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...

- The Independent Samples
- 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

- IV = Treatment (with two groups).

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 )

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 ()

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 ).

**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**.

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.