What is a Factor?
A Factor - is an independent categorical variable.
- ordinal or nominal.
- eg Gender, Diet, Forest Type.
What is a Level?
A Level - is the different categories of the factor.
- Different samples.
- e.g. Gender - Male, Female; Diet - A, B, C; Forest Type. - rainforest, wet sclerophyll, dry sclerophyll.
What is an Independent two sample t-test?
- An inferential statistical test that determines whether there is a statistically significant difference between the means in two unrelated groups
What is involved in an Independent two sample t-test?
An independent two sample t-Test has 1 factor - 2 levels – levels are independent.
What is a Paired difference t-test?
- This t‐test compares one set of measurements with a second set from the same sample.
What does a Paired Difference t-test involve?
A paired difference t-Test includes 1 factor - 2 levels – levels are paired (or blocked).
Give some examples of an Independent t-test.
Example1 - heights of men and women: - factor = gender - levels = male, female - independently sampled Example2 - length of fish from polluted/non polluted locations: - factor = pollution impact - level = polluted, not polluted - independently sampled
Give some examples of Paired Difference t-tests.
Example1 - heights of men and women (couples): - factor = gender - levels = male, female - paired by co-habitation Example2 - length of fish from polluted/non polluted locations-within estuaries - factor = pollution impact - level = polluted, not polluted - paired by estuaries
How should the data for Independent t-test samples be displayed?
- For example if we are testing the height of men and women then there should be no pairing of males and females
- They are independent of each other and should be analysed using an independent two sample t-Test.
- The table should be laid out with two columns, one for the gender and one for the height
How should the data for Paired Difference t-test samples be displayed?
- If we are testing the height of men and women as couples then the data should be paired by family and analysed using a paired t-test
- The table is set out in three columns, the first is for the family (1,2,3,etc), the second and third are for the gender as they should be paired. The two gender columns will contain the heights.
Study the examples of how data is presented for Independent samples and Paired samples.
https://docs.google.com/document/d/1r_ttbYs-4jXdkBbVGPH9vk1swjRRmRJUWdllcJXdaAI/edit?usp=sharing
What is the paired t-test theory?
- A paired t-Test uses the difference between the levels in each pair.
- By creating a new variable called difference we have accounted for the pairing of the observations.
- Theoretically if the numbers were the same (i.e. there was no difference due to gender) the mean difference will be 0.
- We will be testing this new derived variable (difference) against the fixed value of 0 in exactly the same way that we did in the last lecture for a one sample t-Test
How does the paired t-test theory work?
- Each sample has a certain amount of variation or ‘error’ associated with it
- By pairing the data points, we are acknowledging that they have something in common (eg if people come from the same family, you would expect them to share similar characteristics).
- By subtracting one from the other, we can account for this dependency between observations.
What is the procedure for a paired t-test?
- Data - ensure the data is appropriate for the test.
- Hypotheses - define H0 and H1 (1 or 2 tailed).
- Assumptions
- Test Statistic
- Decision Rule - set α and apply decision rule: Method 1 - determine TC (critical value) from t-Tables and compare the this value with test statistic (Tt); Method 2 - use the sig (2 tailed) value from SPSS output and compare this value to α.
- Conclusions - in English with p values (no jargon).
What is the structure of the data required for a paired t-test?
- One independent variable - nominal or ordinal.
- One dependant variable - scale (quantitative) .
- Paired observations (samples are not independent).
What are we trying to find in the data for a paired t-test?
- A paired t-Test uses the difference between the samples (levels) in each pair.
- We what to know whether our samples come from the same population - are the two samples the same?
- That is, is the mean differences between the pairs equal to zero?
What would the research question for a two tailed paired t-test be?
- Is their a difference between the two samples (levels of the factor)?
What would the estimated H0 and H1 values be for a two tailed paired t-test?
Statistical Hypotheses (English):
- H0: The mean population difference is equal to 0.
- H1: The mean population difference is not equal to 0.
Statistical Hypotheses (Symbols):
- H0: μD = 0
- H1: μD does not = 0
- μD (mean difference)
What would the research question for a one tailed paired t-test be?
- Is the mean difference between the two samples (levels of the factor) greater (or less) than 0?
What would the estimated H0 and H1 values be for a one tailed paired t-test?
- If the research question is asking is the mean difference between between the levels of a factor are grater than 0, then they would be:
Statistical Hypotheses (English): - H0: The mean population difference is equal to or less than zero.
- H1: The mean population difference is greater than zero.
Statistical Hypothesis (Symbols): - H0: μD < 0
- H1: μD > 0
However, if the research question was asking if the mean diffrence between the levels of a factor are less than 0 then they would change to:
Statistical Hypotheses (English): - H0: The mean population difference is equal to or greater than zero.
- H1: The mean population difference is less than zero.
Statistical Hypothesis (Symbols): - H0: μD > 0
- H1: μD < 0
What assumptions need to be satisfied for a paired t-test to be valid?
- Each pair of observations were taken randomly and without bias.
- The differences are
Normally Distributed - this test is reasonably robust to departures from this assumption
Study the equation for how to find the Test Statistic for a paired t-test.
https://docs.google.com/document/d/1r_ttbYs-4jXdkBbVGPH9vk1swjRRmRJUWdllcJXdaAI/edit?usp=sharing
What do you need to do to create the conclusion for a paired t-test?
- Should always state your conclusion in plain English (not statistical jargon) and give the associated probability.
- For a two tailed test sig = 0.001: There was a significant mean difference difference between the two samples (p = 0.001 or p < 0.05).
- For a two tailed test sig = 0.100: There was not a significant mean difference difference between the two samples (p = 0.100 or p > 0.05)
Study the example for carrying our a paired t-test.
- Google doc
