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# Independent Samples t test

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.

• 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 )

or

• 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 ()
http://www.math.unb.ca/~knight/utility/t-table.htm

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.
or

or
• 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 ).
http://www.math.unb.ca/~knight/utility/t-table.htm

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.
• So, , 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.

• 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   Contents
jds0282 2010-10-15