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3.1. Independent Samples t test
Independent Samples t Test
- It too goes by many names...
- The Independent Samples t Test
- Independent Means t Test
- Between Groups t Test
- The t test for Independent Means...
- Essentially, it is used for comparing two sample means which are not related in some known or meaningful way.
- Two Independent groups of scores.
Methodological Application
- The Independent Samples t Test is applicable when you have a dichotomous Independent Variable (IV) and an interval or ratio scaled Dependent Variable (DV).
- One IV with two categories (sometimes called conditions).
- One DV which is continuous or nearly continuous.
- Evaluating two treatments for Schizophrenia.
- IV = Treatment (with two groups).
- Electro-Convulsive Therapy (ECT)
- Insulin Shock Therapy (IST)
- DV = Frequency of Hallucinations
Application Distinctions
- Dependent Samples t Test: two groups of scores from the same people or people related in some meaningful, known way.
- Same people at time 1 vs. time 2 or twin 1 vs. twin 2.
- Comparison distribution: Distribution of Difference Scores.
- Independent Samples t Test: two independent groups of people, each with a set of scores (i.e., group 1's scores vs. group 2's scores).
- A group of people exposed to one treatment vs. a group of people exposed to another treatment.
- Comparison distribution: Distribution of Differences Between Means.
A Quick note about notation...
- With the Independent Samples t Test, we have two groups, identified with the notation:
- Group 1:
- Group 2:
- So this:
becomes:
for group 1.
- Use subscripts to identify each group with either a 1 or a 2 subscript.
Getting to the Distribution of Differences Between Means (part 1).
- Each group has a population distribution.
- We can estimate those populations' variances with the sample variances: and
- Each group can be used to create a distribution of means.
- We can estimate the distribution of means' variances with the sample variances divided by the number of individuals in the samples and
- Using those two distributions of means, we can create a Distribution of Differences Between Means.
BUT...
Getting to the Distribution of Differences Between Means (part 2).
- Because we assume both and are equal; we must come up with an average of the two estimates and to get the best overall estimate of the population variance*.
- *This is especially crucial when the size of each group is different.
- This best estimate is called the pooled estimate of the population variance.
- Symbol:
Getting to the Distribution of Differences Between Means (part 3).
- To get you must get , , and (also called )
- So, then we can get using:
- But wait...there's more...
Getting to the Distribution of Differences Between Means (part 4).
- Now, we take and figure and by removing the influence of sample size.
- Which leads to...the variance of the distribution of differences between means.
- Which then leads to the standard deviation of the distribution of differences between means.
Finally...the t test.
- Now that we have we can calculate t or
- We would then use the and our significance level to look in the t table to find our cutoff sample score ()
3.2. Hypothesis Testing Example
NHST Example
- Examine whether viewers of John Stewart's The Daily Show know significantly more about world affairs than viewers of Bill O'Reilly's The O'Reilly Factor show.
- Randomly sample 16 cable viewers, randomly assign them to one of two show groups; Daily and Factor.
- Have the participants watch 20 recent episodes of one show or the other, depending on their group assignment.
- Assess their knowledge of Current World Events using the CWE questionnaire, which has a range of 1 to 10.
Step 1
- Define the populations and restate the research question as null and alternative hypotheses.
- Population 1: Americans who watch The Daily Show.
- Population 2: Americans who watch The O'Reilly Factor.
- In terms of knowledge about current events.
- Notice the directional alternative hypothesis () which indicates a one-tailed test.
Table 3: Daily Show Group
|
|
|
|
6 |
6.75 |
-0.750 |
0.563 |
6 |
6.75 |
-0.750 |
0.563 |
9 |
6.75 |
2.250 |
5.063 |
8 |
6.75 |
1.250 |
1.563 |
4 |
6.75 |
-2.275 |
7.563 |
6 |
6.75 |
-0.750 |
0.563 |
7 |
6.75 |
0.250 |
0.063 |
8 |
6.75 |
1.250 |
1.563 |
|
|
|
|
|
|
|
|
Table 4: Factor Show Group
|
|
|
|
5 |
3.875 |
1.125 |
1.266 |
4 |
3.875 |
0.125 |
0.016 |
3 |
3.875 |
-0.875 |
0.766 |
1 |
3.875 |
-2.875 |
8.266 |
5 |
3.875 |
1.125 |
1.266 |
6 |
3.875 |
2.125 |
4.516 |
3 |
3.875 |
-0.875 |
0.766 |
4 |
3.875 |
0.125 |
0.016 |
|
|
|
|
|
|
|
|
Step 2(a)
- 2. Determine the characteristics of the comparison distribution.
-
- And from above, and
- (a) Calculate the pooled estimate of the population variance.
- So,
Step 2(b)
- (b) Calculate the variance of each distribution of means:
- Please note; if the groups were different sizes, the variances of each distribution of means would be different.
Step 2(c) and Step 2(d)
- (c) Calculate the variance of the distribution of differences between means:
- So,
- (d) Calculate the standard deviation of the distribution of differences between means:
- So,
Step 3
- 3. Determine the critical sample score on the comparison distribution at which the null hypothesis should be rejected.
- Significance level = .05
- Two-tailed test (based on ).
-
Step 4
- 4. Determine the sample's score on the comparison distribution:
- Compute
- So,
Step 5
- 5. Compare the scores from Step 3 and Step 4, and make a decision to reject the null hypothesis or fail to reject the null hypothesis.
- Because;
we reject the null hypothesis and conclude there was a statistically significant difference between the two show groups.
- But, you should know by now, that's not the whole story.
3.3. Effect Size
Calculating Effect Size for two Independent Groups
- Recall, the general formula for Cohen's d.
- In the current (independent groups) situation, we have:
- So, the effect size is fairly large;
3.4. Statistical Power
Using Delta () for Statistical Power
- As was done with the Dependent Samples t Test situation, here again we calculate as a combination of sample size and effect size and use it to look up the power in the table.
- We use the exact same formula as was used with Dependent Samples (Section 2 above).
- So, for our current example, we get the following (where is the number per group):
- The table shows that with a (note: it is best to round down) we have a power of 0.98.
Using Delta to calculate appropriate sample size
- The more useful way to use is for calculating adequate sample size during the planning of the study.
- First, we need to calculate for a desired power with a one-tailed test at .05 significance level.
, now we can calculate the sample size for a given effect size .
As you can see, this is exactly as we did for the dependent samples situation. Just remember that the refers to the number of each group.
An Additional comments on Power
- t tests with evenly distributed participants have greater power than those where the participants are unevenly distributed.
- The dependent samples design has greater power (all else being equal, such as sample size, effect size, etc.) than the independent samples design.
- And, as always, the larger the sample size, the greater the power.
3.5.
Calculating a Confidence Interval
- Note, we are calculating the interval on the difference between means.
- Recall there are two parts of a confidence interval, the upper limit (UL) and the lower limit (LL).
- The general form of the equations for each limit are:
- In the current situation for the differences between means:
Interpretation of Confidence Interval
- Recall, we had a significance level of .05 (), so we conducted a 95% confidence interval () on the differences between means.
- The Lower Limit was 1.50 and the Upper Limit was 4.25.
- So, if we drew an infinite number of random samples of viewers of each show, 95% of the differences between means would be between 1.50 and 4.25.
- Remember, the mean of the population of differences between means is fixed (but unknown); while each sample has its own differences between means (samples fluctuate).
3.6. Summary of Section 3
Summary of Section 3
Section 3 covered the following topics:
- The Independent Samples t Test.
- An NHST example of the Independent Samples t Test.
- Cohen's d Effect Size
- Use of delta () for Statistical Power
- Use of delta () for calculating a-priori sample size
- Calculation of Confidence Intervals.
Next: t Summary
Up: Module 8: Introduction to
Previous: Dependent
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2010-10-15