGNIFICANT difference between the 2 data sets.

If U is greater than the critical value, then ACCEPT the null hypothesis. There is NOT a significant difference between the 2 data sets.

Worked example

A biologist is investigating whether there is a difference in woodland flora between two contrasting areas of woodland, Site A and Site B. Eight randomly placed frame quadrat samples in each of two contrasting areas of woodland produced the following % cover of dog’s mercury, a common woodland plant. Here are the results.

Quadrat	1	2	3	4	5	6	7	8
Site A	26	14	8	6	26	20	11	13
Site B	28	16	26	25	25	24	16	28

Step 1. State the null hypothesis

There is no significant difference in species richness between Site A and Site B.

Step 2. Calculate Mann Whitney U statistic

(a) Give each result a rank. Calculate the sum of the ranks for the two columns.

Now arrange the data values in order, and give each value a rank in order from smallest to highest

Site A	6	8	11	13	14	20	26	26
rank	1	2	3	4	5	8	13	13
Site B	16	16	24	25	25	26	28	28
rank	6.5	6.5	9	10.5	10.5	13	15.5	15.5

(b) Calculate [latex]∑R_1\) and \(∑R_2\)

∑R_1 is the sum of the ranks in the first column (Site A) = 49

\(∑R_2\) is the sum of the ranks in the first column (Site B) = 87

\(n_1\) = 8 and \(n_2 = 8\)

\(U_1 = 8\times 8 + 0.5 \times 8 \times (8 + 1) – 87 = 13\) \(U_2 = 8 \times8 + 0.5 \times 8 \times (8 + 1) – 49 = 51\)

Step 3. Test the significance of the result

In this example, \(U_1\) = 13 and \(U_2\) = 51

Select the smaller or the values. In this case \(U_1\) is the smaller of the two values, so U=13

The critical value at p=0.05 significance level for \(n_1=8\) and \(n_2=8\) is 13. Since our calculated value of 13 = 13, the null hypothesis can be rejected.

In conclusion, there is a significant difference in species richness between Site A and Site B.