This section shows how you can use the Spearman’s Rank Correlation Coefficient to investigate changes with distance from open water. Information about other statistical tests can be found here.

## 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

### Worked example

A biologist is comparing species diversity at two sites within at Woodmere. Raw data from point quadrats is used. The number means the total hits per species per site.

Site 1 | Site 2 | |
---|---|---|

\(n\) | \(n\) | |

Common reed | 22 | 0 |

Saw sedge | 14 | 0 |

Alder | 10 | 0 |

Sheep’s fescue | 0 | 7 |

Common bent | 0 | 18 |

Creeping buttercup | 0 | 8 |

Soft rush | 0 | 10 |

Calculate \(n\), \(n\;\times\;(n\;-\;1)\), \(N\) and \(D\) for each site

Site 1 | Site 1 | Site 2 | Site 2 | |
---|---|---|---|---|

\(n\) | \(n\;\times\;(n\;-\;1)\) | \(n\) | \(n\;\times\;(n\;-\;1)\) | |

Common reed | 22 | 462 | 0 | 0 |

Saw sedge | 14 | 182 | 0 | 0 |

Alder | 10 | 90 | 0 | 0 |

Sheep’s fescue | 0 | 0 | 7 | 42 |

Common bent | 0 | 0 | 18 | 306 |

Creeping buttercup | 0 | 0 | 8 | 56 |

Soft rush | 0 | 0 | 10 | 90 |

TOTAL | 46 | 734 | 43 | 494 |

D = 46 (45) / 734 | D = 43 (42) / 494 | |||

D = 2.82 | D = 3.66 |

The larger the value of D, the higher the species diversity. A low value of D could be due to low overall species richness or to the dominance of one species.

## Spearman’s Rank Correlation Test

Spearman’s Rank Correlation is a statistical test to test whether there is a significant relationship between two sets of data.

The Spearman’s Rank Correlation test can only be used if there are at least 10 (ideally at least 15-15) pairs of data.

There are 3 steps to take when using the Spearman’s Rank Correlation Test

### Step 1. State the null hypothesis

There is no significant relationship between _______ and _______

### Step 2. Calculate the Spearman’s Rank Correlation Coefficient

\(r_s = 1-\frac{(6∑D^2)}{n(n^2-1)}\)- \(r_s\) = Spearman’s Rank correlation coefficient
- \(D\) = differences between ranks
- \(n\) = number of pairs of measurements

Step 3. Test the significance of the result

Compare the value of \(r_s\) that you have calculated *against* the critical value for \(r_s\) at a confidence level of 95% / significance value of p = 0.05.

If \(r_s\) is equal to or above the critical value (p=0.05) the REJECT the null hypothesis. There is a SIGNIFICANT relationship between the 2 variables.

A positive sign for \(r_s\) indicates a significant positive relationship and a negative sign indicates a significant negative relationship.

If \(r_s\) (ignoring any sign) is less than the critical value, ACCEPT the null hypothesis. There is NO SIGNIFICANT relationship between the 2 variables.

### Worked example

A biologist is investigating whether soil moisture decreases with distance from open water at Woodmere. A systematic transect is taken with readings every 4m from the edge of the open water. A soil pin is used to measure soil moisture levels.

Distance from open water (m) | Soil pin depth (cm) |
---|---|

0 | 13.0 |

3 | 14.5 |

6 | 12.8 |

9 | 15.5 |

12 | 14.4 |

15 | 11.0 |

18 | 8.4 |

21 | 7.0 |

24 | 8.5 |

27 | 6.5 |

30 | 8.5 |

33 | 6.5 |

36 | 5.6 |

39 | 4.0 |

42 | 5.2 |

### Step 1. State the null hypothesis

There is no significant relationship between soil moisture and distance from open water at Woodmere

### Step 2. Calculate the Spearman’s Rank Correlation Coefficient

(a) Rank the measurements

Distance from open water (m) | Distance from open water (m) | Soil pin depth (cm) | Soil pin depth (cm) |
---|---|---|---|

Data | Rank | Data | Rank |

0 | 1 | 13.0 | 12 |

3 | 2 | 14.5 | 14 |

6 | 3 | 12.8 | 11 |

9 | 4 | 15.5 | 15 |

12 | 5 | 14.4 | 13 |

15 | 6 | 11.0 | 10 |

18 | 7 | 8.4 | 7 |

21 | 8 | 7.0 | 6 |

24 | 9 | 8.5 | 8.5 |

27 | 10 | 6.5 | 4.5 |

30 | 11 | 8.5 | 8.5 |

33 | 12 | 6.5 | 4.5 |

36 | 13 | 5.6 | 3 |

39 | 14 | 4.0 | 1 |

42 | 15 | 5.2 | 2 |

(b) Calculate \(D\) and \(D^2\)

Distance from open water (m) | Soil pin depth (cm) | ||||
---|---|---|---|---|---|

Data | Rank | Data | Rank | \(D\) | \(D^2\) |

0 | 1 | 13.0 | 12 | 11 | 121 |

3 | 2 | 14.5 | 14 | 12 | 144 |

6 | 3 | 12.8 | 11 | 8 | 64 |

9 | 4 | 15.5 | 15 | 11 | 121 |

12 | 5 | 14.4 | 13 | 8 | 64 |

15 | 6 | 11.0 | 10 | 4 | 16 |

18 | 7 | 8.4 | 7 | 0 | 0 |

21 | 8 | 7.0 | 6 | 2 | 4 |

24 | 9 | 8.5 | 8.5 | 0.5 | 0.25 |

27 | 10 | 6.5 | 4.5 | 5.5 | 30.25 |

30 | 11 | 8.5 | 8.5 | 2.5 | 6.25 |

33 | 12 | 6.5 | 4.5 | 7.5 | 56.25 |

36 | 13 | 5.6 | 3 | 10 | 100 |

39 | 14 | 4.0 | 1 | 13 | 169 |

42 | 15 | 5.2 | 2 | 13 | 169 |

(c) Calculate \(∑D^2\)i.e. the sum of the \(D^2\)column= 1065

(d) Calculate `r_s`

\(r_s = 1-\frac{(6∑D^2)}{n(n^2-1)}\)\(r_s = 1\;-\;\frac{(6\;\times\;1065)}{ (15\times(225\;-\;1))}\)\(r_s = -0.90\)### Step 3. Test the significance of the result

The critical value at \(p=0.05\) significance level for \(15\) pairs of measurements is \(0.521\)

Since our calculated value of \(0.90<0.521\) (ignore the minus sign), the null hypothesis is rejected.

In conclusion, there is a significant relationship between soil moisture and distance from open water at Woodmere.