1. Quadrat Sampling Chi-squared Test
Skills 4.1
Quadrat Sampling
Chi-squared Test

2. Testing for association between two species using the chi-squared test with data obtained by quadrat sampling. 

3. General Set Up
To obtain data for the chi-squared test, an ecosystem should be chosen in which one or more factors affecting the distribution of the chosen species varies.
Sampling should be based on random numbers.
In each quadrat the presence or absence of the chosen species should be recorded.
The collection of raw data through quadrat sampling will be done in Bamfield on the Beach

4. Quadrat sampling 101
The presence of two species within a given environment can be determined using quadrat sampling
In Workbook Activity 116 you learned about Sampling as a way to measure the Diversity of an Ecosystem. CAN YOU RECALL THE FOUR SAMPLE TYPES?

5. In each quadrat, the presence or absence of each species is identified
A quadrat is a rectangular frame of known dimensions that can be used to establish population densities
Quadrats are placed inside a defined area in either a random arrangement or according to a design (e.g. belted transect)
The number of individuals of a given species is either counted or estimated via percentage coverage
The sampling process is repeated many times in order to gain a representative data set
Quadrat sampling is not an effective method for counting motile organisms – it is used for counting plants and sessile animals
In each quadrat, the presence or absence of each species is identified
This allows for the number of quadrats where both species were present to be compared against the total number of quadrats

7. Plot-based (quadrat) methods are often used to study populations of different species within a certain area.
Quadrats are generally square sample areas marked out using a framed structure.
Quadrats are placed in a marked out habitat according to random numbers obtained using a random number table or a random number generator on a calculator

8. Quadrats for Population Estimations
Activity 118
Discuss the questions with your group and share your responses
Activity 119
Do this in a group of 4 and compare your answers when you are finished.
What did you find?

9. What is the Chi-Squared Test
Statistics is one of the most hated subjects by biologists around the globe. In spite of its daily dose of abuse, knowledge of statistics can be a life-saver.

10. T-test or Chi Square??? χ2 test T-Test
Looks at the difference between two groups on some variable of interest
Used for many applications
χ2 test
Compares observed frequencies to expected frequencies
Used in Genetics and Ecology

11. 1. Goodness of Fit - Genetics
The goodness of fit test is normally used in genetics where the genotypic and phenotypic ratios have already been established for a given test and population.
Ie. when the expected outcome has already been established.
For example: You want to understand the outcome of an experiment that you set in your field based on the test cross given by Mendel.
We will use this later in the year

12. Chi-Squared Tests in Ecology
We will use this now and in Bamfield

13. The presence of two species within a given environment will be dependent upon potential interactions between them

14. Types of Association POSITIVE
If two species are typically found within the same habitat, they show a positive association
Species that show a positive association include those that exhibit predator-prey or symbiotic relationships
NEGATIVE
If two species tend not to occur within the same habitat, they show a negative association
Species will typically show a negative association if there is competition for the same resources
One species may utilise the resources more efficiently, precluding survival of the other species (competitive exclusion)
Both species may alter their use of the environment to avoid direct competition (resource partitioning)
NO ASSOCIATION - If two species do not interact, there will be no association between them and their distribution will be independent of one another

15. The Chi-Squared Test (2)
The chi-squared test is used to study differences between data sets.
It is only used for frequencies (counts), never for measurements.
It is used to compare an experimental result with an expected theoretical outcome.
It is not a valid test for small sample sizes (n<20)
It tests the validity of the null hypothesis: no difference between groups of data.
In ecology, chi-squared tests are used to study habitat preference.

16. A chi-squared test can be completed by following five simple steps:
Identify hypotheses (null versus alternative)
Construct a table of frequencies (observed versus expected)
Apply the chi-squared formula
Determine the degree of freedom (df)
Identify the p value (should be <0.05)
Lets Try It

17. The presence or absence of two species of scallop was recorded in fifty quadrats (1m2) on a rocky sea shore 
The following distribution pattern was observed:
6 quadrats = both species  ;  15 quadrats = king scallop only  ;  20 quadrats = queen scallop only  ;  9 quadrats = neither species 

18. Step 1: Identify hypotheses
A chi-squared test seeks to distinguish between two distinct possibilities and hence requires two contrasting hypotheses:
Null hypothesis (H0): There is no significant difference between the distribution of two species (i.e. distribution is random)
Alternative hypothesis (H1): There is a significant difference between the distribution of species (i.e. species are associated)

19. Step 2: Construct a table of frequencies
A table must be constructed that identifies expected distribution frequencies for each species (for comparison against observed)
Expected frequencies are calculated according to the following formula: 
Expected frequency = (Row total × Column total) ÷ Grand total

21. Step 3: Apply the chi-squared formula
The formula used to calculate a statistical value for the chi-squared test is as follows:
Where:  ∑ = Sum  ;  O = Observed frequency  ;  E = Expected frequency These calculations can be broken down for each part of the distribution pattern to make the final summation easier

22. Based on these results the statistical value calculated by the chi-squared test is as follows:
𝝌2  =  ( )  =  7.90

23. Step 4: Determine the degree of freedom (df)
In order to determine if the chi-squared value is statistically significant a degree of freedom must first be identified
The degree of freedom is a mathematical restriction that designates what range of values fall within each significance level 
The degree of freedom is calculated from the table of frequencies according to the following formula:
df = (m – 1) (n – 1) 
Where:  m = number of rows  ;  n = number of columns
When the distribution patterns for two species are being compared, the degree of freedom should always be 1

24. Step 5: Identify the p value
The final step is to apply the value generated to a chi-squared distribution table to determine if results are statistically significant
A value is considered significant if there is less than a 5% probability (p < 0.05) the results are attributable to chance

25. When df = 1, a value of greater than 3
When df = 1, a value of greater than is required for results to be considered statistically significant (p < 0.05)
A value of 7.90 lies above a p value of 0.01, meaning there is less than a 1% probability results are caused by chance
Hence, the difference between observed and expected frequencies are statistically significant
As the results are statistically significant, the null hypothesis is rejected and the alternate hypothesis accepted:
Alternate hypothesis (H1): There is a significant difference between observed and expected frequencies
Because the two species do not tend to be present in the same area, we can infer there is a negative association between them

26. The Flat Periwinkle (Littorina littoralis)

27. Periwinkles feed on a number of seaweed species

28. Food preference is a form of animal behavior
Using quadrats, the number of periwinkles associated with each seaweed species was recorded.

29. State your null hypothesis for this investigation (H0)
H0: There is no difference between the numbers of periwinkles associated with different species.
What is the alternative hypothesis (HA)?
HA : There is a real difference between numbers of periwinkles associated with different species.

30. Use the chi-squared test to determine if the observed differences are significant or if they can be attributed to chance alone.
Enter the observed values and calculate the chi-squared value

31. Here’s how you do it…
The expected value (E) would be the mean number of periwinkles associated with the four seaweed species.

32. Now Complete the Chart…
Calculate the degrees of freedom:
4-1 = 3

33. Check your Chi-square table for 3 degrees of freedom.
57.4 >> 7.82, 11.34
Is H0 accepted or rejected?
There is a significant difference in feeding preferences of periwinkles.