Making Decisions for the Difference between
Making Decisions for the Difference between
Two Dependent Population Means
Question:
How do I know if the measurements have some dependence (or lack of independence) structure?
Answer:
Ask yourself the following question: Do the observations from the 1st group have any significant relationship with the observations from the 2nd group?
|Group 1 | |Group 2 |
|Observation 1 |[pic] |Observation 1 |
|Observation 2 |[pic] |Observation 2 |
|: | |: |
|Observation n |[pic] |Observation n |
If your answer is NO, then
Use the Difference between Two Independent Population Means hypothesis test.
Otherwise,
Use the Difference between Two Dependent Population Means hypothesis test.
Question:
How do we work with Two Dependent Population Means?
Answer:
Somehow make them independent... We can remove the dependence structure by subtracting the observations from the 2nd group from the observations in the 1st group.
|Group 1 |Group 2 |Group1 - Group 2 |
|Observation 1 |Observation 1 |Difference 1 |
|Observation 2 |Observation 2 |Difference 2 |
|: |: |: |
|Observation n |Observation n |Difference n |
Now, realize that Difference 1 is independent of Difference 2, which is independent of Difference 3, etc... Essentially, we have n independent observations and the parameter of interest is μD = mean paired difference.
[pic]
| Data File: |Reading Comprehension |
|Background: |These data come from of a course designed to improve a participants reading comprehension as |
| |measured by a standardized test. |
|Variables: |Before: initial or baseline reading comprehension score |
| |After: reading comprehension score after taking the course. |
|Goal: |To be able to complete (and interpret the output) for a test of differences for two groups |
| |that are dependent of each other.. |
|Question of Interest: |It is believed that the course will increase the reading comprehension scores of the |
| |participants. |
| |First, realize these two groups are dependent of each other because the reading |
| |comprehensions scores before and after completion of the course come from the same |
| |individuals. As a result, the analysis is done 'within' or 'by' individuals. |
| | |
| |If we let D = Score After – Score Before then |
| | |
| |[pic] |
| | |
| |In words we have, |
| | |
| | |
| | |
| | |
| | |
| | |
Obtaining the Differences:
To calculate a column of differences, we need to add a new column. Double-click to the right of the last column and name the column scorediff. Then double-click at the top of the column to obtain the column info window and select Formula from the New Property pull-down menu. Then click Edit Formula which will open the JMP calculator and enter the formula shown below:
After – Before
To do this first click on the variable After from the list that appears in the top left corner of the calculator window which will enter that variable into the calculator window. Next select the minus sign from the buttons located in the middle of the window and finally add the variable Before to the expression by double-clicking on it in the variable list. Your expression should now look like the one above. When you are finished click OK.
Intuitive Decision
In order to determine whether or not the null or alternative hypothesis is true, you should first review the summary statistics for the differences. Remember, all summary statistics are for the observations you sampled. In order to make decisions about all observations of interest, we must apply some inferential technique (i.e. hypothesis tests or confidence intervals). Recall, to get summary statistics for a numerical variable, select Analyze > Distribution. The variable to summarize is the differences just created, scorediff.
[pic]
Assumptions
The observations, i.e. the paired differences, should follow a normal distribution.
To check this assumption, use a normal quantile plot which can be obtained by checking 'Normal Quantile Plot' from the scorediff pull-down menu.
Do you think normality is being satisfied? Explain.
Performing the test
To perform the paired t-test using these differences select Test Mean ... from the scorediff pull-down menu located at the top of the resulting window, entering 0 for the hypothesized value of the mean. After clicking OK the following output will be obtained.
[pic]
There are three p-values reported for each test along with the value of the t-test statistic. The p-values are for a two-tailed, upper-tailed, and lower-tailed, respectively.
What type of test do we have here? Explain.
What is the appropriate p-value?
What is our decision for the test?
Write a conclusion for your findings.
From the Moments box, we see that the likely range for the average change of reading comprehension scores is between (3.5 , 31.92). Again, this is a 95% confidence interval for μD. Interpret the meaning of this interval.
Does this agree with what you found in above using the hypothesis test? Explain.
[pic]
Another way we could examine the changes in reading comprehension scores is in terms of percent improvement. The percent change in reading comprehension score for an individual looks at the size of the change relative to their baseline score as follows:
[pic]
In JMP this done by adding a column and building the formula shown below.
[pic]
[pic]
As with the raw scores we find that there has been a significant increase in the reading comprehension scores of course participants (p-value=.0092). Furthermore, we estimate with 95% confidence that an individuals reading score will improve by between 3.4% and 28.57% on average.
-----------------------
Null Hypothesis: The mean change in scores is 0
Research Hypothesis: The mean change in scores is greater than 0, indicating an improvement in reading comprehension on average.
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